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Demystifying Sampling Theorem in Digital Signal Processing

Explore the importance of discrete sampling in digital signal processing, from Shannon's theorem to aliasing and Nyquist frequency. Understand the ideal sampling rate for accurate signal reconstruction. Learn about impulse sampling and aliasing effects on signal reconstruction. Discover insights into band-limited signals and filtering techniques in digital domains.

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Demystifying Sampling Theorem in Digital Signal Processing

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  1. ECE310 – Lecture 17 Sampling Theorem 03/31/01 – 04/02/01

  2. Why Discrete? • The use of digital computers • Analog filters -> digital filters • Cellular phone: analog & digital mode • Digital television

  3. Sample • A question • For a certain continuous-time signal, how many samples are enough to describe the signal accurately? • Shannon’s theorem • The sampling rate required to exactly reconstruct a signal from its samples is more than twice the highest frequency at which the FT of the signal is non-zero • Band-limited signal

  4. How Does It Come From? • Impulse sampling – the product of a signal and a comb function • FT of the impulse-sampled signal • fs frequency domain period

  5. Cont’d - Example

  6. The Alias Page 9-9, 9-12, 9-13

  7. Nyquist Frequency and Rate • fs: frequency domain period • fm: the highest frequency is called the Nyquist frequency (folding frequency) • 2fm: the minimum rate at which a signal can be sampled and still be reconstructed from its samples is called Nyquist rate • If fs > 2fm, then it’s oversampled • If fs < 2fm, then it’s undersampled • Alias: shifted versions of the original spectrum • If the alias overlap, the discrete-time signal is said to be ‘aliased’

  8. Reconstruct Time-Domain Signal • Filter the impulse-sampled signal using • an ideal lowpass filter with a cutoff frequency at fc, where fm<fc<fs-fm • and a gain of Ts

  9. Cont’d • When fs=2fm, fc must be equal to fm. This works only when the signal’s spectrum does not have an impulse at fm

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