Introduction to Digital Signal Processing and Filter Design

CHAPTER 3

Frequency-Domain Analysis

3.1 INTRODUCTION

In the previous chapter, we derived the definition for the z transform of a discrete-time signal by impulse-sampling a continuous-time signal x_a(t) with a sampling period T and using the transformation z = e^sT. The signal x_a(t) has another equivalent representation in the form of its Fourier transform X(jω). It contains the same amount of information as x_a(t) because we can obtain x_a(t) from X(jω) as the inverse Fourier transform of X(jω). When the signal x_a(t) is sampled with a sampling period T, to generate the discrete-time signal represented by , the following questions need to be answered:

Is there an equivalent representation for the discrete-time signal in the frequency domain?

Does it contain the same amount of information as that found in x_a(t)? If so, how do we reconstruct x_a(t) from its sample values x_a(nT)?

Does the Fourier transform represent the frequency response of the system when the unit impulse response h(t) of the continuous-time system is sampled? Can we choose any value for the sampling period, or is there a limit that is determined by the input signal or any other considerations?

We address these questions in this chapter, arrive at the definition for the discrete-time Fourier transform (DTFT) of the discrete-time system, and describe its properties and applications. In the second half of the chapter, we ...