Signals and Systems

Define signal. Differentiate between discrete time signal and continuous time signal.

Discuss the terms: (i) Linearity (ii) Causality

Discuss the relationship between z-transform and DTFT.

How can aliasing be eliminated? Explain.

What do you understand by reconstruction?

What are the advantages of DFT in comparison of DTFT?

Define circular convolution.

Differentiate between impulse response and step response.

Discuss state transition matrix and its role.

Define magnitude and phase response.

For the following input-output relationship, determine whether the corresponding system is linear, time invariant or both. (a) $y(t)=t^{2}x(t-1)$ (b) $y[n]=x^2[n-2]$

State and prove the initial value and final value theorem for Laplace transform.

Find the z-transform of the expression $x(n)=a^{n}u(n)+b^{n}u(-n-1)$, also draw the ROC.

Explain sampling of continuous time signal by impulse function with diagram and appropriate mathematical expressions.

Find the Nyquest rate and discrete signal x(n) for the following continuous time signals: (a) $x(t)=\sin (4\times 10^3\pi t)/\pi t$ (b) $x(t)=1+\cos (2000\pi t)+\sin (400\pi t)$ (c) $x(t)=\text{sin c}(50\pi t)+\text{sin c}(100\pi t)$

Find the 4-point DFT of sequence $x(n)=\cos \frac{n\pi }{4}$

Determine the convolution of the given signals: $x(t)=u(t-3)-u(t-5)$ and $h(t)=e^{-3t}u(t)$ Where u(t) is unit step function.

Describe all the properties of DFT in detail. Also, define the feedback control systems.

(a) Find the Laplace transform of following signal $x(t)=e^{-2t}u(-t)+e^{-3t}u(-t)$. Mark the poles and zero of x(s) and also the ROC in the s-plane. (b) Explain filtering.

Find the inverse z-transform of $x(z)=\frac{z^3-10z^2-4z+4}{2z^2+2z-4}$ with ROC $|z|<1$

A signal x[n] with Fourier transform $x(e^{j\omega })$ has the property that: $$\left(x[n]\sum _{k=-\infty }^{\infty }\delta [n-3k]\right)\ast \left(\frac{\sin \frac{\pi }{3}n}{\frac{\pi }{3}n}\right)=x[n]$$ For what values of w is it guaranteed that $x(e^{j\omega })=0$.

For a continuous time periodic signal: $$x(t)=2+\cos \left[\frac{2\pi }{3}\right]t+4\sin \left[\frac{5\pi t}{3}\right]$$ Determine the fundamental frequency $w_0$ and the Fourier series coefficient $a_k$ such that $$x[t]=\sum _{k=-\infty }^{\infty }{a}_k\cdot e^{jk{w}_{0}t}$$