A Sequential Partial Optimization Algorithm with Guaranteed Convergence for Minimax Design of IIR Digital Filters

Abstract

Challenges for optimal design of infinite impulse response digital filters include the high nonconvexity of design problem and inevitable stability constraints on the filters. To reduce the nonconvexity and tackle the stability constraints, a sequential partial optimization (SPO) algorithm was recently developed to divide the design problem into a sequence of subproblems, each updating only two second-order denominator factors. But the convergence of that algorithm is not guaranteed. By applying an incremental update with an optimized step length in each subproblem, this paper presents an improved SPO algorithm which is guaranteed to converge to a Karush–Kuhn–Tucker (not necessarily global) solution of the design problem. This paper also extends the SPO algorithm to a more general case where the number of denominator factors optimized in the subproblems can be any positive number smaller than half of the denominator order. Convergence performance of the algorithm is shown by the design of two example filters with typical specifications widely adopted in the literature. Comparisons with state-of-the-art methods demonstrate that the improved SPO algorithm obtains better filters than the competing methods in terms of the maximum magnitude of frequency-response error.

Appendix A. Proof of Theorem 2

Proof

Firstly, from Lemma 1, the unique solution to problem (12) is a solution to problem (11) when the regularization parameter ${\lambda }>0$ is sufficiently small. Then, the solution ${\delta }{{\varvec{x}}}_\ell (\ell )={[\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )}^{\mathrm{T}} (\ell ), \delta {{\varvec{b}}}^{\mathrm {T}}(\ell )]^{\mathrm {T}}$ of problem (12) should satisfy the KKT conditions of problem (11). We now rewrite problem (11) as

$$\begin{aligned}&\mathop {\hbox {minimize}}\limits _{\mathrm {\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}},\theta } \,\theta \end{aligned}$$

(A1a)

$$\begin{aligned}&\hbox {subject to constraints ((11b))-((11e)), } \end{aligned}$$

(A1b)

and

$$\begin{aligned}&\quad \left| {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})} \right| ^{2}\le \theta \hbox { for all }\omega \in \varOmega . \end{aligned}$$

(A1c)

Note that

$$\begin{aligned}&\begin{array}{ll} &{}\frac{\partial }{\partial {\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )} }\left| {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})} \right| ^{2}=\frac{\partial }{\partial {\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )} }\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})\overline{\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})} } \right\} \\ &{}\quad =2\hbox { Re }\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})\overline{\varvec{\xi } _{1,\ell } (\omega ,{{\varvec{a}}}(\ell ),{{\varvec{b}}}(\ell ))} } \right\} , \\ \end{array}\\&\begin{array}{ll} &{}\frac{\partial }{\partial {\delta }{{\varvec{b}}}}\left| {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})} \right| ^{2}=\frac{\partial }{\partial {\delta }{{\varvec{b}}}}\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})\overline{\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})} } \right\} \\ &{}\quad =2\hbox { Re }\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )},{\delta }{{\varvec{b}}})\overline{\varvec{\xi } _2 (\omega ,{{\varvec{a}}}(\ell ),{{\varvec{b}}}(\ell ))} } \right\} , \\ \end{array} \end{aligned}$$

where the bar over a complex-valued quantity represents the complex conjugate of that quantity. Then, by the KKT conditions of problem (A1), there must exist nonnegative Lagrangian multipliers ${\mu }(\ell ,{\omega })\ge 0$ for ${\omega }\in \varOmega$ and ${\nu }_{q(\ell ),n,m}(\ell )\ge 0$, for $n=1,2,\ldots , n(\ell )$ and $m=1,2, \ldots ,7$ such that

$$\begin{aligned}&\sum _{\omega \in \varOmega } {\mu \hbox {(}\ell \hbox {,}\omega \hbox {)}} = 1 ; \quad \sum _{\omega \in \varOmega } {2\mu \hbox {(}\ell \hbox {,}\omega \hbox {)Re}\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )} (\ell ),{\delta }{{\varvec{b}}}(\ell ))\overline{\varvec{\xi } _2 (\omega ,{{\varvec{a}}}(\ell ),{{\varvec{b}}}(\ell ))} } \right\} } =0; \end{aligned}$$

