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About the Author
Robert B. Northrop was born in White Plains, NY, in 1935. After graduating from
Staples High School in Westport, CT, he majored in electrical engineering (EE) at the
Massachusetts Institute of Technology, graduating with a bachelorâs degree in 1956. At
the University of Connecticut (UCONN), he received a masterâs degree in systems engi-
neering in 1958. As the result of a long-standing interest in physiology, he entered a PhD
program at UCONN in physiology, doing research on the neuromuscular physiology of
molluskan catch muscles. He received his PhD in 1964.
In 1963, he rejoined the UCONN EE department as a lecturer, and was hired as an
assistant professor of EE in 1964. In collaboration with his PhD advisor, Dr. Edward
G. Boettiger, he secured a 5-year training grant in 1965 from the National Institute of
General Medical Sciences [National Institutes of Health (NIH)], and started one of the
first, interdisciplinary, biomedical engineering (BME) graduate training programs in
New England. UCONN currently awards MS and PhD degrees in this field of study, as
well as BS degrees in engineering under the BME area of concentration.
Throughout his career, Dr. Northropâs research interests have been broad and inter-
disciplinary and have been centered on BME. He has conducted sponsored research on
the neurophysiology of insect and frog vision and devised theoretical models for visual
neural signal processing. He has also conducted sponsored research on electrofishing
and developed, in collaboration with Northeast Utilities, effective, working systems for
fish guidance and control in hydroelectric plant waterways on the Connecticut River at
Holyoke, MA, using underwater electric fields.
Another area of his sponsored research has been in the design and simulation of
nonlinear, adaptive, digital controllers to regulate in vivo drug concentrations or physi-
ological parameters, such as pain, blood pressure, or blood glucose in diabetics. An out-
growth of this research has led him to develop mathematical models for the dynamics
of the human immune system, which were used to investigate theoretical therapies for
autoimmune diseases, cancer, and human immunodeficiency virus infection.
Biomedical instrumentation has also been an active research area for Dr. Northrop
and his graduate students: an NIH grantâsupported study on the use of the ocular pulse
to detect obstructions in the carotid arteries. Minute pulsations of the cornea from arte-
rial circulation in the eyeball were sensed using a no-touch, phase-locked, ultrasound](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-15-320.jpg)
![1-6 Signals and Systems Analysis in Biomedical Engineering
In the limit as T â â, u â E(u), the âtrueâ mean of u. Suppose the ensemble consists of
N responses to a sensory stimulus given repetitively to an animal. We can then pick a
time t1 after each stimulus, and find the sample mean of the ensemble, {uk(t1)}
u t
N
u t
k
k
N
( ) ( )
1 1
1
1
=
=
â (1.3)
As will be shown in Section 8.4, the ensemble average shown in Equation (1.3), with 0 â¤
t1 ⤠T, is a means of extracting a consistent evoked response signal from additive random
noise with zero mean.
Another measure of a 1-D signal is its intensity. Definitions of intensity can vary,
depending on the type of signal. For example, both sound intensity and light intensity
both have the units of power (W) per unit area (m2). The power can be instantaneous
power or average power. For sound, the average intensity is given by
I p c pu pcu
= = =
2 2
( )
Ď W/m2
(1.4)
where p is p(t), the sound pressure expressed in N/m2 (=Pa) over atmospheric pressure,
u(t) is the particle velocity (m/s), Ď is the density of the medium (kg/m3), and c is the
velocity of propagation (m/s). That is, I is proportional to the mean squared (ms) sound
pressure [note that the overbars in Equation (1.4) denote taking the time average]. When
p(t) = Posin(2Ďft), it is easy to show that the ms sound pressure is Po
2/2, that is, ½ the
peak pressure squared. For light, radiant intensity is also called the irradiance (W/m2)
incident on a surface.
A voltage signal, such as an EEG recording, can be described by its average power,
given by its mean squared voltage divided by R = 1
P
T
v t t
T
T
=
ââ âŤ
lim ( )
1 2
0
d W (1.5)
In Chapter 7, it will be shown that nonstationary signals can be characterized by JTF
plots. Special transforms are used to decompose a signal, u(t), into its spectral compo-
nents as a function of time. A JTF spectrogram gives signal mean squared volts (or rms
volts) as a function of time and frequency, creating a 2-D or 3-D display.
In Chapter 8, we present many statistical parameters that can be attributed to sig-
nals, in addition to their sample means. These include the signalâs probability density
function, and statistical measures such as its variance. Another important measure of
a signal is its frequency spectrum, also known as power density spectrum, with units
of mean squared volts (or ms units) per hertz. A signalâs root power spectrum is also
used to describe noisy signals; it is expressed in root mean squared volts per root hertz.
The root power spectrum is simply the square root of the power density spectrum. The
power spectrum describes the contribution to the total power of u(t) at each incremental
frequency. It implies a superposition of spectral power.](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-22-320.jpg)
![Introduction to Biomedical Signals and Systems 1-7
1.1.6â
Introduction to Modulation and
Demodulation of Physiological Signals
In general, modulation is a process whereby a signal of interest is combined mathemati-
cally with a high-frequency carrier wave to form a modulated carrier suitable for trans-
mission to a receiver/demodulator where the signal is recovered by demodulation, then
processed and stored.
Recorded physiological signals are modulated for two important reasons: (1) for robust
transmission by wire or coaxial cable, fiber optic cable, or radio (telemetry by electro-
magnetic waves) from the recording site to the site where the signal will be processed
and stored; (2) for effective data storage. As an example, in biotelemetry, physiological
signals such as ECG and blood pressure modulate a carrier that is sent as a frequency
modulation (FM) radio signal to a remote receiver, where the signals are demodulated,
digitized, filtered, analyzed, and stored.
Modulation generally involves encoding a high-frequency carrier wave, which can
be a continuous sine wave or a square wave (logic signal) with the low-frequency signal
being measured, vm(t).
There are five major types of modulation involving sinusoidal carrier waves: (1) ampli-
tude modulation (AM), (2) single-sideband AM (SSBAM), (3) FM, (4) phase moduÂ
lation
(PhM),and(5)double-sideband,suppressed-carriermodulation(DSBSCM).Modulation
can also be done using a square wave (or TTL) carrier, and can be by FM, PhM, or delta
modulation (DM), or can use pulse position modulation or pulse width modulation at
constant frequency. AM, FM, and DSBSCM are expressed mathematically below for a
sinusoidal carrier. m(t) is the normalized modulating signal in each case, whereas vm(t)
is the actual physiological signal. The maximum frequency of m(t) must be Ďc, the
carrier frequency. The normalized modulating signal, m(t), is defined by
m t
v t
V
m t
( )
( )
, ( )
= ⤠â¤
m
mmax
0 Normalized modulatin
1 g
g signal. (1.6a)
Thus, we have
y t A m t t
m c AM
( ) [ ( )]cos( )
= +
1 Ď (1.6b)
y t A t K m t t
t
m c f d FM
( ) cos[ ( ) ]
= +
âŤ
Ď (1.6c)
y t A t K m t
m c p PhM
( ) cos ( )
= +

 

Ď (1.6d)
y t Am t t
m c DSBSCM
( ) ( )cos( )
= Ď (1.6e)
It is instructive to examine the frequency spectrums of modulated signals. The extent
of the spectrum of ym(t) determines the required signal processing BW. For illustrative
purposes, let m(t) be a pure cosine wave, [mocos(Ďmt)], 0 mo ⤠1. Thus, by trig identity,
the AM signal can be rewritten as](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-23-320.jpg)
![1-8 Signals and Systems Analysis in Biomedical Engineering
ym(t) = Acos(Ďct) + (Amo/2)[cos((Ďc + Ďm)t) + cos((Ďc â Ďm)t)] (1.7)
We see that the AM signal has a carrier component and two sidebands, each spaced
by the amount of the modulating frequency above and below the carrier frequency. In
SSBAM, a sharp cutoff filter is used to eliminate either the upper or lower sideband; the
information in both sidebands is redundant, so removing one sideband means a lower
BW is required to transmit the SSBAM signal.
FM and PhM are subsets of angle modulation. FM can be further classified as broad-
band or narrowband FM (NBFM). In FM, Kf ⥠2Ďfd, where fd denotes the frequency
deviation constant. In NBFM, fd/fmmax 1; (fmmax is the highest expected frequency in
m(t), which is BW-limited. Unlike AM, the frequency spectrum of an FM carrier is more
difficult to derive. Using m(t) = mocos(Ďmt), we can write the FM carrier as
y t A t K m t
m c f m o m
( ) cos ( / ) sin( )
= +






Ď Ď Ď
ι β
(1.8)
Using the trig identity, cos(Îą + β) = cos(Îą)cos(β) â sin(Îą)sin(β), Equation (1.8) can be
written as
y t A t K m t A t
m c f m o m c
( ) cos( )cos[( / ) sin( )] sin( )
= â
Ď Ď Ď Ď s
sin[( / ) sin( )]
K m t
f m o m
Ď Ď
(1.9)
Now the cos[(Kf/Ďm)mosin(Ďmt)] and sin[(Kf/Ďm)mosin(Ďmt)] terms can be expressed as
two Fourier series whose coefficients are ordinary Bessel functions of the first type and
argument β (Clarke and Hess, 1971); note that β ⥠mo2Ďfd/Ďm
cos[ sin( )] ( ) ( )cos( )
β Ď Î˛ β Ď
m m
t J J n t
n
n
= +
=
â
â
0 2
1
2 2 (1.10a)
sin[ sin( )] ( )sin[ ( ) ]
β Ď Î˛ Ď
m m
t J n t
n
n
= +
+
=
â
â
2 2 1
2 1
0
(1.10b)
The two Bessel sum relations for cos[β sin(Ďmt)] and sin[β sin(Ďmt)] can be recombined
with Equation (1.9) using the trig identities, cosxcosy = ½[cos(x + y) + cos(x â y)], and
sinxsiny = ½[cos(x â y) â cos(x + y)], and we can finally write for the FM carrier spec-
trum, letting mo = 1
y t A J t J t
m c c m
( ) ( )[cos( ) ( )cos(( ) ) cos((
= + + â
0 1
β Ď Î˛ Ď Ď Ďc
c m
c m c m
â
{
+ + â â
Ď
β Ď Ď Ď Ď
) )]
( )[cos(( ) ) cos(( ) )]
t
J t t
2 2 2
+
+ + â â
+
J t t
J
3
4
3 3
( )[cos(( ) ) cos(( ) )]
( )[co
β Ď Ď Ď Ď
β
c m c m
s
s(( ) ) cos(( ) )]
( )[cos((
Ď Ď Ď Ď
β Ď Ď
c m c m
c m
+ â â
+ +
4 4
5
5
t t
J )
) ) cos(( ) )]
t t
â â + }
Ď Ď
c m
5 ď (1.11)](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-24-320.jpg)
![Introduction to Biomedical Signals and Systems 1-9
At first inspection, this result appears quite messy. However, it is evident that the numer-
ical values of the Bessel terms tend to zero as n becomes large. For example, let β = 2Ďfd/
Ďm = 1, then J0(1) = 0.7852, J1(1) = 0.4401, J2(1) = 0.1149, J3(1) = 0.01956, J4(1) = 0.002477,
J5(1) = 0.0002498, J6(1) = 0.00002094, etc. Bessel constants Jn(1) for n ⼠4 contribute less
than 1% each to the ym(t) spectrum, so we can generally neglect them. Thus, the practical
BW of the FM carrier, ym(t), for β = 1 is ca. Âą3Ďm around the carrier frequency, Ďc. In
general, as β increases, so does the effective BW of the frequency modulated ym(t). For
example, when β = 5, the BW becomes ca. Âą8Ďm around Ďc, and when β = 10, the BW
required is ca. Âą14Ďm around Ďc (Clarke and Hess, 1971).
In the case of NBFM, β 1. Thus the modulated carrier can be written
y t A t K t
m c f m m
( ) cos[ ( / )sin( )]
= +
â
Ď Ď Ď
ι β
(1.12a)
y t A t t t
m c m c
( ) {cos( )cos[ sin( )] sin( )sin[ sin
= â
Ď Î˛ Ď Ď Î˛ (
( )]}
{cos( )( ) sin( )[ sin( )]}
{c
Ď
Ď Ď Î˛ Ď
m
c c m
t
A t t t
A
â
â
=
1
o
os( ) ( / )[cos(( ) ) cos(( ) )]}
Ď Î˛ Ď Ď Ď Ď
c c m c m
t t t
â â â +
2 (1.12b)
With the exception of signs of the sideband terms, the NBFM spectrum is very similar
to the spectrum of an AM carrier (Zeimer and Tranter, 1990); sum and difference fre-
quency sidebands are produced around a central carrier.
The spectrum of a DSBSCM signal is given by
ym(t) = Amocos(Ďmt) ⢠cos(Ďct) = (Amo/2)[cos((Ďc + Ďm)t) + cos((Ďc â Ďm)t)] (1.13)
that is, the information is contained in the two sidebands; there is no carrier. DSBSCM is
widely used in instrumentation and measurement systems. For example, it is the natural
result when a light beam is chopped in a photonic instrument such as a spectropho-
tometer, and also results when a Wheatstone bridge is given ac (carrier) excitation and
nulled, then one (or more) arm resistances is slowly varied in time around its null value.
DSBSCM is also present at the output of a linear variable differential transformer length
sensor as the core is moved in and out (Northrop, 1997).
Of equal importance in the discussion of modulation is the process of demodulation
or detection, in which the modulating signal, vm(t), is recovered from ym(t). There are,
in general, several ways to modulate a given type of modulated signal, and several ways
to demodulate it. In the case of AM (or SSBAM), the signal recovery process is called
detection.
There are several practical means of AM detection (Clarke and Hess, 1971, Chapter
10). One simple form of AM detection is to rectify and low-pass filter ym(t). Another
form of AM detection passes ym(t) through a square-law nonlinearity followed by a low-
pass filter. However, the square-law detector suffers from the disadvantage that it gener-
ates a second harmonic component of the recovered modulating signal. A third means](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-25-320.jpg)
![Dresden, and Berlin. All promised on a year from that day to meet
again at the door of the blessed temple, either with neophytes or
with brethren now absent. They then folded their cloaks to conceal
their elegant costumes, and silently dispersed by the shadowy walks
of the park.
Albert and Consuelo, guided by Marcus, went down the ravine to the
stream. Karl received them in his closed gondola, and took them to
the door of the pavilion. There they paused for a moment to
contemplate the majesty of the orb of day which rose in the sky.
Until now, Consuelo, when she replied to Albert had called him by
his true name; when, however, she was awakened from the musing
in which she seemed delighted to lose herself, as she pressed her
burning cheek on his shoulder, she could only say:
Oh Leverani!
