Chapter 4 Fourier transform Prepared by Dr. Taha MAhdy

Fourier transform FT extends the basic idea of the Fourier series to incorporate nonperiodic signals. Think of it as a kind of generalization of the Fourier series concept-because it will allow us to obtain a frequency representation of periodic and nonperiodic signals.

FT lets us convert the representation of a signal in time into a representation of the signal in frequency. The inverse Fourier transform does the opposite. It lets us convert the representation of a signal in frequency into a representation of the signal in time.

Fourier transform of continuous time signals

Fourier transform of the discrete time signals Note: The Fourier transform of a discrete signal is always periodic with a period = 2π

Examples

THE SINC FUNCTION

Example, find the Fourier transform of the following pulse

Solution

PROPERTIES OF THE FOURIER TRANSFORM Time shifting

Frequency shifting

Linearity

Convolution (very important) Convolution in time domain equals multiplication in the frequency domain and vice versa.

Spectrum Plots

Problems Schaums’s Book 5.19, 5.20, 5.21, , 6.12, 6.22, 6.25, 6.31, 6.32, 6.34