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Signals and Systems Fall 2003 Lecture #13 21 October 2003

Signals and Systems Fall 2003 Lecture #13 21 October 2003. 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

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Signals and Systems Fall 2003 Lecture #13 21 October 2003

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  1. Signals and Systems Fall 2003 Lecture #1321 October 2003 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

  2. 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?

  3. 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 reconstructthe original CT signal x(t) from its samples?

  4. Impulse Sampling—Multiplying x(t) by the sampling function

  5. Analysis of Sampling in the Frequency Domain Multiplication Property => =Sampling Frequency Important to note:

  6. Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for |ω|> ωM) signal

  7. Reconstruction of x(t) from sampled signals If there is no overlap between shifted spectra, a LPF can reproduce x(t) from xp(t)

  8. Suppose x(t) is bandlimited, so that X(jω)=0 for |ω|> ωM Then x(t) is uniquely determined by its samples {x(nT)} if where ωs = 2π/T ωs> 2ωM = The Nyquist rate

  9. (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

  10. Observations (Continued) (2) Sampling is fundamentally a time varyingoperation, 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 (ωs> 2ωM). (3) What if ωs<= 2ωM? Something different: more later.

  11. Time-Domain Interpretation of Reconstruction of Sampled Signals—Band-Limited Interpolation The lowpass filter interpolates the samples assuming x(t) containsno energy at frequencies >= ωc

  12. Graphic Illustration of Time-Domain Interpolation Original CT signal After Sampling After passing the LPF

  13. Interpolation Methods • Bandlimited Interpolation • Zero-Order Hold • First-Order Hold —Linear interpolation

  14. Undersampling and Aliasing When ωs ≦ 2ωM=>Undersampling

  15. Undersampling and Aliasing (continued) Xr (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, xr(nT) = x(nT)

  16. A Simple Example X(t) = cos(wot + Φ) Picturewould be Modified… Demo: Sampling and reconstruction of coswot

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