# Questions & Answers of Signals and Systems

#### Topics of Signals and Systems 89 Question(s) | Weightage 10 (Marks)

A continuous-time input signal $\style{font-family:'Times New Roman'}{x(t)}$ is an eigenfunction of an LTI system, if the output is

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 $e$ )

Let a casual LTI system be characterized by the following differential equation, with initial rest condition

$\frac{{d}^{2}y}{d{t}^{2}}+7\frac{dy}{dt}+10y\left(t\right)=4x\left(t\right)+5\frac{dx\left(t\right)}{dt}$

Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is (u(t) is the unit step function)

Let the signal

$x(t)=\sum_{k=-\infty}^\limits{+\infty}(-1)^k\;\delta(t-\frac k{2000})$

Be passed through an LTI system with frequency response H($\infty$), as given in the figure below

The Fourier series representation of the output is given as

The pole-zero plots of three discrete-time system P, Q and R on the z-plane are shown below.

Which one of the following is TRUE about the frequency selectivity of these system?

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

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