Signals and Systems — Winter 2020

Define following terms, (1) Power Signal (2) Deterministic Signal (3) Causal system

Comment on stability of system y(n) = ∑n x(k) k=0

(1) Check the following systems for time invariance and linearity.

y(n) = n[x(n)]2 (ii) y(n) = x(n)cos(nπ/4) (2) Determine period of following signal, x(n)=e−jπn/4

Determine whether the discrete signal x(n) = 6cos 2𝜋n is power signal or energy signal? Prove it.

Find even and odd parts of following signals, (1) x(t) = ei2t (2) x(t) = cos⁡(w t+ 𝜋 )0

(1) Find convolution of two sequences x(n) = [1,⏟2,4] and ↑ h(n) = [1,⏟1,1,1] using graphical method. ↑ (2) Compute convolution for the following signals03 n n x(n) = (1/5) u(n), h(n) = (1/2) u(n) .

Decide whether system with following impulse response, is causal and stable? h(n)=(−1)nu(−n)

Determine and prove the system y(t)=2x(t)+3 is linear or nonlinear?

What is stable system? Derive necessary and sufficient condition for stable LTI system.

Find DTFT of x(n) = 𝛿(n−2)−⁡𝛿(n+2).

Prove that a DT LTI system is causal if and only if h(n)=0 for n<0.

Verify that the impulse response h[n] for this system is h[n] = an u[n], |a|<1. Is the system

memoryless? (ii) causal? (iii) stable?1

Find Fourier transform of cos(w t)u(t).

State and prove differentiation in time property of Fourier Transform.

State and prove time convolution theorem for Fourier transform.

jw Find inverse Fourier transform of X(w) = (2+jw)2

State and prove frequency differentiation property of Fourier Transform.

Discuss the relationship between Fourier transform and Laplace transform.

Prove linearity and time shifting properties of Fourier series.

Determine the Z-transform and ROC of x(n) = anu(n)−bnu(−n−1).

Discuss the properties of ROC for Z-Transform.

Find DTFT of (i) 𝛿(n−m) (ii) anu(n)

z−1 Find the inverse Z-transform of X(z) = ; ROC;|z|>1 3−4z−1+z−2

Find the Fourier series expansion of the half wave rectified sine wave shown in below figure,