[15]The harmonica, when first invented, created such a sensation
in Germany, that poetical imaginations fancied they heard in it
supernatural voices, evoked by the consecrators of certain
mysteries. This instrument, which, before it became popular, was
thought to be magical, was elevated by the adepts of German
theosophy, to the same honor with the lyre among the ancients,
and many other instruments among the primitive people of
Himalaya. They made it one of the hieroglyphic figures of their
mysterious iconography. They represented it under the form of a
fantastic chimera. The neophytes of secret societies, hearing it for
the first time after the rude shocks of their terrible ordeals, were
so much impressed by it that many of them fell into ecstacies.
They fancied they heard the song of invisibile powers, for both
the instrument and the performer were concealed from them
most carefully. There are extremely curious stories told of the
employment of the harmonica in the reception of adepts of
illuminatism.
EPILOGUE
Had we been able to procure faithful documents in relation to Albert
and Consuelo after their marriage, like those which have guided us](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-42-320.jpg)
![This magical symbol was, in fact, well adapted to the object of the
Invisibles. For many years, the new Templists hoped to make Saint
Graal accessible to all mankind. Albert toiled constantly to diffuse the
true ideas of his doctrine. He reached the highest grades of the
order, for we find the list of his titles showing that he had time
enough to reach them. Now all know that eighty-one months are
needed to pass through the twenty-three degrees of masonry, and
we think it certain that a much longer time was required for the
higher grades of Saint Graal. The number of masonic degrees are
now a mystery to no one; yet it will not be out of place here to
recount a few, as they paint the enthusiastic genius and smiling
fancy which presided over their first creation:
Apprentice and Master Mason, Secret and Perfect Master, Provost
and Judge, English and Irish Master, Master in Israel, Master Elect of
the Nine and Fifteen, Elect of the Unknown, Grand Master Architect,
Royal Arch, Grand Scotch Master of the Sublime or Master Masons,
Knight of the Sword, Prince of Jerusalem, Knight of Orient and
Occident, Rose-Cross of France, Heredom and Kilwinning, Grand
Pontiff or Sublime Scot, Architect of the Sacred Roof, Pontiff of
Jerusalem, Sovereign Prince of Masonry and Master ad vitam,
Naochite, Prince of Libon, Chief of the Tabernacle and Knight of the
Iron Serpent, Trinitarian Scot or Prince of Mercy, Grand Commander
of the Temple, Knight of the Gun, Patriarch of the Crusades, Grand
Master of Light, Knight Kadosch, Knight of the White Eagle and of
the Black Eagle, Knight of the PhĹnix and Knight of the Argonauts,
Knight of the Golden Fleece, Grand-Inspector-Inquisitor-
Commander-Sublime, Prince of the Royal Secret and Sublime Master
of the Luminous King,c.[16]
These titles, or at least the majority of them, we find connected with
the name of Albert Podiebrad, in the most illegible rolls of the
freemasons. There are also many less known; such as Knight of St
John, Sublime Johannite, Master of the New Apocalypse, Doctor of
the Gospel, and Elect of the Holy Ghost, Templist, Areopagite,
Magus, and Man of the People, Man-Pontiff, Man-King, and New-](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-45-320.jpg)
![man,c.[17] We have been surprised here to find some titles which
seemed anticipated from the illuminatism of Weishaupt: this
peculiarity, however, was explained at a later day, and will not, when
this story is concluded, need any explanation to our readers.
Amid this labyrinth of obscure factsâwhich, however, are profound,
and connected with the labor, success, and apparent extinction of
the Invisiblesâwe can with difficulty follow the adventurous story of
the young couple. Yet by supplying what we need by a prudent
imagination, the following is nearly the abridged commentary of the
chief events of their lives. The fancy of the reader will supply the
deficiency of the text, and following our experience, we doubt not
that the best dĂŠnoĂťements are those for which the reader and not
the narrator will be responsible.[18]
Probably, after leaving Saint Graal, Consuelo went to the little court
of Bareith, where the Margravine, sister of Frederick, had palaces,
gardens, kiosques, and cascades, in the same style as those of
Count Hoditz at Roswald, though less sumptuous and less expensive.
This intellectual princess had been married without a dower to a
very poor prince; and not long before she had worn robes with trains
of reasonable length, and had pages whose doublets were not
threadbare, her gardens, or rather her garden, to speak without
metaphor, was situated amid a beautiful country, and she indulged in
the Italian Opera in an antique temple Ă la Pompadour. The
margravine was fond of philosophyâthat is to say, she was a disciple
of Voltaire. The young hereditary margrave, her husband, was the
zealous head of a masonic lodge. I am not sure whether Albert was
connected with him, or whether his incognito was observed by the
secresy of the brothers, or whether he remained away from this
court and joined his wife afterwards. Certainly Consuelo had some
secret mission there. Perhaps, also, for the purpose of preventing
attention from being attracted to her husband, she did not live
publicly with him for some time. Their loves, then, had all the
attraction of mystery; and if the publicity of their union, consecrated
by the fraternal sanction of the Templists, seemed gentle and](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-46-320.jpg)
![last twenty years of the maestro's works are so completely
unknown, that we have in vain sought to discover the name of his
last productions. We only know Porporina had the principal part, that
she was most successful, and wrung tears from the whole court. The
empress was satisfied. On the night after this triumph, Porporina
received from an invisible messenger news that filled her with terror
and consternation. At seven in the morningâthat is to say, just at
the hour when the empress was awakened by the faithful valet
known as the sweeper[19] of her majesty, (for his duty consisted in
opening the blinds, making the fire, and cleaning the room, while
the empress was awaking,) Porporina, by eloquence or gold, passed
through every avenue of the palace, and reached the door of the
royal bed-chamber.
'My friend,' said she to the servant, 'I must throw myself at the
empress's feet. The life of an honest man is in danger. A great crime
will be committed in a few days, if I do not see her majesty at once.
I know that you cannot be bribed, but also know you to be generous
and magnanimous. Everybody says so. You have obtained favors
which the greatest courtiers dared not ask.'
'Kind heaven! my dear mistress! I will do anything for you,' said the
servant, clasping his hands and letting his duster fall.
'Karl!' said Consuelo. 'Thank God I am saved! Albert has a
protecting angel in the palace!'
'Albert! Albert!' said Karl. 'Is he in danger? Go In, madame, if I
should lose my place. God knows I shall be sorry; for I am enabled
to do some good and serve our holy cause better than I could do
anywhere else. Listen! The empress is a good soul, when she is not
a queen. Go in: you will be thought to have preceded me. Let those
scoundrels bear the burden of it, for they do not deserve to serve a
queen. They speak lies.
Consuelo went in; and when the empress opened her eyes, she saw
her kneeling at the foot of the bed.](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-53-320.jpg)
![About a month after the meeting at St. Graal, of which in his
memoirs Trenck says nothing, he was recaptured and imprisoned at
Magdenbourg, where he passed ten years of his life, loaded with
eighty pounds of irons. The stone to which he was bound bears the
inscription Here lies Trenck. All know his terrible fate, and the
sufferings he underwent, as also his wonderful attempts at escape,
and his incredible energy, which never left him, but which his
chivalric imprudence counteracted. His sister was subjected to the
cruelty of paying for the erection of a dungeon for him, because she
afforded him a refuge in his flight. Trenck's works of art in prison,
the wonderful engravings he made with the point of a nail on the tin
cups, which are allegories or verses of great beauty, are also well
known.[20] In fine, from his secret relations with the princess Amelia
âthe despair in which she wasted away, and her care to disfigure
her face by means of a corrosive fluid, which almost destroyed her
sightâthe deplorable state of health to which she reduced herself to
avoid marriageâthe remarkable change effected on her characterâ
the ten years of agony, which made him a martyr, and her an old
woman, ugly and malicious, instead of the angelic creature she was,
and would have been had she been happy[21]âthe misfortunes of
the lovers are historical; but they are generally forgotten when the
character of Frederick the Great is written. These crimes, committed
with such refined cruelty, are indelible spots on the character of that
monarch.
At length Trenck was released, as is known, by the intervention of
Maria Theresa, who claimed him as her subject. This was
accomplished by the influence of Karl, her majesty's valet. In relation
to the curious intrigues of this magnanimous man with his sovereign,
some of the strangest, most touching and pathetic pages of the
memoirs of the age have been written.
During the first part of the captivity of Trenck, his cousin, the famous
Pandour, a victim of truer though not less hateful accusations, died it
is said at Spielberg of poison. As soon as Trenck was free, the
Prussian came to ask for his cousin's vast estate; but Maria Theresa](https://image.slidesharecdn.com/23897-250417043817-7dff299c/85/Signals-and-systems-analysis-in-biomedical-engineering-Second-Edition-Northrop-59-320.jpg)
Signals and systems analysis in biomedical engineering Second Edition Northrop
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- 5. Signals and systemsanalysis in biomedical engineering Second Edition Northrop Digital Instant Download Author(s): Northrop, Robert B ISBN(s): 9781439812532, 1439812535 Edition: Second edition File Details: PDF, 15.17 MB Year: 2011 Language: english
- 7. SIGNALS and SYSTEMS ANALYSISin BIOMEDICAL ENGINEERING Second Edition
- 8. Published Titles Electromagnetic Analysisand Design in Magnetic Resonance Imaging, Jianming Jin Endogenous and Exogenous Regulation and Control of Physiological Systems, Robert B. Northrop Artificial Neural Networks in Cancer Diagnosis, Prognosis, and Treatment, Raouf N.G. Naguib and Gajanan V. Sherbet Medical Image Registration, Joseph V. Hajnal, Derek Hill, and David J. Hawkes Introduction to Dynamic Modeling of Neuro-Sensory Systems, Robert B. Northrop Noninvasive Instrumentation and Measurement in Medical Diagnosis, Robert B. Northrop Handbook of Neuroprosthetic Methods, Warren E. Finn and Peter G. LoPresti Angiography and Plaque Imaging: Advanced Segmentation Techniques, Jasjit S. Suri and Swamy Laxminarayan Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation, Robert B. Northrop Biomedical Image Analysis, Rangaraj M. Rangayyan An Introduction to Biomaterials, Scott A. Guelcher and Jeffrey O. Hollinger Foot and Ankle Motion Analysis: Clinical Treatment and Technology, Gerald F. Harris, Peter A. Smith, Richard M. Marks Introduction to Molecular Biology, Genomics and Proteomic for Biomedical Engineers, Robert B. Northrop and Anne N. Connor Signals and Systems Analysis in Biomedical Engineering, Second Edition, Robert B. Northrop 4PJOHLS95LTHU:LYPLZ,KP[VY
- 9. Robert B. Northrop CRCPress is an imprint of the Taylor Francis Group, an informa business Boca Raton London NewYork Michael R. Neuman, Series Editor
- 10. MATLABÂŽ and SimulinkÂŽare trademarks of the MathWorks, Inc. and are used with permission. The Math- Works does not warrant the accuracy of the text or exercises in this book. This bookâs use or discussion of MATLABÂŽ and SimulinkÂŽ software or related products does not constitute endorsement or sponsorship by the MathWorks of a particular pedagogical approach or particular use of the MATLABÂŽand SimulinkÂŽ software. CRC Press Taylor Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 Š 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4398-1253-2 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 11. Contents About the Author................................................................................................................ix 1 Introduction to Biomedical Signals and Systems 1.1 General Characteristics of Biomedical Signals..............................................1-2 1.2 General Properties of PSs................................................................................1-21 1.3 Chapter Summary............................................................................................1-24 2 Review of Linear Systems Theory 2.1 Linearity, Causality, and Stationarity..............................................................2-1 2.2 Analog Systems...................................................................................................2-2 2.3 Systems Described by Sets of ODEs...............................................................2-10 2.4 Linear System Characterization.....................................................................2-21 2.5 Discrete Signals and Systems..........................................................................2-39 2.6 Stability of Systems...........................................................................................2-61 2.7 Chapter Summary............................................................................................2-63 3 The Laplace Transform and Its Applications 3.1 Introduction........................................................................................................3-1 3.2 Properties of the Laplace Transform...............................................................3-3 3.3 Some Examples of Finding Laplace Transforms...........................................3-5 3.4 The Inverse Laplace Transform........................................................................3-6 3.5 Applications of the Laplace Transform...........................................................3-7 3.6 Chapter Summary............................................................................................3-19 4 Fourier Series Analysis of Periodic Signals 4.1 Introduction........................................................................................................4-1 4.2 Properties of the FS............................................................................................4-3 4.3 FS Examples.........................................................................................................4-4 4.4 Chapter Summary..............................................................................................4-9 5 The Continuous Fourier Transform 5.1 Introduction........................................................................................................5-1 5.2 Properties of the CFT.........................................................................................5-2
- 12. vi Contents 5.3 ADCand the Sampling Theorem.....................................................................5-5 5.4 The Analytical Signal and the HT...................................................................5-8 5.5 MTF in Imaging. ...............................................................................................5-18 5.6 Chapter Summary............................................................................................5-25 6 The Discrete Fourier Transform 6.1 Introduction........................................................................................................6-1 6.2 The CFT, ICFT, DFT, and IDFT.......................................................................6-2 6.3 Data Window Functions. ...................................................................................6-7 6.4 The FFT..............................................................................................................6-13 6.5 Chapter Summary............................................................................................6-21 7 Introduction to Joint Time-Frequency Analysis of Biomedical Signals 7.1 Introduction........................................................................................................ 7-1 7.2 The Short-Term Fourier Transform.................................................................7-4 7.3 The Gabor and Adaptive Gabor Transforms..................................................7-6 7.4 The WignerâVille and Pseudo-Wigner Transforms.....................................7-7 7.5 Cohenâs General Class of JTF Distributions................................................. 7-11 7.6 Introduction to JTFA Using Wavelets........................................................... 7-14 7.7 Applications of JTFA to Physiological Signals............................................. 7-21 7.8 JTFA Software...................................................................................................7-32 7.9 Chapter Summary............................................................................................7-34 8 Introduction to the Analysis of Stationary Noise and Signals Contaminated with Noise 8.1 Introduction........................................................................................................8-1 8.2 Noise Descriptors and Noise in Systems.........................................................8-2 8.3 Calculation of Noise Descriptors with Finite Discrete Data.....................8-30 8.4 Signal Averaging and Filtering for SNR Improvement..............................8-32 8.5 Introduction to the Application of Statistics and IT to Genomics.......... 8-46 8.6 Chapter Summary........................................................................................... 8-66 9 Basic Mathematical Tools Used in the Characterization of Physiological Systems 9.1 Introduction........................................................................................................9-1 9.2 Some General Properties of PSs.......................................................................9-1 9.3 Some Properties of Nonlinear Systems...........................................................9-5 9.4 Physical Factors Determining the Dynamic Behavior of PSs.....................9-7 9.5 Means of Characterizing PSs..........................................................................9-14 9.6 Chapter Summary........................................................................................... 9-40 10 Introduction to the Mathematics of Tomographic Imaging 10.1 Introduction......................................................................................................10-1 10.2 Algebraic Reconstruction................................................................................10-5 10.3 The Radon Transform......................................................................................10-8 10.4 The Fourier Slice Theorem............................................................................10-14
