Signal Analysis & Synthesis

Analogue Modulation

Analogue signal processing

Digital signal processing

Digital modulation

Communication Systems

Periodic Functions

A function f(t) is said to have a period T or to be periodic with a period T, if for all; where T is a constant (positive). The least value of T,

which is greater than zero, is called the least period or simply the period of f(t).

(i) The function sin t has periods 2, 4, 6,...... since sin (t + 2), sin (t + 4),
sin (t + 6), all are equal to sin t. Thus 2 is the least period or the period of sin t.

(ii) Similarly the function tan t is periodic and the least period is .

Fourier's Theorem

The theorem states that any single valued, periodic function f(t) which is continuous or has a finite number of discontinuities.

Fourier's Theorem Illustration

The function following Dirichlet- condition may be expressed as a summation of simple harmonic terms, having frequencies which are multiples of the frequency of the given function.

The above series, known as Fourier series, can also be written as a series of only sine terms or only cosine terms

if
and

Therefore, equation (1) may be written as

Evaluation of Constants:

By integrating both sides of the equation (1) with respect to t from t = 0 to t = T, where T = 2 / , we have.

But,

=0

Also,

=0

Therefore,

Again multiplying both sides of equation (1) by cos (n t) and integrating from 0 to T, we get,

Consider, where k being an integer which is equal to:

The first integral is zero for all integral values of k. The second integral is also zero for all integral values of k, except when k = n and then it is equal to T

Again,

For all integral values of k including k = n.

Thus,

Therefore,

Similarly, multiplying both sides of the equation (1) by sin (n t) and then integrating, we have

Form of Fourier's Theorem

I. Square wave:

The wave function which has been illustrated as below:

Known as Square wave, may be expressed mathematically as

Then,

Thus, all the cosine terms of the series vanish.

Again,

When n is even, cos n = 1; therefore, bn = 0. Hence all even terms are absent.

If n is odd, cos n = -1 and then evidently

Hence, the complete series which will be equivalent to square wave is given by

II. Saw-tooth wave:

The curves drawn in above illustration show that the approach towards saw tooth wave is more pronounced if more terms of the Fourier series are considered.

The wave function which has been shown in below illustration known as saw-tooth wave, may be expressed mathematically as

Then,

= 0

Again,

= 0

Hence, the complete series which will be equivalent to saw tooth wave is given by

III. Triangular wave:

The wave function which has been shown in above illustration known as triangular wave may be expressed mathematically as:

Then, =A

Again,

=

The even terms are zero, therefore the amplitude coefficients of cosine terms are

and the periodic time respectively.

Similarly,

= 0, i.e. all the sine terms disappear.

The Fourier series thus becomes