# Questions & Answers of Signals and Systems

#### Topics of Signals and Systems 83 Question(s) | Weightage 10 (Marks)

Question No. 13

The Laplace Transform of $f\left(t\right)={e}^{2t}\mathrm{sin}\left(5t\right)u\left(t\right)$ is

Question No. 16

The transfer function of a system is The steady state output y(t) is for the input $\mathrm{cos}\left(2t\right)$ The values of respectvely are

Question No. 18

Consider a continuous-time system with input $x\left(t\right)$ and output $y\left(t\right)$ given by

$y\left(t\right)=x\left(t\right)\mathrm{cos}\left(t\right)$

This system is

Question No. 19

The value of $\int_{-\infty}^{+\infty}e^{-t}\delta\left(2t-2\right)\mathrm{dt}$ where $\delta \left(t\right)$ is the Dirac delta function, is

Question No. 37

Let $S=\sum_{n=0}^\infty\limits n\alpha^n\;\mathrm{where}\;\left|\mathrm\alpha\right|<1$. The value of $\alpha$ in the range $0<\alpha <1$ such that $S=2\alpha$ is _______.

Question No. 41

Consider the following state-space representation of a linear time-invariant system.

$\overset.x\left(t\right)=\begin{bmatrix}1&0\\0&2\end{bmatrix}x\left(t\right),y\left(t\right)=c^Tx\left(t\right),\;c=\begin{bmatrix}1\\1\end{bmatrix}\;\mathrm{and}\;x\left(0\right)=\begin{bmatrix}1\\1\end{bmatrix}$

The value of $y\left(t\right)$ for $t={\mathrm{log}}_{e}2$ is __________.

Question No. 44

Suppose  have the Fourier transforms as shown below.

Which one of the following statements is TRUE?

Question No. 45

The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where $x\left(t\right)$ is the input signal. A signal $z\left(t\right)$ is called eigen-signal of the system T , when T{z(t)}= y z(t), where $\gamma$ is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE

Question No. 114

Consider a causal LTI system characterized by differential equation $\frac{dy\left(t\right)}{dt}+\frac{1}{6}y\left(t\right)=3x\left(t\right)$. The response of the system to the input $x\left(t\right)=3{e}^{-\frac{t}{3}}u\left(t\right)$, where u(t) denotes the unit step function, is

Question No. 115

Suppose the maximum frequency in a band-limited signal $x\left(t\right)$ is 5 kHz. Then, the maximum frequency in $x\left(t\right)\mathrm{cos}\left(2000\mathrm{\pi t}\right)$, in kHz, is ________.

Question No. 118

The solution of the differential equation, for $t>0,y\text{'}\text{'}\left(t\right)+2y\text{'}\left(t\right)+y\left(t\right)=0$ with initial conditions $y\left(0\right)=0$ and $y\text{'}\left(0\right)=1$ is($u\left(t\right)$ denotes the unit step function),

Question No. 120

Let f(x) be a real, periodic function satisfying $f\left(-x\right)=-f\left(x\right)$. The general form of its Fourier series representation would be

Question No. 128

Consider a linear time-invariant system with transfer function

$H\left(s\right)=\frac{1}{\left(s+1\right)}$

If the input is $\mathrm{cos}\left(t\right)$ and the steady state output is $A\mathrm{cos}\left(t+\alpha \right)$, then the value of $A$ is _________.

Question No. 138

Let ${x}_{1}\left(t\right)↔{X}_{1}\left(\omega \right)$ and ${x}_{2}\left(t\right)↔{X}_{2}\left(\omega \right)$ be two signals whose Fourier Transforms are as shown in the figure below. In the figure, $h\left(t\right)={e}^{-2\left|t\right|}$ denotes the impulse response.
For the system shown above, the minimum sampling rate required to sample y(t), so that y(t) can be uniquely reconstructed from its samples, is

Question No. 139

The value of the integral $2{\int }_{-\infty }^{\infty }\left(\frac{sin\mathit{2}\pi t}{\pi t}\right)dt$ is equal to

Question No. 19

A moving average function is given by $y\left(t\right)=\;\frac1T\int\limits_{t-T}^tu\left(\zeta\right)\operatorname d\zeta$. If the input u is a sinusoidal signal of frequency $\frac{1}{2T}\mathrm{Hz},$ then in steady state, the output y will lag u (in degree) by ________.

