Signal Processing and Integrated Circuits

Chapter 6

The Fast Fourier Transform and its Applications

6.1 Introduction

In representing an analog periodic signal f(t), with period T, by a Fourier series we write

6.1

where

6.2a

or

6.2b

and

6.3

In contrast, a non-periodic signal f(t) has a Fourier transform given by

6.4

In the case of the Fourier series, the evaluation of the integral in (6.2) which defines the coefficients c_k, is usually performed using numerical techniques. This allows the use of efficient high speed computational methods. Therefore, the integral in (6.2) must be approximated by a summation, since the computer can only process numbers at discrete values of the variable t.

Furthermore, the representation of f(t) by the Fourier series must be done using only a finite number of terms so that we use the nth partial sum, or truncated series

6.5

to approximate the function.

In the case of the Fourier transform F(ω) in ...