# GATE Questions & Answers of Signals and Systems Electronics and Communication Engg

#### Signals and Systems 134 Question(s) | Weightage 12 (Marks)

Let the input be $u$ and the output be $y$ of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

A discrete-time all-pass system has two of its poles at $0.25\angle0^\circ$ and $2\angle30^\circ$.  Which one of the following statements about the system is TRUE?

Let $x\left(t\right)$ be a periodic function with period $T$ = 10. The Fourier series coefficients for this series are denoted by $a_k$ , that is

$x\left(t\right)=\sum\limits_{k=-\infty}^\infty a_ke^{jk\frac{2\mathrm\pi}Tt}$

The same function $x\left(t\right)$ can also be considered as a periodic function with period $T′$ = 40. Let $b_k$ be the Fourier series coefficients when period is taken as $T′$. If $\textstyle\sum\limits_{k=-\infty}^\infty\left|a_k\right|=16$, then $\textstyle\sum\limits_{k=-\infty}^\infty\left|b_k\right|$ is equal to

Let $X\left[k\right]=k+1,\;0\leq k\leq7$ be 8-point DFT of a sequence $x\left[n\right]$,

where $X\left[k\right]={\textstyle\sum_{n=0}^{N-1}}x\left[n\right]e^{-j2\pi nk/N}$ .

The value (correct to two decimal places) of $\textstyle\sum_{n=0}^3x\left[2n\right]$ is ______.

Consider the following statements for continuous-time linear time invariant (LTI) system.

I.       There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane.

II.       There is no casual and BIBO stable system with a pole in the right half of the complex plane.

Which one among the following is correct?

Consider a single input single output discrete-time system with $x\left[n\right]$ as input and $y\left[n\right]$ as output, where the two are related as

$y\left[n\right]=\left\{\begin{array}{l}\;\;\;\;\;\;\;\;\;\;\;\;n\left|x\left[n\right]\right|,\;\;\;\;\;\;\mathrm{for}\;0\leq\mathrm n\leq10\\x\left[n\right]-x\left[n-1\right],\;\;\;\;\;\;\;\mathrm{otherwise}.\end{array}\right.$

Which one of the following statement is true about the system?

A periodic signal x(t) has a trigonometric Fourier aeries expansion

$x\left(t\right)=a_0+\sum\limits_{n=1}^\infty\left(a_n\;\cos\omega_0t\;+\;b_n\sin\;n\omega_0t\right)$

If $x\left(t\right)=-x\left(-t\right)=\left(t-\pi/\omega_0\right)$, we can conclude that

Let x(t) be a continuous time periodic signal with fundamental period T=1 seconds. Let {ak} be the complex Fourier series coefficients of x (t), where k is integer valued. Consider the following statements about x (3t):

I.                  The complex Fourier series coefficients of x (3t) are {ak} where k is integer valued

II.                 The complex Fourier series coefficients of x (3t) are {3ak} where k is integer valued

III.                The fundamental angular frequency of x (3t) is 6$\pi$ rad/s

Two discrete-time signals $x\left[n\right]$ and $h\left[n\right]$ are both non-zero only for $n=0,\;1,\;2,$ and are zero otherwise. It is given that

Let $y\left[n\right]$ be linear convolution of $x\left[n\right]$ and $h\left[n\right]$. Given that $y\left[1\right]=3$ and $y\left[2\right]=4$, the value of the expression $\left(10y\left[3\right]+y\left[4\right]\right)$ is_________

Let h[n] be the impulse response of a discrete-time linear time invariant (LTI) filter. The impulse response is given by and h[n]=0 for n <0 and n >2.

Let H[ω] be the discrete-time Fourier transform (DTFT) of h[n], where ω is the normalization angular frequency in radians. Given that $H\left(\omega_0\right)=0$ and $0<\omega_0<\pi$, the value of $\omega_0$ (in radians) is equal to________.

A continuous time signal , where t is in seconds, is the input to a linear time invariant(LTI) filter with the impulse response

Let y(t) be the output of this filter. The maximum value of |y(t)| is____________.

