Signal Basics

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 $\frac{1}{T} \int_0^T|x(t)|^2 d t=$________(round off to the nearest integer).

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 function. The value of $$ \int_{-\infty}^{\in...

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)=\left\{\begin{array}{cc}t, & t \in[0,1]...

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\{0,1,2\} \\ 0, & \text { otherwise } \en...

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 signal $u(t)$. Which one of the following s...

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

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

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).

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 + \frac{1}{2^n} x[ n ] $} Which one of the...

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

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 $h_2[n]=\delta[n]+\delta[n-1]$ are connected in cascade. Where $\delta[n]...

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 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 system is

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)}= yz(t), where $$\gamma$$ is a complex...

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

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

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

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 following statements is TRUE?

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

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

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

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] equals

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

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 response of this system for a unit step input u...

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

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

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

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 to a system with response $$h\left( {t -...

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

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 system will be of the form $$K{e^{ - \beta t}...

The impulse response of a causal linear time-invariant system is given as $$h(t)$$. Now consider the following two statements: Statement-$$\left( {\rm I} \right)$$: Principle of superposition holds Statement-$$\left( {\r...

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 system will be of the form $$K{e^{ - \beta t}...

If u(t), r(t) denote the unit step and unit ramp functions respectively and u(t)*r(t) their convolution, then the function u(t+1)*r(t-2) is given by

Let a signal $${a_1}\,\sin \left( {{\omega _1}t + {\phi _1}} \right)$$ be applied to a stable linear time-invariant system. Let the corresponding steady state output be represented as $${a_2}F\left( {{\omega _2}t + {\phi...

A continuous-time system is described by $$y\left( t \right) = {e^{ - |x\left( t \right)|}},$$ where $$y(t)$$ is the output and $$x(t)$$ is the input. $$y(t)$$ is bounded

$$x\left[ n \right] = 0;\,n < - 1,\,n > 0,\,x\left[ { - 1} \right] = - 1,\,x\left[ 0 \right]$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2$$ is the input and $$y\left[ n \right] = 0;\,n < - 1,\...

The RMS value of the voltage v(t) = 3 + 4cos(3t) is

The rms value of the resultant current in a wire which carries a dc current of 10 A and a sinusoidal alternating current of peak value 20 A is

$$s(t)$$ is step response and $$h(t)$$ is impulse response of a system. Its response $$y(t)$$ for any input $$u(t)$$ is given by

The convolution of the function $$f_1\left(t\right)=e^{-2t}\;u\left(t\right)$$ and $$f_2\left(t\right)=e^t\;u\left(t\right)$$ is equal to __________.

If $$f(t)$$ is the step-response of a linear time-invariant system, then its impulse response is given by ___________

The value of the integral $$\int_{-5}^{+6}e^{-2t}\delta\left(t-1\right)dt$$ is equal to ________.

$$s(t)$$ is step response and $$h(t)$$ is impulse response of a system. Its response $$y(t)$$ for any input $$u(t)$$ is given by

The impulse response of a network is $$h\left( t \right) = 1$$ for $$0 \le t < 1$$ and zero otherwise. Sketch the impulse response of two such networks in cascade, neglecting loading effects.