1. Signals and Systems Chapter 7: Sampling
The Concept and Representation of Periodic Sampling of a CT Signal
Analysis of Sampling in the Frequency Domain
The Sampling Theorem - the Nyquist Rate
In the Time Domain: Interpolation
Undersampling and Aliasing
Review/Examples of Sampling/Aliasing
DT Processing of CT Signals

2. Book Chapter#: Section#
SAMPLING
We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. x(t).
How do we convert them into DT signals x[n]?
Sampling, taking snap shots of x(t) every T seconds
T sampling period
x [n] ≡x (nT), n = ..., -1, 0, 1, 2, regularly spaced samples
Applications and Examples
Digital Processing of Signals
Strobe
Images in Newspapers
Sampling Oscilloscope …
How do we perform sampling?
Computer Engineering Department, Signal and Systems

3. Why/When Would a Set of Samples Be Adequate?
Book Chapter#: Section#
Why/When Would a Set of Samples Be Adequate?
Observation: Lots of signals have the same samples
By sampling we throw out lots of information (all values of x(t) between sampling points are lost).
Key Question for Sampling:
Under what conditions can we reconstruct the original CT signal x(t) from its samples?
Computer Engineering Department, Signal and Systems

4. Impulse Sampling: Multiplying x(t) by the sampling function
Book Chapter#: Section#
Impulse Sampling: Multiplying x(t) by the sampling function
Computer Engineering Department, Signal and Systems

5. Analysis of Sampling in the Frequency Domain
Book Chapter#: Section#
Analysis of Sampling in the Frequency Domain
Multiplication Property:
Important to note:
Sampling Frequency
Computer Engineering Department, Signal and Systems

6. Book Chapter#: Section#
Illustration of sampling in the frequency-domain for a band-limited signal (X(jω)=0 for |ω| > ωM)
drawn assuming:
No overlap between shifted spectra
Computer Engineering Department, Signal and Systems

7. Reconstruction of x(t) from sampled signals
Book Chapter#: Section#
Reconstruction of x(t) from sampled signals
If there is no overlap between shifted spectra, a LPF can reproduce x(t) from xp(t)
Computer Engineering Department, Signal and Systems

8. Reconstruction of x(t) from sampled signals
Book Chapter#: Section#
Reconstruction of x(t) from sampled signals
Suppose x(t) is band-limited, so that :
Then x(t) is uniquely determined by its samples {x(nT)} if :
where ωs = 2π/T
X(jω)=0 for |ω| > ωM
ωs > 2ωM = The Nyquist rate
Computer Engineering Department, Signal and Systems

9. Observations on Sampling
Book Chapter#: Section#
Observations on Sampling
In practice, we obviously don’t sample with impulses or implement ideal low-pass filters.
One practical example: The Zero-Order Hold
Computer Engineering Department, Signal and Systems

10. Observations (Continued)
Book Chapter#: Section#
Observations (Continued)
Sampling is fundamentally a time varying operation, since we multiply x(t) with a time-varying function p(t). However,
is the identity system (which is TI) for band-limited x(t) satisfying the sampling theorem (ωs > 2ωM ).
What if ωs ≤ 2ωM? Something different: more later.
Computer Engineering Department, Signal and Systems

11. Book Chapter#: Section#
Time-Domain Interpretation of Reconstruction of Sampled Signals: Band-Limited Interpolation
The low-pass filter interpolates the samples assuming x(t) contains no energy at frequencies ≥ ωc
Computer Engineering Department, Signal and Systems

12. Original CT signal After Sampling After passing the LPF
Book Chapter#: Section#
Original
CT signal
After Sampling
After passing the LPF
Computer Engineering Department, Signal and Systems

13. Interpolation Methods
Book Chapter#: Section#
Interpolation Methods
Band-limited Interpolation
Zero-Order Hold
First-Order Hold : Linear interpolation
Computer Engineering Department, Signal and Systems

14. Undersampling and Aliasing
Book Chapter#: Section#
Undersampling and Aliasing
When ωs ≤ 2ωM => Undersampling
Computer Engineering Department, Signal and Systems