An LTI system with unit sample response $h\left[n\right]=5\delta \left[n\right]-7\delta \left[n-1\right]+7\delta \left[n-3\right]-5\delta \left[n-4\right]$ is a

The input x(t) and the output y(t) of a continuous-time system are related as $y\left(t\right)={\int }_{t-T}^{t}x\left(u\right)du$

The system is

Consider an LTI system with magnitude response

$\left|H\left(f\right)\right|=\left\{\begin{array}{ll}1-\frac{\left|f\right|}{20},& \left|f\right|\le 20\\ 0,& \left|f\right|\ge 20\end{array}\right\$

and phase response

$arg\left\{H\left(f\right)\right\}=-2f$

If the input to the system is

then the average power of the output signal () is___________

The transfer function of a casual LTI system is () = 1/s. If the input to the system is $x\left(t\right)=\left[\sin\left(t\right)\pi t\right]u\left(t\right)$ is a unit step function, the system output () as  $t\rightarrow\infty$ is ___________

Consider the parallel combination of two LTI system shown in the figure.

The impulse responses of the system are

${h}_{1}\left(t\right)=2\delta \left(t+2\right)-3\delta \left(t+1\right)\phantom{\rule{0ex}{0ex}}{h}_{2}\left(t\right)=\delta \left(t-2\right)$

If the input x(t) is a unit step signal, then the energy of y(t) is__________

The signal $x\left(t\right)=\sin\left(14000\pi t\right)$, where t is in seconds, is sampled at a rate of 9000 samples per seconds. The sampled signal is the input to an ideal lowpass filter with frequency response () as follows:

What is the number of sinusoids in the output and their frequencies in kHz?

Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems ($u\left(t\right)$ denotes the unit-step function)?

A continuous-time function $x(t)$ is periodic with period T. The function is sampled uniformly with a sampling period Ts. In which one of the following cases is the sampled signal periodic?

Consider the sequence $x\left[n\right]={a}^{n}u\left[n\right]+{b}^{n}u\left[n\right]$, where $u\left[n\right]$ denotes the unit-step sequence and $0<\left|a\right|<\left|b\right|<1$. The region of convergence (ROC) of the z-transform of $x\left[n\right]$ is

A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________

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

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

$\sum\limits_{k=1}^3a_k\;\cos\left(k\omega_0t\right),\;\mathrm{where}\;a_k\neq0\;,\omega_0\neq0.$

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to $t=0$). Match the excitation signals X, Y, Z with the corresponding time responses for $t\geq0;$
 X: Impulse $P:1-{e}^{-t/T}$ Y: Unit step $Q:t-T\left(1-{e}^{-t/T}\right)$ Z: Ramp ${\mathrm{R:e}}^{-t/T}$

Consider the signal

$x\left[n\right]=6\;\delta\left[n+2\right]+3\;\delta\left[n+1\right]+8\;\delta\left[n\right]+7\;\delta\left[n-1\right]+4\;\delta\left[n-2\right]$

If $X\left({e}^{j\omega }\right)$ is the discrete-time Fourier transform of $x\left[\mathrm n\right],$

then $\frac1n\int_{-\mathrm\pi}^\mathrm\pi X\left(e^{j\omega}\right)\sin^2\left(2\omega\right)d\omega$ is equal to _________

The energy of the signal is ________

A continuous-time filter with transfer function $H\left(s\right)=\frac{2s+6}{{s}^{2}+6s+8}$ is converted to a discretetime filter with transfer function so that the impulse response of the continuous-time filter, sampled at 2 Hz, is identical at the sampling instants to the impulse response of the discrete time filter. The value of k is ________

The Discrete Fourier Transform (DFT) of the 4-point sequence

$X\left[k\right]=\left\{X\left[0\right],X\left[1\right],X\left[2\right],X\left[3\right]\right\}=\left\{12,2j,0-2j\right\}.$

If${X}_{1}\left[k\right]$ is the DTF of 12-point sequence ${x}_{1}\left[n\right]=\left\{3,0,0,2,0,0,3,0,0,4,0,0\right\}$

the value of $\left|\frac{{X}_{1}\left[8\right]}{{X}_{1}\left[11\right]}\right|$ is ________

