Theory and Design of Linear-Phase Minimax FIR Filters with Mixed Constraints in the Frequency Domain

Abstract

The alternation theorem is the core of efficient approximation algorithms for the minimax design of finite-impulse response (FIR) filters. In this paper, an extended alternation theorem with additional mixed constraints, i.e., equality-and-inequality constraints, is obtained. Then, an efficient multiple-exchange algorithm based on the extended theorem is presented for designing linear-phase FIR filters with frequency mixed constraints in the minimax sense. Further, convergence of the algorithm is established. Several design examples and comparisons with existing techniques are presented, and the simulation results show that the proposed algorithm is numerically more efficient and guaranteed to converge to the optimal solution.

Appendices

Appendix A: Proof of Theorem 4

Proof

The theorem is trivial if δ ^∗=0. Then, assume δ ^∗>0.

(i) The proof of necessity. Assume that α ^∗ is the solution to problem (12). It is obvious that α ^∗⊂(F _e ∩F _i ). Then, α ^∗ is necessarily the solution to the following problem:

(39)

where δ ^∗ is given by (14).

Otherwise, there should exist some α ^∘∈F _e satisfying

(40)

Considering α ^∗⊂F _i , we have

and then

(41)

Combining (14) and (41), we further have

(42)

According to the definition (19) of the weighting function $\tilde{W}(\omega,\delta)$, (42) can be equivalently written as

(43)

Then it follows from (40) and (43) that

or

(44)

Equation (44) implies that α ^∘ also belongs to F _i , i.e., α ^∘∈F _i , and

It is contrary to the assumption that α ^∗ is the solution to problem (12). So, α ^∗ is also the solution to the problem (39).

By applying Theorem 2 to the problem (39), we assert that there exist r+1 frequencies A≡{ω _k ,k=0,1,…,r}⊂B such that Ω _e ⊂A and E(ω,α ^∗) satisfies

which is equivalent to the condition (13)–(14).

(ii) The proof of sufficiency. Let α ^∗∈F _e ∩F _i and assume that there exist r+1 frequencies {ω _k ,k=0,1,…,r}⊂B satisfying (13)–(14). If α ^∗ is not the unique solution to the problem (12), then there would exist some α ^∘∈F _e ∩F _i satisfying

Consequently, the following function

should have the same sign as E(ω,α ^∗) or be equal to 0 at frequencies ω _k for k=0,1,…,r. It follows that $\tilde{E}(\omega)$ has at least r zeros (counting repeated zeros) in [0,π]. However, $\tilde{E}(\omega)$ is a trigonometric polynomial of degree at most r−1, which means its zeros are no more than r−1 in [0,π], a contradiction. Therefore, such α ^∘ does not exist and the proof is complete. □

Appendix B: Proof of Theorem 5

Proof

First, we show that {δ(l)} is a monotone increasing sequence, i.e., δ(l+1)>δ(l). From (20) and (32), and noting that b(l) and c(l) have the same sign, we obtain

or

(45)

Since δ(l+1) and A(l+1) satisfy

we have from (20) that

(46)

From (45) and (46), it is easy to see that δ(l+1)>δ(l).

Next, we show that δ ^∗ is an upper bound of the sequence {δ(l)}. Suppose that δ(l)>δ ^∗ for some l. Noticing that δ(l) is the weighted error norm of problem (29) and using Corollary 1, we have

(47)

where ω _k (l),k=0,1,…,r are the frequency points arranged in increasing order in A(l). In addition, according to the definitions of α ^∗ and δ ^∗, we have

(48)

From (47) and (48) and the assumption of δ(l)>δ ^∗, it is easy to see that

which implies that H(ω,α ^∘(l))−H(ω,α ^∗) has at least r zeros (counting repeated zeros) in [0,π]. This is impossible because its degree is at most r−1. Therefore, δ(l)≤δ ^∗ for all l.

Since the sequence {δ(l)} is bounded and monotonically increasing, then it is convergent. Let limδ(l)=δ ^∘ as l→∞, and $A^{\circ}\equiv\{\omega_{k}^{\circ},k=0,1,\dots ,r\}$ be a corresponding limit point of {A(l)}. Considering the equivalence of (16) and (18) and using Theorem 2, the optimal solution α ^∘(A ^∘,δ ^∘) of problem

satisfies

Then, α ^∘(A ^∘,δ ^∘) and A ^∘ satisfy Theorem 4, i.e., the sequences {A(l)}, {δ(l)} and {α ^∘(l)} converge to A ^∗, δ ^∗ and α ^∗, respectively. The proof is complete. □