Sources - Separation methods allows the possibility to transmit many signals through a given channel without multiplexing them. These signals may be frequency overlapped. As an extension, the paper presents a method to reconstruct the transmitted signals from amplitudemodulation radio frequency (AM-RF) channels that are partially overlapped in frequency. The cross talks due to the overlapping procedure leads to a lost of information but increase the number of channels in the RF band. The signals reconstruction implies the use of Volterra-integral equations. Data transmissions imply also the improvement of the receiver equalizer. Simulations are presented. The presented method lead then to a diminishing of the equivalent frequency bandwidth of the transmitting signals.

1. ACEEE International Journal on Communication, Vol 1, No. 1, Jan 2010
AM-RF partially overlapped channels separation
by Volterra integral equations.
Dan Ciulin.
E-I-A/Lausanne, Switzerland
Email: dan.ciulin@gmail.com
Abstract—Sources - Separation methods allows the function of the band-pass filter with its central frequency
possibility to transmit many signals through a given channel shifted to zero.
without multiplexing them. These signals may be frequency
overlapped. As an extension, the paper presents a method to Channel 1 Channel 2
reconstruct the transmitted signals from amplitude- Δω1
modulation radio frequency (AM-RF) channels that are
partially overlapped in frequency. The cross talks due to the
Carrier ω1
Carrier ω2
overlapping procedure leads to a lost of information but
increase the number of channels in the RF band. The signals
reconstruction implies the use of Volterra-integral
equations. Data transmissions imply also the improvement
of the receiver equalizer. Simulations are presented. The
presented method lead then to a diminishing of the angular frequency
equivalent frequency bandwidth of the transmitting signals. a
Index Terms—Source Separation, R.F. Signals.
Channel 1 Channel 2
Δω
I. INTRODUCTION.
Many signals may be transmitted through a given
Carrier ω1
Carrier ω2
channel without multiplexing them by using the Source -
Separation methods [5], [6]. In this case, only the
receivers have to be modified. Such methods lead then to
a diminishing of the equivalent bandwidth of the
transmitting signals. An other method to diminish the angular frequency
bandwidth of an audio signal based on a Volterra- b
Figures 1: Two AM-RF channels: (a) non-overlapped and (b) partially
Hammerstein integral equation [7] has been presented in overlapped in frequency.
[1], [3] and [4]. To diminish the necessary bandwidth of
AM-RF channels placed into a RF transmission band, one Similarly, we can shift to zero the carrier angular
cans partially overlap these channels in frequency. This frequency ω 2 and denote by s 2 (t ) the transmitted signal
method leads to cross talk errors but increases the number on channel 2 with its carrier shifted to zero. For similar
of transmitted channels into the band. As an extension of RF channels, the time responses h1 (t ) and h2 (t ) of the
[5], [6] and [2], a method to separates these frequency-
band-pass filters corresponding to the first and second
overlapped signals will be presented.
channel will differ only by their central frequency. One
Let us first consider the simple case of only 2
denote by h(t ) the time response function of the band-
transmitting AM-RF channels like in figure 1 a.
Diminishing the channel bandwidth may be realized by pass filter with its central frequency shifted to zero. The
partially overlapping these transmitting channels as in received signals r1 (t ) and r2 (t ) (considered also with
figure 1 b. One can observe that the shift Δω1 between their carrier shifted to zero) will be distorted by cross talk
the angular frequency carriers ω1 and ω2 is bigger than due to the overlapping procedure for the first channel:
Δω due to the overlapping procedure and thus, the total r1 (t ) = [ s1 (t ) + Re{ s 2 (t ).e [ j .( ω 2 − ω 1 ). t ]
}] * h (t ). (1)
bandwidth of the AM-RF channels is smaller.
