Show that for any periodic signal of period T0 the Fourier t

Show that for any periodic signal of period T_0, the Fourier transform is equal to the Fourier series coefficients at the harmonics (integer multiples) of the fundamental frequency omega_0 = 2 pi/T_0 and zero at all other frequencies.

Solution

A)We discussed how certain classes of things can be built usingcertain kinds of basis functions.In this lecture we will consider specically functions that are periodic,and basic functions which are trigonometric.Then the series is said to be a Fourier series.

he frequency representation of periodic and aperiodic signals indicates

how their power or energy is allocated to different frequencies. Such a distribution over frequency

is called the

spectrum of the signal

. For a periodic signal the spectrum is discrete, as its power

is concentrated at frequencies multiples of a so-called

fundamental frequency

, directly related to

the period of the signal. On the other hand, the spectrum of an aperiodic signal is a contin-

uous function of frequency. The concept of spectrum is similar to the one used in optics for

light, or in material science for metals, each indicating the distribution of power or energy over

frequency. The Fourier representation is also useful in finding the frequency response of linear

time-invariant systems, which is related to the transfer function obtained with the Laplace trans-

form. The frequency response of a system indicates how an LTI system responds to sinusoids of

different frequencies. Such a response characterizes the system and permits easy computation of

its steady-state response, and will be equally important in the synthesis of systems.

his can be seen by finding the output corresponding to

x

.

t

/

D

e

j

0

t

by means of the convolution

integral