GATE Questions & Answers of Signals and Systems Electrical Engineering

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 ft=e2tsin5tut is

The transfer function of a system is YSRS=SS+ The steady state output $ y(t) $ is A cos2t+φ for the input cos2t The values of A and φ respectvely are

Consider a continuous-time system with input xt and output yt given by


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 δt 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 α in the range 0<α<1 such that S=2α is _______.

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


The value of yt for t=loge2 is __________.

Suppose x1t and x2t 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 xt is the input signal. A signal zt is called eigen-signal of the system T , when $ \{z(t)\}=y\;z(t) $, where γ 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 dytdt+16yt=3xt. The response of the system to the input xt=3e-t3ut, where u(t) denotes the unit step function, is

Suppose the maximum frequency in a band-limited signal xt is 5 kHz. Then, the maximum frequency in xtcos2000πt, in kHz, is ________.

The solution of the differential equation, for t>0,y''t+2y't+yt=0 with initial conditions y0=0 and y'0=1 is(ut denotes the unit step function),

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

Consider a linear time-invariant system with transfer function


If the input is cost and the steady state output is Acost+α, then the value of A is _________.

Let x1tX1ω and x2tX2ω be two signals whose Fourier Transforms are as shown in the figure below. In the figure, ht=e-2t 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-sin2πtπtdt 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 12THz, 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

sgnx=xx;x00; x=0

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

xt=ej10tfor t10for t>1

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

xt is nonzero only for Tx<t<T'x , and similarly, yt is nonzero only for Ty<t<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:[0,):[0,) be a function defined by gx=x-x, where x 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 gx is _______

The function shown in the figure can be represented as

Let XZ=11-z-3 be the Z-transform of a causal signal xn. Then, the values of x2 and x3 are

Let ft be a continuous time signal and let Fω be its Fourier Transform defined by

Define gt 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 ft and gt?

Consider an LTI system with transfer function


If the input to the system is cos3t and the steady state output is Asin3t+α, then the value of A is

Consider an LTI system with impulse response ht=e-5tut. If the output of the system is yt=e-3tut-e-5tut then the input, xt, is given by

A discrete system is represented by the difference equation

X1K+1X2K+1=aa-1a+1a X1KX2K

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


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


The Fourier transform of the convolved signal yt=x2t*xt/2 is

For the signal ft=3 sin 8πt+6 sin 12πt +sin 14πt , the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is _________.

A continuous-time LTI system with system function Hω 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 Xω and Yω. Which of the following statements is TRUE?

The transfer function V2(s)V1(s) 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(t)=30sin100t+10cos300t+6sin(500t+π/4), 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) =δ(t -1) +δ(t - 3) . The value of the step response at t = 2 is

If xn=1/3n-1/2nun, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

The unilateral Laplace transform of f (t) is 1s2+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 Hjω=2cosωsin2ω/ω. The value of h(0) is

The input x(t) and output y(t) of a system are related as yt=-t xτcos3τdτ.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(i)=e-t and y(i)=e-2t which exist for t>0, the convolution z(t)=x(t)*y(t) is

Let the Laplace transform of a function f(t) which exists for t>0 be F1s and the Laplace transform of its delayed version f(t-τ) be F2s. Let F1*s be the complex conjugate of F1s with the Laplace variable set as s=σ+jω. If Gs=F2s.F1*sF1s2, then the inverse Laplace transform of Gs is

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

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

The second harmonic component of the periodic waveform given in the figure has an amplitude of