Signals and Systems Prof. H. Sameti Chapter 7: 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

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? Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

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? Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

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

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

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

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

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 ω s = 2π/T Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8 X(jω)=0 for |ω| > ω M ω s > 2ω M = The Nyquist rate

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 Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

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 (ω s > 2ω M ).  What if ω s ≤ 2ω M ? Something different: more later. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

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 ≥ ω c Book Chapter#: Section# Computer Engineering Department, Signal and Systems 11

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

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

Undersampling and Aliasing  When ω s ≤ 2ω M => Undersampling Book Chapter#: Section# Computer Engineering Department, Signal and Systems 14

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, x r (nT) = x (nT) Book Chapter#: Section# Computer Engineering Department, Signal and Systems 15 X r (jω) ≠ X(jω) Distortion because of aliasing

A Simple Example X(t) = cos(ω o t + Φ) Picture would be Modified… Demo: Sampling and reconstruction of cosω o t

Sampling Review Demo: Effect of aliasing on music.

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

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 x c (t) and x d [n], or X c (jω) and X d (e jΩ )

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

Frequency-Domain Interpretation of C/D Conversion

Illustration of C/D Conversion in the Frequency-Domain

D/C Conversion y d [n] →y c (t) Reverse of the process of C/D conversion Again, Ω = ωΤ

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 x c (t), with  When the input x c (t) is band-limited (X(jω) = 0 at |ω| > ω M )and the sampling theorem is satisfied (ω S > 2ω M ), then

Frequency-Domain Illustration of DT Processing of CT Signals

Assuming No Aliasing In practice, first specify the desired H c (jω), then design H d (e jΩ ).

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

Construction of Digital Differentiator Bandlimited Differentiator Desired: Set Assume (Nyquist rate met) Choice for H d (e jΩ ):

Band-Limited Digital Differentiator (continued)

Changing the Sampling Rate 30

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

Downsampling -3T -2T –T 0 T 2T 3T A continuous-time signal Suppose we now sample at two rates: C/D T (1) C/D MT (2) Q: Relationship between the DTFT’s of these two signals? 32 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

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

Downsampling Change of variable: 35

Downsampling If M=2, 36

Downsampling (example) 37

Downsampling (example) What could go wrong here? 38

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

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

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

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

Upsampling n – a b cd e f b df 43 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Upsampling 44

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

Upsampling n – a b cd e f b df n df 46 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

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

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

Upsampling L 50