(A2a)

$$\begin{aligned}&\quad \sum _{\omega \in \varOmega } {2\mu \hbox {(}\ell \hbox {,}\omega \hbox {)Re}\left\{ {\hat{{E}}(\ell ,\omega ,{\delta }{\tilde{{{\varvec{a}}}}}_{q(\ell )} (\ell ),{\delta }{{\varvec{b}}}(\ell ))\overline{\varvec{\xi } _{1,\ell } (\omega ,{{\varvec{a}}}(\ell ),{{\varvec{b}}}(\ell ))} } \right\} }\nonumber \\&\quad +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1,1} (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ),1} (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} \rho \\ {-1} \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1,2} (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ),2} (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} {-\rho } \\ {-1} \\ \end{array} }} \right] \nonumber \\&\quad +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1,3} (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ), 3 } (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} 0 \\ 1 \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1, 4 } (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ), 4 } (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} 1 \\ 0 \\ \end{array} }} \right] \nonumber \\&\quad +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1, 5 } (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ), 5 } (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} {-1} \\ 0 \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1, 6 } (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ), 6 } (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} 0 \\ 1 \\ \end{array} }} \right] \nonumber \\&\quad +\left[ {{\begin{array}{l} {\nu _{q(\ell ), 1, 7 } (\ell )} \\ \vdots \\ {\nu _{q(\ell ),n(\ell ), 7 } (\ell )} \\ \end{array} }} \right] \otimes \left[ {{\begin{array}{l} 0 \\ {-1} \\ \end{array} }} \right] =\quad 0; \end{aligned}$$

(A2b)

$$\begin{aligned}&\quad \nu _{q(\ell ),n,1} (\ell )[\rho {\delta }a_{q(\ell )P-P+n,1} (\ell )-{\delta }a_{q(\ell )P-P+n,2} (\ell )-\rho ^{2}\nonumber \\&\quad +\rho a_{q(\ell )P-P+n,1} (\ell )-a_{q(\ell )P-P+n,2} (\ell )]=0, n=1,2,\ldots ,n(\ell ); \end{aligned}$$

(A2c)

$$\begin{aligned}&\quad \nu _{q(\ell ),n,2} (\ell )[-\rho {\delta }a_{q(\ell )P-P+n,1} (\ell )-{\delta }a_{q(\ell )P-P+n,2} (\ell )-\rho ^{2}\nonumber \\&\quad -\rho a_{q(\ell )P-P+n,1} (\ell )-a_{q(\ell )P-P+n,2} (\ell )]=0, n=1,2,\ldots ,n(\ell ); \end{aligned}$$

(A2d)

$$\begin{aligned}&\quad \nu _{q(\ell ),n,3} (\ell )[{\delta }a_{q(\ell )P-P+n,2} -\rho ^{2}+a_{q(\ell )P-P+n,2} (\ell )]=0, n=1,2,\ldots ,n(\ell ); \end{aligned}$$

(A2e)

$$\begin{aligned}&\quad \nu _{q(\ell ),n,4} (\ell )[{\delta }a_{q(\ell )P-P+n,1} -h]=0,\nonumber \\&\quad \nu _{q(\ell ),n,5} (\ell )[{\delta }a_{q(\ell )P-P+n,1} +h]=0, n=1,2,\ldots ,n(\ell ); \end{aligned}$$

(A2f)

$$\begin{aligned}&\quad \nu _{q(\ell ),n,6} (\ell )[{\delta }a_{q(\ell )P-P+n,2} -h]=0,\nonumber \\&\quad \nu _{q(\ell ),n,7} (\ell )[{\delta }a_{q(\ell )P-P+n,2} +h]=0, n=1,2,\ldots ,n(\ell ). \end{aligned}$$

(A2g)

