The magnitude and phase of the complex Fourier series coefficient ak 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. <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">

$$x\left( t \right) \in R$$

$$x\left( t \right) \in P$$

$$x\left( t \right) \in \left( {C - R} \right)$$

The information given is not sufficient to draw any conclusion about x(t)

Correct Answer: $$x\left( t \right) \in R$$

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

|n|-3 and |n|-2

|n|-1 and |n|-3

|n|-1 and |n|-2

|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

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

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

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

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

The Laplace transform of the causal periodic square wave of period T shown in the figure below is It

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

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

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

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

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

12 kHz sinusoid

8 kHz sinusoid

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?

Only 2 and 3 are true

Only 1 and 3 are true

Only 3 is true

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,

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

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

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

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

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

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

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

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

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

For the discrete-time system shown in the figure, the poles of the system transfer function are located at

2, 3

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

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

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

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

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

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

The pole-zero pattern of a certain filter is shown in figure. The filter must be of the following type

Low-pass

High-pass

All-pass

Band-pass