Signals and Systems — Summer 2021

Define discrete time signal. Enlist detailed classification of Discrete time signals.

Sketch each of the following signals.

x[n] = 2u[n+2] - 2u[n - 3] (ii) x(t) = r(-t). u(t +3)

Classify following systems as : (a) Causal or non-causal; (b) Linear or nonlinear and (c) Time invariant or time variant

y(n) = x(n) (ii) y(n)= 2x(n+2) - x(n-2)

For LTI systems , state and prove Time shifting property of Laplace transform.

For LTI system, if input sequence is x(n) and impulse response is defined as h(n), derive equation for discrete time convolution sum y(n).

1 Consider a causal LTI system with frequency response H(w)  . For jw3 a particular input x(t), this system observes output y(t)et.u(t)e2tu(t). Determine x (t).

LTI system is described by difference equation51 y(n) y(n1) y(n2) x(n) With y(-1)=1 and y(-2) =0.66 Determine forced response for an inputx(n)( )nu(n).4

Enlist different conditions for existence of Fourier transform.

Find convolution of following two signals. x(t)e2tu(t) with h(t)e4tu(t)

Compute the Fourier transform for the signal x(t) in following figure:01 Figure :011

Explain differentiation in time domain property of Fourier transform,

An LTI system has impulse response given by h(n)={1,2,3,4} . Find its response to input x(n)= {1,1,1}.

Compute the Fourier transform for the signal x(t) in following figure:02. Draw its magnitude spectrum. Figure :02

Prove that for causal sequences, the ROC of Ztransform is exterior of a circle.

Find the Fourier transform of sine wave signal sinw t. Draw its magnitude0 spectrum.

State and prove (a) time reversal and (b) time scaling properties of LTI systems using Fourier transform.

Explain with suitable mathematical equations, relation between Laplace Transform and Fourier Transform,

Using properties of Ztransform, compute Ztransform for following signals.

x(n)= u(-n) (ii) x(n)= u(-n+2)

t Find fourier transforms of rectangular pulse (gate pulse) defined as rect( ).  Draw its magnitude spectrum.

Find inverse Ztransform of z1 X(z)  ;RoCz 1 34z1 z2

Enlist and prove necessary and sufficient condition for stability of LTI system.

Determine steady state (forced ) response for the system with the step input and characterized by difference equation y(n)0.75y(n1)0.125y(n2)  x(n) x(n1) With initial conditions y(-1)=0 and y(-2)=-1

Find inverse Ztransform of X(z)2z3 3z2 z 32z1 3z2 2z3

Enlist useful properties of ROC of X(z).

An LTI system is described by the difference equation y(n)  x(n)0.81x(n1)0.81x(n2)0.45y(n2) Determine transfer function of the system. Draw pole zero plot and assess stability.2