GATE Electronics & Communication

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=$$\mathop f\limits_{ - \infty }^\infty $$ f(t) \exp (-j \omega t) d t$ exists. Which one of...

Let $x[n]$ be a discrete-time signal whose $z$-transform is $X(z)$. Which of the following statements is/are TRUE?

Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$. The Fourier series expansions of $f(t)$ and $y(t)$ are gi...

Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $\left[-\omega_c, \omega_c\right]$, where $\omega_c$ is in ra...

A causal and stable LTI system with impulse response h(t) produces an output y(t) for an input signal x(t) . A signal x(0.5t) is applied to another causal and stable LTI system wit...

For a causal discrete-time LTI system with transfer function $H(z) = \frac{2z^2 + 3}{\left(z + \frac{1}{3}\right)\left(z - \frac{1}{3}\right)}$ which of the following statements is...

A continuous time signal $x(t) = 2 \cos(8 \pi t + \frac{\pi}{3})$ is sampled at a rate of 15 Hz. The sampled signal $x_s(t)$ when passed through an LTI system with impulse response...

The radian frequency value(s) for which the discrete time sinusoidal signal $x[n] = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are __.

The relationship between any N-length sequence $x[n]$ and its corresponding N-point discrete Fourier transform $X[k]$ is defined as $X[k] = \mathcal{F}\{x[n]\}$. Another sequence $...

Let $${w^4} = 16j$$. Which of the following cannot be a value of $$w$$?

Consider a system with input $$x(t)$$ and output $$y(t) = x({e^t})$$. The system is

Let $$m(t)$$ be a strictly band-limited signal with bandwidth B and energy E. Assuming $${\omega _0} = 10B$$, the energy in the signal $$m(t)\cos {\omega _0}t$$ is

The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$

In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic (a) Continuous, aperiodic (ii) Con...

Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over...

Let an input $$x[n]$$ having discrete time Fourier transform $$x({e^{j\Omega }}) = 1 - {e^{ - j\Omega }} + 2{e^{ - 3j\Omega }}$$ be passed through an LTI system. The frequency resp...

Let $$x(t) = 100\cos (10.5Wt)$$ be passed through an LTI system having impulse response $$h(t) = \pi {\left( {{{\sin Wt} \over {\pi t}}} \right)^2}\cos 10Wt$$. The output of the sy...

The Fourier transform X(j$$\omega$$) of the signal $$x(t) = {t \over {{{(1 + {t^2})}^2}}}$$ is ____________.

Let x 1 (t) = e $$-$$t u(t) and x 2 (t) = u(t) $$-$$ u(t $$-$$ 2), where u( . ) denotes the unit step function. If y(t) denotes the convolution of x 1 (t) and x 2 (t), then $$\math...

For a vector $$\overline x $$ = [x[0], x[1], ....., x[7]], the 8-point discrete Fourier transform (DFT) is denoted by $$\overline X $$ = DFT($$\overline x $$) = [X[0], X[1], .....,...

Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point inverse discrete Fourier transform (I...

Consider the signals $x[n]=2^{n-1} u[-n+2]$ and $y[n]=2^{-n+2} u[n+1]$, where $u[n]$ is the unit step sequence. Let $X\left(e^{j \omega}\right)$ and $Y\left(e^{j \omega}\right)$ be...

For a unit step input $u[n]$, a discreate-time $L T I$ system produces an output signal $(2 \delta[n+1]+\delta[n]+\delta[n-1])$. Let $y[n]$ be the output of the system for an input...

The exponential Fourier series representation of a continu-ous-time periodic signal $X(t)$ is defined as $$ x(t)=\sum\limits_{k=-\infty}^{\infty} a_k e^{j k w_0 t} $$ Where $\omega...

Consider a real-valued base-band signal $x(t)$. band limited to 10 kHz . The Nyquist rate for the signal $y(t)=x(t) \times \left(1+\frac{t}{2}\right)$ is

The output $y[n]$ of a discrete - time system for an input $x[n]$ is $$ y[n]=\max\limits_{-\infty \leq k \leq n}|x[k]| $$ The unit impulse response of the system is

The transfer function of a stable discrete - time LTI system is $H(z)=\frac{K(z-\alpha)}{(z+0.5)}$ where $K$ and $\alpha$ are real numbers. The value of $\alpha$ (rounded off to on...

A finite duration discrete-time signal $x[n]$ is obtained by sampling a continuous - time signal $x(t)=\cos (200 \pi t)$ at sampling instants $t=\frac{n}{400}, n=0,1, \ldots ., 7$....

Which one of the following pole-zero plots corresponds to the transfer function of an LTI system characterized by the input-output difference equation given below? $$ y[n]=\sum_{k=...

A discrete time all-pass system has two of its poles at 0.25$$\angle 0^\circ $$ and $$\angle 30^\circ $$. Which one of the following statements about the system is TRUE?

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:

Let 𝑥(𝑡) 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 = - \inft...

Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n], where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}} $$. The value (correct to two decim...

A periodic signal x(t) has a trigonometric Fourier series expansion $$$x\left(t\right)=a_0\;+\;\sum_{n=1}^\infty\left(a_n\cos\;n\omega_0t\;+\;b_n\sin\;n\omega_0t\right)$$$ If $$x\l...

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

Consider an LTI system with magnitude response $$$\left| {H(f)} \right| = \left\{ {\matrix{ {1 - \,{{\left| f \right|} \over {20}},} & {\left| f \right| \le 20} \cr {0,} & {\left|...

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...

Two discrete-time signals x [n] and h [n] 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 [n] be the lin...

