GATE Electrical Engineering

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t)=\left\{\begin{array}{cc}1, & t \in[0,1] \\ 2-t, & t \in[1,2] \\ 0, & \text { otherwise }\end{array}\right.$ and $x_2(t)...

Consider a discrete-time linear time-invariant (LTI) system, $\boldsymbol{S}$, where $$ y[n]=S\{x(\mathrm{n})\} $$ $$Let\,\,\,\, S\{\delta[n]\}=\left\{\begin{array}{lc} 1, & n \in\...

An ideal low pass filter has frequency response given by $$ H(j \omega)= \begin{cases}1, & |\omega| \leq 200 \pi \\ 0, & \text { otherwise }\end{cases} $$ Let $h(t)$ be its time do...

A continuous time periodic signal $x(t)$ is $$ x(t)=1+2 \cos 2 \pi t+2 \cos 4 \pi t+2 \cos 6 \pi t $$ If $T$ is the period of $x(t)$, then $\frac{1}{T} \int_0^T|x(t)|^2 d t=$______...

Consider a continuous-time signal $$ x(t)=-t^2\{u(t+4)-u(t-4)\} $$ where $u(t)$ is the continuous-time unit step function. Let $\delta(t)$ be the continuous-time unit impulse funct...

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system $S$. The output is observed to be the continuous-time unit step sig...

If the energy of a continuous-time signal $x(t)$ is $E$ and the energy of the signal $2x(2t - 1)$ is $cE$, then $c$ is _____ (rounded off to 1 decimal place).

If $u(t)$ is the unit step function, then the region of convergence (ROC) of the Laplace transform of the signal $x(t) = e^{t^2}[u(t-1)-u(t-10)]$ is

The input $x(t)$ and the output $y(t)$ of a system are related as $$ y(t) = e^{-t} \int\limits_{-\infty}^{t} e^{\tau} x(\tau) d\tau, \quad - \infty The system is

Consider the discrete-time systems $T_1$ and $T_2$ defined as follows: { $T_1 x[ n ] = x[ 0 ] + x[ 1 ] + \cdots + x[ n ] $} { $T_2 x[ n ] = x[ 0 ] + \frac{1}{2} x[ 1 ] + \cdots + \...

Let $X(\omega)$ be the Fourier transform of the signal $x(t) = e^{-t^4} \cos t, \quad -\infty The value of the derivative of $X(\omega)$ at $\, \omega = 0$ is ______ (rounded off t...

Suppose signal $y(t)$ is obtained by the time-reversal of signal $x(t)$, i.e., $y(t) = x(-t)$, $-\infty

If the Z-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the Z-transform of the signal $y[n] = x[2n]$ is

The Z-transform of a discrete signal $$x[n]$$ is $$X(z) = {{4z} \over {(z - {1 \over 5})(z - {2 \over 3})(z - 3)}}$$ with $$ROC = R$$. Which one of the following statements is true...

The discrete-time Fourier transform of a signal $$x[n]$$ is $$X(\Omega ) = (1 + \cos \Omega ){e^{ - j\Omega }}$$. Consider that $${x_p}[n]$$ is a periodic signal of period N = 5 su...

A signal $$x(t) = 2\cos (180\pi t)\cos (60\pi t)$$ is sampled at 200 Hz and then passed through an ideal low pass filter having cut-off frequency of 100 Hz. The maximum frequency p...

A continuous-time system that is initially at rest is described by $${{dy(t)} \over {dt}} + 3y(t) = 2x(t)$$, where $$x(t)$$ is the input voltage and $$y(t)$$ is the output voltage....

The Fourier transform $$X(\omega)$$ of the signal $$x(t)$$ is given by $$X(\omega ) = 1$$, for $$|\omega | $$ = 0$$, for $$|\omega | > {W_0}$$ Which one of the following statements...

Which of the following statement(s) is/are true?

The period of the discrete-time signal $$x[n]$$ described by the equation below is $$N=$$ __________ (Round off to the nearest integer). $$x[n] = 1 + 3\sin \left( {{{15\pi } \over...

Let a causal LTI system be governed by the following differential equation $$y(t) + {1 \over 4}{{dy} \over {dt}} = 2x(t)$$, where x(t) and y(t) are the input and output respectivel...

Let an input x(t) = 2 sin(10$$\pi$$t) + 5 cos(15$$\pi$$t) + 7 sin(42$$\pi$$t) + 4 cos(45$$\pi$$t) is passed through an LTI system having an impulse response, $$h(t) = 2\left( {{{\s...

The discrete time Fourier series representation of a signal x[n] with period N is written as $$x[n] = \sum\nolimits_{k = 0}^{N - 1} {{a_k}{e^{j(2kn\pi /N)}}} $$. A discrete time pe...

