SIGNALS AND SYSTEMS - B.Tech 3rd Semester Special Exam., 2020

If the Z-transform of \( x(n) \) is \( X(z) \) then show that \( Z[x_{1}(n)*x_{2}(n)]=X_{1}(z)X_{2}(z) \)

If the impulse response for a system is given by \( h(n)=a^{n}u(n) \), then what is the condition for the system to be BIBO stable?

A voltage having the laplace transform \( \frac{4s^{2}+3s+2}{7s^{2}+6s+5} \) is applied across a 2H inductor. What is the current in inductor at \( t \to \infty \) assuming zero initial condition?

Differentiate between Kronecker delta function and Direc delta function.

The step response of an LTI system when the impulse response \( h(n) \) is unit step \( u(n) \) is _________.

Find the Laplace transform \( f(t)=e^{3t}\cos(2t)u(t) \) where symbols have their usual meanings.

An LTI system is described as \( 0.5\frac{d^{2}y(t)}{dt^{2}}+2.5\frac{dy(t)}{dt}+2y(t)=\delta(t) \). Find the final value of the output response where \( y(t) \) is output and \( x(t) \) is input.

The period of a sequence \( x(n)=\cos(\frac{2\pi n}{3}) \) is _________.

The final value of step response of a causal LTI system with \( H(s)=\frac{s+1}{s+4} \) is

1. 0.5
2. 0.25
3. 1
4. 10

Consider two functions \( f(t)=h(t)h(3-t) \) and \( g(t)=h(t)-h(t-3) \). Are these two functions identical? Show that \( L[f(t)]=L[g(t)] \) where L is the Laplace operator.

Let a system is described by the differential equation as \( \ddot{y}+3\dot{y}+2y=e^{-t} \); with initial condition \( y(0)=\dot{y}(0)=0 \). Compute the solution of the equation.

Let \( f(t) \) is a periodic function with periodicity T for \( t\ge0 \), then show that \( L[f(t)]=\frac{L[f_{T}(t)]}{1-e^{-sT}} \), \( s>0 \)

Find the Laplace transform of Fig. 1.

State why ROC does not include any pole. Find the Z-transform of \( x(n)=\begin{cases}(0.5)^{n}u(n),&n>0\\ (0.25)^{-n},&n < 0\end{cases} \)

Find the inverse Z-transform of \( X(z)=\frac{1-\frac{1}{4}z^{-1}}{1-\frac{1}{9}z^{-1}} \) where ROC: \( |z|>\frac{1}{3} \)

Show that \( Z[nx(n)]=-z\frac{dX(z)}{dz} \) where \( X(z)\leftrightarrow x[n] \).

Briefly explain the causality of a system.

Find whether the signal \( x[n]=\sin(\frac{3\pi}{4}n)+\sin(\frac{\pi}{3}n) \) is periodic or aperiodic. If periodic, then what is the periodicity of x[n]?

Write down the Dirichlet condition.

Find the Fourier transform of \( x(t)=e^{-|t|}u(t) \), and hence draw the magnitude and phase spectrums.

Compute the Fourier transform of signal shown in Fig. 2: \( R_{x}(\tau)=\begin{cases}\frac{N}{2},&-B\le\tau\le B\\ 0,&elsewhere\end{cases} \)

Find the convolution of the following discrete sequences: \( x(n)=\frac{1}{3}u(n) \) and \( h(n)=\frac{1}{5}u(n) \)

State why the realization of an ideal low-pass filter is not possible, with proper justification.

A system is defined as \( y(n)=x(n^{2}) \). Check whether the system is linear or non-linear, time-varying or time-invariant, causal or non-causal, and memoryless or memory type.

State Parseval's theorem.

Sketch the signal \( x(t)=-2u(t-1) \).

Compute the Nyquist sampling rate for the signal \( g(t)=10\cos(50\pi)\cos^{2}(150\pi t) \)

Show that \( u(n)=\sum_{k=-\infty}^{n}\delta(n) \) where symbols have their usual meanings.

What is unit doublet? Prove that \( \int_{-\infty}^{\infty}x(t)\delta^{k}(t)dt=(-1)^{k}\frac{d^{k}x(t)}{dt^{k}}|_{t=0} \) where \( x^{k} \) is k-th derivative of function \( x(t) \) and \( \delta(t) \) is Dirac delta function.

A system is described by its input-output relationship as \( y[n]=\sum_{k=-\infty}^{n}x[n-k] \). Is the system memoryless, stable, causal, time-invariant and linear?

Find the fundamental period of signal \( x[n]=e^{j7.351\pi n} \)

Let \( x[n] \) be an arbitrary function with even and odd part as \( x_{e}[n] \) and \( x_{o}[n] \) respectively. Show that \( \sum_{n=-\infty}^{\infty}x^{2}[n]=\sum_{n=-\infty}^{\infty}x_{e}^{2}[n]+\sum_{n=-\infty}^{\infty}x_{o}^{2}[n] \)

Perform the convolution operation between \( x[n]=\{0,0,0,0,(2),-3,1,0,0\} \) and \( h[n]=\{0,0,0,1,(2),2,0,0,0\} \) using graphical method.

Calculate the Fourier transform of \( x[n]=u[n] \).