What window function would you choose to truncate the ideal

Question

Question 1. A system that is known to be linear shift invariant and causal is described by the system function given below:

H(z) = Y(z)/X(z) =  1-z^-1/1-0.25z^-1 -0.125z^-2

a) By inspection and from the given information, state whether this more likely represents an FIR or IIR. system and why.

b) Write the difference equation corresponding to the system ( y(n) = ).

Question 2. A discrete time system is described by the difference equation below:

y(n) = 2y(n -1) - 3y(n-2) + 3x(n)+2x(n -1) + 0.25x(n - 2)

a) If you assume all calculations are real, how many multiplies and how many additions (two answers) are needed for the computation of each output value? Justify your answer.

b) If the sampling rate for the system were given as F_s =8000H_z how many multiplies and adds would be required per second to implement this system? Justify your answer.

Question 3. Consider the design of an FIR filter using the window design method discussed in class. An FIR filter meeting the following specifications is desired:

i) Lowpass with passband from dc to 8,000Hz with no more than 2dB attenuation,

ii) Stopband from 12,000Hz upward with at least 50dB of attenuation,

iii) Sampling frequency of 48,000Hz, and

iv) Linear phase in the passband.

a) What window function would you choose to truncate the ideal impulse response and why?

b) Compute an estimate of the required filter order (length). Justify your answer.

c) What characteristic of the impulse response is required in order for the filter to have linear phase in the passband? Be specific.

Question 4. Consider the design of an analog filter using the techniques of Chapter 3 and having the specifications below (you do not need to actually design the filter):

i) Lowpass with passband from dc to 4.000Hz with no more than 2dB attenuation,

ii) Stopband from 10,000Hz upward with at least 40dB of attenuation, and

iii) Sampling frequency of 40,000Hz.

a) What type of filter prototype (Butterworth, Chebyshev or Elliptic) should be chosen on which to base the design if a smooth response is desired in the passband and stopband? Why?

b) In general, what are the advantages and disadvantages of choosing an elliptic filter prototype versus a butterworth filter prototype?

Question 5. Consider computing the 1000-point DFT of a 1,000 point long sine wave having a frequency of 200 Hz sampled at a rate of 10,000 samples per second.

X(k) = DFT[x(n)]

a) What is the digital frequency resolution (frequency spacing between adjacent values of k, in radians/sample)? What is the equivalent analog frequency resolution (in Hertz)?

b) At what value of k would we expect to see evidence of the 200 Hz sinusoid in the DFT?