# GATE Questions & Answers of Signals and Systems Electrical Engineering

#### Signals and Systems 87 Question(s) | Weightage 10 (Marks)

Let $f$ be a real-valued function of a real variable defined as $\style{font-family:'Times New Roman'}{f(x)=x^2}$ for $\style{font-family:'Times New Roman'}{x\geq0}$ , and $\style{font-family:'Times New Roman'}{f(x)=-x^2}$ for $\style{font-family:'Times New Roman'}{x<0}$ . Which one of the following statements is true?

Let $f$  be a real-valued function of a real variable defined as $\style{font-family:Arial}{f(x)=x-\lbrack x\rbrack}$, where $\style{font-family:Arial}{\lbrack x\rbrack}$ denotes the largest integer less than or equal to $x$. The value of  $\style{font-family:'Times New Roman'}{\int\limits_{0.25}^{1.25}f(x)\operatorname dx}$ is _______ (up to 2 decimal places).

Consider the two continuous-time signals defined below:

$x_1(t)=\left\{\begin{array}{l}\left|t\right|,-1\leq t\leq1\\0,\;\;otherwise\end{array},\;\;\;\;\;\;x_2\left(t\right)=\left\{\begin{array}{l}1-\left|t\right|,\;\;-1\leq t\leq1\\0,\;\;\;\;\;\;\;\;\;\;otherwise\;\end{array}\right.\right.$

These signals are sampled with a sampling period of $T=0.25$ seconds to obtain discrete time signals $x_1\left[n\right]$ and $x_2\left[n\right]$ , respectively. Which one of the following statements is true?

The signal energy of the continuous-time signal

$x(t)=\lbrack(t-1)u(t-1)\rbrack-\lbrack(t-2)u(t-2)\rbrack-\lbrack(t-3)u(t-3)\rbrack+\lbrack(t-4)u(t-4)\rbrack$ is

The Fourier transform of a continuous-time signal $x(t)$ is given by $X(w)=\frac1{(10+jw)^2}\;,-\infty<w<\infty,\;$ where $j=\sqrt{-1}$ and  $w$ denotes frequency. Then the value of | ln $x(t) |$ at $t=1$ is ___________ (up to 1 decimal place). ( ln denotes the logarithm to base $e$ )

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

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

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

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

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 _______.

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 __________.

Suppose  have the Fourier transforms as shown below.

Which one of the following statements is TRUE?

The output of a continuous-time, linear time-invariant system is denoted by $\{x(t)\}$ where $x\left(t\right)$ is the input signal. A signal $z\left(t\right)$ is called eigen-signal of the system T , when $\{z(t)\}=y\;z(t)$, 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

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

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 ________.

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),

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

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 _________.

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

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

A moving average function is given by $\style{font-family:'Times New Roman'}{y\left(t\right)=\;\frac1t\int_{t-T}^tu\left(\tau\right)}$. 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 ________.

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)?

The signum function is given by

The Fourier series expansion of $sgn\left(\cos\left(t\right)\right)$ has

Consider a discrete time signal given by

The region of convergence of its Z-transform would be

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

Consider a signal defined by

Its Fourier Transform is

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

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

$x\left(t\right)$ 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?

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

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 _______

The function shown in the figure can be represented as

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

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 $g\left(t\right)$?

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

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

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

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

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

The Fourier transform F(ω) of f(t) is

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

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

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?

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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