Unit-2

Fourier transform & Z – Transform

Question-1: Using complex form,find the Fourier series of the function

f(x) = sinx =

Solution:

We calculate the coefficients

  =

=

Hence the Fourier series of the function in complex form is

We can transform the series and write it in the real form by renaming it as

 n=2k-1,n=

                                    =

Question-2: Using complex form find the Fourier series of the function f(x) = x2, defined on the interval [-1,1]

Solution:

Here the half-period is L=1. Therefore, the coefficient c0 is,

For n

Integrating by parts twice, we obtain

=

=

= .

= .

Question-3: Find the Fourier integral representation of the function

Solution:

The graph of the function is shown in the below figure satisfies the hypothesis of

Theorem -1. Hence from Eqn,(5) and (6), we have

Substituting these coefficients in Eqn. (4) we obtain

This is the Fourier integral representation of the given function.

Question-4: Find the Fourier cosine integral of , where x>0, k>0 hence show that

Solution:

The Fourier cosine integral of f(x) is given by:

Question-5: Find the Fourier transform of-

Hence evaluate

Sol. As we know that the Fourier transform of f(x) will be-

So that-

For s = 0, we get- F(s) = 2

Hence by the inverse formula, we get-

Putting x = 0, we get

So-

Question-6: Find the Fourier transform of

Sol. As we know that the Fourier transform of f(x) will be-

   So that-

Now put

So that-

Question-7: Find the Fourier sine transform of

Sol. Here x being positive in the interval (0, ∞)

Fourier sine transform of will be-

Question-8: Find the Fourier cosine transform of-

Sol. We know that the Fourier cosine transform of f(x)-

                     =

                     = 

                     =

Question-9: Find the Fourier sine transform of

Sol. Let                    

Then the Fourier sine transform will be-

Now suppose,

Differentiate both sides with respect to x, we get-

……. (1)

On integrating (1), we get-

Question-10: Find Z-transform of the following functions-

(i)

(ii)

Sol. (i) 

(ii)

Question-11: Find Z-transform of the following functions-

(i)

(ii)

Sol. (i) As we know that-

So that-

(ii) we know that-

So that-

Question-12:
 

Solution:

Long division method to obtain

2

Now x(z) can be written as,

X(z) = 2-

Question-13: Solve the differential equation  by the z-transformation method.

Solution:

Given,

Let y(z) be the z-transform of

Taking z-transforms of both sides of eq(1) we get,

ie.

using the given condition, it reduces to

(z+1)y(z) =

Ie.

Y(z) =

Or Y(Z) =

On taking inverse Z-transforms, we obtain

Question-14: Solve the following by using Z-transform

Sol. If then

And

Now taking the Z-transform of both sides, we get

z[

It becomes-

So,

Now-

On inversion, we get-