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

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                                                                                                                                           x[0]=1,     x[1]=2,      x[2]=1,      h[0]=1.

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 (10y[3]+y[4]) is_________

Let h[n] be the impulse response of a discrete-time linear time invariant (LTI) filter. The impulse response is given by h[0]=13; h[1]=13; h[2]=13; 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 x(t)=4 cos(200πt)+8 cos(400πt), where t is in seconds, is the input to a linear time invariant(LTI) filter with the impulse response h(t)=2 sin(300πt)πt,t0600,t=0

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

An LTI system with unit sample response h[n]=5δ[n]-7δ[n-1]+7δ[n-3]-5δ[n-4] is a

The input x(t) and the output y(t) of a continuous-time system are related as y(t)=t-Ttx(u)du

The system is 

Consider an LTI system with magnitude response


and phase response


If the input to the system is

x(t)=8 cos20πt+π4+16 sin 40πt+π8+24 cos80πt+π16

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


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:

H(f)=1,f12 kHz.0,f>12 kHz.

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 (ut 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 xn=anun+bnun, where un denotes the unit-step sequence and 0<a<b<1. The region of convergence (ROC) of the z-transform of xn 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


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-T1-e-t/T
Z: Ramp   R:e-t/T


Consider the signal


If Xejω 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 xt = sin4πt4πt is ________

A continuous-time filter with transfer function Hs=2s+6s2+6s+8 is converted to a discretetime filter with transfer function Gz=2z2-0.5032 zz2-0.5032 z+k 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

xn=x0,x1,x2,x3=3,2,3,4 is


IfX1k is the DTF of 12-point sequence x1n=3,0,0,2,0,0,3,0,0,4,0,0

the value of X18X111 is ________

Consider the signal xt=cos6πt+sin8πt, where t is in seconds. The Nyquist sampling rate (in samples/second) for the signal yt=x2t+5 is

If the signal xt=sintπt*sintπt with *  denoting the convolution operation, then x(t) is equal to

A discreate-time signal  xn=δn-3+2δn-5 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


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 xn=2.0n,-<n<+ , 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 gt=xt-12 . The average power of $ g(t) $ is ______.

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

ABC=1111W3-1W3-21W3-2W3-4abcWhere W3=ej2π3

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


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 ft=1if atb0Otherwise 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, ft< . 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 yn=m=0nxm . If the z-transform of yn is 2zz-12 , the value of $ x\lbrack2\rbrack $ is _______.

The signal cos10πt+π4 is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response sinπtπtcos40πt-π2 . 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

Xs = 16s2- 16-4 <Re{s}<4;

Then the value of $ \beta $ is______.

Consider the function gt=e-tsin2πtut 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 yn=i=03αxn-i . 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 x~n=1+cosπn8 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 x1n and x2n have the same energy. Suppose x1n=α 0.5n un, where α is a positive real number and un is the unit step sequence. Assume

x2n=1.5 for n=0,10 otherwise.

Then the value of α is_______.

The complex envelope of the bandpass signal xt=-2sinπt/5πt/5sin(πt-π4),centered about f=12Hz, 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 xn=sinπ2n,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

ht=3for 0t30otherwise

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 -+gte-jwtdt=ωe-2ω2 for any real value ω.
If yt=-tgtdτ, , then -tytdt  is

Let  xn=-19nun--13nu-n-1 The Region of Convergence (ROC) of the z-transform of x[n]

Consider a discrete time periodic signal xn=sinπn5. Let ak be the complex Fourier series coefficients of xn. The coefficients ak are non-zero when k=BM±1 where M is any integer. The value of B is______.

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system


When y(0) =1 and