Let us make a frequency shift so that the angular We remark that the second RF channel is shifted by
frequency ω1 carrier becomes zero. We denote by s1 (t ) Δω 1before having been filtered by the first channel filter
the transmitted the transmitted signal on channel l with its h(t ) . One gets also for the second channel:
carrier shifted to zero. Similarly, we can shift to zero the r2 (t ) = [ s 2 (t ) + Re{ s1 (t ).e [ j .( ω 2 − ω 1 ). t ]
}] * h (t ). (2)
carrier angular frequency ω 2 and denote by s 2 (t ) the
transmitted signal on channel 2 with its carrier shifted to Of course:
zero. For similar RF channels, the time responses h1 (t ) ⎧ s1 (t ) = s1 (t ) * h (t ).
⎨ s (t ) = s (t ) * h (t ). (3)
and h2 (t ) of the band-pass filters corresponding to the ⎩ 2 2
first and second channel will differ only by their central
Taking into account (1), (2) and (3) we get finally:
frequency. One denote by h(t ) the time response
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2. ACEEE International Journal on Communication, Vol 1, No. 1, Jan 2010
⎧ r1 (t ) = s1 (t ) + s 2 (t ). cos[( Δω .t ] * h (t ). (4) product’. Using an oscillator which frequency is Δω , the
⎨ r (t ) = s (t ) + s (t ). cos[ Δω .t ] * h(t ).
⎩ 2 2 1 multipliers M 1 and M 2 , two channel filters ( h(t ) ) and
where ‘*’ stand for convolution. two devices to subtract, we get the signals f 1 (t ) and
f 2 (t ) .
II. SIGNALS RECONSTRUCTION BASED ON
VOLTERRA INTEGRAL EQUATIONS.
M1
Equations (4) represent a system of 2 equations where r1 (t) h(t) Γ1(t,τ) ~
s1 (t)
f1 (t)
only the transmitted signals s1 (t ) and s 2 (t ) are unknown.
Cos(Δωt)
Solving this system results in: M2
r2 (t) h(t) Γ2(t,τ) ~
s2 (t)
s 2 (t ) = r2 (t ) − s1 (t ). cos[ Δω .t ] * h (t ). f2 (t)
Separation System
r1 ( t ) = s 1 (t ) + { r2 (t ) − s 1 (t ). cos[ Δ ω .t ] * (5)
Figures 2: Reconstruction time-varying filter.
h (t )}. cos[ Δ ω .t ] * h (t ).
A pair of generalized filters which kernel is Γ(t ,τ )
This is an integral equation that can also be written as: and two adders leads to the reconstructed signals. The
reconstruction system implies thus only filters and
r1 ( t ) − r 2 ( t ). cos[ Δ ω .t ] * h ( t ) =
mixers. This corresponds to a linear time-varying system.
As for ordinary AM signals, shifting their carrier to zero
s1 (t ) − {[ s1 (t ). cos[Δω .t ] * h(t )]}. cos[Δω .t ] * h(t ). (5’) correspond to a product demodulation, the resulting
With: signals ~1 (t ) and ~2 (t ) are in this case the reconstructed
s s
demodulated signals.
f1 (t ) = r1 (t ) − r 2 (t ). cos[ Δ ω .t ] * h (t ). (6)
one observes that this function can be easily calculated as III. EXTENSION TO 3 FREQUENCIES
r1 (t ) , r2 (t ) are the received signals and the filter h (t ) is OVERLAPPED RF CHANNELS.
known. The model presented before works only for 2 RF
The term {[ s1 (t ).Cos [Δω .t ] * h2 (t )]}. cos[ Δω .t ] * h2 (t )} can channels overlapped in frequency. The case of 3 RF
be written as: channels partially overlapped in frequency is shown in
figure 3. In the following we suppose that the angular
⎧ t t
{[s1 (u ).Cos [Δω .u ]. h 2 (τ − u )]}. frequency gap between all the adjacent channels is the
⎪ ∫∞ −∫∞cos[Δω .τ ].h (t − τ )}.du .d τ = same and equal to Δω .
⎪ − 2
⎪ t
⎨ ∫ s1 (u ). cos[Δω .u ].H (t , u ).du . (7)
Channel 1 Channel 2 Channel 3
Δω Δω
⎪ −∞
⎪
t
⎪ H (t , u ) = h(t − u ). ∫ h(τ − u ). cos[Δω .τ ]}.d τ .