- 13. Contents vii 10.5 FilteredBack-Projection Algorithm...............................................................10-15 10.6 Chapter Summary. ............................................................................................10-18 11 Introduction to the Analysis of Nonlinear Biochemical Systems and Biochemical Oscillators 11.1 Introduction: Some General Properties of Nonlinear Systems.................11-1 11.2 All Living Systems Are Nonlinear.................................................................11-8 11.3 Parametric Regulation in Nonlinear Biological Systems...........................11-8 11.4 Approaches to Nonlinear Analysis: The Phase Plane............................... 11-14 11.5 Chaos, Stability, and Limit Cycles in Nonlinear Biological Systems. .....11-19 11.6 Chapter Summary..........................................................................................11-66 12 Introduction to Complex Systems in Biology and Medicine 12.1 Introduction to Complex Systems.................................................................12-1 12.2 When Is a System Complex?...........................................................................12-2 12.3 Some Examples.................................................................................................12-3 12.4 Properties of Complex Systems: Chaos and Tipping Points......................12-4 12.5 The Law of Unintended Consequences........................................................ 12-6 12.6 Why Study Complex Systems?. .......................................................................12-9 12.7 Human Responses to Complexity................................................................12-10 12.8 Complex Systems Engineering.....................................................................12-13 12.9 Some Complex Physiological Regulatory Systems....................................12-15 12.10 Structure and Function: Some Examples of Complex Physiological Regulatory Systems and Their Simplified Models.....................................12-21 12.11 Examples of When Complex Physiological Systems Fail.........................12-78 12.12 Some Approaches to Dealing with Complexity in an Organized Manner............................................................................................................ 12-86 12.13 Chapter Summary..........................................................................................12-93 Appendix A......................................................................................................Appendix A-1 Appendix B...................................................................................................... Appendix B-1 Appendix C......................................................................................................Appendix C-1 Appendix D......................................................................................................Appendix D-1 Glossary...................................................................................................................Glossary-1 Bibliography.................................................................................................Bibliography-1 Index............................................................................................................................... Index-1
- 15. ix About the Author RobertB. Northrop was born in White Plains, NY, in 1935. After graduating from Staples High School in Westport, CT, he majored in electrical engineering (EE) at the Massachusetts Institute of Technology, graduating with a bachelorâs degree in 1956. At the University of Connecticut (UCONN), he received a masterâs degree in systems engi- neering in 1958. As the result of a long-standing interest in physiology, he entered a PhD program at UCONN in physiology, doing research on the neuromuscular physiology of molluskan catch muscles. He received his PhD in 1964. In 1963, he rejoined the UCONN EE department as a lecturer, and was hired as an assistant professor of EE in 1964. In collaboration with his PhD advisor, Dr. Edward G. Boettiger, he secured a 5-year training grant in 1965 from the National Institute of General Medical Sciences [National Institutes of Health (NIH)], and started one of the first, interdisciplinary, biomedical engineering (BME) graduate training programs in New England. UCONN currently awards MS and PhD degrees in this field of study, as well as BS degrees in engineering under the BME area of concentration. Throughout his career, Dr. Northropâs research interests have been broad and inter- disciplinary and have been centered on BME. He has conducted sponsored research on the neurophysiology of insect and frog vision and devised theoretical models for visual neural signal processing. He has also conducted sponsored research on electrofishing and developed, in collaboration with Northeast Utilities, effective, working systems for fish guidance and control in hydroelectric plant waterways on the Connecticut River at Holyoke, MA, using underwater electric fields. Another area of his sponsored research has been in the design and simulation of nonlinear, adaptive, digital controllers to regulate in vivo drug concentrations or physi- ological parameters, such as pain, blood pressure, or blood glucose in diabetics. An out- growth of this research has led him to develop mathematical models for the dynamics of the human immune system, which were used to investigate theoretical therapies for autoimmune diseases, cancer, and human immunodeficiency virus infection. Biomedical instrumentation has also been an active research area for Dr. Northrop and his graduate students: an NIH grantâsupported study on the use of the ocular pulse to detect obstructions in the carotid arteries. Minute pulsations of the cornea from arte- rial circulation in the eyeball were sensed using a no-touch, phase-locked, ultrasound
- 16. x About theAuthor technique. Ocular pulse waveforms were shown to be related to cerebral blood flow in rabbits and humans. More recently, Dr. Northrop addressed the problem of noninvasive blood glucose measurement for diabetics. Starting with a Phase I Small Business Innovation Research grant, he developed a means of estimating blood glucose by reflecting a beam of polar- ized light off the front surface of the lens of the eye, and measuring the very small optical rotation resulting from glucose in the aqueous humor, which in turn is proportional to blood glucose. As an offshoot of techniques developed in micropolarimetry, he devel- oped a magnetic sample chamber for glucose measurement in biotechnology applica- tions. The water solvent was used as the Faraday optical medium. He has written eight textbooks: Analog Electronic Circuits; Introduction to Instru mentation and Measurements (2nd ed.); Endogenous and Exogenous Regulation and Control of Physiological Systems; Dynamic Modeling of Neuro-Sensory Systems; Signals and Systems Analysis in Biomedical Engineering (2nd ed.); Noninvasive Instrumentation and Measurements in Medical Diagnosis; Analysis and Application of Analog Electronic Circuits in Biomedical Engineering; and Introduction to Molecular Biology, Genomics Proteomics for Biomedical Engineers (with Anne N. Connor). His current research interest lies in complex systems. Dr. Northrop was on the electrical and computer engineering faculty at UCONN until his retirement in June 1997. Throughout this time, he was director of the BME graduate program. As emeritus professor, he still teaches courses in BME, writes texts, sails, and travels. He lives in Chaplin, CT, with his wife, and a smooth fox terrier. MATLABŽ is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508 647 7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: www.mathworks.com
- 17. 1-1 1 Introduction to Biomedical Signalsand Systems In this second edition of Signals and Systems Analysis in Biomedical Engineering, the author has preserved and updated Chapters 1 through 10, dealing with the powerful mathematical tools of linear systems analysis applied to physiological and biochemical systems. Chapter 7 deals with joint time-frequency (JTF) analysis of linear, time-variant (nonstationary) systems and emphasizes the utility of this method in the description and analysis of nonstationary biomedical signals. Chapter 8 is all about noise, how to describe it, and how it propagates through linear and nonlinear systems. Chapter 8 also describes signal averaging, the lock-in amplifier, and data âscrubbing,â and introduces the application of statistics to genomic systems. In Chapter 9, physiological systems (PSs) are considered as examples of nonstationary nonlinear systems, and analysis of PSs are reviewed by describing function analysis and Gaussian noise-based techniques. In Chapter 11, an all-new chapter, nonlinear biochemical systems, mass action equa- tions for chemical kinetics, and parametric regulation of biochemical systems are intro- duced. The phase plane is presented as a means of describing the dynamics of nonlinear biochemical systems, including chemical oscillators. Transport lags are shown to be destabilizing in both linear and nonlinear, negative- and positive-feedback systems. Several important models for biochemical oscillators (BCOs) are presented as well as models for the synchronization of large groups of cellular BCOs. This chapter includes the following topics: general properties of nonlinear systems, parametric regulation in nonlinear biological systems, approaches to nonlinear analysis, the phase plane, as well as chaos, stability, and limit cycles in nonlinear biological systems. Because many biochemical, physiological, and ecological systems are complex, I have written a new Chapter 12 to introduce the reader to the analysis and modeling strategies for complex systems. This chapter includes various topics relating to complex systems. In addition to the 12 chapters described above, this text has extensive appendices: SimnonÂŽ Programs Used in Chapters 11 and 12 (Appendix A), How to Use Root Locus to Determine the Stability of SISO Linear Systems (Appendix B), Signal Flow Graphs and Masonâs Rule (Appendix C), and Computational Tools for Biomedical Signal Processing and Systems Analysis (Appendix D). There is also an extensive Glossary, which includes a list of widely used acronyms in biochemical, physiological, and genomic systems, as
- 18. 1-2 Signals andSystems Analysis in Biomedical Engineering well as important, defined terms in the same areas. These are supplemented by a robust listing of sources (both bound and Internet) in the References as well as a detailed Index. To extract maximum information from this text, the reader should have had college courses in algebra, calculus, and introductory differential equations, as well as basic courses in physics and chemistry. Background courses in engineering systems analysis, biochemistry, and introductory cell biology (or equivalent) are also important. 1.1â General Characteristics of Biomedical Signals 1.1.1âIntroduction In this chapter, we introduce key definitions and concepts about biomedical signals and systems. The concepts of signal bandwidth (BW), stationarity, nonstationarity, and the diversity of biomedical and physiological signal modalities are presented. In Section 1.1.6, we introduce a number of important schemes for modulation and demodulation of signals. System is defined for our purposes in Section 1.2.1. The concepts of system lin- earity, nonlinearity, and stationarity are set forth in Section 1.2.2. The general properties of PSs are enumerated in Section 1.2.3, and discrete signals and systems are described in Sections 1.1.4 and 1.2.4, respectively. Intuitively, we think of a biomedical signal as some natural (endogenous) or man- made (exogenous), continuous, time-varying record that carries information about the internal functioning of a biomedical system (system is defined below). A signal can be a system input, or a system output as the result of one or more inputs, or can carry information about a system state variable. In PSs, a signal can be an electrical potential, a force, a torque, a length, a pressure, a chemical concentration of ions or of molecules, including hormones or cytokines. A signal can also be in the form of nerve impulses that lead to the release of neurotransmitters or hormones, or the contraction of muscles (force, length). In an optical system, a signalâs intensity can vary with position (x,y,z) as well as with time t and the wavelength of light Îť. A biomedical signal is generally acquired by a sensor or a transducer, and is converted to a proportional voltage or current for processing and storage, or is recorded directly from an electrode in contact with tissues. Naturally acquired, endogenous signals are continuous (analog) because nature viewed on a macroscale is continuous. One example of a biomedical signal is the electrocardiogram (ECG) waveform recorded from the skin surface; another example is the blood velocity in an artery measured by continuous wave (CW) Doppler ultrasound using piezoelectric transducers. Biomedical signals are gen- erally noisy because of interfering signals from the body, noise picked up from the envi- ronment, and noise arising in electrodes and from signal conditioning amplifiers. As noted above, hormone concentrations are a type of physiological signal; they carry information about the health of the body. A hormone is generally quantified by its con- centration in a compartment, such as the blood, urine, cerebrospinal fluid, or extracel- lular fluid. Hormones are generally used physiologically as control substances as part of natural, closed-loop regulatory systems. For example, the protein hormone insulin,
- 19. Introduction to BiomedicalSignals and Systems 1-3 secreted by the pancreatic beta cells, acts on cells carrying insulin receptor molecules on their cell membranes to increase the rate of diffusion of glucose from the blood or extra- cellular fluid into those cells. Insulin also causes the liver to uptake glucose and store it intracellularly as the polymer, glycogen (Northrop and Connor, 2008). After the initial acquisition and conditioning, an analog biomedical signal may be converted to discrete form by (periodic) analog-to-digital (A/D) conversion. In discrete form, a signal can be more easily stored, and can be further numerically processed by discrete filtering, or other nonlinear, discrete transforms. Discrete signals are very important because of the ability of digital signal processing algorithms to reveal their characteristics in the time, frequency, and JTF domains. Both continuous and discrete signals can be characterized by a number of functions, which will be described in later sections of this text. For example, if the signal is random in nature, the mean, standard deviation, root-mean-square value, probability density function, the joint probability density function, and the auto- and cross-correlation functions can serve as descriptors. In the frequency domain, we have the auto- and cross-power density spectra, and time-frequency spectrograms. Deterministic continu- ous and discrete signals can be characterized by certain transforms: Fourier, Laplace, Hilbert, Radon, Wigner, etc., which will be described in this text. Because biomedical signals are generally contaminated with noise, their signal-to- noise ratios (SNRs) can often be improved via filtering by analog or discrete filters, or by signal averaging. How filtering and averaging can improve the SNRs is explored in Section 8.4. 1.1.2â Signals from PSs Endogenous biomedical signals from PSs are acquired for a number of reasons: for pur- poses of diagnosis, for postsurgical/posttrauma intensive care monitoring, for neona- tal monitoring, to guide therapy, and for research. Such signals include, but certainly are not limited to, ECG, electroencephalogram (EEG), electromyogram (EMG), nerve action potentials, muscle force, blood pressure, temperature, respiration, hemoglobin psO2, blood pCO2, blood glucose concentration, the concentrations of various hormones and ions in body fluids, the density of various blood cells and antibodies, heart sounds, breath sounds, otoacoustic emissions, etc. Signals can also be rates or frequencies derived from other signals (e.g., heart rate, respiratory rate). In general, the BWs (equivalent frequency content) of endogenous physiological sig- nals can range from nearly dc (ca. 1.2 Ă 10â5 Hz or 12 ÂľHz, a period of 24 h) to sev- eral kilohertz (Stefanovska, 2007). Many hormones, neuromodulators, and behavioral parameters have circadian (24 h) cycles. Cardiac pumping spans 0.6â2 Hz, respiratory effort lies in a 0.145- to 0.6-Hz BW. Other neurogenic mechanical rhythms in the diges- tive system have even lower BWs. Bioelectric signals recorded from the skin surface have higher BWs: these include the EEG, electrooculography, electrocorticography, EMG, ECG, etc. Sounds emitted by organs recorded through the skin with microphones include heart valves and blood vessel turbulence sounds, sounds from the respiratory tract (e.g., rales), and sounds from arthritic joints. Such sounds generally require an audio signal conditioning BW, typically from 10 Hz to 3 kHz or so. Details on the BWs
- 20. 1-4 Signals andSystems Analysis in Biomedical Engineering required to condition and reproduce physiological signals and their amplitudes are given by Northrop (2002). Signals from PSs have another important property that we will encounter throughout this textâthey are nonstationary. This means that the physical, biochemical, and physi- ological processes contributing to their origins change in time. Take, for example, the arterial blood pressure (ABP), which has a waveform with the almost-periodic rhythm of the heartbeat. However, many physiological factors affect the heart rate and the heartâs stroke volume; the bodyâs vasomotor tone is under control by the autonomic nervous system. The time of day (diurnal rhythm), emotional state, the blood concentration of hormones such as epinephrine and norepinephrine, blood pH, exercise, respiratory rate, diet, drugs, blood volume, and water intake all affect the ABP. Over a short interval of several minutes, the ABP waveform is relatively invariant in shape and period, and can be said to be short-term stationary (STS). In fact, many physiological signals can be treated as STS; others change so rapidly that the STS assumption is not valid. For example, certain breath sounds that change from breath to breath should be treated as nonstationary. 1.1.3â Signals from Man-Made Instruments In a number of instances, energy (photons, sound, radioactivity) is put into the body in order to measure physiological parameters and structures. For example, CW Doppler ultrasound, used to estimate blood velocity in arteries and veins, generally operates from 3 to 10 MHz (it has a sinusoidal waveform). Thus, the transducers, filters, ampli- fiers, mixers, etc. used in a CW Doppler system must operate in the 3- to 10-MHz range. The blood velocity Doppler signal itself lies in the audio frequency range (Northrop, 2002). An example of a prototype medical measurement system used angle-modulated, lin- early polarized light (LPL) to measure the optical rotation caused by glucose in the aque- ous humor (AH) of the eye. The typical wavelength of the LPL used is 512 nm; the angle modulation is done by a Faraday rotator at ca. 2 kHz. The glucose concentration in the AH follows the blood glucose concentration, which varies slowly over a 24-h period with peaks and valleys associated with meals. I estimate its BW to be from microhertz to ca. 6 cycles/h (1.7 mHz) (Northrop, 2002). AH glucose concentration is an example of a very important physiological signal with a very low BW. Indeed, many hormonal systems in the body, such as that controlling the secretion rate of the hormone melatonin, have diurnal (24 h) periods, as do parameters such as body temperature, and the concentra- tions of certain ions in the blood, etc. Clearly, living systems have slow rhythms as well as fast. In certain imaging systems, the radioactive decay of radioisotopes provides the signals that are processed to form a tomographic (slice) image. Tens of thousands of random decay events/second can be processed to form a positron emission tomography image over a counting period of tens of minutes. The times of individual decay events are completely random, however, generally following a Poisson probability distribution (Northrop, 2002).
