18.1 Introduction

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18.1 Introduction

Fourier series enable us to represent a periodic function as a sum of sinusoids and to obtain the frequenc y spectrum from the series. The Fourier transform allows us to e xtend the concept of a frequenc y spectrum to nonperiodic functions. The transform assumes that a nonperiodic function is a periodic function with an infinite period. Thus, the Fourier transform is an inte gral representation of a nonperiodic function that is analogous to a Fourier series representation of a periodic function.

The F ourier transform is an integral tr ansform lik e the Laplace transform. It transforms a function in the time domain into the frequency domain. The Fourier transform is very useful in communications systems and digital signal processing, in situations where the Laplace transform does not apply . While the Laplace transform can only handle circuits with inputs for t > 0 with initial conditions, the F ourier transform can handle circuits with inputs for t < 0 as well as those for t > 0.

We begin by using a Fourier series as a stepping stone in defining the Fourier transform. Then we de velop some of the properties of the Fourier transform. Next, we apply the Fourier transform in analyzing circuits. We discuss Parseval’s theorem, compare the Laplace and F ourier transforms, and see ho w the F ourier transform is applied in amplitude modulation and sampling.

Figure 18.1

(a) A nonperiodic function, (b) increasing T to infinity makes f(t) become the nonperiodic function in (a).