New Insights into Convergence Theory of Constrained Frequency-Domain Adaptive Filters

Abstract

Two kinds of update equations are commonly used for the constrained frequency-domain adaptive filter (FDAF), namely the gradient-constrained version and the weight-constrained version. The constraint is imposed only on the stochastic gradient vector in the first version, while it is imposed on the whole weight vector in the second version. It was already found that the two versions have different convergence behaviors, but a rigors analysis of the convergence behavior of the gradient-constrained FDAF is still lacking so far. This paper presents a comprehensive statistical analysis of the gradient-constrained FDAF. We set up an equivalent update equation of the gradient-constrained FDAF, which provides a close link with that of the weight-constrained version. Then, the mean and mean-square convergence behaviors of the gradient-constrained FDAF are analyzed using the new update equation, and the corresponding steady-state solutions are provided. Theoretical results confirm that the gradient-constrained FDAF will converge to a biased solution and exhibits a larger mean-square error than the weight-constrained version when, for instance, the weight vector is not initialized properly. Simulation results agree with our theoretical predictions very well.

Appdenidx A

In this section, we attempt to analyze the mean-square convergence of the gradient-constrained FDAF using the recursion (6). Our objective is to highlight the inconvenience of performing such an analysis based on (6).

Subtracting ${\mathbf{W}}$ from both sides of (6), we obtain

$$\begin{aligned} {\tilde{\mathbf{W}}}(k + 1) = {\tilde{\mathbf{W}}}(k) -\mu {{\mathbf{G}}^{10}}{{\varvec{\Lambda }}^{ - 1}}{{\mathbf{X}}^H}(k){\mathbf{E}}(k). \end{aligned}$$

(40)

Substituting (16) into (40), we have

$$\begin{aligned} {\tilde{\mathbf{W}}}(k + 1) = \left[ {{{\mathbf{I}}_N} -\mu {{\mathbf{G}}^{10}}{{\mathbf{A}}^H}(k)} \right] {\tilde{\mathbf{W}}}(k) -\mu {{\mathbf{G}}^{10}}{{\varvec{\Lambda }}^{ - 1}}{{\mathbf{X}}^H}(k){\mathbf{V}}(k). \end{aligned}$$

(41)

Postmultiplying both sides of (40) by ${{\tilde{\mathbf{W}}}^H}(k + 1)$ and taking mathematical expectation, after some manipulations, it has

$$\begin{aligned}&E\left[ {\tilde{\mathbf{W}}}(k + 1){{{\tilde{\mathbf{W}}}}^H}(k + 1)\right] \nonumber \\&\quad = E\left[ {\tilde{\mathbf{W}}}(k){{{\tilde{\mathbf{W}}}}^H}(k)\right] \nonumber \\&\qquad -\, \mu {{\mathbf{G}}^{10}}E\left[ {{\mathbf{A}}^H}(k){\tilde{\mathbf{W}}} (k){{{\tilde{\mathbf{W}}}}^H}(k)\right] \nonumber \\&\qquad -\, \mu E\left[ {\tilde{\mathbf{W}}}(k){{{\tilde{\mathbf{W}}}}^H} (k){\mathbf{A}}(k)\right] {{\mathbf{G}}^{10}}\nonumber \\&\qquad +\, {\mu ^2}{{\mathbf{G}}^{10}}E\left[ {{\mathbf{A}}^H}(k){\tilde{\mathbf{W}}} (k){{{\tilde{\mathbf{W}}}}^H}(k){\mathbf{A}}(k)\right] {{\mathbf{G}}^{10}}\nonumber \\&\qquad +\, {\mu ^2}E\left[ {{\mathbf{G}}^{10}}{{\varvec{\Lambda }}^{ - 1}}{{\mathbf{X}}^H} (k){{\varvec{\Phi }}_v}{\mathbf{X}}(k){{\varvec{\Lambda }}^{ - 1}}{{\mathbf{G}}^{10}}\right] . \end{aligned}$$

(42)

Taking the vectorization operation on both sides of (42) and also using the identity (25), we get

$$\begin{aligned} {\mathbf{z}}(k + 1) = \bar{{\varvec{\Theta }}}{} \mathbf{z}(k) + \bar{{\varvec{\Psi }}}(k), \end{aligned}$$

(43)

where

$$\begin{aligned} \bar{{\varvec{\Theta }}}= & {} {{\mathbf{I}}_{{N^2}}} - \mu \left( {{\mathbf{I}} \otimes {{\mathbf{G}}^{10}}} \right) \left( {{{\mathbf{I}}_N} \otimes E[{{\mathbf{A}}^H}(k)]} \right) \nonumber \\&-\, \mu \left( {{{({{\mathbf{G}}^{10}})}^T} \otimes {\mathbf{I}}} \right) \left( {E[{{\mathbf{A}}^T}(k)] \otimes {{\mathbf{I}}_N}} \right) \nonumber \\&+\, {\mu ^2}\left( {{{({{\mathbf{G}}^{10}})}^T} \otimes {{\mathbf{G}}^{10}}} \right) E\left[ {{{\mathbf{A}}^T}(k) \otimes {{\mathbf{A}}^H}(k)} \right] , \end{aligned}$$

(44)

$$\begin{aligned} \bar{{\varvec{\Psi }}}(k)= & {} {\mathrm{vec}}\left\{ {E\left[ {{{\mathbf{G}}^{10}} {{\varvec{\Lambda }}^{ - 1}}{{\mathbf{X}}^H}(k){{\varvec{\Phi }}_v}{\mathbf{X}}(k) {{\varvec{\Lambda }}^{ - 1}}{{\mathbf{G}}^{10}}} \right] } \right\} . \end{aligned}$$

(45)

Actually, Eqs. (43) and (27) are indeed mathematically equivalent, and hence both of them can be utilized to describe the mean-square behavior of the gradient-constrained FDAF. However, we found that the matrix ${{\mathbf{I}}_{{N^2}}} -\bar{{\varvec{\Theta }}}$ is not full-rank, and thus the steady-state solution ${\mathbf{z}}(\infty )$ could not be directly solved using (43). This is why we have derived a new update equation (12) for the statistical analysis of the gradient-constrained FDAF.