Consider the signal $x\left(t\right)=\mathrm{cos}\left(6\mathrm{\pi t}\right)+\mathrm{sin}\left(8\mathrm{\pi t}\right)$, where t is in seconds. The Nyquist sampling rate (in samples/second) for the signal $y\left(t\right)=x\left(2t+5\right)$ is

If the signal $x\left(t\right)=\frac{\mathrm{sin}\left(t\right)}{\mathrm{\pi t}}*\frac{\mathrm{sin}\left(t\right)}{\mathrm{\pi t}}$ with *  denoting the convolution operation, then x(t) is equal to

A discreate-time signal  $x\left[n\right]=\delta \left[n-3\right]+2\delta \left[n-5\right]$ has z-transform $X(z).$ If $Y(z)=X(-z)$ is the z-transform of another signal $y[n]$,then

A signal $2\cos\left(\frac{2\mathrm\pi}3t\right)-\;\cos\;\left(\mathrm{πt}\right)$ is the input to an LTI system with the transfer function

$H\left(s\right)=e^s+e^{-s}$

If Ck denotes the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to

The ROC (region of convergence) of the z-transform of a discrete-time signal is represented by the shaded region in the z-plane. If the signal $x\left[n\right]={\left(2.0\right)}^{\left|n\right|},-\infty , then the ROC of its z-transform is represented by

A continuous-time speech signal xa(t) is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes 20 μs for computing each complex multiplication (including multiplications by 1 and −1) and the time required for addition/subtraction is negligible, then the maximum value of N is __________

The direct form structure of an FIR (finite impulse response) filter is shown in the figure.

The filter can be used to approximate a

The result of the convolution $\mathrm x(-\mathrm t)\;\ast\;\mathrm\delta(-\mathrm t-{\mathrm t}_0)$ is

The waveform of a periodic signal $x(t)$ is shown in the figure.

A signal g(t) is defined by $g\left(t\right)=x\left(\frac{t-1}{2}\right)$ . The average power of $g(t)$ is ______.

Two sequences $\lbrack a,b,c\rbrack$ and $\lbrack A,B,C\rbrack$ are related as,

If another sequence $\lbrack p,q,r\rbrack$ is derived as,

$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& {W}_{3}^{1}& {W}_{3}^{2}\\ 1& {W}_{3}^{2}& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& {W}_{3}^{2}& 0\\ 0& 0& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{c}A/3\\ B/3\\ C/3\end{array}\right],$

then the relationship between the sequences $\lbrack p,q,r\rbrack$ and $\lbrack a,b,c\rbrack$ is

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

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[0] = 1, we can conclude.

The bilateral Laplace transform of a function is

The magnitude and phase of the complex Fourier series coefficients 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 numbers, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

Let the signal $f(t)=0$ outside the interval $\lbrack T_1,T_2\rbrack$, where $T_1$ and $T_2$ are finite. Furthermore, $\left|f\left(t\right)\right|<\infty$ . The region of convergence (RoC) of the signal’s bilateral Laplace transform $F(s)$ is

Two casual discrete-time signals $x\lbrack n\rbrack$ and $y\lbrack n\rbrack$ are related as $y\left[n\right]=\sum _{m=0}^{n}x\left[m\right]$ . If the z-transform of , the value of $x\lbrack2\rbrack$ is _______.

The signal $\mathrm{cos}\left(10\mathrm{\pi t}+\frac{\mathrm{\pi }}{4}\right)$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $\left(\frac{\mathrm{sin}\left(\mathrm{\pi t}\right)}{\mathrm{\pi t}}\right)\mathrm{cos}\left(40\mathrm{\pi t}-\frac{\mathrm{\pi }}{2}\right)$ . The filter output is

Consider the differential equation $\frac{dx}{dt}=10-0.2x$ with initial condition $x(0)=1$. The response $x(t)$ for $t>0$ is

Input $x(t)$ and output $y(t)$ of an LTI system are related by the differential equation $y"(t)-y'(t)-6y(t)=x(t)$. If the system is neither causal nor stable, the impulse response $h(t)$ of the system is

Consider two real sequences with time – origin marked by the bold value,

x1[n] ={1,2,3,0} , x2[n] ={1,3,2,1}

Let X1(k) and X2(k) be 4-point DFTs of x1[n] and x2[n] , respectively .

