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

H(f)=1-f20,f200,f20

and phase response

arg{H(f)}=-2f

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

h1(t)=2δ(t+2)-3δ(t+1)h2(t)=δ(t-2)

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

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

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-T1-e-t/T
Z: Ramp   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 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

Xk=X0,X1,X2,X3=12,2j,0-2j.

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

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

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 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,

pqr=1111W31W321W32W341000W32000W34A/3B/3C/3,

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

y.t+5yt=ut

When y(0) =1 and