Sampling Theorem, Nyquist, Sampling Examples, Sine Spectrum

1. UNIT 1
DSP Preliminaries

2. UNIT 1 SYLLABUS
• Sampling
• DT signals
• Sampling theorem in time domain
• Sampling of analog signals
• Recovery of analogue signals
• Analytical treatment with examples
• Mapping between analog frequencies to digital frequency
• Representation of signals as vectors
• Concept of Basis function and orthogonality
• Eigen value and Eigen vector
• Basic elements of DSP and its requirements
• Advantages of Digital over Analog signal processing

3. SAMPLING
• What is sampling?
• Continuous time  Discrete time

4. SOMETHING ABOUT HARRY NYQUIST
• As an engineer at Bell Laboratories,
Nyquist did important work on
• thermal noise ("Johnson–Nyquist noise")
• the stability of feedback amplifiers,
• telegraphy, facsimile,
• television, and other important
communications problems.
• With Herbert E. Ives, he helped to
develop AT&T's first facsimile machines
that were made public in 1924.
• In 1932, he published a classic paper on
stability of feedback amplifiers.
• The Nyquist stability criterion can now be
found in all textbooks on feedback
control theory.

5. AND CLAUDE SHANNON
• Shannon developed information
entropy as a measure of the
information content in a message,
which is a measure of uncertainty
reduced by the message, while
essentially inventing the field
of information theory.
• In 1949 Claude Shannon and Robert
Fano devised a systematic way to
assign code words based on
probabilities of blocks.
• This technique, known as Shannon–
Fano coding, was first proposed in the
1948 article.

6. SHANNON GOT INSPIRED BY ALAN TURING
• For two months early in 1943, Shannon came into contact with the leading British
mathematician Alan Turing.
• Turing had been posted to Washington to share with the U.S. Navy's cryptanalytic
service the methods used by the British Government Code and Cypher
School at Bletchley Park to break the ciphers used by the Kriegsmarine U-boats in
the north Atlantic Ocean.
• He was also interested in the encipherment of speech and to this end spent time at
Bell Labs.
• Shannon and Turing met at teatime in the cafeteria.
• Turing showed Shannon his 1936 paper that defined what is now known as the
"Universal Turing machine".This impressed Shannon, as many of its ideas
complemented his own.

7. SAMPLING THEOREM
• A continuous time signal can be represented in its samples and can be recovered
back when sampling frequency fs is greater than or equal to the twice the highest
frequency component of message signal. i. e.
• fs = 2* fmax
• X(t)
• X(w)
• Y(W) output spectrum - replicas

8. THREE CASES OF SAMPLING FREQUENCY
• Case 1
• fs > 2*fmax
• Case 2
• fs = 2*fmax
• Case 3
• fs < 2*fmax
• Case 1
• fs > 2*fmax
• Oversampling
• Case 2
• fs = 2*fmax
• Nyquist Plot
• Case 3
• fs < 2*fmax
• Undersampling

9. SPECTRUM OF SAMPLED SIGNAL

10. ALIASED SIGNAL AND
AFFECTED SPECTRUM

11. CONCEPT OF ALIASING
• The overlapped region in case of under sampling represents aliasing effect, which
can be removed by
• Making fs >2fm
• By using anti aliasing filters

12. ASPECTS OF ALIASING
• Spectrum exhibits
spectral peaks at
harmonics, i.e., integer
multiples, of 440 Hz.
• The amplitude of the
harmonics is quite small
above 4 kHz.
• The spectral peak near 0
Hz occurs at 60 Hz and
is due to power-line
noise contaminating the
recording.
Saxophone Note Frequencies

13. SPECTRUM OF SAMPLED SAXOPHONE NOTE WITH ANTIALIASING FILTER USED TO
PREVENT ALIASING
• We sample this signal at 2756
Hz with and without an
antialiasing filter.
• At this sampling frequency
the maximum continuous-
time sinusoid frequency that
can be represented without
aliasing is 1378 Hz.
• Thus, the antialiasing filter
limits the signal to the first
three harmonics.
• The third harmonic occurs at
1320 Hz.

