THE FOURIER SERIES AND FOURIER TRANSFORM

Signals Spectra, and Signal Processing

Topics Outline:

The Fourier series provides spectral representations for periodic continuous-time signals. The Fourier transform is derived by applying a limiting process to the Fourier series.

Most of the signals occurring in nature are essentially continuous in time, and it is, therefore, reasonable to expect the spectrums of discrete-time signals to be closely related to those of the continuous-time signals from which they are derived.

1. Fourier Series

The frequency spectrum of a signal is an alternative to the its time-domain representation.

A periodic signal that comprises a weighted sum of sinusoids can be represented completely in the frequency domain in terms of the amplitudes and phase angles of its individual sinusoidal components.

Through the use of the Fourier series, the concept of frequency spectrum can be applied to arbitrary periodic signals.

Periodic signals are typically represented in terms of nonperiodic signals.

To avoid possible confusion between the two types of signals we will use the notation ˜x(t ) for a periodic signal and simply x(t ) for a nonperiodic one.

Signals will be assumed to be real unless otherwise stated.