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QUESTION: 1

If x(n) = (1/2)^{n} u(n), y(n) = x^{2}(n) and (e^{iω}) be the fourier transform of y(n), then y(e^{0}) is

Solution:

and

⇒

then,

QUESTION: 2

For a signal x(t), the fourier transform is X(f), then inverse fourier transform of x(3f + 2) is given by

Solution:

Apply scaling and shifting property.

QUESTION: 3

The Fourier transform of given signal x(t)

Solution:

QUESTION: 4

Fourier transform of x(t) = e^{at} u (-t), a > 0 is

Solution:

QUESTION: 5

Determine the fourier transform of the signal x(t) shown in figure.

Solution:

⇒

⇒

QUESTION: 6

Match List-I [Function f(t)] with List-II [Fourier transform F(ω)] and select the correct answer using the codes given below the lists:

List- I

A. f(t – t_{0})

B. f(t) e^{jω0t}

C. f_{1}(t) . f_{2}(t)

List-ll

1. f(ω – ω_{0})

2.

3.

4.

5.

Codes

Solution:

QUESTION: 7

Consider a continuous time low pass filter whose impulse response h(t) is known to be real and whouse frequency response magnitude is given by

Determine the value of h(t) if the group delay function is specified as t(ω) = 5.

Solution:

Since,

0; otherwise

using duality property,

Since group delay is constant and

So,

QUESTION: 8

Consider a signal x_{1}(t) having a fourier transform x_{1}(jω). An another signal x_{2}(t) having fourier transform x_{2}(jω) is related to x_{1 }(t) by x_{2}(jω) = [1 +sgn(ω)] x_{1}(jω) x_{2}(t) in terms of x_{1}(t) is equal to

Solution:

Since,

or,

or,

because,

using, Hilbert transform

QUESTION: 9

A Fourier transform pair is as follows:

The Fourier transform of given signal y(t) is

Solution:

⇒

QUESTION: 10

Suppose; y(t) = x(t) cost

and

then x(t) will be

Solution:

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