- 21. Introduction to BiomedicalSignals and Systems 1-5 In all medical imaging systems, it is appropriate to describe image resolution in terms of the spatial frequency response of the system. The spatial frequency response describes how an imaging system reproduces an object that is a one-dimensional (1-D), spatial sinusoid in intensity as a 1-D spatial sinusoidal image. The normalized, 1-D, spa- tial frequency response is called the modulation transfer function (MTF) (Spring and Davidson, 2001). The higher the cutoff frequency of the MTF, the greater the objectâs detail that is visible in the image. (The MTF is discussed in detail in Section 5.5.) 1.1.4â Introduction to Discrete Signals A discrete signal can be formed by periodically sampling an analog signal, x(t), and con- verting each sample to a (digital) number, x*(kT). The sequence of numbers (samples) is considered only to exist at the sampling instants (t = kT), and thus can be represented by a periodic train of impulses or delta functions, each with an area equal to the value of x(t) at the sampling instant, x(kT). Mathematically, this can be stated as: x kT x t t kT k *( ) ( ) ( ) = â = â â δ 0 (1.1) where T is the sampling period and δ(Îź) only exists for zero argument. Note that the sampling and A/D process is equivalent to impulse modulation, that is, multiplication of the analog signal, x(t), by a train of unit impulses occurring at a frequency fs = 1/T. The properties of x*(kT) in the frequency domain will be discussed in detail in Section 4.6. Note that other notations exist for the members of a sequence of sampled data, for example, x(k) or xk (where k denotes the kth sample from a local time origin). 1.1.5â Several Ways to Describe Signals There are many ways to characterize 1-D and 2-D signals, that is, signals that vary as a function of time, or spatial dimensions, x and y. A signal can be described in terms of its statistical amplitude properties, its frequency properties, and, if nonstationary, its JTF properties. To begin this section, consider a stationary, continuous signal, u(t). In fact, let us collect an ensemble of N signals, {uk(t)}, all recorded from the same source under identical conditions, but at different times. The signal itself can be a voltage (e.g., an ECG record), a chemical concentration (e.g., calcium ions in the blood), a fluid pressure (e.g., blood pressure), a sound pressure (e.g., the first heart sound), etc. There are many descriptors we can apply to a signal that will characterize it quantita- tively. For example, the signalâs expected or true mean value, E(u). E(u) can be estimated by the finite time average ĹŤ of one, typical ensemble member u T u t t T = ⍠1 0 ( )d (1.2)
- 22. 1-6 Signals andSystems Analysis in Biomedical Engineering In the limit as T â â, u â E(u), the âtrueâ mean of u. Suppose the ensemble consists of N responses to a sensory stimulus given repetitively to an animal. We can then pick a time t1 after each stimulus, and find the sample mean of the ensemble, {uk(t1)} u t N u t k k N ( ) ( ) 1 1 1 1 = = â (1.3) As will be shown in Section 8.4, the ensemble average shown in Equation (1.3), with 0 ⤠t1 ⤠T, is a means of extracting a consistent evoked response signal from additive random noise with zero mean. Another measure of a 1-D signal is its intensity. Definitions of intensity can vary, depending on the type of signal. For example, both sound intensity and light intensity both have the units of power (W) per unit area (m2). The power can be instantaneous power or average power. For sound, the average intensity is given by I p c pu pcu = = = 2 2 ( ) Ď W/m2 (1.4) where p is p(t), the sound pressure expressed in N/m2 (=Pa) over atmospheric pressure, u(t) is the particle velocity (m/s), Ď is the density of the medium (kg/m3), and c is the velocity of propagation (m/s). That is, I is proportional to the mean squared (ms) sound pressure [note that the overbars in Equation (1.4) denote taking the time average]. When p(t) = Posin(2Ďft), it is easy to show that the ms sound pressure is Po 2/2, that is, ½ the peak pressure squared. For light, radiant intensity is also called the irradiance (W/m2) incident on a surface. A voltage signal, such as an EEG recording, can be described by its average power, given by its mean squared voltage divided by R = 1 P T v t t T T = ââ ⍠lim ( ) 1 2 0 d W (1.5) In Chapter 7, it will be shown that nonstationary signals can be characterized by JTF plots. Special transforms are used to decompose a signal, u(t), into its spectral compo- nents as a function of time. A JTF spectrogram gives signal mean squared volts (or rms volts) as a function of time and frequency, creating a 2-D or 3-D display. In Chapter 8, we present many statistical parameters that can be attributed to sig- nals, in addition to their sample means. These include the signalâs probability density function, and statistical measures such as its variance. Another important measure of a signal is its frequency spectrum, also known as power density spectrum, with units of mean squared volts (or ms units) per hertz. A signalâs root power spectrum is also used to describe noisy signals; it is expressed in root mean squared volts per root hertz. The root power spectrum is simply the square root of the power density spectrum. The power spectrum describes the contribution to the total power of u(t) at each incremental frequency. It implies a superposition of spectral power.
- 23. Introduction to BiomedicalSignals and Systems 1-7 1.1.6â Introduction to Modulation and Demodulation of Physiological Signals In general, modulation is a process whereby a signal of interest is combined mathemati- cally with a high-frequency carrier wave to form a modulated carrier suitable for trans- mission to a receiver/demodulator where the signal is recovered by demodulation, then processed and stored. Recorded physiological signals are modulated for two important reasons: (1) for robust transmission by wire or coaxial cable, fiber optic cable, or radio (telemetry by electro- magnetic waves) from the recording site to the site where the signal will be processed and stored; (2) for effective data storage. As an example, in biotelemetry, physiological signals such as ECG and blood pressure modulate a carrier that is sent as a frequency modulation (FM) radio signal to a remote receiver, where the signals are demodulated, digitized, filtered, analyzed, and stored. Modulation generally involves encoding a high-frequency carrier wave, which can be a continuous sine wave or a square wave (logic signal) with the low-frequency signal being measured, vm(t). There are five major types of modulation involving sinusoidal carrier waves: (1) ampli- tude modulation (AM), (2) single-sideband AM (SSBAM), (3) FM, (4) phase modu lation (PhM),and(5)double-sideband,suppressed-carriermodulation(DSBSCM).Modulation can also be done using a square wave (or TTL) carrier, and can be by FM, PhM, or delta modulation (DM), or can use pulse position modulation or pulse width modulation at constant frequency. AM, FM, and DSBSCM are expressed mathematically below for a sinusoidal carrier. m(t) is the normalized modulating signal in each case, whereas vm(t) is the actual physiological signal. The maximum frequency of m(t) must be Ďc, the carrier frequency. The normalized modulating signal, m(t), is defined by m t v t V m t ( ) ( ) , ( ) = ⤠⤠m mmax 0 Normalized modulatin 1 g g signal. (1.6a) Thus, we have y t A m t t m c AM ( ) [ ( )]cos( ) = + 1 Ď (1.6b) y t A t K m t t t m c f d FM ( ) cos[ ( ) ] = + âŤ Ď (1.6c) y t A t K m t m c p PhM ( ) cos ( ) = +     Ď (1.6d) y t Am t t m c DSBSCM ( ) ( )cos( ) = Ď (1.6e) It is instructive to examine the frequency spectrums of modulated signals. The extent of the spectrum of ym(t) determines the required signal processing BW. For illustrative purposes, let m(t) be a pure cosine wave, [mocos(Ďmt)], 0 mo ⤠1. Thus, by trig identity, the AM signal can be rewritten as
- 24. 1-8 Signals andSystems Analysis in Biomedical Engineering ym(t) = Acos(Ďct) + (Amo/2)[cos((Ďc + Ďm)t) + cos((Ďc â Ďm)t)] (1.7) We see that the AM signal has a carrier component and two sidebands, each spaced by the amount of the modulating frequency above and below the carrier frequency. In SSBAM, a sharp cutoff filter is used to eliminate either the upper or lower sideband; the information in both sidebands is redundant, so removing one sideband means a lower BW is required to transmit the SSBAM signal. FM and PhM are subsets of angle modulation. FM can be further classified as broad- band or narrowband FM (NBFM). In FM, Kf ⥠2Ďfd, where fd denotes the frequency deviation constant. In NBFM, fd/fmmax 1; (fmmax is the highest expected frequency in m(t), which is BW-limited. Unlike AM, the frequency spectrum of an FM carrier is more difficult to derive. Using m(t) = mocos(Ďmt), we can write the FM carrier as y t A t K m t m c f m o m ( ) cos ( / ) sin( ) = +       Ď Ď Ď Îą β (1.8) Using the trig identity, cos(Îą + β) = cos(Îą)cos(β) â sin(Îą)sin(β), Equation (1.8) can be written as y t A t K m t A t m c f m o m c ( ) cos( )cos[( / ) sin( )] sin( ) = â Ď Ď Ď Ď s sin[( / ) sin( )] K m t f m o m Ď Ď (1.9) Now the cos[(Kf/Ďm)mosin(Ďmt)] and sin[(Kf/Ďm)mosin(Ďmt)] terms can be expressed as two Fourier series whose coefficients are ordinary Bessel functions of the first type and argument β (Clarke and Hess, 1971); note that β ⥠mo2Ďfd/Ďm cos[ sin( )] ( ) ( )cos( ) β Ď Î˛ β Ď m m t J J n t n n = + = â â 0 2 1 2 2 (1.10a) sin[ sin( )] ( )sin[ ( ) ] β Ď Î˛ Ď m m t J n t n n = + + = â â 2 2 1 2 1 0 (1.10b) The two Bessel sum relations for cos[β sin(Ďmt)] and sin[β sin(Ďmt)] can be recombined with Equation (1.9) using the trig identities, cosxcosy = ½[cos(x + y) + cos(x â y)], and sinxsiny = ½[cos(x â y) â cos(x + y)], and we can finally write for the FM carrier spec- trum, letting mo = 1 y t A J t J t m c c m ( ) ( )[cos( ) ( )cos(( ) ) cos(( = + + â 0 1 β Ď Î˛ Ď Ď Ďc c m c m c m â { + + â â Ď Î˛ Ď Ď Ď Ď ) )] ( )[cos(( ) ) cos(( ) )] t J t t 2 2 2 + + + â â + J t t J 3 4 3 3 ( )[cos(( ) ) cos(( ) )] ( )[co β Ď Ď Ď Ď Î˛ c m c m s s(( ) ) cos(( ) )] ( )[cos(( Ď Ď Ď Ď Î˛ Ď Ď c m c m c m + â â + + 4 4 5 5 t t J ) ) ) cos(( ) )] t t â â + } Ď Ď c m 5 ď (1.11)
- 25. Introduction to BiomedicalSignals and Systems 1-9 At first inspection, this result appears quite messy. However, it is evident that the numer- ical values of the Bessel terms tend to zero as n becomes large. For example, let β = 2Ďfd/ Ďm = 1, then J0(1) = 0.7852, J1(1) = 0.4401, J2(1) = 0.1149, J3(1) = 0.01956, J4(1) = 0.002477, J5(1) = 0.0002498, J6(1) = 0.00002094, etc. Bessel constants Jn(1) for n ⼠4 contribute less than 1% each to the ym(t) spectrum, so we can generally neglect them. Thus, the practical BW of the FM carrier, ym(t), for β = 1 is ca. Âą3Ďm around the carrier frequency, Ďc. In general, as β increases, so does the effective BW of the frequency modulated ym(t). For example, when β = 5, the BW becomes ca. Âą8Ďm around Ďc, and when β = 10, the BW required is ca. Âą14Ďm around Ďc (Clarke and Hess, 1971). In the case of NBFM, β 1. Thus the modulated carrier can be written y t A t K t m c f m m ( ) cos[ ( / )sin( )] = + â Ď Ď Ď Îą β (1.12a) y t A t t t m c m c ( ) {cos( )cos[ sin( )] sin( )sin[ sin = â Ď Î˛ Ď Ď Î˛ ( ( )]} {cos( )( ) sin( )[ sin( )]} {c Ď Ď Ď Î˛ Ď m c c m t A t t t A â â = 1 o os( ) ( / )[cos(( ) ) cos(( ) )]} Ď Î˛ Ď Ď Ď Ď c c m c m t t t â â â + 2 (1.12b) With the exception of signs of the sideband terms, the NBFM spectrum is very similar to the spectrum of an AM carrier (Zeimer and Tranter, 1990); sum and difference fre- quency sidebands are produced around a central carrier. The spectrum of a DSBSCM signal is given by ym(t) = Amocos(Ďmt) ⢠cos(Ďct) = (Amo/2)[cos((Ďc + Ďm)t) + cos((Ďc â Ďm)t)] (1.13) that is, the information is contained in the two sidebands; there is no carrier. DSBSCM is widely used in instrumentation and measurement systems. For example, it is the natural result when a light beam is chopped in a photonic instrument such as a spectropho- tometer, and also results when a Wheatstone bridge is given ac (carrier) excitation and nulled, then one (or more) arm resistances is slowly varied in time around its null value. DSBSCM is also present at the output of a linear variable differential transformer length sensor as the core is moved in and out (Northrop, 1997). Of equal importance in the discussion of modulation is the process of demodulation or detection, in which the modulating signal, vm(t), is recovered from ym(t). There are, in general, several ways to modulate a given type of modulated signal, and several ways to demodulate it. In the case of AM (or SSBAM), the signal recovery process is called detection. There are several practical means of AM detection (Clarke and Hess, 1971, Chapter 10). One simple form of AM detection is to rectify and low-pass filter ym(t). Another form of AM detection passes ym(t) through a square-law nonlinearity followed by a low- pass filter. However, the square-law detector suffers from the disadvantage that it gener- ates a second harmonic component of the recovered modulating signal. A third means
- 26. Discovering Diverse ContentThrough Random Scribd Documents
- 27. had been wellpaid, without being aware to whom he was under obligations. As Mozart, just before he died, was to receive the sublimest inspiration for a requiem mysteriously required, old Porpora regained all his youthful genius to write an epithalamium the poetic mystery of which had aroused his imagination. In the very first passage, Consuelo remembered her old master's style, and looking around, she sought for her adopted father among the choristers. Among those who were its interpreters, Consuelo recognised many friendsâFrederick Von Trenck, Porporino, Young Benda, Count Golowkin, Schubert, the Chevalier D'Eon, (whom she had met at Berlin, but of whose sex she, like all Europe, was ignorant,) the Count St. Germain, the Chancellor Coccei, (husband of Barberini,) the bookseller Nicolai, Gottlieb, (whose voice predominated above all the others,) and Marcus, whom a gesture of Wanda pointed out to her, and whom, from some instinctive sympathy, she had recognised in her guide, and who discharged the functions of putative father or sponsor. All the Invisibles had opened and thrown back on their shoulders their long melancholy robes, and a neat white costume, which was elegant and simple, relieved by a chain of gold, to which hung the insignia of the order, gave to the whole scene the appearance of a festival. Their masks hung around their wrists, ready to be replaced at the slightest signal of the watcher, who was on the dome of the edifice. The orator who communicated between the adepts and chief of the order, unmasked, and came to wish the couple happiness. This was the Duke of ****, who had consecrated his enthusiasm and immense fortune to the undertaking of the Invisibles. He was owner of their place of meeting, and at his house Wanda and Albert had frequent interviews, unseen by any profane eyes. This house was also the head-quarters of the operations of the chief of the order, though there were other places at which there were smaller gatherings. Initiated into all the secrets of the order, the duke acted with and for them. He did not betray their incognito, but assumed all the dangers of the enterprise, being himself their visible means of contact with the members of the association.