⎩ −∞
Carrier ω3
Carrier ω1
Carrier ω2
Using (6) and (7), the equation (5') becomes:
⎧ t
⎪s1 (t ) = f1 (t ) + ∫ s1 (u ). cos[Δω .u ].H (t , u ).du =
⎪ (8)
⎨ −∞
t angular frequency
⎪ s1 (t ) = f1 (t ) + ∫ s1 (u ).K (t , u ).du .
⎪
⎩ −∞
Figures 3: Three partially-frequency overlapped channels.
which is a linear Volterra integral equation of second
The received signals r1 (t ) , r2 (t ) and r3 (t ) for each
kind and has the solution:
channel, where each signal has its carrier shifted to zero,
⎧ ~ t
will be given by:
⎪ s1 (t ) = f1 (t ) + ∫ f (u ).Γ1 (t , u ).du .
⎪ −∞
(9)
⎪ ∞
⎨ Γ1 (t , u ) = ∑ K n + 1 (t , u ).
⎪∞ n =1 ⎧ r1 (t ) = s1 (t ) + s 2 (t ). cos[ Δω .t ] * h(t ).
⎪ r2 (t ) = s 2 (t ) + { s1 (t ). cos[ Δω .t ] +
t
⎪ K (t , u ) = K ( z , u ). K (t , z ).d .;
⎪∑ n + 1 ∫∞ n ⎪ (10)
⎩ n =1 ⎨
⎪ s 3 (t ). cos[ Δω .t ]} * h(t ).
−
⎪ r3 (t ) = s 3 (t ) + s 2 (t ). cos[ Δω .t ] * h(t ).
⎩
Taking into account (9) it results that the kernel Γ(t, u)
Solving the system (10) with respect to s 3 (t ) will
does not depend on the received signals. So, it can be a
leads to:
priori computed. The separation will imply only a single
generalized convolution (a single special filter) by r3 (t ) = s 3 (t ) + ( r2 (t ) − {[ 2.s 3 (t ) + r1 (t ) −
(11)
channel. One observes that due to the symmetry of the r3 (t )]. cos[ Δω .t ]} * h (t )}). cos[ Δω .t ] * h (t ).
equations (4), an equation similar to (5’) may be obtained
for ~2 (t ) .
s which is a Volterra integral equation in s 3 ( t ) as all the
This process is schematized in figure 2. The received other functions are known. The solution of this equation
signals r1 (t ) and r2 (t ) are in the base-band. For ordinary
AM, they correspond to the signals ‘demodulated by
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3. ACEEE International Journal on Communication, Vol 1, No. 1, Jan 2010
is the reconstructed signal ~ 3 (t ) . The signals ~1 (t ) and
s s X(k)
~ (t ) can also be found out in a similar manner. T/2 T/2 T/2 T/2 T/2
s2
It can be observed that this method may also be
c0 c1 c2 c N-1 cN
extended to ‘N’ partially overlapped AM-RF channels.
Σ
IV. EQUALIZER FOR PARTIALLY OVERLAPPED
DATA QAM-RF (QUADRATURE AMPLITUDE a n-i
y(k)
MODULATION) CHANNELS.
In a data transmission, the equalizer chooses the en
sampling instants that minimize the intersymbol Figures 5: Transversal equalizer.
interference (ISI) of the received signals. For partially
frequency-overlapped channels, one has to minimize also The coefficients of the transversal equalizer should be
the interference from the spectrally overlapping signals. ~
chosen such that the mean squared error E is minimized.
For 2 partially frequency-overlapped channels, as in the For frequency-overlapped channels, this mean squared
figure 1b, one side band of each channel has partially error is a function of two variables because it depends on
spectral interference coming from the other channel as 2 statistical independent input signals. The extreme value
shown in the figure 4. This new spectral interference of the mean squared error may be then computed using
implies an improved equalizer. Thus, we chose to sample the Monge conditions. As the system implies 2
all the signals at twice the symbol frequency and use only frequency-overlapped receivers, we have to realize an
finite impulse response (FIR) filters. extreme value for both receivers at the same time. The
extreme minimum (minimum minimorum) of a function
Channel1 Channel2 f (u , v ) of 2 variables is given by the following
conditions (Monge):
Carrier ω2
Carrier ω1
⎧ ∂f ( u , v ) ∂f (u , v )
⎪ = 0;. = 0.
