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QUESTION: 1

The signals x_{1}(t) and x_{2}(t) are given by

If the periods of x_{1}(t) and x_{2}(t) be T_{1} and T_{2} respectively, then T_{1}/T_{2} is equal to

Solution:

x_{1}(t) = sin15πt is a periodic signal having period

x_{2}(t) = sin√2 πt is a periodic signal having period T_{2} equal to

∴

QUESTION: 2

The Fourier transform of a unit step function is

Solution:

∴

QUESTION: 3

Assertion (A): A frequency domain representation of a signal shows the frequency content and indicates the shape of the waveform or the combined amplitude of all the input components at any specific time.

Reason (R): In frequency domain, a signal is represented by its frequency spectrum.

Solution:

A frequency domain representation of a signal shows the frequency content but not necessarily indicate the shape of the waveform or the combined amplitude of all the input components at any specific time. Hence, assertion is a false statement. Reason is a true statement because in frequency domain, a signal is represented by its frequency spectrum and to obtain frequency spectrum of a signal, Fourier series and Fourier transform are used.

QUESTION: 4

Which of the following option is not correct for obtaining the Fourier transform of a signal x(f)7

Solution:

For obtaining the Fourier transform of a signal x(t), it must have infinite number of discontinuities in any finite interval T.

QUESTION: 5

Match List-I (Operations on x(t)) with List-ll (X(ω)/Fourier transform) and select the correct answer using the codes given below the lists:

Solution:

QUESTION: 6

Assertion (A): The Parseval’s theorem implies superposition of the average powers.

Reason (R): The interpretation of the Parseval’s theorem is that the total average power of the signal x(t) can be found by squaring and adding the heights of the amplitude lines in the spectrum of the periodic signal x(t).

Solution:

QUESTION: 7

Fourier series analysis is a tool used to

1. analyse any periodic and non-periodic signal.

2. find how many frequency components are present in the signal.

3. find the relative phase difference between various frequency components.

4. represents a waveform in the form of sum of infinite number of exponential terms.

5. find the amplitude of various terms in the series.

Which of the statements given above are correct?

Solution:

Fourier series analysis is a tool used to

1. analyse any periodic signal only.

2. find how many frequency components are present in the signal.

3. find the relative phase difference between various frequency components.

4. represents a periodic waveform in the form of sum of infinite number of sine and cosine terms.

5. find the amplitude of various frequency components of the series.

Hence, statements 2, 3 and 5 are only correct.

QUESTION: 8

The power spectral density of a power signal x(t) is shown in figure below.

The average power of the signal is

Solution:

The total average power of a signal is the area of curve under s(f).

Hence,

QUESTION: 9

The energy of a causal exponential pulse shown in figure is

Solution:

Given,

∴ Normalized energy of given signal is

QUESTION: 10

The Fourier transform of the decaying exponential pulse shown in figure is

Solution:

The given signal can be represented as

(Time constant,

∴ Fourier transform of x(t) is

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