TABLE OF CONTENTS This chapter introduces the important bridge between continuous-time and discrete-time signals provided by the sampling process. Sampling records discrete values of a continuous-time signal at periodic instants of time, either for real-time processing or for storage and subsequent off-line processing. Sampling opens up a world of possibilities for the processing of continuous-time signals through the use of discrete-time systems such as infinite impulse response (IIR) and finite impulse response (FIR) filters.
Recall that the Fourier transform of the continuous-time signal x( t) is given by
| (15.1) |  |
Under certain conditions, a continuous-time signal can be completely represented by (and recovered from) its samples taken at periodic instants of time. The sampling operation can be seen as the multiplication of a continuous-time signal with a periodic impulse train of period T s, as depicted in Figure 15.1. The spectrum of the sampled signal is the convolution of the Fourier transform of the signal with the spectrum of the impulse train, which is itself a frequency-domain impulse train of period equal to the sampling frequency ? s = 2 ?/ T s. This frequency-domain analysis of sampling is illustrated in Figure 15.2.
The resulting sampled signal in the time domain is a sequence of impulses given by
| (15.2) |  |
where the impulse at time t = nT
