The Discrete Time Fourier Transform - Synthesis Lectures on Signal Processing - page 1

CHAPTER 1

The Discrete Time Fourier
Transform

1.1 OVERVIEW

1.1.1 IN THE PREVIOUS VOLUME
The previous volume of the series, Volume I, covered DSP fundamentals such as basic signals and LTI
systems, difference equations, sampling, the Nyquist rate, normalized frequency, correlation, convolution,
the real DFT, matched filtering, and basic IIR and FIR filters.

1.1.2 IN THIS VOLUME
In this volume, Volume II of the series, we take up discrete frequency transforms in detail, including an
overview of many transforms, both continuous-domain and discrete-domain, followed in sequence by
detailed discussions of a number of discrete transforms, knowledge of which is generally deemed essential
in the signal processing field.

1.1.3 IN THIS CHAPTER
We are now prepared in this chapter to begin a detailed exploration of discrete frequency transforms. A
number of such transforms exist, and we'll begin by summarizing all of the basic facts and comparing each
to the others to better emphasize the important characteristics of each distinct transform. We include a
brief mention of continuous signal domain transforms for background and perspective, but concentrate
most of our effort on discrete signal transforms. All of the transforms we'll investigate, both the continuous
and the discrete domain types, work on the same general concept-summing (or integrating in the case of
continuous time signals) the product of the signal and orthogonal-pair correlating waveforms of different
frequencies.
The transforms covered in detail in this topic section are the Discrete Time Fourier Transform
(DTF T) , which is covered in this chapter, the z -Transform ( z -T) , covered in the following chapter, and
the Discrete Fourier Transform (DF T) , covered in the third and final chapter of this volume.
By the end of this chapter, the reader will have learned how to evaluate the frequency response
of an LTI system using the DTFT. This sets the stage for the following chapter, which discusses the
more generalized z -transform, which is in widespread use in industry and academia as the standard
method to describe the transfer function of an LTI system. This will be followed in the succeeding
chapter by a detailed look at the workhorse of practical frequency domain work, the DFT and a fast
implementation, the decimation-in-time FFT, as well as time domain convolution using the frequency
domain, the Goertzel algorithm, computing the DTFT using the DFT, etc.

1.2 SOF TWARE FOR USE WITH THIS Topic
The software files needed for use with this topic (consisting of m-code (.m) files, VI files (.vi), and related
support files) are available for download from the following website: