Signals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R.

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1 Signals & Systems Chapter 7: Sampling Adapted from: Lecture notes from MIT, Binghamton University, and Purdue Dr. Hamid R. Rabiee Fall 2013

2 Outline 1. The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem the Nyquist Rate 4. In the Time Domain: Interpolation 5. Undersampling and Aliasing 6. Review/Examples of Sampling/Aliasing 7. DT Processing of CT Signals 2

3 SAMPLING o most of the signals we encounter are CT signals, e.g. x(t). Question: How do we convert them into DT signals x[n]? Sampling, taking snap shots of x(t) every T seconds. o T sampling period x[n] x(nt), n=..., -1, 0, 1, 2,... regularly spaced samples o Applications and Examples Digital Processing of Signals Images in Newspapers Sampling Oscilloscope How do we perform sampling? 3

4 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. Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x(t) from its samples? 4

5 Impulse Sampling Multiplying x(t) by the sampling function p( t) ( t nt ) n x ( ) ( ) ( ) ( ) ( ) ( ) ( ) p t x t p t t nt x t t nt x nt n n Lecture 12 (Chapter 7) p(t) x(t) x p (t) 5

6 Analysis of Sampling in the Frequency Domain Multiplication Property => x ( t) x( t). p( t) p 1 X p ( j) X ( j)* P( j) 2 2 P( j) ( ks ) T s 2 T k = Sampling Frequency Important to note: 1 1 X p( j) X ( j)* ( ks ) X ( j( ks)) T T k k 6

7 Illustration of sampling in the frequency-domain for a bandlimited (X(jω)=0 for ω > ω M ) signal 2 T 2 s s 0 s s s X ( ) p j drawn assuming s M M ie.. 2 s M X ( ) ( )* ( ) / 2 p j X j P j No overlap between shifted spectra 7

8 Reconstruction of x(t) from sampled signals p( t) ( t nt ) n x (t) x p (t) H( j) x r (t) If there is no overlap between shifted spectra, a LPF can reproduce x(t) from x p (t) 8

9 Reconstruction (Continued) Suppose x(t) is bandlimited, so that X(jω)=0 for ω > ω M Then x(t) is uniquely determined by its samples {x(nt)} if ω s > 2ω M = The Nyquist rate where ω s = 2π/T 9

10 Observations on Sampling (1) In practice, we obviously don t sample with impulses or implement ideal lowpass filters. One practical example:the Zero-Order Hold 10

11 Observations (Continued) (2) Sampling is fundamentally a time varying operation, since we multiply x(t) with a time-varying function p(t). However, p( t) ( t nt ) n is the identity system (which is TI) for bandlimited x(t) satisfying the sampling theorem (ω s > 2ω M ). (3) What if ω s <= 2ω M? Something different: more later. c H( j) c ( ) M c s M 11

12 Time-Domain Interpretation of Reconstruction of Sampled Signals Band-Limited Interpolation p( t) ( t nt ) n x ( t) x ( t)* h( t), where h( t) r p ( x( nt ) ( t nt ))* h( t) n T sin[ c ( t nt )] x( nt ) h( t nt ) x( nt) ( t nt ) n n Tsin ct t The lowpass filter interpolates the samples assuming x(t) contains no energy at frequencies >= ω c 12

13 Graphic Illustration of Time-Domain Interpolation Lecture 12 (Chapter 7) Original CT signal After Sampling After passing the LPF 13

14 Interpolation Methods Bandlimited Interpolation Zero-Order Hold First-Order Hold Linear interpolation ht () 1 t 14

15 Undersampling and Aliasing When ω s 2ω M => Undersampling 2 s T 15

16 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) s 2 H( j) s 2 X r (jω) X(jω) Distortion because of aliasing 16

17 Aliasing: What if the signal is NOT BANDLIMITED? For Non-BL Signals, Aliasing always happens regardless of Fs value All practical signal are Non-BL! so we choose Fs to minimize aliasing. So we use a CT low-pass BEFORE the ADC! (demonstrated on next slide) 17

18 Aliasing: What if the signal is NOT BANDLIMITED? (continued) Lecture 12 (Chapter 7) 18

19 A Simple Example X(t) = cos(w o t + Φ) X( j) 0 0 X P ( j) 3 2 s 0 M Picture would be Modified 20

20 Sampling Review 2 If X ( j) 0, M and s 2M T then, assumi ng wechoose : M c s M x ( t) x( t) r 21

21 Strobe Demo Lecture 12 (Chapter 7) 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 Demo: Effect of aliasing on music. 22

22 DT Processing of Band-Limited CT Signals X () c t C/D conversion X [ n] X ( nt) d c Discrete-Time System y [ n] y ( nt ) d c D/C conversion ` y () c t T j He ( ) Why do this? Inexpensive, versatile, and higher noise margin. T 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 ) 24

23 Time-Domain Interpretation of C/D Conversion Lecture 12 (Chapter 7) Note: Not full analog/digital(a/d) conversion not quantizing the x[n] values x () C t pt () C/D conversion x () P t Conversion of impulse train to discrete-time sequence x[ n] x ( nt ) C 25

24 Frequency-Domain Interpretation of C/D Conversion x ( t) x ( t). ( t nt ) x ( nt ) ( t nt ) p c c n n 1 jnt X p ( j) X c( j( ks )) xc ( nt ) e (1) T n CT n 2 Periodic with period T j jn jn X ( e ) x [ n] e x ( nt ) e (2) d d c n n DT Periodic with period 2 26

25 Frequency-Domain Interpretation of C/D Conversion Compare Eqs.(1)&(2) and note j X d( e ) X p( j( )) T Note : 2 s CT DT 27

26 Illustration of C/D Conversion in the Frequency-Domain X ( ) c j 1 X ( ) c j 1 28

27 D/C Conversion y d [n] y c (t) Reverse of the process of C/D conversion Lecture 12 (Chapter 7) jt Y ( j) Y ( e ) reverses frequency scaling p d Again, Ω = ω jt s TYd ( e ), bandli mited Yc ( j) 2 0, otherwise 29

28 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 M 2 s When the input xc(t) is band-limited (X(jω) = 0 at ω > ω M )and the sampling theorem is satisfied (ω S > 2ω M ), then Y ( j) H ( j) X ( j) y ( t) h ( t)* x ( t) LTI c c c c c c 30

29 Frequency-Domain Illustration of DT Processing of CT Signals Lecture 12 (Chapter 7) DT filter Sampling DT freq->ct freq CT freq->dt freq Interpolate(LPF) -> Equivalent CT filter 31

30 Assuming No Aliasing Step1- C/D: j p - periodic 1 T X c j T, if no aliasing T Step 2 - DT Filter: Step 3 - D/C: X e X j T d Y c T j j j Y e H e X e d d d 1 j Hd e X c j T, T jt jt s s, 2 2 TYd e Hd e X c j j 0, otherwise H c In practice, first specify the desired H c (jω), then design H d (e jω ). jt s e, 2 Hd j 0, otherwise 32

31 Example: Digital Differentiator Applications: Edge Enhancement 33

32 Construction of Digital Differentiator Bandlimited Differentiator Desired: j, Hc( j) 0, 2 s 2c T M c Set Assume (Nyquist rate met) s c c c Choice for H d (e jω ): H d Hc( j / T ), j ( e ) periodic, j( ) jc ( ) T 34

33 Band-Limited Digital Differentiator(continued) 35