Problems Based on Fourier Transform

Problems Based on Fourier Transform

1. Obtain the Fourier transform of unit step signal u(t).

Solution:

Fourier transform of x(t)

2. Obtain the Fourier transform of unit impulse function δ(t)

Solution:

Fourier transform of x(t)

3. Obtain the Fourier transform of x(t) = 1

Solution:

Fourier transform of x(t)

4. Obtain the Fourier transform of following functions (i) x(t) = cos ω_ct (ii) x(t) = sin ω_ct

(i) x(t) = cos ω_ct (or) cos 2π f_ct

Solution:

Fourier transform of x(t)

(ii) x(t) = sin ω_ct

Solution :

5. Obtain the Fourier transform of following signals (i) x (t) = cos ω₀t (ii) x(t) = sin ω₀t

(i) x(t) = cos ω₀t

Solution:

From frequency shifting property

Apply (2) in (1), we get

(ii) x(t) = sin ω₀t

Solution:

From frequency shifting property

Apply (2) in (i), we get

6. Obtain the Fourier transform of x(t) = sgn(t)

Solution:

Signum function sgn(t) is defined as

Differentiate on both sides of (1).

By taking Fourier transform on both sides

7. Obtain the Fourier transform of the double exponential function shown in

Solution :

By linearity property of Fourier transform

Fourier transform of x₂(t)

8. Find the Fourier transform of the signal x(t) = e ^{– a t} u(t) and sketch the magnitude and phase spectrum.

Solution:

x(t) = e ^{– a t} u(t)

Fourier transform of x(t)

So, limits of integration changes from 0 to ∞

Magnitude of

|X(ω)| and < X(ω) are tabulated for various values of @ to sketch the magnitude and phase spectrum.

For negative Values of ω

9. Find the Fourier transform of x(t) = e ^{- |t|}

Solution:

x(t) can be expressed as

Fourier transform of x(t) is

Magnitude of X(ω)

For positive values of ω

10. Obtain the Fourier transform of x(t) = e^at u(-t) and plot the magnitude and phase spectrum.

Solution:

Fourier transform of x(t)

But u(-t) = 1, for 0 to -∞, So integration limits change from 0 to -∞.

11. Obtain the Fourier transform of a rectangular pulse, as shown in Fig.

Solution :

Rectangular pulse is defined as

Fourier transform of x(t) is

12. Obtain the Fourier transform of the antisymmetric exponential pulse as shown in figure.

Solution:

x(t) can be represented as

Fourier transform of x(t)

13. Obtain the Fourier transform of x(t) = e ^{j 2 π f}_c ^t

Solution:

14. Obtain the Fourier transform of x(t) = t e ^{– a t} u(t)

Solution:

Fourier transform of x(t) is

So limits of integration of (1) changes into 0 to ∞.

Integrating by parts

15. Find the Fourier transform of x(t) = e ^{– 4 t} u(t-2)

Solution:

By rearranging x(t)

By time shifting property

16. Find the Fourier transform of x(t) = 5 sin²3(t)

Solution :

By using linearity property

Inverse Fourier Transform

Steps to find inverse Fourier transform of X(ω)

Step 1:

Convert X(ω) by using partial fraction expansion in the form of

Step 2:

Then obtain inverse Fourier transform using standard Fourier transform pairs. Problems based on inverse Fourier transform:

1. Obtain inverse Fourier transform of the following signal.

Solution:

By taking the inverse Fourier transform using standard relation.

2. Obtain the inverse F.T of 

Solution :