ECE310 – Lecture 17

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