Suppose that we have a program that finds the set of coefficients a[n], n = 0, 1, . . ....

Suppose that we have a program that finds the set of coefficients a[n], n = 0, 1, . . . , L, that minimizes given L, F, W(ω), and H_d (e^jω).We have shown that the solution to this optimization problem implies a noncausal FIR zero-phase system with impulse response satisfying he[n] = he[−n]. By delaying he[n] by L samples, we obtain a causal type I FIR linear-phase system with frequency response where the impulse response is related to the coefficients a[n] by and M = 2L is the order of the system function polynomial. (The length of the impulse response is M + 1.) The other three types (II, III, and IV) of linear-phase FIR filters can be designed by the available program if we make suitable modifications to the weighting function W(ω) and the desired frequency response H_d (e^jω). To see how to do this, it is necessary to manipulate the expressions for the frequency response into the standard form assumed by the program. Assume that we wish to design a causal type II FIR linear-phase system such that h[n] = h[M − n] for n = 0, 1, . . . , M, where M is an odd integer. Show that the frequency response of this type of system can be expressed as and determine the relationship between the coefficients b[n] and h[n]. Show that the summation can be written as by obtaining an expression for b[n] for n = 1, 2, . . . , (M + 1)/2 in terms of ˜b [n] for n = 0, 1, . . . , (M − 1)/2. Hint: Note carefully that b[n] is to be expressed in terms of ˜b [n]. Also, use the trigonometric identity cos α cos β = 1 2 cos(α + β) + 1 2 cos(α − β). (c) If we wish to use the given program to design type II systems (M odd) for a given F, W(ω), and H_d (e_jω), show how to obtain W(ω), and d (e_jω) in terms of M, F, W(ω), and H_d (e_jω) such that if we run the program using W(ω), and d (e_jω), we may use the resulting set of coefficients to determine the impulse response of the desired type II system. (d) Parts (a)–(c) can be repeated for types III and IV causal linear-phase FIR systems where h[n] = −h[M −n]. For these cases, you must show that, for type III systems (M even), the frequency response can be expressed as and for type IV systems (M odd),

As in part (b), it is necessary to express c[n] in terms of ˜ c[n] and d[n] in terms of ˜d [n] using the trigonometric identity sin α cos β = 1 2 sin(α + β) + 1 2 sin(α − β). McClellan and Parks (1973) and Rabiner and Gold (1975) give more details on issues raised in this problem.