- 28. When Albert andConsuelo had exchanged the gentle evidence of joy and affection with their brethren, all took their places, and the duke having resumed his functions of brother orator, thus spoke, as with crowns of flowers they knelt before the altar:â Very dear and beloved childrenâIn the name of the true Godâall power, love, and intelligence; and after him, in the name of the three virtues which reflect divinity in the human soul, Activity, Charity, and Justice, translated in effect by our formula, Liberty, Fraternity, and Equality; finally, in the name of the tribunal of the Invisibles, devoted to the triple duty of zeal, faith, and studyâthat is, to the triple search of the three divine moral and political virtues âAlbert Podiebrad, Consuelo Porporina, I pronounce the ratification and confirmation of the marriage already contracted before God and your kindred, and before a priest of the Christian religion, at the Giants' Castle, 175â. Three things however were wanting: first, the absolute wish of the wife to live with the husband, seemingly in extremis; second, the sanction of a moral and religious society received and acknowledged by the husband; third, the consent of a person here present, the name of whom I am not permitted to mention, but who is closely bound to one of the party by the ties of blood. If now these three conditions be fulfilled, and neither of you have aught to object, join your hands, and, rising, call on heaven to testify to the liberty of your act and the holiness of your love. Wanda, who continued unknown among the brothers of the order, took the hands of the two children. An impulse of tenderness and enthusiasm made all three rise, as if they had been but one. The formulĂŚ of marriage were pronounced, and the simple and touching rite of the new church performed quietly but fervently. This engagement of mutual love was not an isolated part amid indifferent strangers who were careless of what passed. Those present were called to sanction the religious consecration of two beings bound together by one faith. They extended their arms over the couple and blessed them; then, taking hold of each other's hands, they made a living circle, a chain of paternal love, swearing to protect and defend
- 29. their honor andlife, to preserve them as much as possible from seduction and persecution, on all occasions and under all circumstances: in fine, to love them purely, cordially, and seriously, as if they were united to them by name and blood. The handsome Trenck pronounced this formula for all the others, in elegant and simple terms. He then added, as he spoke to the husbandâ Albert, the profane and guilty law of old society, from which we separate ourselves, some day to lead it back to us, wills that the husband impose fidelity on his wife by humiliation and despotic authority. If she fail, he must kill his rival; he has even the right to kill his wife; and this is called washing out the stain of his honor in blood. In the blind and corrupt world, every man is the enemy of happiness thus savagely and sternly guarded. The friend, the brother even, arrogates to himself a right to wrest honor and happiness from his friend or brother; or, at least, a base pleasure is experienced in exciting his jealousy and sowing distrust and trouble between him and the object of his love. Here, you know that we have a better understanding of honor and family pride. We are brothers in the sight of God; and any one who would look impurely on the wife of his brother has in his heart already committed the crime of incest. All the brothers, moved and excited, then drew their swords, and were about to swear to use their weapons on themselves, rather than violate the oath they had just sworn at Trenck's dictation. The sibylâagitated by one of those enthusiastic impulses which gave her so much influence over their imaginations, and which often modified the opinions and decisions of the chiefs themselvesâbroke the circle, and rushed into the midst. Her language, always energetic and burning; her tall form, her floating drapery, her thin frame trembling yet majestic, the convulsive tremor of her ever veiled head, and withal, a grace which at once betokened the former existence of beauty which moves the mind when it ceases to appeal to the senses;âin fine, even her broken voice, which at once assumed a strange expression, had conspired to make her a
- 30. mysterious being, andinvested her with persuasive power and irresistible prestige. All were silent to hear the voice of her inspiration. Consuelo was perhaps more moved than others, because she was aware of her singular story. She asked herself, shuddering with strange emotion, if this spectre, escaped from the tomb, really belonged to the world, and if, after having spoken, she would not disappear in the air, like the flame on the tripod, which made her appear so blue and transparent. Hide from the light these affirmations, said Wanda, with a shudder. They are impious oaths when what is invoked is an instrument of hatred and murder. I know the old world attached the sword to the side of all reputed free, as a mark of independence and virtue. I well know that, in obedience to the ideas you have here preserved in spite of yourselves, the sword is the symbol of honorâthat you deem you make holy engagements when, like citizens of old Rome, you swear on the sword. But here you would profane a solemn vow. Swear, rather, by this flame and tripodâthe symbol of life, light, and divine love. Do you yet need emblems and visible signs? Are you yet idolators? Do the figures around this temple represent aught but ideas? O! swear rather by your own sentiments, by your better instincts, by your own heart; and if you dare not swear by the living God, the true, eternal, and holy religion, swear by pure humanity, by the glorious promptings of your courage, by the chastity of this young woman and her husband's loveâswear by the genius and beauty of Consuelo, that your desire, that even your thoughts will never profane this holy arc of matrimony, this invisible and mystic altar on which the hand of an angel engraves the vow of love. Do you know what love is? said the sibyl, after having paused for an instant, in a voice which every moment became more clear and penetrating. If you did, oh! you venerable chiefs of our order and priests of our worship, you would never suffer that formula, which God alone can ratify, to be pronounced before you; and which, consecrated by men, is a kind of profanation of the divinest of
- 31. mysteries. What powercan you give to an engagement which in its very nature is miraculous? Yes, the confounding of two wills in one is in itself a miracle, for every heart is in itself free by virtue of a divine right. Yet when two souls yield and become bound to each other, their mutual possession becomes sacred, and as much a divine right as individual liberty. You see this is a miracleâthat God reserves its mystery to himself, as he does that of life and death. You are about to ask this man and woman if during their lives they will belong respectively to each other. Their fervor is such that they will reply, 'Not only for life, but forever.' God then inspires them, by the miracle of love, with more faith, power, virtue, than you can or dare to ask. Away, then, with sacrilegious oaths and gross laws. Leave them their ideal, and do not bind them to reality by chains of gold. Leave the care of the continuation of the miracle to God. Prepare their souls for its accomplishment; form the ideal of love in them; exhort, instruct; extol and demonstrate the glory of fidelity, without which there is no moral honor, no sublime love. Do not come between, however, like Catholic priests, like magistrates, to interfere by the imposition of an oath. I tell you again, men cannot make themselves responsible, or be guardians of the perpetuity of a miracle. What know you of the secrets of the Eternal? Have we already penetrated the temple of the future, in that celestial world where, beneath sacred groves, man will converse with God as one friend does with another? Has a law for indissoluble marriage emanated from the mouth of God? Have his designs been proclaimed on earth? Have you, children of men, promulgated this law unanimously? Have the Roman pontiffs never dissolved marriage? They call themselves infallible! Under the pretext of the nullity of certain engagements, have they not pronounced real divorces, the scandal of which history has preserved in its records? The Christian societies, the reformed sects, the Greek church, following the example of the Mosaic dispensation, and all ancient religions, frankly introduced divorce into modern law. What then becomes of the holiness and efficacy of a vow to God, when it is maintained that man can release us from it? Touch not love by the profanation of marriage. You cannot stifle it in pure hearts. Consecrate the conjugal tie by exhortations, by
- 32. prayers, by apublicity which will make it respectable, by touching ceremonies. You should do so, if you be our priestsâthat is to say, our aids, our guides, our advisers, our consolers, our lights. Prepare souls for the sanctity of a sacrament; and, as a father of a family seeks to establish his children in positions of prosperity, dignity, and security, occupy yourselvesâour spiritual fathersâassiduously in fixing your sons and daughters in circumstances favorable to the development of true love, virtue, and sublime fidelity. When you shall have analysed them by religious ordeals, and ascertained that in their mutual attraction there is neither cupidity, vanity, nor frivolous intoxication, nor that sensual blindness that is without idealityâwhen you have convinced yourselves that they appreciate the grandeur of their sentiments, the holiness of their duty, and the liberty of their choice, then permit them to endow each other with their own inalienable liberty. Let their families, their friends, and the vast family of the faithful, unite to ratify this sacrament. Attend to my words! Let the sacrament be a religious permission, a paternal and social permission, an encouragement, an exhortation to perpetuate the engagement. Let it not be a command, an obligation, a law, with menaces and punishmentsâa forced slavery, with scandal, prisons and chains if it be violated; for in this way you would reverse the whole miracle in all its entirety accomplished on earth. Eternally fruitful providenceâGod, the indefatigable dispenser of grace, always will conduct before you young, fervent, and innocent couples, ready to bind themselves for time and eternity. Your anti-religious law and your inhuman sacrament will always abrogate the effect of grace in them. The inequality of conjugal rights between the sexesâimpiety made venerable by social lawsâ the difference of duty in public opinionâall the absurd prejudices following in the wake of bad institutions, will ever extinguish the faith and enthusiasm of husband and wife. Those who are most sincere, who are most inclined to fidelity, will be the first to grow sad, and become terrified at the duration of the engagement, and thus disenchant each other. The abjuration of individual liberty is in effect contrary to the will of nature and the dictates of conscience when men participate in it, for they oppress it with the yoke of
- 33. ignorance and brutality.It is in conformity with the will of generous hearts, and necessary to the religious instincts of strong minds, when God gives us the means to contend against the various snares man has placed around marriage, so as to make it the tomb of love, happiness, and virtue, and a sworn prostitution, as our fathers the Lollards, whom you know and often invoke, called it. Give to God what is God's, and take from CĂŚsar what is not his. And you, my children, said she, turning towards Albert and Consuelo, you, who have sworn to reverence the conjugal tie, did not, perhaps, know the true meaning of what you did. You obeyed a generous impulse, and replied with enthusiasm to the appeal of honor. That is worthy of you, disciples of a victorious faith! You have performed more than an act of individual virtueâyou have consecrated a principle without which there can be neither chastity nor conjugal fidelity. O love! sublime flameâso powerful and so fragile, so sudden and so fugitive! light from heaven, seemingly passing through our existence, to die before we do, for fear of consuming and annihilating us, we feel you are a vivifying fire, emanating from God himself, and that whoever would fix it in his bosom and retain it to his last hour, always ardent, always in its pristine vigor, would be the happiest and noblest of men. Thus the disciples of the ideal will always seek to prepare sanctuaries for you in their bosoms, that you may not hasten to return to heaven. But alas! you whom we have made it a virtue to honor, have declined to be renewed at the dictate of our institutions, and have remained free as the bird of the air, capricious as the flame on the altar. You seem to laugh at our oaths, our contracts and our will. You fly from us in spite of all we have invented to fix you in your manners. You no longer inhabit the harem, guarded by the vigilant sentinels which Christian society places between the sentence of the magistrate and the yoke of public opinions. Whence, then, comes your inconstancy and your ingratitude? Oh! mysterious influence! oh, love! cruelly symbolised under the form of an infant and blind god! what tenderness and what contempt inspire human hearts you enkindle with your blaze;
- 34. and whom youdesert, leaving them to wither amid the anguish of repentance, and, more frightful yet, of disgust! Why is it that man kneels to you in every portion of the globeâthat you are exalted and deifiedâthat divine poets call you the soul of the worldâthat barbarous nations sacrifice human victims to you, precipitating wives on the fire at the husband's funeralâthat young hearts call you in their gentlest dreams, and that old men curse life when you abandon them to the horror of solitude? Whence comes that adorationâsometimes sublime, sometimes fanaticalâwhich has been decreed you from the golden infancy of humanity to our age of iron, if you be but a chimera, the dream of a moment of intoxication, an error of the imagination, excited by the delirium of the sense. Ah! it is not a vulgar instinct, a mere animal want. You are not the blind child of Paganism, but the true son of God, and very essence of the divinity. You have not yet revealed yourself to us, except through the mist of errors; and you would not make your abode among us, because you were unwilling to be profaned. You will return to us, as in the days of the fabulous Astrea, as in the visions of poets, to fix your abode in our terrestrial paradise, when we shall, by our sublime virtues, have merited the presence of such a guest. How blessed then will this abode be to man! and then it will be well to have been born. We will then be brothers and sisters, and unions, freely contracted, will be maintained by your own power. When, in place of this terrible contest, whose continuance is impossibleâconjugal fidelity being forced to resist infamous attempts at debauch, hypocritical seduction or mad violence, hypocritical friendship and wise corruptionâevery husband will find around him chaste sisters, himself the jealous and delicate guardian of the happiness of a sister confided to him as a companion; while every wife will find in other men so many brothers of her husband, proud of her happiness and protectors of her peace; then the faithful wife will no longer be the fragile flower that hides herself to maintain the treasures of her chastity, often a deserted victim, wasting in solitude and tears, unable to revive in her husband's mind the flame she has preserved in purity in her own.
