# 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*. The signal

^{sT}*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 ...

Get *Introduction to Digital Signal Processing and Filter Design* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.