Signals and Systems. Spring Room 324, Geology Palace, ,

Transcription

1 Signals and Systems Spring 2013 Room 324, Geology Palace, ,

2 Chapter 7 Sampling 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) Under sampling and Aliasing Signals

3 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?

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 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?

5 Impulse Sampling Multiplying x(t) by the sampling function

6 Analysis of Sampling in the Frequency Domain Important to note: ω s 1/T

7 Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for ω > ω M ) signal No overlap between shifted spectra

8 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)

9 The Sampling Theorem Suppose x(t) is bandlimited, so that Then x(t) is uniquely determined by its samples {x(nt)} if

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

11 Observations (Continued) (2) 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 bandlimited x(t) satisfying the sampling theorem (ω s > 2ω M ). (3) What if ω s 2ω M? Something different: more later.

12 Time-Domain Interpretation of Reconstruction of Sampled Signals Band-Limited Interpolation The lowpass filter interpolates the samples assuming x(t) contains no energy at frequencies ω c

13 Original CT signal Graphic Illustration of Time-Domain Interpolation h(t) After sampling T After passing the LPF

14 Interpolation Methods Bandlimited Interpolation Zero-Order Hold First-Order Hold Linear interpolation

15 When ω s 2 ω M Undersampling Undersampling and Aliasing

16 Undersampling and Aliasing (continued) X r (jω) X(jω) Distortion because of aliasing 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)

17 A Simple Example Picture would be Modified Demo: Sampling and reconstruction of cosω o t

18 Chapter 7 Sampling 1) Review/Examples of Sampling/Aliasing 2) DT Processing of CT Signals

19 Sampling Review Demo: Effect of aliasing on music.

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

21 DT Processing of Band-Limited CT 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 )

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

23 Frequency-Domain Interpretation of C/D Conversion Note: ω s 2π CT DT

24 Illustration of C/D Conversion in the Frequency-Domain X d (e jω ) X d (e jω ) Ω = ωt 1 Ω = ωt2

25 D/C Conversion y d [n] y c (t) Reverse of the process of C/D conversion

26 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 ω M < ω s 2 When the input x c (t) is band-limited (X(jω) = 0 at ω > ω Μ ) and the sampling theorem is satisfied (ω s > 2ω M ), then DT omege needs to changed

27 Frequency-Domain Illustration of DT Processing of CT Signals DT filter Sampling DT freq CT freq CT freq DT freq Interpolate (LPF) equivalent CT filter

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

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

30 Bandlimited Differentiator Construction of Digital Differentiator

31 Band-Limited Digital Differentiator (continued) CT DT