15. Undersampling and Aliasing (continued)
Book Chapter#: Section#
Undersampling and Aliasing (continued)
Higher frequencies of x(t) are “folded back” and take on the “aliases” of lower frequencies
Note that at the sample times, xr (nT) = x (nT)
Xr (jω) ≠ X(jω)
Distortion because of aliasing
Computer Engineering Department, Signal and Systems

16. A Simple Example X(t) = cos(ωot + Φ) Picture would be Modified…
Demo: Sampling and reconstruction of cosωot

17. Sampling Review
Demo: Effect of aliasing on music.

18. Strobe Demo
Δ > 0, strobed image moves forward, but at a slower pace
Δ = 0, strobed image still
Δ < 0, strobed image moves backward.
Applications of the strobe effect (aliasing can be useful sometimes):—
E.g., Sampling oscilloscope

19. DT Processing of Band-LimitedCT Signals
Why do this?
—Inexpensive, versatile, and higher noise margin.
How do we analyze this system?
—We will need to do it in the frequency domain in both CT and DT
—In order to avoid confusion about notations, specify
ω—CT frequency variable
Ω—DT frequency variable (Ω = ωΤ)
Step 1:Find the relation between xc(t) and xd[n], or Xc (jω) and Xd (ejΩ)

20. Time-Domain Interpretation of C/D Conversion
Note: Not full
analog/digital (A/D)
conversion – not
quantizing the x[n]
values

21. Frequency-Domain Interpretation of C/D Conversion

22. Illustration of C/D Conversion in the Frequency-Domain

23. D/C Conversion yd[n] →yc(t) Reverse of the process of C/D conversion
Again, Ω = ωΤ

24. Now the whole picture
Overall system is time-varying if sampling theorem is not satisfied
It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs xc (t), with
When the input xc(t) is band-limited (X(jω) = 0 at |ω| > ωM)and the sampling theorem is satisfied (ωS> 2ωM), then

25. Frequency-Domain Illustration of DT Processing of CT Signals

26. Assuming No Aliasing
In practice, first specify the desired Hc(jω), then design Hd(ejΩ).

27. Example:Digital Differentiator
Applications: Edge Enhancement
Courtesy of Jason
Oppenheim. Used
with permission.

28. Construction of Digital Differentiator
Bandlimited Differentiator
Desired:
Set Assume (Nyquist rate met)
Choice for Hd(ejΩ):

29. Band-Limited Digital Differentiator (continued)

30. Changing the Sampling Rate

31. Downsampling b e c d f a n Downsample by a factor of 2 b d f n– n
Downsample by a factor of 2 b d f n
Downsample by a factor of N: Keep one sample, throw away (N-1) samples
Advantage?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

32. Downsampling A continuous-time signal
-3T -2T –T T 2T 3T
A continuous-time signal
Suppose we now sample at two rates:
C/D
C/D
(1)
(2) T MT
Q: Relationship between the DTFT’s of these two signals?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. Downsampling (2) (1) MT T Change of variable: C/D C/D
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. Downsampling Change of variable:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35. Downsampling
Change of variable:

36. Downsampling
If M=2,

37. Downsampling (example)

38. Downsampling (example)
What could go wrong here?

39. Downsampling with aliasing (illustration)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

40. Downsampling (preventing aliasing)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41. Downsampling with aliasing (illustration)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42. Upsampling
It is the process of increasing the sampling rate by an integer factor.
Application?
C/D
(1) T
C/D
(2)
T/L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43. Upsampling b d f b e c d f a n -3 -2 –1 0 1 2– n
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

44. Upsampling

45. gain L and cut-off frequency
Upsampling
Question: How can we obtain from ?
Proposed Solution:
Low-pass filter with
gain L and cut-off frequency L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

46. Upsampling b d f b e c d f a n d f n -3 -2 –1 0 1 2 -3 -2 –1 0 1 2– n d f – n
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47. Upsampling Shifting property:
Q: Relationship between the DTFT’s of these three signals?
Shifting property:

48. Upsampling (Eq.1) On the other hand: (Eq.2) (Eq.1) and (Eq.2)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49. Upsampling In order to get from we thus need an ideal low-pass filter.

50. Upsampling L