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Test: Time Shifting & Scaling Property - Electrical Engineering (EE) MCQ


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10 Questions MCQ Test - Test: Time Shifting & Scaling Property

Test: Time Shifting & Scaling Property for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Test: Time Shifting & Scaling Property questions and answers have been prepared according to the Electrical Engineering (EE) exam syllabus.The Test: Time Shifting & Scaling Property MCQs are made for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Time Shifting & Scaling Property below.
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Test: Time Shifting & Scaling Property - Question 1

The given mathematical representation belongs to:
y(t) = x(t - T)

Detailed Solution for Test: Time Shifting & Scaling Property - Question 1

Time-shifting property: When a signal is shifted in time domain it is said to be delayed or advanced based on whether the signal is shifted to the right or left.
For example:

Test: Time Shifting & Scaling Property - Question 2

The Fourier transform of a signal x(t), denoted by X(jω), is shown in the figure.

Let y(t) = x(t) + ejtx(t). The value of Fourier transforms of y(t) evaluated at the angular frequency ω = 0.5 rad/s is

Detailed Solution for Test: Time Shifting & Scaling Property - Question 2

y(t) = x(t) + ejtx(t)
x(t) ↔ X(jω)
ejtx(t) ↔ X(j(ω - 1))

y(jω) at ω = 0.5 rad/sec = X(jω) + X(j(ω - 1))
= 1 + 0.5 = 1.5

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Test: Time Shifting & Scaling Property - Question 3

For a signal x(t) = e-t+1 u(t-1) Fourier transform is

Detailed Solution for Test: Time Shifting & Scaling Property - Question 3

Fourier transform of e−tu(t) 

Fourier transform of e-(t-1)u(t-1)

Test: Time Shifting & Scaling Property - Question 4

A real valued signal x(t) limited to frequency band  |f| < ω/2 is passed through an LTI system whose frequency response is

The output of the system is

Detailed Solution for Test: Time Shifting & Scaling Property - Question 4

y(t) = x(t)∗h(t)
Taking the Fourier Transform, we get:
Y(f) = X(f)H(f)
Y(f) = X(f)e−j4πf
Taking the inverse Fourier Transform of Y(f), we get:
y(t) = x(x - 2)

Test: Time Shifting & Scaling Property - Question 5

The Fourier transform of x(t) is X(jω). Then, the Fourier transform of 

Detailed Solution for Test: Time Shifting & Scaling Property - Question 5

Let x1(t) = x(t − 1)

Now,

Taking Fourier transform
 X2(jω)=(jω)2X1(jω)
⇒X2(jω) = −ω2X(jω)e−jω
Thus,

 

Test: Time Shifting & Scaling Property - Question 6

Let x(t) and y(t) (with Fourier transform X(ω) and Y(ω) be related as shown in figure below

Then Y(ω) in terms of X(ω) is

Detailed Solution for Test: Time Shifting & Scaling Property - Question 6

From the given pictures of x(t) and y(t)
We get,
y(t) = - x(2t + 2)
y(t) is time scaled and time shifted version of x(t)
Step 1:
If x(t) ↔ X(ω)
Then, x(t + 2) ↔ ej.2.ω X(ω)
Step 2:
Using time shifting property
x(2t + 2) ↔ ½ e X(ω/2)
Step 3:
Using time scaling property:

Test: Time Shifting & Scaling Property - Question 7

The Fourier series coefficients of signal x(t) shown in Figure (A) are given as:
c0 = 1/π, c1 = −j/4, cn = 1/π(1 − n2), for k even.


Which of the following Fourier series coefficients of y(t) shown in Figure (B) is/are correct?

Detailed Solution for Test: Time Shifting & Scaling Property - Question 7

Concept:
Time shifting property of Fourier series states that:

Since ω0 = 2π/T0, the above expression can be written as:

Application:
Observing the two figures, we can write:
y(t) = x(t - 1)

Where cn = Fourier series coefficient of x(t)
Since, T0 = 2

Using the above expression, we get:

Also for even values of n, e-jπn = 1

Test: Time Shifting & Scaling Property - Question 8

The magnitude of Fourier transform X(ω) of a function x(t) is shown below in figure (a). The magnitude of Fourier transform Y(ω) of another function y(t) is shown in figure (b). The phases of X(ω) and Y(ω) are zero for all ω. The magnitude and frequency units are identical in both the figures. The function y(t) can expressed in terms of x(t) as

Detailed Solution for Test: Time Shifting & Scaling Property - Question 8

We know that, expansion in frequency domain result in compression in the time domain and vice versa.
In the given question, compression is done frequency domain. So there will be expansion in time domain by same amount.
A x(t/2) ↔ 2A X(2f)
2A = 3 ⇒ A = 3/2

Test: Time Shifting & Scaling Property - Question 9

Let X(t) be Wide Sense Stationary random process with power spectral density Sx(f). If Y(t) is a random process defined as Y(t) = X(2t − 1), the power spectral density Sy(f) is:

Detailed Solution for Test: Time Shifting & Scaling Property - Question 9

Power density has no effect of shifting. It is affected only by scaling
We know,
E[X(t)X(t + τ)] = Rx(τ)
and
E[X(2t + 1)X(2(t + τ) + 1)] = E[X(2t + 1)X(2t + 2τ + 1)] = Rx(2τ)

then using the scaling property of Fourier transforms we have,

Thus, the power spectral density of 

Test: Time Shifting & Scaling Property - Question 10

Let x(t)<-> X(jω) be Fourier Transform pair. The Fourier Transform of the signal x(5t-3) in terms of X(jω) is given as

Detailed Solution for Test: Time Shifting & Scaling Property - Question 10

using the properties of fourier transform

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