- 35. The brother thenwill not be forced to avenge his sister, and slay him she loves and regrets, in obedience to the dictates of false honor. The mother will not tremble for her daughter, nor the child blush for its parent. The husband then will be neither suspicious nor despotic; and, on her part, the wife will escape the bitterness of the victim and the rancor of the slave; atrocious suffering and abominable injustice will cease to sully the peace of the domestic hearth. It may be some day, that the priest and the magistrate, relying with reason on the permanent miracle of love, will consecrate in God's name indissoluble unions, with as much wisdom and justice as they now ignorantly display impiety and folly. But these glorious days are not yet come. Here, in this mysterious temple, where we are now united in obedience to the evangelists, three or four in the name of the Lord, we can only dream of divinest joys. It is an oracle which then escapes from their bosoms. Eternity is the ideal of love, as it is of faith. The human soul never comes nearer to the apex of its power and lucidity than in the enthusiasm of a great love. The always of lovers is an eternal revelation, a divine manifestation, casting its sovereign light and blessed warmth over every instant of their union. Woe to whoever profanes this sacred formula! He falls from grace to sinâextinguishes the faith, power and light in his heart. Albert, said Consuelo, I receive your promise, and adjure you to accept mine. I feel myself under the power of a miracle, and the always of our brief lives does not resemble the eternity for which I give myself to you. Sublime and rash Consuelo, said Wanda, with a smile of enthusiasm, which seemed to pass through her veil, ask God for eternity with him you love, as a recompense of your fidelity to him in this brief life. Ah! yes, said Albert, lifting his wife's hand, clasped in his own, to heaven, that is our end, hope, and rewardâto love truly in this phase of existence, to meet and unite in others. Ah! I feel that this is not the first day of our unionâthat we have already loved, and loved
- 36. in other lives.Such bliss is not the work of chance. The hand of God reunites us, like two parts of one being inseparable in eternity. After the celebration of the marriage, though the night was far advanced, they proceeded to the final initiation of Consuelo in the order of the Invisibles, and, then, the members of the tribunal having dispersed amid the shadows of the holy wood, soon reassembled at the castle of fraternal communion. The prince (Brother Orator) presided, and took care to explain to Consuelo the deep and touching symbols. The repast was served by faithful domestics, affiliated with a certain grade of the order. Karl introduced Matteus to Consuelo, and she then saw bare his gentle and expressive face; she observed with admiration that these respectable servants were not treated as inferiors by their brothers of the other grades. No personal distinction separated them from the higher grades of the order, of whatever rank. The brother servitors, as they were called, discharged willingly the duty of waiters and butlers. It was for them to make all arrangements for the festivity, as being best prepared to do so; and this duty they considered a kind of religious observanceâa sort of eucharistic festival. They were then no more degraded than the Levites of a temple who preside over the details of sacrifice. When they arranged the table, they sat at it themselves, not at peculiar isolated places, but in chairs retained among the others for them. All seemed anxious to be civil to them, and to fill their cups and plates. As at masonic banquets, the cup was never raised to the lip without invoking some noble idea, some generous sentiment, some august patronage. The cadenced noises, the puerile conduct of the freemasons, the mallet, the jargon of the toasts, and the vocabulary of tools, were excluded from this grave yet costly entertainment. The servitors were respectful without constraint, and modest without baseness. Karl sat during one of the services between Albert and Consuelo. The latter saw with emotion that besides his sobriety and good behavior, he had made progress in healthy religious notions, by means of the admirable education of sentiment.
- 37. Ah, my friend,said she to her husband, when the deserter had changed his place, and her husband drew near to her, this is the slave beaten by the Prussian corporals, the savage woodman of Boehmer-wald, and the would-be murderer of Frederick the Great. Enlightenment and charity have in a few days converted into a sensible, pious, and just man, a bandit, whom the precocious justice of nations pushed to murder, and would have corrected with the lash and gallows. Noble sister, said the Prince, who had placed himself on Consuelo's right, you gave at Roswald, to this mind crazed by despair, great lessons on religion and prudence. He was gifted with instinct. His education has since been rapid and easy; and when we've essayed to teach him, his reply was, 'So the signora said.' Be sure the rudest men may be enlightened more easily than is thought. To improve their conditionâto inoculate them with self-respect by esteeming and encouraging them, requires but sincere charity and human dignity. You see that as yet they have been initiated merely in the inferior degrees. The reason is, we consult the extent of their minds and progress in virtue when we admit them into our mysteries. Old Matteus has taken two degrees more than Karl; and if he does not pass those he now occupies, it is because his mind and heart can go no farther. No baseness of extraction, no humility of condition, will ever stop them. You see here Gottlieb the cobbler, son of the jailer at Spandau, admitted to a grade equal to your own, though in my house, from habit and inclination, he discharges his subordinate functions. His imagination, fondness for study and enthusiasm for virtueâin a word, the incomparable beauty of soul inhabiting that distorted body, renders him almost fit to be treated, in the interior of the temple, as a brother and as an equal. We had scarcely any ideas and virtues to impart to him. On the contrary, mind and heart were too teeming, and it became necessary to repress them and soothe his excitement, treating at the same time the moral and physical causes which would have led him to folly. The immorality of those among whom he lived, and the perversity of the official world, would have irritated without corrupting him. We alone, armed with the
- 38. mind of JamesBoehm and the true explanation of his sacred symbols, were able to undeceive and convince him, and to direct his poetic fancy without chilling his zeal and faith. Remark how the cure of his mind has reacted on his body, and that he has regained health as if by magic. His strange face is already transformed. After the repast they resumed their cloaks, and walked along the gentle slope of the hill, which was shaded by the sacred wood. The ruins of the old castle, reserved for ordeals, was above it; and gradually Consuelo remembered the path she had passed so rapidly over, on a night of storm, not long before. The plenteous streamâ which ran from a cavern rudely cut in the rock, and once reserved for superstitious devotionâmurmured amid the undergrowth towards the valley, where it formed the brook the prisoner in the pavilion knew so well. Alleys covered by nature with fine sand, crossed under the luxuriant shade where the various groups met and talked together. High barriers, but which did not intercept the river, shut in the enclosure, the kiosque of which might be considered the study. This was a favorite retreat of the duke, and was forbidden to the idle and indiscreet. The servitors also walked in groups around the barriers, watching to prevent the approach of any profane being. Of this there was no great danger. The duke seemed merely occupied with masonic mysteries; as was the case, in a manner. Free masonry was then tolerated by the law and protected by the princes who were, or thought themselves, initiated in it. No one suspected the importance of the superior grades; which, after many degrees, ended in the tribunal of the Invisibles. Besides, at this moment the ostensible festival which lighted up the façade of the palace too completely absorbed the attention of the numerous guests of the prince, for any to think of leaving his brilliant halls and the new gardens, for the rocks and ruins of the old park. The young Margravine of Bareith, an intimate friend of the duke, presided over the honors of the entertainment. To avoid appearing, he had feigned sick, and after the banquet of the Invisibles supped with his numerous guests in the palace. As she saw the glare of the lights in the distance, Consuelo, who leaned on
- 39. Albert's arm, rememberedAnzoleto and accused herself innocently in presence of her husband, who charged her with having become too ironical and stern to the companion of her childhood. Yes, it was a guilty idea, but then I was most unhappy. I had resolved to sacrifice myself to Count Albert, and the malicious and cruel Invisibles again cast me into the arms of the dangerous Leverani. Wrath was in my heart; gladly I met him from whom I was to separate in despair, and Marcus wished to soothe my sorrow by a glance at the handsome Anzoleto. Ah! I never expected to be so indifferent to him. I fancied I was about to be doomed to sing with him, and could have hated him for thus depriving me of my last dream of happiness. Now, my friend, I could see him without bitterness and treat him kindly; happiness makes us so merciful. May I be useful to him some day, and inspire him with, a serious love of art, if not virtue. Why despair? Let us wait for him in the scene of want and misery. Now, amid his triumphs, he would be deaf to the voice of reason. Let him lose his voice and his beauty, and we will take possession of his soul. Do you take charge of this conversion, Albert? Not without you, my Consuelo. Then you do not fear the past? No; I am presumptuous enough to fear nothing. I am under the power of a miracle. I, too, Albert, cannot doubt myself. Day began to break, and the pure morning air to exhale a thousand exquisite perfumes. It was the most delicious period of the summer; the birds singing amid the trees and flying from hill to valley. Groups formed every moment around the couple and far from being importunate, added to the pleasure of their fraternal friendship, to their pure happiness. All the Invisibles present were introduced to Consuelo as members of her family. They were the most eminent in virtue, talent, and intelligence in the order. Some were illustrious,
- 40. and others obscurein the world, but were known in the temple by their labors. The noble and the peasant mingled together in close intimacy. Consuelo had to learn their true names, and the more poetical titles of their fraternal association. They were Vesper, Ellops, Peon, Hyas, Euryalus, Bellerophon, etc. Never had she around her so many pure and noble souls, so many interesting characters. The stories told of their conversion, the dangers they had run, and what they had done, charmed her as poems, the tenor of which she could not have reconciled with actual life, they appeared so touching and moving. There was, however, no portion of the common-place gallantry, and not the slightest approach to dangerous familiarity. Lofty language, inspired by equality and fraternity, was realised in its purest phase. The beautiful golden dawn rising over their souls as over the world, was, as it were, a dream in the existence of Consuelo and Albert. Enlaced in each other's arms, they did not think of leaving their beloved brethren. A moral intoxication, gentle and bland as the morning air, filled their souls. Love had expanded their hearts too amply to make them tremble. Trenck told them the dangers of his captivity and escape in Glatz. Like Consuelo and Haydn in the Boehmer-wald, he had crossed Poland, but in the midst of cold, covered with rags, with a wounded companionâthe amiable SHELLES, whom his memoirs make known to us as an affectionate friend. To earn his bread, he had played on the violin, and, like Consuelo on the Danube, had been a minstrel. He then spoke in a low tone of the Princess Amelia, his love and hope. Poor Trenck! the terrible storm which overhung him, neither he nor his happy friends foresaw. He was doomed to pass from the midsummer's night's dream to a life of combat, deception, and suffering. Porporino sang beneath the cypress-trees an admirable hymn composed by Albert, to the memory of the martyrs of their cause. Young Benda accompanied him on the violin; Albert took the instrument and delighted his hearers with a few notes; Consuelo could not sing, but wept with joy and enthusiasm; Count Saint Germain told of conversations with John Huss and Jerome of Prague, with such warmth, eloquence, and probability, that it was impossible
- 41. not to havefaith in him. In such seasons of emotion and delight, reason does not prohibit poetry. The Chevalier d'Eon described with refined taste the miseries and absurdities of the great tyrants of Europe, the vices of courts, and the weakness of the scaffolding of the social system that enthusiasm fancied so easy to break. Count Golowkin described the great soul and strange contradictions of his friend, Jean Jacques Rousseau. This philosophical noble (they will to-day call him eccentric) had a very beautiful daughter, whom he educated according to his ideas, and who was at once Emile and Sophie, now as handsome a boy, then as charming a girl as possible. He wished to have her initiated, and for Consuelo to instruct her. The illustrious Zinzendorf explained the evangelical constitution of his colony of Moravian Hernhuters.â He consulted Albert with deference about many particulars, and wisdom seemed to speak by Albert's mouth. He was inspired by the presence and smile of his mistress. To Consuelo he seemed divine. All advantages to her seemed to deck him. He was a philosopher, an artist, a martyr, who had survived the ordeal; grave as a sage of the Portico, beautiful as an angel, joyous and innocent as a child or happy loverâperfect, in fine, as the one we love always is. Consuelo, when she knocked at the door of the temple, had expected to die of fatigue and emotion. Now she felt herself aroused and animated as when, on the shore of the Adriatic, she used to sport in the sands in full health beneath a bright sun moderated by the evening breeze. It seemed that life in all its power, happiness in all its intensity, had taken possession of her, and that she breathed them at every pore. Why cannot the sun be stopped in the sky over certain valleys, where we feel all the plentitude of being, and where the dreams of imagination seem realised, or about to be? The sky at last became purple and gold, and a silver bell warned the Invisibles that night withdrew its protecting cloak. They sang a hymn to the rising sun, emblematical to them of the day they dreamed of, and prepared for the world. All then made them adieux, promising to meet, some at Paris, others at London, Madrid, Vienna, Petersburg,
- 42. Dresden, and Berlin.All promised on a year from that day to meet again at the door of the blessed temple, either with neophytes or with brethren now absent. They then folded their cloaks to conceal their elegant costumes, and silently dispersed by the shadowy walks of the park. Albert and Consuelo, guided by Marcus, went down the ravine to the stream. Karl received them in his closed gondola, and took them to the door of the pavilion. There they paused for a moment to contemplate the majesty of the orb of day which rose in the sky. Until now, Consuelo, when she replied to Albert had called him by his true name; when, however, she was awakened from the musing in which she seemed delighted to lose herself, as she pressed her burning cheek on his shoulder, she could only say: Oh Leverani! [15]The harmonica, when first invented, created such a sensation in Germany, that poetical imaginations fancied they heard in it supernatural voices, evoked by the consecrators of certain mysteries. This instrument, which, before it became popular, was thought to be magical, was elevated by the adepts of German theosophy, to the same honor with the lyre among the ancients, and many other instruments among the primitive people of Himalaya. They made it one of the hieroglyphic figures of their mysterious iconography. They represented it under the form of a fantastic chimera. The neophytes of secret societies, hearing it for the first time after the rude shocks of their terrible ordeals, were so much impressed by it that many of them fell into ecstacies. They fancied they heard the song of invisibile powers, for both the instrument and the performer were concealed from them most carefully. There are extremely curious stories told of the employment of the harmonica in the reception of adepts of illuminatism. EPILOGUE Had we been able to procure faithful documents in relation to Albert and Consuelo after their marriage, like those which have guided us
- 43. up to thispoint, we might, doubtless, have written a long history, telling of all their adventures and journeys. But, most persevering readers, we cannot satisfy you; and of you, weary reader, we only ask a few moments of patience. Let neither of you reproach nor praise us. The truth is, that the materials by means of which we have so far been able to connect the items of this story, entirely disappear from the dates of the romantic night which blessed and consecrated the union of the two great characters of our story amid the Invisibles. Whether the engagements contracted by them in the temple prevented them from yielding to friendship in their letters; or that their friends, being affiliated in the same mysteries, in the days of persecution thought it proper to destroy their correspondence, we cannot say; but henceforth we see them through the maze of a cloud, under the veil of the temple or the mask of adepts. Without examining the traces of their existence which we find in manuscripts, it would often have been difficult to follow them; contradictory evidence shows both to have been at the same time at two different geographical points, or following different objects. However, we can easily understand the possibility of their voluntarily creating such errors, from the fact that they were secretly devoted to the plans of the Invisibles, and often were forced, amid a thousand perils, to avoid the inquisitorial policy of governments. In relation to the existence of this one soul, with two persons, called Consuelo and Albert, we cannot say whether love fulfilled all its promises, or if fate contradicted those which it had seemed to make during the intoxication of what they called The Midsummer Night's Dream. They were not, however, ungrateful to Providence, which had conferred this rapid happiness, in all its plentitude, and which, amid reverses, continued the miracle of love Wanda had announced. Amid misery, suffering and persecution, they always remembered that happy life, which seemed to them a celestial union, and, as it were, a bargain made with the divinity, for the enjoyment of a better existence after many toils, ordeals, and sacrifices. In other respects, all becomes so mysterious to us that we have been quite unable to discover in what part of Germany this
- 44. enchanted residence was,in which, protected by the tumult of the chase and festivals, a prince unknown in documents became a rallying point and a principal mover of the social and philosophical conspiracy of the Invisibles. This prince had received a symbolical name, which, after a thousand efforts to discover the cypher used by the adepts, we presume to be Christopher, or Chrysostom, or the Golden Mouth. The temple where Consuelo was married and initiated was particularly called Saint Graal, and the chiefs of the tribunal Templists. These were Romanesque emblems, renewed from the old legends of the age of gold and chivalry. All the world knows that in these charming fictions, Saint Graal was hidden in a mysterious sanctuary, amid a grotto unknown to men. There the Templists, illustrious saints of primitive Christianity, devoted even in this world to immortality, kept the precious cup which Jesus had used in the consecration of the Eucharist, when he kept the passover with his disciples. This cup doubtless contained the celestial grace, represented sometimes by blood and then by the tears of Christ; a divine ichor or eucharistic substance, the mystic influence of which was inexplicable, but which it was sufficient merely to see, to be transformed, both morally and physically, so as to be forever sheltered from death and sin. The pious paladins, who, after terrible macerations and exploits sufficient to make the earth tremble, devoted themselves to the career of knight-errantry, had the idea of reaching Saint Graal at the end of their peregrinations. They looked for it amid the ices of the north, on the shores of Armorica, and in the depths of the forests of Germany. To realise this sublime conquest, it was necessary to confront danger, equal to those of the Hesperidesâto overcome monsters, elements, barbarous people, hunger, thirst, and even death. Some of these Christian Argonauts discovered, it is said, the sanctuary, and were regenerated by the divine cup; they never, however, betrayed the terrible secret. Their triumph was known by the power of their invincible arm, by the transfiguration of all their existence: few, however, survived this glorious initiation. They disappeared from among men as Jesus did after his resurrection, and passed from earth to heaven without undergoing the bitter transition of death.