14. SPECTRUM OF SAMPLED SAXOPHONE NOTE WITH
NO ANTIALIASING FILTER.
• The aliased fourth, fifth,
and sixth harmonics are
labeled in Figure 4.
• The spectral peaks at 324
Hz and 764 Hz are
aliases of the seventh
and eighth harmonics

15. WHAT EXACTLY HAPPENS WHEN
ALIASING OCCURS
• A wind instrument like a saxophone creates a very tonal sound
• because the only significant contributors are harmonics of the
fundamental.
• Aliasing introduces energy at frequencies which are not
harmonically related to the fundamental, and introduces a muddy
character to the sound, or a buzzing aspect.
https://allsignalprocessing.com/2015/04/25/aliasing-of-
signals-identity-theft-in-the-frequency-domain/

16. ANALOG FREQUENCY AND DIGITAL
FREQUENCY
• Analog Signal
• t
• x(t)
• T
• F analog freq
• Omega Ώ
• Digital Signal/ discrete signal
• n
• x[n]
• N time period for DT signals
• f digital freq
• w

17. ANALOG TO DIGITAL---HOW DO WE
DO IT?
•t= n*Ts
•X(t) = sin(2*pi*50*t)
•x[t/fs] = sin(2*pi*50*n/fs)
•Fs = 2*50 =100
•X[n] = sin(2*pi*50*n/100) = sin(pi*n)

18. ANALOG TO DIGITAL---HOW DO WE
DO IT?
•t= n*Ts
•X(t) = sin(2*pi*50*t)
•x[n/fs] = sin(2*pi*50*n/fs)
•If not given, consider the sampling
frequency to be 2*fmax

19. SAMPLING SUMS
• Important points to remember:
• Find maximum frequency, fmax in the signal
• Convert x(t) into x[n] by using the relation
• ( t = n*Ts OR t = n/fs )
• If sampling frequency is given, substitute the value
• If not given, take fs = 2*fmax
• Plot the spectrum up to given frequency

20. SPECIAL CASE OF SINE

21. SINE: CASE 1

22. SINE: CASE 1
We have non-overlapping
replicas, and a simple
reconstruction filter Hr(w)
as seen previously can
reconstruct.
This is not surprising, since
we have obeyed the rules of
the sampling theorem.

23. SINE: CASE 2

24. SINE: CASE 2
Here, our replicas overlap.
When we apply the
reconstruction filter, we do
not recover the same signal.
In fact, in this case, xr(t) =
sin−(ws − wo)t =
−sin (ws − wo)t which is quite
different from xc(t) = sin wot.
Under this sampling scheme,
xr(t) is an alias of xc(t).

25. SINE: CASE 3

26. SINE: CASE 3
In this case, even if weird stuff
didn’t happen due to the
undefined boundary
of the reconstruction filter, the
replicas clearly cancel each other
out.
Thus, the recovered signal will
have no frequency components.
xr(t) will be a DC
signal, which is clearly wrong.

27. Relationship between the
analog and digital
frequency i.e. Ω and ω:
The point 7 gives the
derivation of that
relationship.
It tells that digital freq is
restricted by (-pi to +pi)

28. DSP Block Schematic
and advantages of DSP
over ASP

29. EXAMPLES
Example1: Consider a CT signal xc(t) with a Fourier transform Xc(ω) that is sampled with ideal sampling,
with Ts = 10^−4 seconds. For each of the following scenarios, determine whether the sampling theorem
guarantees that xc(t) can be recovered from its samples.
Scenario1: What if Xc(ω) = 0 for |ω| > 5000π?
Scenario 2: What if Xc(w) = 0 for |w| > 10000?
Scenario 3: What if Xc(w) = 0 for |w| > 15000?
Scenario 4: What if Xc(w) * Xc(w) = 0 for |w| > 15000?

30. Scenario 4

35. TYPICAL DSP BLOCK DIAGRAM
Courtesy: google.com

36. THANK YOU