Note that $q(\ell )$ goes from 1 to $Q+\mathrm{sign}(J)$ repeatedly as the iteration index $\ell$ proceeds. Then after using (7c) and (7d) and taking the limits of (A2a)–(A2g) as $\ell \rightarrow \infty$, there must exist ${\mu }^{*}({\omega })\ge 0$ for all ${\omega }\in \varOmega$ and $\nu _{nm}^*\ge 0$ for $n=1,2,\ldots ,N/2$ and $m=1,2,\ldots ,7$ such that

$$\begin{aligned}&\quad \sum _{\omega \in \varOmega } {\mu ^{*}\hbox {(}\omega \hbox {)}} = 1, \quad \sum _{\omega \in \varOmega } {2\mu ^{*}\hbox {(}\omega \hbox {)Re}\left\{ {E(\omega ,{{\varvec{a}}}^{*},{{\varvec{b}}}^{*})A^{-1}\left( {\mathrm{e}}^{-\mathrm{j}\omega },{{\varvec{a}}}^{*} \right) {\varvec{\psi }} ({\mathrm{e}}^{-\mathrm{j}\omega })} \right\} } =0, \end{aligned}$$

(A3a)

$$\begin{aligned}&\quad -\sum _{\omega \in \varOmega } {2\mu ^{*}\hbox {(}\omega \hbox {)Re}\left\{ {E(\omega ,{{\varvec{a}}}^{*},{{\varvec{b}}}^{*})\frac{H\hbox {(e}^{-\mathrm{j}\omega },{{\varvec{a}}}^{*},{{\varvec{b}}}^{*})}{A_n \hbox {(e}^{-\mathrm{j}\omega },{{\varvec{a}}}_n^*)}{\varvec{\phi }} (\hbox {e}^{-\mathrm{j}\omega })} \right\} }\nonumber \\&\quad +\nu _{n,1}^*\left[ {{\begin{array}{l} \rho \\ {-1} \\ \end{array} }} \right] +\nu _{n,2}^*\left[ {{\begin{array}{l} {-\rho } \\ {-1} \\ \end{array} }} \right] +\nu _{n,3}^*\left[ {{\begin{array}{l} 0 \\ 1 \\ \end{array} }} \right] =0, \quad n=1,2,\ldots ,N/2,\end{aligned}$$

(A3b)

$$\begin{aligned}&\quad \nu _{n,1}^*[-\rho ^{2}+\rho a_{n,1}^*-a_{n,2}^*]=0,\quad \nu _{n,2}^*[-\rho ^{2}-\rho a_{n,1}^*-a_{n,2}^*]=0,\nonumber \\&\quad \nu _{n,3}^*[-\rho ^{2}+a_{n,2}^*]=0, n=1,2,\ldots ,N/2. \end{aligned}$$

(A3c)

Equations (A3a)–(A3c) are the KKT optimality conditions for the following problem

$$\begin{aligned}&\mathop {\hbox {minimize}}\limits _{\theta ,{{\varvec{b}}},{\tilde{{{\varvec{a}}}}}_1,{\tilde{{{\varvec{a}}}}}_2 ,\ldots ,{\tilde{{{\varvec{a}}}}}_{Q+{\mathrm{sign}}(J)} } \theta , \\&\hbox { s.t}.{:}\quad \left| {E(\omega ,{{\varvec{a}}},{{\varvec{b}}})} \right| ^{2}\le \theta \hbox { for all }\omega \in \varOmega ,\\&{\tilde{{{\varvec{a}}}}}_q \in {\tilde{S}}_q, q=1,2,\ldots ,Q+{\mathrm{sign}}(J), \end{aligned}$$

which is equivalent to problem (5) with the same stability domain ${\tilde{S}}_q$ as defined by (4). It follows that $[({{\varvec{a}}}^{*})^{\mathrm{T}}, ({{\varvec{b}}}^{*})^{\mathrm{T}}]^{\mathrm{T}}$ is a KKT point of problem (5) with ${\tilde{S}}_q$ defined by (4). $\square$