The magnitude and phase of the complex Fourier series coefficient a<sub>k</sub> of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation: C is the set of complex number, R is the set of purely real numbers, and P is the set of purely imaginary numbers.<br><img src="/images/question-image/electronics-and-communications-engineering/signal-processing/1681537378-the-magnitude-and-phase-of-the-complex-fourier-series.jpg" title="Signal Processing mcq question image" alt="Signal Processing mcq question image">

One period (0, T) each of two periodic waveforms W₁ and W₂ are shown in the figure. The magnitudes of the nth Fourier series coefficients of W₁ and W₂, for n ≥ 1, n is odd, are respectively proportional to
Signal Processing mcq question image

A

|n|^-3 and |n|^-2

B

|n|^-1 and |n|^-3

C

|n|^-1 and |n|^-2

D

|n|^-4 and |n|^-2

Let x(t) and y(t) (with Fourier transforms X(f) and Y(f) respectively) be related as shown in the figure. Then Y(f) is
Signal Processing mcq question image

A

$$ - \frac{1}{2}X\left( {\frac{f}{2}} \right){e^{ - j2\pi f}}$$

B

$$ - \frac{1}{2}X\left( {\frac{f}{2}} \right){e^{j2\pi f}}$$

C

$$ - X\left( {\frac{f}{2}} \right){e^{j2\pi f}}$$

D

$$ - X\left( {\frac{f}{2}} \right){e^{ - j2\pi f}}$$

The Laplace transform of the causal periodic square wave of period T shown in the figure below is It
Signal Processing mcq question image

A

$$F\left( s \right) = \frac{1}{{1 + {e^{ - \frac{{sT}}{2}}}}}$$

B

$$F\left( s \right) = \frac{1}{{s\left( {1 + {e^{ - \frac{{sT}}{2}}}} \right)}}$$

C

$$F\left( s \right) = \frac{1}{{s\left( {1 - {e^{ - \frac{{sT}}{2}}}} \right)}}$$

D

$$F\left( s \right) = \frac{1}{{1 - {e^{ - sT}}}}$$

A rectangular pulse train s(t) as shown in the figure is convolved with the signal cos²(4π × 10³t). The convolved signal will be a
Signal Processing mcq question image

A

DC

B

12 kHz sinusoid

C

8 kHz sinusoid

D

14 kHz sinusoid

Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {a_k} be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t):
1. The complex Fourier series coefficients of x(3t) are {a_k} where k is integer valued.
2. The complex Fourier series coefficients of x(3f) are {3a_k} where k is integer valued.
3. The fundamental angular frequency of x(3t) is 6π rad/s.
For the three statements above, which one of the following is correct?

A

Only 2 and 3 are true

B

Only 1 and 3 are true

C

Only 3 is true

D

Only 1 is true

The function x(t) is shown in the figure. Even and odd parts of a unit-step function u(t) are respectively,
Signal Processing mcq question image

A

$$\frac{1}{2},\frac{1}{2}x\left( t \right)$$

B

$$ - \frac{1}{2},\frac{1}{2}x\left( t \right)$$

C

$$\frac{1}{2}, - \frac{1}{2}x\left( t \right)$$

D

$$ - \frac{1}{2}, - \frac{1}{2}x\left( t \right)$$

Consider the system shown in the figure below. The transfer function $$\frac{{Y\left( z \right)}}{{X\left( z \right)}}$$ of the system is
Signal Processing mcq question image

A

$$\frac{{1 + a{z^{ - 1}}}}{{1 + b{z^{ - 1}}}}$$

B

$$\frac{{1 + b{z^{ - 1}}}}{{1 + a{z^{ - 1}}}}$$

C

$$\frac{{1 + a{z^{ - 1}}}}{{1 - b{z^{ - 1}}}}$$

D

$$\frac{{1 - b{z^{ - 1}}}}{{1 - a{z^{ - 1}}}}$$

For the discrete-time system shown in the figure, the poles of the system transfer function are located at
Signal Processing mcq question image

A

2, 3

B

$$\frac{1}{2},3$$

C

$$\frac{1}{2},\frac{1}{3}$$

D

$$2,\frac{1}{3}$$

Consider the 2 Statement: 1. An odd and imaginary signal always has an odd and imaginary Fourier transform. 2. The convolution of an odd Fourier transform with an even Fourier transform is always even. Which of the above statements is/are true:

A

None

B

1 and 2

C

1

D

2

The impulse response and the excitation function of a linear time invariant causal system are shown in figure (a) and (b) respectively. The output of the system at t = 2 sec is equal to
Signal Processing mcq question image

A

0

B

$$\frac{1}{2}$$

C

$$\frac{3}{2}$$

D

1

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