Question No. 20

The impulse response g(t) of a system, G, is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure(b)?

Question No. 38

The signum function is given by

The Fourier series expansion of sgn(cos(t)) has

Question No. 45

Consider a discrete time signal given by

The region of convergence of its Z-transform would be

Question No. 114

The Laplace transform of is ${s}^{-3}{2}}$. The Laplace transform of $g\left(t\right)=\sqrt{1}{\mathrm{\pi t}}}$ is

Question No. 139

Consider a signal defined by

Its Fourier Transform is

Question No. 144

For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?

Question No. 145

The z-Transform of a sequence x[n] is given as X(z) = 2z+4-4/z+3/${z}^{2}$. If y[n] is the first difference of x[n], then Y(z) is given by

Question No. 162

The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integrationtime step is h.
$\frac{{x}_{k+1}-{x}_{k}}{h}={y}_{k}$
$\frac{{y}_{k+1}-{y}_{k}}{h}=-{x}_{k}$
For this discrete-time system, which one of the following statements is TRUE?

Question No. 19

$x\left(t\right)$ is nonzero only for ${T}_{x} , and similarly, $y\left(t\right)$ is nonzero only for ${T}_{y}. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE?

Question No. 20

For a periodic square wave, which one of the following statements is TRUE?

Question No. 36

Let $g:\left[0,\infty \right)\to :\left[0,\infty \right)$ be a function defined by $g\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ represents the integer part of x. (That is, it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of $g\left(x\right)$ is _______

Question No. 43

The function shown in the figure can be represented as

Question No. 44

Let $X\left(Z\right)=\frac{1}{1-{z}^{-3}}$ be the Z-transform of a causal signal $x\left[n\right]$. Then, the values of $x\left[2\right]$ and $x\left[3\right]$ are

Question No. 45

Let $f\left(t\right)$ be a continuous time signal and let $F\left(\omega \right)$ be its Fourier Transform defined by

 Define $g\left(t\right)$ by $F\left(\omega\right)=\int\limits_{-\infty}^\infty f\left(t\right)e^{-j\omega t}dt$ $g\left(t\right)=\int\limits_{-\infty}^\infty F\left(u\right)e^{-jut}du$

What is the relationship between $f\left(t\right)$ and $g\left(t\right)$?

Question No. 119

Consider an LTI system with transfer function

$H\left(S\right)=\frac{1}{s\left(s+4\right)}$

If the input to the system is $\mathrm{cos}\left(3t\right)$ and the steady state output is $A\mathrm{sin}\left(3t+\alpha \right)$, then the value of $A$ is

Question No. 120

Consider an LTI system with impulse response $h\left(t\right)={e}^{-5t}u\left(t\right)$. If the output of the system is $y\left(t\right)={e}^{-3t}u\left(t\right)-{e}^{-5t}u\left(t\right)$ then the input, $x\left(t\right)$, is given by

Question No. 143

A discrete system is represented by the difference equation

It has initial conditions X1(0) = 1; X2(0) = 0. The pole locations of the system for a = 1, are

Question No. 144

An input signal x(t) 2 +  5sin(100πt) is sampled with a sampling frequency of 400 Hz and applied to the system whose transfer function is represented by

$\frac{\mathrm{Y}\left(\mathrm{z}\right)}{\mathrm{X}\left(\mathrm{z}\right)}=\frac{1}{\mathrm{N}}\left(\frac{1-{\mathrm{Z}}^{-\mathrm{N}}}{1-{\mathrm{Z}}^{-1}}\right)$

where, N represents the number of samples per cycle. The output y(n) of the system under steady state is

Question No. 215

A function f(t) is shown in the figure.