The causal signal with $z$-transform $z^2(z-a)^{-2}$ is ( $u[n]$ is the unit step signal)

Let $f(t)$ be an even function, i.e., $f(-t)=f(t)$ for all $t$. Let the Fourier transform of $f(t)$ be defined as $F(\omega)=\int_{-\infty}^{\infty} f(t) e^{-j \omega t} d t$. Supp...

Two discrete-time linear time-invariant systems with impulse responses $h_1\lfloor n\rfloor=\delta\lfloor n-1\rfloor+\delta\lfloor n+1\rfloor$ and $h_2[n]=\delta[n]+\delta[n-1]$ ar...

If the input $x(t)$ and output $y(t)$ of a system are related as $y(t)=\max [0, x(t)]$, then the system is

Consider a continuous time signal $x(t)$ defined by $x(t)=0$ for $|t|>1$ and $x(t)=1-|t|$ for $|t| \leq 1$ Let the Fourier transform of $x(t)$ be defined as $X(\omega)=\int_{-\inft...

Consider the system with following input-output relation $$y\left[n\right]=\left(1+\left(-1\right)^n\right)x\left[n\right]$$ where, x[n] is the input and y[n] is the output. The sy...

Consider $$$g\left(t\right)=\left\{\begin{array}{l}t-\left\lfloor t\right\rfloor,\\t-\left\lceil t\right\rceil,\end{array}\right.\left.\begin{array}{r}t\geq0\\otherwise\end{array}\...

Let $$z\left(t\right)=x\left(t\right)\ast y\left(t\right)$$, where "*" denotes convolution. Let C be a positive real-valued constant. Choose the correct expression for z(ct).

The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where x(t) is the input signal. A signal z(t) is called eigen-signal of the system T, when T{z(t...

The value of $$\int_{-\infty}^{+\infty}e^{-t}\partial\left(2t-2\right)dt$$. where $$\partial\left(t\right)$$ is the Dirac delta function, is

Consider a continuous-time system with input x(t) and output y(t) given by $$y\left(t\right)=x\left(t\right)\cos\left(t\right)$$. This system is

For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?

A moving average function is given by $$y\left(t\right)=\frac1T\int_{t-T}^tu\left(\tau\right)d\tau$$. If the input u is a sinusoidal signal of frequency $$\frac1{2T}Hz$$, then in s...

Consider a signal defined by $$$x\left(t\right)=\left\{\begin{array}{l}e^{j10t}\;\;\;for\;\left|t\right|\leq1\\0\;\;\;\;\;\;\;for\;\;\left|t\right|>1\end{array}\right.$$$ Its Fouri...

The z-Transform of a sequence x[n] is given as X(z) = 2z+4−4/z+3/z 2 . If y[n] is the first difference of x[n], then Y(Z) is given by

The Laplace transform of f(t)=$$2\sqrt{t/\mathrm\pi}$$ is $$s^{-3/2}$$. The Laplace transform of g(t)=$$\sqrt{1/\mathrm{πt}}$$ is

The signum function is given by $$$\mathrm{sgn}\left(\mathrm x\right)=\left\{\begin{array}{l}\frac{\mathrm x}{\left|\mathrm x\right|};\;\mathrm x\neq0\\0\;;\;\;\mathrm x=0\end{arra...

Consider a discrete time signal given by x[n]=(-0.25) n u[n]+(0.5) n u[-n-1] The region of convergence of its Z-transform would be

For a periodic square wave, which one of the following statements is TRUE?

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\ri...

x(t) is nonzero only for $$T_x\;<\;t\;<\;T_x^1$$ , and similarly, y(t) is nonzero only for $$T_y\;<\;t\;<\;T_y^1$$ . Let z(t) be convolution of x(t) and y(t). Which one of the foll...

Let $$X\left(z\right)=\frac1{1-z^{-3}}$$ be the Z–transform of a causal signal x[n]. Then, the values of x[2] and x[3] are

A discrete system is represented by the difference equation $$$\begin{bmatrix}X_1\left(k+1\right)\\X_2\left(k+2\right)\end{bmatrix}=\begin{bmatrix}a&a-1\\a+1&a\end{bmatrix}\begin{b...

A signal is represented by $$$x\left(t\right)=\left\{\begin{array}{l}1\;\;\;\left|t\right|\;<\;1\\0\;\;\;\left|t\right|\;>\;1\end{array}\right.$$$ The Fourier transform of the conv...

A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are X($$\omega$$) and Y($$\omega$$). Which of the following statemen...

A 10 kHz even-symmetric square wave is passed through a bandpass filter with centre frequency at 30 kHz and 3 dB passband of 6 kHz. The filter output is

Let f(t) be a continuous time signal and let F($$\omega$$) be its Fourier Transform defined by $$F\left(\omega\right)=\int_{-\infty}^\infty f\left(t\right)e^{-j\omega t}dt$$. Defin...