Another sequence x3[n] is derived by taking 4-point inverse DFT of X3(k) =X1(k)X2(k) .

The value of x3[2] is_____.

Let $x(t)=\alpha\;s(t)+s(-t)$ with $s(t)=\beta e^{4t}u(t)$ , where $u(t)$ is unit step function . If the bilateral Laplace transform of $x(t)$ is

Then the value of $\beta$ is______.

Consider the function $g\left(t\right)={e}^{-t}\mathrm{sin}\left(2\pi t\right)u\left(t\right)$ where $u\left(t\right)$ is the unit step function. The area under $g\left(t\right)$ is _____.

The value of $\sum\limits_{n=0}^\infty n\left(\frac12\right)^n$ is _____.

The impulse response of an LTI system can be obtained by

Consider a four-point moving average filter defined by the equation . The condition on the filter coefficients that results in a null at zero frequency is

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at $z=±2j$ . Which one of the following statements is TRUE for the signal x[n]?

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input $x\left[n\right]$, the response $y\left[n\right]$ is

Let $\stackrel{~}{x}\left[n\right]=1+\mathrm{cos}\left(\frac{\mathrm{\pi n}}{8}\right)$ be a periodic signal with period 16. Its DFS coefficients are defined by $a_k=\frac1{16}\sum\limits_{n=0}^{15}\widetilde x\left[n\right]exp\left(-j\frac{\mathrm\pi}8kn\right)$ for all $k$. The value of the coefficients a31 is_____.

Consider a continuous-time signal defined as
$x\left(t\right)=\left(\frac{\sin\left(\pi t/2\right)}{\left(\pi t/2\right)}\right)\ast\sum_\limits{n=-\infty}^\infty\delta\left(t-10n\right)$
where '*' denotes the convolution operation and $t$ is in seconds. The Nyquist sampling rate (in samples/sec) for $x\left(t\right)$ is ___________.

Two sequences ${x}_{1}\left[n\right]$ and ${x}_{2}\left[n\right]$ have the same energy. Suppose , where $\alpha$ is a positive real number and $u\left[n\right]$ is the unit step sequence. Assume

Then the value of $\alpha$ is_______.

The complex envelope of the bandpass signal $x\left(t\right)=-\sqrt{2}\left(\frac{\mathrm{sin}\left(\pi t/5\right)}{\pi t/5}\right)\mathrm{sin}\left(\pi t-\frac{\pi }{4}\right)$,centered about $f=\frac{1}{2}\mathrm{Hz}$, is

C is a closed path in the z-plane given by |z|=3. The value of the integral $\oint_C\left(\frac{z^2-z+4j}{z+2j}\right)dz$

A discrete-time signal $x\left[n\right]=\mathrm{sin}\left({\mathrm{\pi }}^{2}n\right),n$ being an integer is

Consider two real valued signals, x(t) band-limited to [ –500 Hz, 500 Hz] and y(t) bandlimited to [ –1 kHz, 1 kHz]. For z(t) = x(t)•y(t), the Nyquist sampling frequency (in kHz) is ______.

A continuous, linear time-invariant filter has an impulse response h(t) described by

When a constant input of value 5 is applied to this filter, the steady state output is_____.

For a function g(t), it is given that ${\int }_{-\infty }^{+\infty }g\left(t\right){e}^{-jwt}dt=\omega {e}^{-2{\omega }^{2}}$ for any real value $\omega$.
If $y\left(t\right)={\int }_{-\infty }^{t}g\left(t\right)d\tau ,$, then is

Let  $x\left[n\right]={\left(-\frac{1}{9}\right)}^{n}u\left(n\right)-{\left(-\frac{1}{3}\right)}^{n}u\left(-n-1\right)$ The Region of Convergence (ROC) of the z-transform of x[n]

Consider a discrete time periodic signal $x\left[n\right]=sin\left(\frac{\mathit{\pi }\mathit{n}}{\mathit{5}}\right)$. Let ak be the complex Fourier series coefficients of $x\left[n\right]$. The coefficients $\left\{{a}_{k}\right\}$ are non-zero when $k=BM±1$ where M is any integer. The value of B is______.
$\stackrel{.}{y}\left(t\right)+5y\left(t\right)=u\left(t\right)$