- 45. This magical symbolwas, in fact, well adapted to the object of the Invisibles. For many years, the new Templists hoped to make Saint Graal accessible to all mankind. Albert toiled constantly to diffuse the true ideas of his doctrine. He reached the highest grades of the order, for we find the list of his titles showing that he had time enough to reach them. Now all know that eighty-one months are needed to pass through the twenty-three degrees of masonry, and we think it certain that a much longer time was required for the higher grades of Saint Graal. The number of masonic degrees are now a mystery to no one; yet it will not be out of place here to recount a few, as they paint the enthusiastic genius and smiling fancy which presided over their first creation: Apprentice and Master Mason, Secret and Perfect Master, Provost and Judge, English and Irish Master, Master in Israel, Master Elect of the Nine and Fifteen, Elect of the Unknown, Grand Master Architect, Royal Arch, Grand Scotch Master of the Sublime or Master Masons, Knight of the Sword, Prince of Jerusalem, Knight of Orient and Occident, Rose-Cross of France, Heredom and Kilwinning, Grand Pontiff or Sublime Scot, Architect of the Sacred Roof, Pontiff of Jerusalem, Sovereign Prince of Masonry and Master ad vitam, Naochite, Prince of Libon, Chief of the Tabernacle and Knight of the Iron Serpent, Trinitarian Scot or Prince of Mercy, Grand Commander of the Temple, Knight of the Gun, Patriarch of the Crusades, Grand Master of Light, Knight Kadosch, Knight of the White Eagle and of the Black Eagle, Knight of the PhĹnix and Knight of the Argonauts, Knight of the Golden Fleece, Grand-Inspector-Inquisitor- Commander-Sublime, Prince of the Royal Secret and Sublime Master of the Luminous King,c.[16] These titles, or at least the majority of them, we find connected with the name of Albert Podiebrad, in the most illegible rolls of the freemasons. There are also many less known; such as Knight of St John, Sublime Johannite, Master of the New Apocalypse, Doctor of the Gospel, and Elect of the Holy Ghost, Templist, Areopagite, Magus, and Man of the People, Man-Pontiff, Man-King, and New-
- 46. man,c.[17] We havebeen surprised here to find some titles which seemed anticipated from the illuminatism of Weishaupt: this peculiarity, however, was explained at a later day, and will not, when this story is concluded, need any explanation to our readers. Amid this labyrinth of obscure factsâwhich, however, are profound, and connected with the labor, success, and apparent extinction of the Invisiblesâwe can with difficulty follow the adventurous story of the young couple. Yet by supplying what we need by a prudent imagination, the following is nearly the abridged commentary of the chief events of their lives. The fancy of the reader will supply the deficiency of the text, and following our experience, we doubt not that the best dĂŠnoĂťements are those for which the reader and not the narrator will be responsible.[18] Probably, after leaving Saint Graal, Consuelo went to the little court of Bareith, where the Margravine, sister of Frederick, had palaces, gardens, kiosques, and cascades, in the same style as those of Count Hoditz at Roswald, though less sumptuous and less expensive. This intellectual princess had been married without a dower to a very poor prince; and not long before she had worn robes with trains of reasonable length, and had pages whose doublets were not threadbare, her gardens, or rather her garden, to speak without metaphor, was situated amid a beautiful country, and she indulged in the Italian Opera in an antique temple Ă la Pompadour. The margravine was fond of philosophyâthat is to say, she was a disciple of Voltaire. The young hereditary margrave, her husband, was the zealous head of a masonic lodge. I am not sure whether Albert was connected with him, or whether his incognito was observed by the secresy of the brothers, or whether he remained away from this court and joined his wife afterwards. Certainly Consuelo had some secret mission there. Perhaps, also, for the purpose of preventing attention from being attracted to her husband, she did not live publicly with him for some time. Their loves, then, had all the attraction of mystery; and if the publicity of their union, consecrated by the fraternal sanction of the Templists, seemed gentle and
- 47. edifying to them,the secrecy they maintained in a hypocritical and licentious world, at first, was a necessary ĂŚgis and kind of mute protestation in which they found their enthusiasm and power. Many male and female Italian singers at that time delighted the little court of Bareith. Corilla and Anzoleto appeared there, and the vain prima donna again became enamored of the traitor she had previously devoted to all the furies of hell. Anzoleto, however, while he cajoled the tigress, sought with a secret and mysterious reserve to find favor with Consuelo, whose talent, enhanced by such profound revelations, now eclipsed all rivalry. Ambition had become the dominant passion of the young tenor; love had been stifled by mortification, and voluptuousness by satiety. He then loved neither the chaste Consuelo nor the passionate Corilla, but kept terms with both, ready to attach himself to either of the two, who would serve his purpose, and make him advantageously known. Consuelo treated him kindly, and neither spared good advice nor such instructions as would enable him to exhibit his talent. She never, though, felt uneasy when she was with him, and the completeness of her pardon exhibited how completely she had mastered her passion. Anzoleto was not re-installed, and having listened with emotion to the advice of his friend, lost all patience when he lost all hope, and his deep mortification and sorrow, in spite of himself, became evident in his words. Under these circumstances, it appears that Amelia of Rudolstadt came to Bareith with the Princess of Culmbach, daughter of the Countess Von Hoditz. If we may believe some exaggerating and indiscreet witnesses, some strange scenes took place between Consuelo, Amelia, Corilla, and Anzoleto. When she saw the handsome tenor appear unexpectedly on the boards of the opera of Bareith, the young baroness fainted. No one observed the coincidence, but the lynx-eyed Corilla discovered on the brow of Anzoleto a peculiar expression of gratified vanity. He missed his point; the court, disturbed by the accident, did not applaud the singer, and instead of growling between his teeth, as was his fashion
- 48. on such occasions,there was an unequivocal smile of triumph on his face. See, said Corilla, in an angry voice to Consuelo, as she went behind the scenes, he loves neither you nor me, but that little fool who has been playing her part in the boxes. Do you know her? who is she? I do not know, said Consuelo, who had observed nothing: I can assure you, however, neither you, nor she, nor I, occupy him. Who then does? Himself al solito, said Consuelo with a smile. The story goes on to say that on the next day Consuelo was sent for to come to a retired wood to talk with Amelia. I know all, said the latter, angrily, before she permitted Consuelo to open her mouth; he loves you, unfortunate scourge of my lifeâyou, who have robbed me of Albert's love and his. His, madame? I do not knowââ Do not pretend. Anzoleto loves you. You were his mistress at Venice, and yet areââ It is either a base slander, or a suspicion unworthy of you. It is the truth. I assure you; he confessed it to me last night. Last night! What do you say, madame? said Consuelo, blushing with shame and chagrin. Amelia shed tears; and when the kind Consuelo had succeeded in calming her jealousy, she obtained in spite of her diffidence, the confession of this unfortunate passion. Amelia had heard Anzoleto sing at Prague, and became intoxicated with his beauty and success. Being ignorant of music, she took him for one of the first musicians in the world. At Prague he was decidedly popular. She sent for him as her singing-master, and while her father the old Baron Frederick, paralysed by inactivity, slept in his chair dreaming of wild boars, she yielded to a seducer. Ennui and vanity ruined her. Anzoleto, flattered
- 49. by this illustriousconquest, and wishing to make the scandal public in order to secure popularity, persuaded her that she might become the greatest singer of the age, that an artist's life was a paradise on earth, and that she could not do better than fly with him, and make her dĂŠbut at the Haymarket Theatre in Handel's operas. Amelia at first viewed with horror the idea of deserting her old father, but when Anzoleto was about to leave Prague, feigning a despair he did not feel, she yielded to his solicitations, and fled with him. The intoxication of her love for Anzoleto was but of brief duration. His insolence and coarse manners, when he no longer played the part of seducer, recalled her to her senses; and it was not without a feeling of pleasure mingled with remorse at her conduct, that, three months after her escape, she was arrested at Hamburg, and brought back to Prussia, where, at the instance of her Saxon kin, she was incarcerated in the fortress of Spandau. Her punishment was both long and severe, and in a measure rendered her mind callous to the agony she would otherwise have felt at hearing of her father's death. At last her freedom was granted, and it was not till then that she heard of all the misfortunes which had afflicted her family. She did not dare to return to the canoness, and feeling utterly incapable of leading a life of retirement and repose, she implored the protection of the Margravine of Bareith; and the Princess of Culmbach, who was then at Dresden, assumed the responsibility of taking her to her kinswoman. In this frivolous yet philosophical court she found that amiable toleration of vice which then was the only virtue. Here she again met with Anzoleto, and again submitted to the ascendancy which he seemed to have acquired over the fair sex, and which the chaste Consuelo found so difficult to resist. At first she avoided him, but gradually became again fascinated, and made an appointment to meet him one evening in the garden, and once more yielded to his solicitations. She confessed to Consuelo that she yet loved him, and related all her faults to her old singing mistress with a mixture of feminine
- 50. modesty and philosophicalcoolness. It seems certain that Consuelo by her earnest appeals found the way to her heart, and that she made up her mind to return to the Giants' Castle, and to shake off her dangerous passion in solitude, by soothing her old aunt in her decline. After this adventure Consuelo could remain at Bareith no longer. The haughty jealousy of Corilla, who was always imprudent, yet at the same time kind-hearted, induced the prima donna sometimes to find fault, and then to humble herself. Anzoleto, who had fancied that he could avenge for her disdain by casting himself at Amelia's feet, never pardoned her for having removed the young baroness from danger. He did her a thousand unkind offices, contriving to make her miss the cue on the stage, preventing her from taking up the key in a duo, and by a self-sufficient air attempting to make the unwary audience think she was in error. If he had a stage effect to perform with her, he went to her right instead of her left hand, and tried to make her stumble amid the properties. All these ill-natured tricks failed, in consequence of Consuelo's calmness. She was, however, less stoical when he began to calumniate her, and when she knew that there were persons, who could not believe in the chastity of an actress, to listen to him. Hence libertines of every age were rude towards her, refusing to believe in her innocence; and she had to bear with Anzoleto's defamation, influenced as he was by mortification and revenge. This base and narrow-minded persecution was the commencement of a long martyrdom which the unfortunate prima donna submitted to during all her theatrical career. As often as she met Anzoleto, he annoyed her in a thousand ways. Corilla, too, from envy and ill- feeling, gave her trouble. Of her two rivals, the female was the least in the way, and most capable of a kind emotion. Whatever may be said of the misconduct and jealous vanity of actresses, Consuelo discovered that when her male companions were influenced by the same vices, they became even more degraded, and less worthy of their relative position. Arrogant and dissipated nobles, managers and
- 51. people of thepress, depraved by such connection, fine ladies, curious and whimsical patronesses, ready to deceive, yet offended at finding in an actress more virtue than they could themselves boast ofâin fact, and most unjust of all, the public rose en masse against the wife of Leverani, and subjected her to perpetual mortification. Persevering and faithful in her profession as she was in love, she never yielded, but pursued the tenor of her way, always increasing in musical knowledge, and her virtuous conduct remaining unaltered. Sometimes she failed in the thorny path of success, yet often won a just triumph. She became the priestess of a purer art than even Porpora himself was acquainted with; and found immense resources in her religious faith, and vast consolation in her ardent and devoted love to her husband. The career of her husband, though a parallel to her own, for he accompanied her in her wanderings, is enwrapped in much mystery. It may be presumed that he was not sentenced to be the slave of her fortune and the book-keeper of her receipts and disbursements. Consuelo's profession was not very lucrative. At that time the public did not reward artists with as much munificence as it does now. Then they were remunerated by the presents they received from princes and nobles, and women who knew how to take advantage of their position had already begun to amass large fortunes. Chastity and disinterestedness are, however, the greatest enemies an actress can have. Consuelo was successful, respected, and excited enthusiasm in some, when those who were about her did not interfere with her position before the true public. She owed no triumph to gallantry, however, and infamy never crowned her with diamonds or gems. Her laurels were spotless, and were not thrown on the stage by interested hands. After ten years of toil and labor, she was no richer than when she began her career. She had made no speculations, for she neither could nor would do so. She had not even saved the fruit of her labors, to get which she often had much trouble, but had expended it in charity, or for the purposes of secret but active propagandism, for which her own means had not always
- 52. sufficed. The centralpower of the Invisibles had often provided for her. What may have been the real success of the ardent and tireless pilgrimage of Albert and Consuelo, in France, Spain, England and Italy, there is nothing to tell the world; and I think we must look twenty years later, and then use induction, to form an idea of the result of the secret labors of the societies of the Invisibles. Had they a greater effect in France than in the bosom of that Germany where they were produced? The French Revolution loudly says Yes. Yet the European conspiracy of Illuminism, and the gigantic conceptions of Weishaupt, prove that the divine dream of Saint Graal did not cease to agitate the German mind for thirty years, in spite of the dispersion and defection of the chief adepts. Old newspapers tell us that Porporina sang with great success in Pergolese's operas at Paris, in the oratorios and operas of Handel at London, with Farinelli at Madrid, with La Faustina at Dresden, and with Mergotti at Venice. At Rome and Naples she sang the church music of Porpora and other great masters, with triumphant applause. Every item of Albert's career is lost. A few notes to Trenck or Wanda prove this mysterious personage to have been full of faith, confidence, and activity, and enjoying in the highest degree lucidity of mind. At a certain epoch all documentary information fails. We have heard the following story told, in a coterie of persons almost all of whom are now dead, relative to Consuelo's last appearance on the stage. It was about 1760, at Vienna. The actress was then about thirty years old, and it was said was handsomer than she had been in her youth. A pure life, moral and calm habits, and physical prudence, had preserved all the grace of her beauty and talent. Handsome children accompanied her, but no one knew their father, though common report said that she had a husband, and was irrevocably faithful to him. Porpora having gone several times to Italy, was with her, and was producing a new opera at the Imperial Theatre. The
- 53. last twenty yearsof the maestro's works are so completely unknown, that we have in vain sought to discover the name of his last productions. We only know Porporina had the principal part, that she was most successful, and wrung tears from the whole court. The empress was satisfied. On the night after this triumph, Porporina received from an invisible messenger news that filled her with terror and consternation. At seven in the morningâthat is to say, just at the hour when the empress was awakened by the faithful valet known as the sweeper[19] of her majesty, (for his duty consisted in opening the blinds, making the fire, and cleaning the room, while the empress was awaking,) Porporina, by eloquence or gold, passed through every avenue of the palace, and reached the door of the royal bed-chamber. 'My friend,' said she to the servant, 'I must throw myself at the empress's feet. The life of an honest man is in danger. A great crime will be committed in a few days, if I do not see her majesty at once. I know that you cannot be bribed, but also know you to be generous and magnanimous. Everybody says so. You have obtained favors which the greatest courtiers dared not ask.' 'Kind heaven! my dear mistress! I will do anything for you,' said the servant, clasping his hands and letting his duster fall. 'Karl!' said Consuelo. 'Thank God I am saved! Albert has a protecting angel in the palace!' 'Albert! Albert!' said Karl. 'Is he in danger? Go In, madame, if I should lose my place. God knows I shall be sorry; for I am enabled to do some good and serve our holy cause better than I could do anywhere else. Listen! The empress is a good soul, when she is not a queen. Go in: you will be thought to have preceded me. Let those scoundrels bear the burden of it, for they do not deserve to serve a queen. They speak lies. Consuelo went in; and when the empress opened her eyes, she saw her kneeling at the foot of the bed.
