An improved adaptive constrained constant modulus reduced-rank algorithm with sparse updates for beamforming

Abstract

In this work, we propose an adaptive set-membership (SM) reduced-rank filtering algorithm using the constrained constant modulus criterion for beamforming. We develop a stochastic gradient type algorithm based on the concept of SM techniques for adaptive beamforming. The filter weights are updated only if the bounded constraint cannot be satisfied. We also propose a scheme of time-varying bound and incorporate parameter dependence to characterize the environment for improving the tracking performance. A detailed analysis of the proposed algorithm in terms of computational complexity and stability is carried out. Simulation results verify the analytical results and show that the proposed adaptive SM reduced-rank beamforming algorithms with a dynamic bound achieve superior performance to previously reported methods at a reduced update rate.

Appendix: Derivation for (20) and (21)

In this appendix, we derive the expressions in (20) and (21). Note that each constraint set comprises of two parallel hyperstrips in the parameter space. Based on the constraint $e^2(i)>\gamma ^2(i)$, we consider the following two cases for update: (1) $|y(i)|>\sqrt{1+\gamma (i)}$ and (2) $|y(i)|<\sqrt{1-\gamma (i)}$. Firstly, let us focus on the variable step-size $\mu _{p}(i)$. We obtain $\mu _{p}(i)$ for computing $\mathbf {p}(i+1)$ by projecting $\mathbf {p}(i)$ onto $\mathcal {H}_{p}(i)$, i.e., the set of all $\mathbf {p}$ that satisfy:

$$\begin{aligned} \sqrt{1-\gamma (i)}\le |\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}|\le \sqrt{1+\gamma (i)}. \end{aligned}$$

(30)

For case 1) $|\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}(i)|>\sqrt{1+\gamma (i)}$, $\mathbf {p}(i)$ is closer to the hyperplanes defined by

$$\begin{aligned} |\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}|=\sqrt{1+\gamma (i)} \end{aligned}$$

(31)

than to the ones defined by

$$\begin{aligned} |\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}|=\sqrt{1-\gamma (i)}. \end{aligned}$$

(32)

By substituting (18) into (31) and using the constraint $\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})\mathbf {p}^{*}(i)=\nu$ we have

$$\begin{aligned}&\bigg |\underbrace{\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}(i)}_{y(i)}-\mu _{p}(i)e(i)y(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\bigg (\mathbf {I}-\frac{\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})}{\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)} \bigg ) \nonumber \\&\quad \times \,\mathbf {R}^{'H}(i)\mathbf {T}^{T}\bar{\mathbf {w}}(i)\bigg |=\sqrt{1+\gamma (i)}. \end{aligned}$$

(33)

By making an arrangement to (33) we have the following expression:

$$\begin{aligned} \mu _{p}(i)=\frac{\big (1-\frac{\sqrt{1+\gamma (i)}}{|y(i)|} \big )}{e(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\left( \mathbf {I}-\frac{\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})}{\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)} \right) \mathbf {R}^{'H}(i)\mathbf {T}^{T}\bar{\mathbf {w}}(i)}. \end{aligned}$$

(34)

For case 2) $|\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}(i)|<\sqrt{1-\gamma (i)}$, $\mathbf {p}(i)$ is closer to the hyperplanes defined by (32). By following the same approach we have:

$$\begin{aligned} \mu _{p}(i)=\frac{\big (1-\frac{\sqrt{1-\gamma (i)}}{|y(i)|} \big )}{e(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\left( \mathbf {I}-\frac{\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})}{\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})\mathbf {D}^{H}(\theta _{0})\mathbf {T}^{T}\bar{\mathbf {w}}(i)} \right) \mathbf {R}^{'H}(i)\mathbf {T}^{T}\bar{\mathbf {w}}(i)}. \end{aligned}$$

(35)

Therefore, the expressions for $\mu _{p}(i)$ can be summarized as (20).

Then, we derive the expression for the variable step-size $\mu _{w}(i)$. By employing (19) we have the following expression

$$\begin{aligned} \big |\bar{\mathbf {w}}^{H}(i+1)\bar{\mathbf {r}}(i)\big |= \bigg |\underbrace{\bar{\mathbf {w}}^{H}(i)\bar{\mathbf {r}}(i)}_{y(i)}-\mu _{w}(i)e(i)y(i)\bar{\mathbf {r}}^{H}(i)\varvec{\varOmega }(i)\bar{\mathbf {r}}(i)\bigg |. \end{aligned}$$

(36)

Similarly, by considering these two cases to update and making the arrangement to (36), we can obtain (21).