⎪ ∂u ∂v
⎪ ∂ f (u , v )
2
∂ f (u , v )
2 (12)
⎨ > 0;.or . > 0.
Δω ⎪ ∂u 2
2
∂v 2
⎪⎛ ∂ f (u , v ) ⎞
2
∂ f (u , v ) ∂ f ( u , v )
2 2
⎪⎜
⎜ ⎟ −
⎟ . < 0.
Channel1 Channel2 ⎩⎝ ∂u .∂v ⎠ ∂u 2 ∂v 2
Carrier ω2
Carrier ω1
At the output of the first channel equalizer we will get
~
the mean squared error E1 and at the output of the second
~
channel equalizer we will get the mean squared error E 2 .
ω Each receiver will receive the same signals, uses the same
ω
filters but these are centered on their own carrier. The
Zone with frequenc y interferenc e function f (u , v ) of 2 variables for which a minimum
~ ~
Figures 4: Frequency interference zones for 2 partially-frequency
must be found may be constituted as a function g ( E1 , E 2 )
overlapped channels. of the mean squared errors of both receivers. The
simplest linear function that may be chosen is:
The bloc diagram of the transversal filter with (T/2)
~ ~ ~ ~
taps spacing and (N+1) complex coefficients g ( E1 , E2 ) = E1 + E2 . (13)
c 0 , c1 , c 2 ,..., c N , used as improved equalizer, is shown in
as the minimum of the sum correspond to the minimum
the figure 5. For practical reasons, the number M=N+1 of its components. Of course, other types of functions
must be smaller than 30. While the output signal y (k ) of may be chosen too. It can also be remarked that the
the transversal filter delivers output data at twice the ~ ~
function g ( E1 , E2 ) depends on the equalizers time-
symbol rate, only every second output sample is used to varying coefficients:
derive a decision ( a n − i ) on the transmitted data signals.
~ ~
Since the equalizer introduces some delay, one cannot g ( E1 , E2 ) = w ({c 0 , c1 ,...c N }1 , {c 0 , c1 ,...c N }2 ). (14)
obtain the symbol a n at timing instant n but a delayed
We therefore compute the partial first and second
symbol a n − i , where i.T is the delay introduced by the derivatives of the function w ({c 0 , c1 ,...c N }1 , {c 0 , c1 ,...c N }2 )
transversal filter. The difference between the data output for the real and imaginary part of its coefficients and use
a n − i and the output signal y (k ) at this instant represents the Monge conditions for extreme. From this we get a
the error en of the equalizer. system of linear equations that can be solved to find out
the optimum coefficients of both equalizers.
V. SIMULATION.
We will consider two channels, partially overlapped in
frequency, as in figure 1b. The simulated transmitted
signals s1 (t ) and s2 (t ) for each channel are shown in
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4. ACEEE International Journal on Communication, Vol 1, No. 1, Jan 2010
figure 6. The resulted cross talk errors due to channels Separation methods presented in [2], [5] and [6]. The
partially overlapped in frequency are given by reconstruction implies a special linear time-varying filter
r1 (t ) − s1 (t ) and r2 (t ) − s 2 (t ) for the first and second that uses also mixers. In a simulation, it is observed that
channel respectively. These cross talk errors are shown in the reconstruction errors are much smaller than the cross
figure 7. Using the equation (9) and a special tool talk errors for each considered channel. Extension to 3
described in (Ciulin 2002) we compute ~1 (t ) and ~2 (t )
s s AM-RF channels partially overlapped in frequency is also
presented and may be generalized to ‘N’ AM-RF
by considering a sum of 20 terms for Γ(t ,τ ) . The
channels partially overlapped in frequency. With some
reconstruction errors are given by s1 (t ) − ~1 (t ) and s modifications, this method may be extended to other
~ (t ) for the first and second channel respectively.
