Applied Digital Signal Processing Theory and Practice

This chapter is primarily concerned with the conversion of continuous-time signals into discrete-time signals using uniform or periodic sampling. The presented theory of sampling provides the conditions for signal sampling and reconstruction from a sequence of sample values. It turns out that a properly sampled bandlimited signal can be perfectly reconstructed from its samples. In practice, the numerical value of each sample is expressed by a finite number of bits, a process known as quantization. The error introduced by quantizing the sample values, known as quantization noise, is unavoidable. The major implication of sampling theory is that it makes possible the processing of continuous-time signals using discrete-time signal processing techniques.

Study objectives

After studying this chapter you should be able to:

• Determine the spectrum of a discrete-time signal from that of the original continuous-time signal, and understand the conditions that allow perfect reconstruction of a continuous-time signal from its samples.

• Understand how to process continuous-time signals by sampling, followed by discrete-time signal processing, and reconstruction of the resulting continuous-time signal.

• Understand how practical limitations affect the sampling and reconstruction of continuous-time signals.

• Apply the theory of sampling to continuous-time bandpass signals and two-dimensional image signals.

Ideal periodic sampling of continuous-time signals

In the most common form of sampling, known as periodic or uniform sampling, a sequence of samples x[n] is obtained from a continuous-time signal xc(t) by taking values at equally spaced points in time.