Signals and Systems

Define the Continuous and Discrete time signals.

Represent the signal x(n) in frequency domain using Fourier series - $$x(n)=2+\cos (2\pi n/3)+4\sin (5\pi n/3)$$

Find the Z-transform of $x(n)=a^{n}u(n)$.

What do you mean by periodic and non-periodic signals?

Find the Z-transform of the given sequences - $\{5,3,-2,0,4,-3\}$.

Determine the convolution of the given signal - $X[n]=[1,2,-1,3]$ and $h[n]=[3,-4,2]$

Elaborate the Dirichlet's condition for the existence of Fourier series.

What is aliasing phenomena? How can aliasing phenomena be eliminated?

Find $e^{-at}$ is an Energy signal or Power signal.

Determine the Discrete Time Fourier Transform of periodic impulse Train. $$x(n)=\sum _{k=-\infty }^{\infty }\delta (n-kN)$$

Determine whether or not the following signal is periodic. If it is periodic determine the fundamental period - $$x(t)=\cos t+\sin \sqrt{2}tx[n]=e^{\left(\frac{j\pi }{4}\right)n}$$

Sketch the following signal and check whether it is an energy signal or power signal or neither. $x(t)=e^{-a|t|}$ for $a>0$.

Analyze all the Fourier Series Coefficient of $x(t)=\sin \omega t$.

Evaluate the periodicity of the following signals - (a) $x(n)=u(n)+u(-n)$ (b) $x(t)=2e^{j\left(t+\pi /4\right)}u(t)$

Determine the Nyquist rate for a continuous time signal. $$x(t)=8\cos 50\pi t+10\sin 300\pi t-10\cos 100\pi t$$

Describe Zero Order Hold Sampling with reconstruction filter.

Consider a discrete-time LTI system with impulse response h[n] given by $h[n]=a^{n}u[n]$ (a) Is this system causal? (b) Is this system BIBO stable?

Consider the system $y(t)=x^2(t-t_0)+2$, evaluate if system is - (i) Linear (ii) Stable (iii) Causal (iv) Time Invariant (v) Deterministic

Design and implement an LTI system with impulse response $h(t)=e^{-at}u(t)$ $a>0$ to the input signal $x(t)=e^{-bt}u(t)$ $b>0$. Find the response using convolution property of Continuous Time Fourier Transform for the cases. (a) $b\ne a$ (b) $b=a$

Using the residue method, find the inverse Z-transform of - $$X(Z)=\frac{1}{(Z-0.25)(Z-0.5)},\text{ROC: }|Z|>0.5$$

Consider the Periodic square wave, determine the Trigonometric Fourier Series of x(t). [Graph description: A periodic square wave $x(t)$ with period $T_0$ and peak amplitude $A$. The pulses of width $T_0/2$ are centered at $t=n{T}_0$ (for integers $n$). The signal is $A$ for $-T_0/4<t<T_0/4$, and $0$ for $T_0/4<t<3T_0/4$ within the first period.]

Explain Ideal Sampling and Practical sampling in details.