Problems Based on Properties of DIFT

PROBLEMS BASED ON PROPERTIES OF DTFT

Problem 1:

Solution:

(a) Using the time shifting property, we have

(b) Using the time shifting property, we have

(d) Using the time shifting property, we have

(e) Using differentiation in the frequency domain property, we have

(f) Using the time reversal property, we have

(g) Using differentiation in frequency domain and time reversal properties, we have

(h) Using the frequency shifting property, we have

Problem 2:

Find the inverse Fourier transform for the first order recursive filter

Solution:

Let h(n) be the inverse Fourier transform of H(ω)

On comparing the two expressions for H(ω). We can say that the samples of h(n) are the co-efficients of e ^{–j ω n}

h(n) = {1, a, a², a³ ..., a^k,...}

Problems 3:

Determine the output sequence from the output spectrum:

Solution:

The output sequence y(n) is the inverse Fourier transform of Y(ω).

Using the time shifting property, we have

Also we know that

Using the time shifting property, we have

Problem 4:

The impulse response of a LTI system is h(n) = {1, 2, 1, -3}. Find the response of the system for the input x(n) = {1, 3, 2, 1}.

Solution:

The response of the system y(n) for an input x(n) and impulse response h(n) is given by

y(n) = x(n) * h(n)

Using the convolution property of Fourier transform, we get

Taking inverse fourier transform on both sides, we get

Problem 5:

Find the Convolution of the signals given below using Fourier transform:

Solution:

Using the convolution property of Fourier transform we get

F[x₁(n) * x₂(n)] = X₁(ω) X₂(ω)

Taking inverse Fourier transform on both sides, we have.

Problem 6:

Consider a discrete time LTI system with impulse response h(n) = (1/2)ⁿ u(n). Use Fourier transform to determine the response to the signal x(n) = (3/4)ⁿ u(n).

Solution:

Given the impulse response h(n) and the input x(n) to the system, the response y(n) in given by

y(n) = x(n) * h(n)

Using the convolution property of Fourier transform we have

Where X(ω) and H(ω) are the Fourier transforms of x(n) and h(n) respectively

Taking inverse Fourier transform on both sides, we get the response