Reference no: EM132307402

Question 1:

a. Generate the signal x(n)= e^iωn for ω= pi/6 and the values of n =0 to 50. For the signal calculate the even component and odd component and plot them in Matlab.

b. Analyse the signal energy and the energy of even and odd components for ω= pi/6 with the values of n =0 to 50 and for ω= pi/3 with the values of n =0 to 100. Can you find any relation between them?

Question 2: Design a causal discrete-time LTI system with the property that if the input is

x(n)=(1/2)ⁿ u(n)-1/4.(1/2)^(n-1) u(n-1)

then the output is

y(n)=(1/3)ⁿ u(n)

a. Determine the impulse response h(n)and the system function H(z)of a system that satisfies the foregoing condition.

b. Find the difference equation that characterizes this system.

c. Determine the realization of the system that requires the minimum possible amount of memory.

d. Determine if the system is stable.

Question 3: A discrete system is characterized by the transfer function

H(z)= (1-4z^-1)/((1-3z^-1+ 2z^-2))

Find all the possible impulse response for the system function andanalyse each impulse response and comment on the stability of the system.

Question 4: a. FFT is used to represent a signal in frequency plane. If frequency spectrum of a signal is taken by FFT and multiply the spectrum by a desired frequency response of a FIR filter, whether we can implement filters efficiently? Justify your answer in terms of merits and demerits of this system.

b. Calculate FFT of a signal x(n)= [7, 3+j, 2, 5, 9, 6,0,0] using the signal flow graph given in figure 2. Show the intermediate values at every nodes of the flow graph.

Figure 2