# 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 ...