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Signal Processing and Integrated Circuits by Hussein Baher

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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 6.1

where

6.2a 6.2a

or

6.2b 6.2b

and

6.3 6.3

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

6.4 6.4

In the case of the Fourier series, the evaluation of the integral in (6.2) which defines the coefficients ck, 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 6.5

to approximate the function.

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

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