1. Sampling of continuous-Time Signal What is sampling? How to describe sampling mathematically? Is sampling arbitrary?

2. What is sampling? Sampling is the process of obtaining a sequence of instantaneous values of a particular signal, usually at a regular time interval.

3. Mathematical model  Input signal x(t) Sampled result X(t) Sampling period T s Sampling rate f s = 1/T s  (1)  (2)  (3)

4. Is sampling arbitrary? Ts=T/10 Ts=T/4 Ts=3/4T

5. Shannon Sampling Theorem A continuous-time signal x(t) with frequencies no higher than f max can be reconstructed from its samples x[k] = x(k T s ) if the samples are taken at a rate f s which is greater than 2 f max. Nyquist rate = 2 f max Nyquist frequency = f s /2.

6. Sampling Theorem Assumptions The continuous-time signal has no frequency content above the frequency f max The sampling time is exactly the same between any two samples The sequence of numbers obtained by sampling is represented in exact precision The conversion of the sequence of numbers to continuous-time is ideal

7. Generalized Sampling Theorem Sampling rate must be greater than twice the bandwidth Bandwidth is defined as the non-zero extent of the spectrum of the continuous-time signal in positive frequencies For a lowpass signal with maximum frequency f max, the bandwidth is f max For a bandpass signal with frequency content on the interval [ f 1, f 2 ], the bandwidth is f 2 - f 1