Steady-state behavior of the improved normalized subband adaptive filter algorithm and its improvement in under-modeling

Abstract

In practice, adaptive filter could work in an under-modeling scenario, meaning that its length is less than that of the unknown system. In this realistic situation, therefore, the existing analysis for the improved normalized subband adaptive filter (INSAF) algorithm is not applicable. To this end, this paper analyzes the mean square steady-state performance of the INSAF for under-modeling. In addition, we propose a variable step size INSAF algorithm suitable for under-modeling scenario, to obtain fast convergence rate and low steady-state error. Simulation results have supported our theoretical analysis and proposed algorithm.

Appendix: derivation of (13)

Postmutiplying (9) by its transpose ${{\varvec{A}}}^{T}(k)$ gives

$$\begin{aligned} {{\varvec{A}}}^{T}(k){{\varvec{A}}}(k)= & {} \mathbf{I}_L -2\sum _{i=0}^{N-1} {\frac{{\varvec{u}}_{L\mathrm{opt},i} (k)\left[ \begin{array}{ll} {{\varvec{u}}_i^T (k)}&{} {\mathbf{0}_{1\times (L-M)} } \\ \end{array} \right] }{\left\| {{\varvec{u}}_i (k)} \right\| ^{2}}} \nonumber \\&+\,\mu ^{2}\sum _{i=0}^{N-1} {\frac{{\varvec{u}}_{L\mathrm{opt},i} (k){\varvec{u}}_{L\mathrm{opt},i}^T (k)}{\left\| {{\varvec{u}}_i (k)} \right\| ^{2}}} . \end{aligned}$$

(34)

To obtain (34), we employ the fact that different subband input signals are mutually orthogonal, i.e., ${\varvec{u}}_i^T (k){\varvec{u}}_j (k)\approx 0,i\ne j$, called the diagonal assumption, which has been used in the derivation of the INSAF algorithm [13]. Taking the expectation of (34) and using the assumption (A.3), we get

$$\begin{aligned} \begin{array}{c} E\left\{ {{{\varvec{A}}}^{T}(k){{\varvec{A}}}(k)} \right\} =\mathbf{I}_L +\mu ^{2}\mathop {\sum }\limits _{i=0}^{N-1} {\frac{E\left\{ {{\varvec{u}}_{L\mathrm{opt},i} (k){\varvec{u}}_{L\mathrm{opt},i}^T (k)} \right\} }{E\left\{ {\left\| {{\varvec{u}}_i (k)} \right\| ^{2}} \right\} }} \\ -\,2\mathop {\sum }\limits _{i=0}^{N-1} \frac{E\left\{ {{\varvec{u}}_{L\mathrm{opt},i} (k)\left[ \begin{array}{ll} {{\varvec{u}}_i^T (k)}&{} {\mathbf{0}_{1\times (L-M)} } \\ \end{array} \right] } \right\} }{E\left\{ {\left\| {{\varvec{u}}_i (k)} \right\| ^{2}} \right\} } \\ \end{array}. \end{aligned}$$

(35)

From the assumption (A.4), we obtain

$$\begin{aligned}&E\left\{ {{\varvec{u}}_{L\mathrm{opt},i} (k)\left[ {{\begin{array}{ll} {{\varvec{u}}_i^T (k)}&{} {\mathbf{0}_{1\times (L-M)} } \\ \end{array} }} \right] } \right\} \nonumber \\&\quad =\left[ {{\begin{array}{ll} {\sigma _{u_i }^2 \mathbf{I}_M }&{} {\mathbf{0}_{1\times (L-M)} } \\ {\mathbf{0}_{1\times (L-M)} }&{} {\mathbf{0}_{L-M} }\\ \end{array} }} \right] , \end{aligned}$$

(36)

$$\begin{aligned}&E\left\{ {{\varvec{u}}_{L\mathrm{opt},i} (k){\varvec{u}}_{L\mathrm{opt},i}^T (k)} \right\} =\sigma _{u_i }^2 \mathbf{I}_L . \end{aligned}$$

(37)

Substituting (36) and (37) into (35) and again using the assumption (A.4), (13) is obtained.