For the signal f(t) = 3 sin8$$\mathrm\pi$$t + 6sin12$$\mathrm\pi$$t + sin14$$\mathrm\pi$$t, the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is _________.

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. T...

An input signal x(t) = 2 + 5sin(100$$\mathrm\pi$$t) is sampled with a sampling frequency of 400 Hz and applied to the system whose transfer function is represented by $$$\frac{Y\le...

Consider an LTI system with transfer function $$H\left(s\right)=\frac1{s\left(s+4\right)}$$.If the input to the system is cos(3t) and the steady state output is $$A\sin\left(3t+\al...

A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency in kHz which is not valid is

Two systems with impulse responses h 1 (t) and h 2 (t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

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

The impulse response of a continuous time system is given by h(t) = $$\delta$$(t − 1) + $$\delta$$(t − 3). The value of the step response at t = 2 is

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

The impulse response of a system is h(t) = tu(t). For an input u(t − 1), the output is

L et y[n] denote the convolution of h[n] and g[n], where $$h\left[n\right]=\left(1/2\right)^nu\left[n\right]$$ and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1]...

The Fourier transform of a signal h(t) is $$H\left(j\omega\right)=\left(2\cos\omega\right)\left(\sin2\omega\right)/\omega$$. The value of h(0) is

The input x(t) and output y(t) of a system are related as $$\int_{-\infty}^tx\left(\tau\right)\cos\left(3\tau\right)d\tau$$.The system is

The unilateral Laplace transform of f(t) is $$\frac1{s^2\;+\;s\;+\;1}$$. The unilateral Laplace transform of tf(t) is

A low–pass filter with a cut-off frequency of 30 Hz is cascaded with a high-pass filter with a cut-off frequency of 20 Hz. The resultant system of filters will function as

Given two continuous time signals $$x\left(t\right)=e^{-t}$$ and $$y\left(t\right)=e^{-2t}$$ which exist for t > 0, the convolution z(t) = x(t)*y(t) is

The response h(t) of a linear time invariant system to an impulse $$\delta\left(t\right)$$, under initially relaxed condition is $$h\left(t\right)=e^{-t}\;+\;e^{-2t}$$. The respons...

Let the Laplace transform of a function f(t) which exists for t > 0 be F 1 (s) and the Laplace transform of its delayed version f(1 - $$\tau$$) be F 2 (s). Let F 1 * (s) be the com...

The system represented by the input-output relationship $$y\left(t\right)=\int_{-\infty}^{5t}x\left(\tau\right)d\tau$$, t > 0 is

The period of the signal $$x\left(t\right)=8\sin\left(0.8\mathrm{πt}+\frac{\mathrm\pi}4\right)$$ is

For the system $$\frac2{\left(s+1\right)}$$, the approximate time taken for a step response to reach 98% of its final value is

A cascade of 3 Linear Time Invariant systems is casual and unstable. From this, we conclude that

The Fourier Series coefficients, of a periodic signal $$x\left( t \right),$$ expressed as $$x\left( t \right) = \sum {_{k = - \infty }^\infty {a_k}{e^{j2\pi kt/T}}} $$ are given by...

A linear Time Invariant system with an impulse response $$h(t)$$ produces output $$y(t)$$ when input $$x(t)$$ is applied. When the input $$x\left( {t - \tau } \right)$$ is applied...

The $$z$$$$-$$ transform of a signal $$x\left[ n \right]$$ is given by $$4{z^{ - 3}} + 3{z^{ - 1}} + 2 - 6{z^2} + 2{z^3}.$$ It is applied to a system, with a transfer function $$H\...

A system with input $$x(t)$$ and output $$y(t)$$ is defined by the input $$-$$ output relation: $$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t} {x\left( \tau \right)} d\tau...

A signal $${e^{ - \alpha t}}\,\sin \left( {\omega t} \right)$$ is the input to a real Linear Time Invariant system. Given $$K$$ and $$\phi $$ are constants, the output of the syste...

Given X(z)=$$\frac z{\left(z-a\right)^2}$$ with $$\left|z\right|$$ > a, the residue of X(z)z n-1 at z = a for n $$\geq$$ 0 will be

A signal $$x\left( t \right) = \sin c\left( {\alpha t} \right)$$ where $$\alpha $$ is a real constant $$\left( {\sin \,c\left( x \right) = {{\sin \left( {\pi x} \right)} \over {\pi...

A signal $${e^{ - \alpha t}}\,\sin \left( {\omega t} \right)$$ is the input to a real Linear Time Invariant system. Given $$K$$ and $$\phi $$ are constants, the output of the syste...