SIGNALS AND SYSTEMS - B.Tech 4th Semester Exam., 2022 (New Course)

Determine the fundamental period of the signal \( x[n]=1+e^{j\frac{4\pi n}{7}}-e^{j\frac{2\pi n}{5}} \)

Check whether the signal \( x(t)=2e^{j(t+\frac{\pi}{4})}u(t) \) is periodic or not. If periodic, then compute the periodicity.

Find the convolution of two signals \( x_{1}(t)=e^{-t^{2}} \) and \( x_{2}(t)=3t^{2} \)

Let \( X(e^{j\omega}) \) be the DTFT of \( x[n] \) prove that \( X(e^{j0})=\sum_{n=-\infty}^{\infty}x[n] \)

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

Use the convolution property of Laplace transform to determine \( y(t)=e^{at}u(t)*e^{bt}u(t) \) where symbols have their usual meanings.

Determine the Laplace transform of the signal \( x(t)=\cos^{3}(3t)u(t) \).

If \( X(z)\leftarrow\underline{z}\rightarrow x[n] \) with ROC: R, then prove that \( Z\{x[-n]\}=X(z^{-1}) \) with ROC: \( \frac{1}{R} \)

List down the properties of region of convergence (ROC).

Determine the conditions on the sampling interval \( T_{s} \) so that \( x(t)=\cos(\pi t)+3\sin(2\pi t)+\sin(4\pi t) \) is uniquely represented by the discrete-time sequence \( x[n]=x(nT_{s}) \)

Consider a causal LTI system that is represented by the difference equation \( y[n]-\frac{3}{4}y[n-1]+\frac{1}{8}y[n-2]=2x[n] \). Find the frequency response \( H(e^{j\omega}) \) and the impulse response \( h[n] \) of the system.

Find the inverse DTFT of \( X(e^{j\omega})=\delta(\omega), -\pi < \omega < \pi \)

Find the Fourier transform of \( x(n)=a^{|n|}, |a|< 1 \)

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 \( X(z)=\frac{1-\frac{1}{4}z^{-1}}{1-\frac{1}{9}z^{-1}} \) ROC: \( |z|>\frac{1}{3} \)

Comment on the causality of the system whose transfer function is given by \( H(z)=\frac{3-4z^{-1}}{1-3.5z^{-1}+1.5z^{-2}}, |z|>3 \)

Derive the condition for BIBO stability.

Consider a continuous-time system with input \( x(t) \) and output \( y(t) \) is related by \( y(t)=x(\sin(t)) \). Is the system (i) causal and (ii) linear?

Sketch the signal \( x(t)=\delta(\cos t) \).

Find even and odd components of signal \( x(t)=e^{-2t}\cos(t) \).

Compute the Fourier transform of the rectangular pulse train signal shown in Fig. 1.

Compute the step response of the LTI system \( H(s)=\frac{6(s+1)}{s(s+3)}; Re\{s\}>0 \)

State and prove Parseval's theorem.

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 memory less or memory type.

Compute the convolution \( y[n]=x[n]*h[n] \) where \( x[n]=u(n-1) \) and \( h[n]=\alpha^{n}u(n-1) \)

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

Consider the causal difference equation \( y[n]-0.8y[n-1]=2x[n] \) where the input signal is \( x[n]=(\frac{1}{2})^{n}u(n) \) with \( y[-1]=0 \). Find the output response \( y[n] \).

Compute the inverse Z-transform of \( X(z)=\frac{z}{(1-0.5z^{-1})}; |z|<0.5 \) using the power series expansion method. Find the signal \( x[n] \).

Consider the stable LTI system defined by its transfer function \( H(z)=\frac{z^{2}+z-2}{z^{2}+z+0.5} \). Sketch the pole-zero plot for this transfer function and give its ROC. Is the system causal? Sketch the direct form realization of this system.

Let \( x[n] \) be an arbitrary function with even and odd parts as \( x_{e}[n] \), \( x_{o}[n] \), 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.

The signal \( x(t) \) is shown in Fig. 2. Sketch the signals for \( \alpha=\frac{1}{2} \) and \( \alpha=2 \).