s 2 (t ) − s 2 types of amplitude modulation. For a data QAM
transmission, due to the spectral overlapping of the
Tra n s m it t e d s ig n a l in t h e firs t c h a n n e l Tra n s m it t e d s ig n a l in t h e s e c o n d c h a n n e l channels, the equalizers of the receivers have to be
10 10 improved too. It can be observed that the coefficients of
the equalizers have to be optimized simultaneously for all
5 5
spectral overlapped channels due to Monge conditions of
amplitude
amplitude
0 0 extrema. Then, for the senders, only their carrier
frequencies has to be modified but the receivers must
-5 -5
includes also the reconstruction time-varying filter and a
-1 0 -1 0
new equalizer. This equivalent method of bandwidth
100 200 300 400 500 100 200 300 400 500
compression is simpler than the method presented in [3],
s a m p le s s a m p le s
[4] but the practical obtained compression factor is
Figures 6: Transmitted signals in partially-frequency overlapped smaller.
channels. Applications may be found in increasing the number of
transmitted channels for a given AM-RF band as, for
C ros s talk errors in the firs t c h ann el C ros s t alk erro rs in t he s ec o nd c han nel
5 5 example, for single side band AM phones channels and
for data QAM transmission channels. The present method
may save frequency spectrum and then, permit the
transmissions of more AM-RF channels in the same
amplitude
amplitude
0 0 frequency bandwidth and/or use it for other applications
too.
REFERENCES
-5 -5
100 200 30 0 400 500 10 0 200 30 0 40 0 500
s am ple s s a m p le s [1] D. Ciulin, “Method of reducing the useful bandwidth of
bandwidth-limited signals by coding and decoding the
Figures 7: Cross talk errors in theses partially-frequency overlapped signals, and system to carry out the method” U.S.A. Patent
channels.
5020104, May 28, 1991.
[2] D. Ciulin, “Some tools for speech processing”, 6th WORLD
These reconstruction errors are shown in figure 8. It MULTICONFERENCE ON SYSTEMICS, CYBERNETICS
can be observed that the cross talk errors are well AND INFORMATICS (SCI2002), Orlando, Florida, U.S.A
diminished by this method even for an approximation of 2002.
only 20 terms for the integral kernel. [3] D. Ciulin, “Bandwidth-compression theorem, a new tool in
signal processing”, 10th International, Signal Processing
R ec ons truc tion errors in the firs t c hannel R ec ons truc tion errors in the s ec ond c hannel
5 5
Conference on Concurrent Engineering: Research and
Application, Madeira, Portugal, 2003.
[4] D. Ciulin, “New improved blocks for signal processing”,
DECOM-TT 2004, AUTOMATIC SYSTEMS FOR
BUILDING THE INFRASTRUCTURE IN
amplitude
amplitude
0 0
DEVELOPING COUNTRIES, Bansko, Bulgaria, 3-5
October 2004.
[5] P. Comon, “Blind separation of sources, Part ll: Problem
stamens”, Signal processing, volume 24, Nr. 1, July 1991.
-5
100 200 300 400 500
-5
100 200 300 400 500
[6] Ch. Jutten, and J. Herault, “Blind separation of sources,
s am ples s am ples Part l: An adaptive algorithm based on neuromimetic
architecture”, Signal processing, volume 24, Nr. 1, July
Figures 8: Reconstruction errors in these partially-frequency overlapped
1991.
channels.
[7] V. Volterra, Theory of Functionals and of Integral and
Integro-differential Equation, Dover Publications, New
VI. CONCLUSION. York, 1959.
A method to diminish the necessary frequency D. Ciulin, “Quantizing Theorem”, 4th IEEE Intelligent Systems
IS’08: Methodology, Models, Applications in Emerging
bandwidth of an AM-RF band is to partially overlap in
Technologies, September 6-8, Varna, Bulgaria, 2008.
frequency their inside transmitted channels. This will
increase the number of inside transmitted channels but
leads to cross talk errors. A method to reconstruct the
cross talk errors of AM-RF signals partially overlapped
in frequency, based on Volterra integral equations, has
been presented. It represents an extension of the Source-
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