A continuous time signal x(t) = $$4\cos (200\pi t)$$ + $$8\cos(400\pi t)$$, where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response $$...

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

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

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

Let h[n] be the impulse response of a discrete time linear time invariant (LTI) filter. The impulse response is given by h(0)= $${1 \over 3};h\left[ 1 \right] = {1 \over 3};h\left[...

The transfer function of a causal LTI system is H(s) = 1/s. If the input to the system is x(t) = $$\left[ {\sin (t)/\pi t} \right]u(t);$$ where u(t) is a unit step function. The sy...

Which one of the following is an eight function of the class of all continuous-time, linear, time- invariant systems u(t) denotes the unit-step function?

The Discrete Fourier Transform (DFT) of the 4-point sequence $$x\left[ n \right]$$= {x[0], x[1], x[2], x[3]} = {3, 2, 3, 4 } is x[k] = {X[0], X[1], X[2], X[3]} = {12, 2j, 0, -2j }...

The energy of the signal x(t) =$${{\sin (4\pi t)} \over {4\pi t}}$$ is ___________.

A discrete-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...

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...

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

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 \lef...

A continuous-time filter with transfer function $$\,H(S) = {{2s + 6} \over {{s^2} + 6s + 8}}$$ is converted to a discrete time filter with transfer function $$G(Z) = {{2{z^2} - 0.5...

Consider the sequence $$x\left[ n \right]$$= $${a^n}u\left[ n \right] + {b^{\partial n}}u\left[ n \right]$$ , where u[n] denotes the unit step sequence and 0<$$\left| a \right| < \...

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 c...

A continuous-time speech signal $${x_a}(t)$$ 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 comput...

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}...

A sequence x$$\left[ n \right]$$ is specified as $$\left[ {\matrix{ {x\left[ n \right]} \cr {x\left[ {n - 1} \right]} \cr } } \right] = {\left[ {\matrix{ 1 \cr 1 \cr } \,\matrix{ 1...

Consider a four-point moving average filter defined by the equation $$y[n] = \sum\limits_{i = 0}^3 {{\alpha _i}x[n - i]} $$. The condition on the filter coefficients that results i...

The solution of the differential equation $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}\,\,\,\,\,{{{d^2}y} \over {d{t^{ \to 2}}}} + {{2\,dy} \over {dt}} + y\, = \,0$$ wit...

The complex envelope of the bandpass signal $$x(t)\, = \, - \sqrt 2 \left( {{{\sin (\pi t/5)} \over {\pi t/5}}} \right)\sin \left( {\pi t - {\pi \over 4}} \right),$$ centered about...

Let $$\widetilde x\left[ n \right]\, = \,1 + \cos \left[ {{{\pi n} \over 8}} \right]$$ be a periodic signal with period 16. Its DFS coefficients are defined by $${a_k}$$ = $${1 \ov...

The impulse response of an LTI system can be obtained by

The signal $$\cos \left( {10\pi t + {\pi \over 4}} \right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse respons...

The value of $$\sum\limits_{n = 0}^\infty n {\left( {{1 \over 2}} \right)^n}$$ is ________________.

The bilateral Laplace transform of a function $$f\left( t \right) = \left\{ {\matrix{ {1\,if\,a \le t \le b} \cr {0\,otherwise} \cr } } \right.$$ is

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

The value of the integral $$\int_{ - \infty }^\infty {12\,\cos (2\pi )\,{{\sin (4\pi t)} \over {4\pi t}}\,dt\,} $$ is

Suppose x $$\left[ n \right]$$ 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. Whic...

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

Two sequences [a, b, c ] and [A, B, C ] are related as, $$\left[ {\matrix{ A \cr B \cr C \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } {\mkern 1mu} \,\matrix{ 1 \cr {W_3^{...

Consider two real sequences with time- origin marked by the bold value, $${x_1}\left[ n \right] = \left\{ {1,\,2,\,3,\,0} \right\}\,,\,{x_2}\left[ n \right] = \left\{ {1,\,3,\,2,\,...

Two sequence $${x_1}\left[ n \right]$$ and $${x_2}\left[ n \right]$$ have the same energy. Suppose $${x_1}\left[ n \right]$$ $$ = \alpha \,{0.5^n}\,u\left[ n \right],$$ where $$\al...

The result of the convolution $$x\left( { - t} \right) * \delta \left( { - t - {t_0}} \right)$$ is

Consider a continuous-time signal defined as $$x(t) = \left( {{{\sin \,(\pi t/2)} \over {(\pi t/2)}}} \right)*\sum\limits_{n = - \infty }^\infty {\delta (t - 10n)} $$ Where ' * ' d...

The complex envelope of the bandpass signal $$x(t) = - \sqrt 2 \left( {{{\sin \,(\pi t/5)} \over {\pi t/5}}} \right)\,\sin \,\left( {\pi t - {\pi \over 5}} \right)$$, centered abou...

Two casual discrete-time signals $$x\left[ n \right]$$ and $$y\left[ n \right]$$ =$$\sum\limits_{m = 0}^n x \left[ m \right]$$. If the z-transform of y$$\left[ n \right]$$=$${2 \ov...

The output of a standrad second-order system for a unit step input is given as $$y(t) = 1 - {2 \over {\sqrt 3 }}{e^{ - t}}\cos \left( {\sqrt 3 t - {\pi \over 6}} \right)$$. The tra...

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 imulse response h...

Consider a discrete time periodic signal x$$\left[ n \right]$$= $$\sin \left( {{{\pi n} \over 5}} \right)$$. Let a k be the complex Fourier serier coefficients of x$$\left[ n \righ...