The Fourier transform F(ω) of f(t) is

Question No. 219

A signal is represented by

$x\left(t\right)=\left\{\begin{array}{ll}1& \left|t\right|<1\\ 0& \left|t\right|>1\end{array}\right\$

The Fourier transform of the convolved signal $\mathrm{y}\left(\mathrm{t}\right)=\mathrm{x}\left(2\mathrm{t}\right)*\mathrm{x}\left(\mathrm{t}/2\right)$ is

Question No. 220

For the signal , the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is _________.

Question No. 243

A continuous-time LTI system with system function $H\left(\omega \right)$ has the following pole-zero plot. For this system, which of the alternatives is TRUE?

Question No. 244

A sinusoid x(t) of unknown frequency is sampled by an impulse train of period 20 ms. The resulting sample train is next applied to an ideal lowpass filter with a cutoff at 25 Hz. The filter output is seen to be a sinusoid of frequency 20 Hz. This means that x(t) has a frequency of

Question No. 245

A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are $X\left(\omega \right)$ and $Y\left(\omega \right)$. Which of the following statements is TRUE?

Question No. 2

The transfer function $\frac{{V}_{2}\left(s\right)}{{V}_{1}\left(s\right)}$ of the circuit shown below is

Question No. 4

The impulse response of a system is h(t) = t u(t) . For an input u(t -1) , the output is

Question No. 5

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

Question No. 6

Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

Question No. 17

For a periodic signal $v\left(t\right)=30sin100t+10cos300t+6sin\left(500t+\mathrm{\pi }/4\right)$, the fundamental frequency in rad/s is

Question No. 18

A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency in kHz which is not valid is

Question No. 41

The impulse response of a continuous time system is given by h(t) =$\delta$(t -1) +$\delta$(t - 3) . The value of the step response at t = 2 is

Question No. 8

If $x\left[n\right]={\left(1/3\right)}^{\left|n\right|}-{\left(1/2\right)}^{n}u\left[n\right]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

Question No. 15

The unilateral Laplace transform of f (t) is $\frac{1}{{s}^{2}+s+1}$. The unilateral Laplace transform of t f (t) is

Question No. 31

Let y[n] denote the convolution of h[n] and g[n], where h[n]=(1/2)n u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1] equals

Question No. 42

The Fourier transform of a signal h(t) is $H\left(j\omega \right)=\left(2\mathrm{cos}\omega \right)\left(\mathrm{sin}2\omega \right)/\omega .$The value of h(0) is

Question No. 44

The input x(t) and output y(t) of a system are related as The system is

Question No. 4

The fourier series expansion $\style{font-size:18px}{f\left(t\right)=a_0+\sum\limits_{n=1}^\infty a_n\cos n\omega t+b_n\sin n\omega t}$ of the periodic signal shown below will contain the following nonzero terms

Question No. 17

Given two continuous time signals $x\left(i\right)={e}^{-t}$ and $y\left(i\right)={e}^{-2t}$ which exist for $t>0,$ the convolution $z\left(t\right)=x\left(t\right)*y\left(t\right)$ is

Question No. 29

Let the Laplace transform of a function $f\left(t\right)$ which exists for $t>0$ be ${F}_{1}\left(s\right)$ and the Laplace transform of its delayed version $f\left(t-\tau \right)$ be ${F}_{2}\left(s\right)$. Let ${F}_{1}^{*}\left(s\right)$ be the complex conjugate of ${F}_{1}\left(s\right)$ with the Laplace variable set as $s=\sigma +j\omega$. If $G\left(s\right)=\frac{{F}_{2}\left(s\right).{F}_{1}*\left(s\right)}{{\left|{F}_{1}\left(s\right)\right|}^{2}},$ then the inverse Laplace transform of $G\left(s\right)$ is

Question No. 30

A zero mean random signal is uniformly distributed between limits −a and +a and its mean square value is equal to its variance. Then the r.m.s value of the signal is

Question No. 4

The system represented by the input-output relationship $\style{font-size:14px}{y\left(t\right)=\int\limits_{-\infty}^{5t}x\left(\tau\right)d\tau,t>0}$ is

x(t) is a positive rectangular pulse from t = -1 to t = +1 with unit height as shown in the figure. The value of $\int\limits_{-\infty}^\infty\left|X\left(\omega\right)\right|^2d\omega$ {where X(ω) is the Fourier transform of x(t)} is