Signals and Systems

a) Determine the convolution integral when f₁(t)=e^2tand f₂(t)=2t.

f) Write down the properties of the ROC for the z-transform. 

e) Prove it: i) 

d) Determine the signal energy and signal power for 

c) Check the dynamicity for the following system 

b) Whether the following system are "Time Variant" or "Time  Invariant" 

a) Determine whether the given signals are periodic and determine the fundamental period 

c) using long division, determine the inverse 2-transform of

     X[z]=1-(3/2)z^-1+(1/2)z^-2

b) State and prove the Parseval's theorem for z-transform.

a) What are the effects of under sampling and oversampling?

b) Find the z-transform of x[n]=n² u[n]. 

a) An analog signal is given below m(t)-4 cos 100  t calculate. 

i) Minimum sampling rate to avoid aliasing.

ii) If the signal is sampled at the Fs=200Hz, what is the discrete time signal after sampling.

iii) If the signal is sampled at the rate Fs=75 Hz. What is the discrete time after aliasing?

b) Compare Fourier Transform and Laplace Transform?

j) Find out the Laplace transfer of the following function: f(t)= cos³3t. 

c) Prove the Initial and Final value theorem of Laplace Transform? 

D) Draw the amplitude and phase spectrum of full wave rectified time domain signal m(t) with the fundamental period 

a) Prove that the convolution of two signals in the time domain will give the multiplication in frequency domain.

b) Obtain the trigonometric Fourier series for the half wave rectified sine wave with the time period of T. 

a) Discuss the properties of discrete time Fourier transform. 

c)  A discrete time system is defining as y[n]+7y(n-1)-5yſn-2]+y[n- 4]=u[n). Draw only the state space model for this system. 

b)Find out the circular convolution of the sequence graphically.

x(n)=[1 2 3 4] and h(n)-13 1 1 3]

a)A discrete time signal x[n) is defined as x[n]=1+n/4 for 

=1 for O  and =0 elsewhere,

Find the value of x[n+2], x[2n], x[n/2], x[n-2).

c) Write down the state equation and output equation for a discrete time systems. Also find out the transfer function out of that equation.

b) Obtain the convolution sum of two D.T. sequence given below x[n]=u[n] and y[n]=0.5)ⁿ u[n].

a)Determine whether the following system is linear or not? Find out the condition to make it linear? y(t)=1/12 x(t)-5/6.