Question

1. In this HW you are asked to perform one round of manual Parks-McClellan optimization. Suppose that we are going to design a type I optimal FIR low- pass filter of length 7 (therefore L=3). The specs are: Wp = 1/4 and ws = 7/3.

a. [10 pts] Assume that we have made an initial guess that there will be L+2 = 5 extremal frequencies at w = 0, Wp, W., 7/2, and at w = 11. Calculate xi = cos w; at these frequencies.

b. [10 pts] Use Eq. (7.115) to find coefficients by, ...,b5.

c. [10 pts] Use Eq. (7.114) to find 8. (Assume that W(w) = 1 for both the passband and the stopband and Ww) = 0 for the transition band.)

d. [10 pts] Use Eq. (7.116b) to find the values of C1, ...,C4.

e. [20 pts] I personally found Eq. (7.116a) to be problematic. Instead, please just find a polynomial p(x) = a, +ajx + azx2 + azx3 that passes through points (x,y) = (xi,C), i = 1,2,3,4.

f. [10 pts] Verify that the polynomial also passes through (X5, C5).

g. [10 pts) Find out where the actual local extrema of p(x) are located, and suggest how the list of (W2, ...,w) should be updated in the next round.

h. [20 pts] Continuing from (e), though the filter is not optimal yet, find the sample values h[0], h[1], h[2], and h[3] such that the frequency response H(ejw) = p(cos w).

Bonus: (10 points) Read Section 7.9 and answer the following question - what is the main advantage of FIR design? What is the main advantage of IIR design?