- 54. 'Who is that?'said Maria Theresa, as, gathering the counterpane over her shoulders, she rose up as proud and as haughty in her night-dress, and on her bed, as if she sat on her throne, decked with the Imperial crown on her brow, and the sword by her side. 'Madame,' said Consuelo, 'I am your humble subject, an unfortunate mother, a despairing wife, who begs on her knees her husband's life and liberty.' Just then Karl came in, pretending to be very angry. 'Wretch,' said he, 'who bade you come hither?' 'I thank you, Karl, for your vigilance and fidelity. Never before was I awakened with such insolence.' 'Let not your majesty say a word, and I will kill this woman at once.' Karl knew the empress. He was aware that she liked to be merciful before others, and that she always played the great queen and the great woman before even her valets. 'You are too zealous,' said she, with a majestic smile. 'Go, and let this poor weeping woman speak. I am not in danger in the company of my subjects. What is the matter, madame? But, are you not the beautiful Porporina? You will spoil your voice, if you weep thus.' 'Madame,' said Consuelo, 'ten years ago I was married in the Catholic Church. I have never once disgraced myself. I have legitimate children, whom I have educated virtuously. I dare to say ââ' 'Virtuously I know you have, but not religiously. You are chaste, they tell me, but you never go to church. Tell me, however, what has befallen you?' 'My husband, from whom I have never been separated, is now in Prague, and I know not by what infamous means he has been arrested in that city on the charge of usurping a name and title not his own, of attempting to appropriate an estate to which he had no claimâin fine, of being a swindler, a spy, and an impostor. Perhaps
- 55. even now hehas been sentenced to perpetual imprisonment, or to death.' 'Prague? and an impostor?' said the empress. 'There is a story of that kind in the reports of the secret police. What is your husband's name? for you actresses do not bear them.' 'Leverani.' 'That is it! My child, I am sorry that you are married to such a wretch. This Leverani is in fact a swindler and a madman, who, taking advantage of a perfect resemblance, attempts to personate the Count of Rudolstadt, who died ten years ago. The fact is proved. He introduced himself into the home of the old Canoness of Rudolstadt, and dared to say he was her nephew, he would have succeeded in getting possession of her inheritance, if just then the old lady had not been relieved of him by friends of the family. He was arrested and very properly. I can conceive your mortification, but do not know how I can help it. If it be shown that this man is mad, and I hope he is, he will be placed in an hospital, where you will be able to see and attend him. If, however, he be a scamp, as I fear, he must be severely treated, to keep him from annoying the true heiress of Rudolstadt, the young Baroness Amelia, who I think, after all her past errors, is about to be married to one of my officers. I hope, mademoiselle, that you are ignorant of your husband's conduct, and are mistaken in relation to his character, otherwise I would be offended at your request. I pity you too much to humiliate you, however. You may retire.' Consuelo saw she had nothing to expect, and that in seeking to establish the identity of Albert and Leverani she would injure his position. She arose and walked towards the door, pale as if she was about to faint. Maria Theresa, however, who followed her with an anxious eye, took pity on her, and called her back. 'You are much to be pitied,' said she, in a less dry tone. 'All this is not your fault, I am sure. Be at ease and be calm. The affair will be conscientiously investigated; and if your husband does not ruin
- 56. himself, I willhave him treated as a kind of madman. If you can communicate with him, have this understood. That is my advice.' 'I will follow it, and thank your majesty, without whose protection I am quite powerless. My husband is imprisoned at Prague, and I am engaged at the Imperial Theatre at Vienna. If your majesty will but give me leave of absence and an order to see my husband, who is in strict confinementââ' 'You ask a great deal. I do not know whether Kaunitz will give you leave of absence, or if your place at the theatre can be supplied. We will see all about it in a few days.' 'A few days!' said Consuelo, boldly. 'Then, perhaps, he will be no more. I must go now!ânow!' 'That is enough,' said the empress. 'Your urgency would injure you in the minds of judges less calm than I. Go, mademoiselle.' Consuelo went to the old Canon ***, and entrusted her children to his charge, at the same time saying she was about to leave for she knew not how long a time. 'If you go for a long time,' said he, 'so much the worse for me. As for the children, they will give me no trouble, for they are perfectly well brought up, and will be company to Angela, who begins to be subject to ennui.' The good canon did not attempt to ascertain her secret. As, however, his quiet easy mind could not conceive a sorrow without a remedy, he attempted to console her. Finding that he did not succeed in inspiring her with hope, he sought at least to make her easy about her children. 'Dear Bertoni,' said he, kindly, and striving in spite of his tears to smile, 'remember, if you do not come back, your children are mine. I take charge of their education. I will marry the girl, and that will diminish Angela's portion a little, and make her more industrious. The boys, I warn you, I will make musicians.'
- 57. 'Joseph Haydn willshare that burden with you,' said Consuelo, 'and old Porpora will yet be able to give them some lessons. My children are docile and seem intellectual; so that their physical existence does not trouble me. They will be able to support themselves honestly. You must replace my love and advice.' 'I promise to do so,' said the canon. 'I hope to live long enough to see them established. I am not very fat, and I can yet walk steadily. I am not more than sixty, although Bridget insists that I should make my will. Then have courage, my daughter, and take care of your health. Come back soon, for God takes care of the pure-hearted.' Consuelo, without any trouble about her leave of absence, had horses put to her carriage. Just as she was about to set out, Porpora came to know whither she was going. She had been unwilling to see him, knowing as she did that he would seek to prevent her departure. He was afraid, notwithstanding her promises, that she would not be back in time for the opera next day. 'Who the devil dreams of going to the country in the winter time,' said he, with a nervous tremor caused as much by fear as old age. 'If you take cold you will endanger my success. I do not understand you. We succeeded yesterday, and you travel to-day.' This conversation made Consuelo lose a quarter of an hour, and enabled the directors to inform the authorities of her intention. She was in consequence forced to submit to a picket of Hulans, who immediately surrounded the house and stood sentinels at her door. She was soon seized with fever caused by this sudden check on her liberty, and frantically paced the room while she replied to the questions of Porpora and the directors. She did not sleep that night, but passed it in prayer. In the morning she was calm, and went to the rehearsal as she was desired. Her voice was never more melodious, but she was so mentally abstracted that Porpora became alarmed. 'Cursed marriage! Cursed lovers' folly!' murmured he to the orchestra, striking the keys of his instrument as if he would break it.
- 58. Porpora was unchanged,and would have willingly said, 'Perish all lovers and husbands in the world, so that my opera succeeds.' At night Consuelo made her toilet as usual, and went on the stage. She placed herself in proper attitude, and she moved her lips, but the voice was goneâshe could not speak! The audience was amazed. The court had heard something vague about her attempt at flight, and pronounced it an unpardonable whim. There were cries, hisses, and applause at every effort she made. Still she was inaudible. She stood erect not thinking of the loss of her voice, nor feeling humiliated by the indignation of her tyrants, but resigned and proud as a martyr condemned to an unjust punishment; while she thanked God for having so afflicted her, that she could leave the stage and join her husband. It was proposed to the empress that the rebellious artist should be imprisoned, there to recover her voice and good temper. Her majesty was angry for a moment, and the courtiers thought to ingratiate themselves with her by advising cruelty; but the empress did not like unnecessary severity, though she could connive at remunerative crime. 'Kaunitz,' said she, 'permit the poor woman to leave, and say nothing more about it. If her loss of voice is feigned, her duty seems to require it. Few actresses would sacrifice professional success at the altar of conjugal affection and duty.' Consuelo thus authorised set out. She was unwell, without being apparently aware of it. Here again we lose the thread of events. The cause of Albert may have been public or secret. It is probable that it was analogous to the suit which Trenck made and lost, after so many years' dispute. Who in France would not know the details of this affair, had not Trenck himself published and spread his complaints abroad for thirty years? Albert left no documents. We must then turn to Trenck's story, he too being one of our heroes. It is probable his troubles may throw some light on those of Albert and Consuelo.
- 59. About a monthafter the meeting at St. Graal, of which in his memoirs Trenck says nothing, he was recaptured and imprisoned at Magdenbourg, where he passed ten years of his life, loaded with eighty pounds of irons. The stone to which he was bound bears the inscription Here lies Trenck. All know his terrible fate, and the sufferings he underwent, as also his wonderful attempts at escape, and his incredible energy, which never left him, but which his chivalric imprudence counteracted. His sister was subjected to the cruelty of paying for the erection of a dungeon for him, because she afforded him a refuge in his flight. Trenck's works of art in prison, the wonderful engravings he made with the point of a nail on the tin cups, which are allegories or verses of great beauty, are also well known.[20] In fine, from his secret relations with the princess Amelia âthe despair in which she wasted away, and her care to disfigure her face by means of a corrosive fluid, which almost destroyed her sightâthe deplorable state of health to which she reduced herself to avoid marriageâthe remarkable change effected on her characterâ the ten years of agony, which made him a martyr, and her an old woman, ugly and malicious, instead of the angelic creature she was, and would have been had she been happy[21]âthe misfortunes of the lovers are historical; but they are generally forgotten when the character of Frederick the Great is written. These crimes, committed with such refined cruelty, are indelible spots on the character of that monarch. At length Trenck was released, as is known, by the intervention of Maria Theresa, who claimed him as her subject. This was accomplished by the influence of Karl, her majesty's valet. In relation to the curious intrigues of this magnanimous man with his sovereign, some of the strangest, most touching and pathetic pages of the memoirs of the age have been written. During the first part of the captivity of Trenck, his cousin, the famous Pandour, a victim of truer though not less hateful accusations, died it is said at Spielberg of poison. As soon as Trenck was free, the Prussian came to ask for his cousin's vast estate; but Maria Theresa
- 60. had no ideaof yielding it. She had taken advantage of the exploits of Pandour, and profited by his death. Like Frederick and other crowned tyrants, while the power of position dazzled the masses, she paid no attention to the secret offences for which God will call her to account at the day of judgment, and which will at least weigh as heavy as her official virtues. The avarice of the empress was exceeded by her agents, the ignoble persons she had made curators of Pandour's estate, and the prevaricating magistrates who decided on the rights of the heir. Each had a share of the spoil, but the empress secured the largest. It was in vain that, years after, she sent to prison and the galleys all her accomplices in this fraud, as she never made complete restoration to Trenck. Nothing describes the character of the empress better than that portion of Trenck's book, in which he speaks of his interviews with her. Without divesting himself of the loyalty which was then a kind of patrician religion, he makes us feel how very avaricious and hypocritical this deceitful woman was. He exhibits an union of contrasts, a character at once base and sublime, innocent and false, like all those naturally pure hearts which become captivated by the corruption of absolute powerâthat great river of evil, on the breakers of which the noblest impulses of the human heart have been dashed to pieces. Resolved to thwart him, she yet afterwards deigned to console and encourage him, and promise him protection against his infamous judges;âand, finally, pretending not to have been able to discover the truth she sought, she bestowed on him the rank of major, and offered the hand of an ugly old woman who was both devout and gallant. On the refusal of Trenck, the royal matrimomaniac told him he was a presumptuous madman, that she had no means of gratifying his ambition, and coldly turned her back upon him. The reasons assigned for the confiscation of his estate varied under circumstances. One court said that Pandour, undergoing an infamous sentence, could make no will. Another, that if there were a will, the claimant, as a Prussian, could not benefit by it; and that the debts of the deceased absorbed everything. Incident
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