Maximum-Output-Power Beamforming Despite No Prior Information About the Signals-of-Interest Nor Possibly About the Interference

Abstract

This paper proposes a new data-dependent beamforming strategy to adaptively maximize the output power, while nulling preset directions-of-arrival. This new beamformer needs no iterative computation and needs no prior information of the signals-of-interest’s incident directions nor temporal waveforms. Surprisingly, this proposed beamformer can “blindly” null any dominant interferences with no prior information of their incident directions nor of their waveforms, even as this beamformer simultaneously preserves the signals-of-interest as mentioned above. Monte Carlo simulations verify this new method’s superior array gain relative to the “linearly constrained minimum-variance” beamformer’s.

Appendices

Appendix A: Wideband Beamforming

The wideband counterpart of Sect. 2.1’s narrowband $L \times M$ data matrix $\mathbf{X}$ is here denoted as $\mathbf{X}_w$, on which the discrete Fourier transform may be applied to give

$$\begin{aligned} {\tilde{\mathbf{X}}}_w= & {} \mathrm{DFT}\left\{ \mathbf{X}_w\right\} \nonumber \\:= & {} \left[ {\tilde{\mathbf{x}}}_{w,1}, {\tilde{\mathbf{x}}}_{w,2}, \ldots , {\tilde{\mathbf{x}}}_{w,P}\right] , \end{aligned}$$

(15)

where $\mathrm{DFT}\left\{ \cdot \right\}$ applies the discrete Fourier transform separately on each individual row of the matrix inside the curly brackets. Moreover, P denotes the number of digital-angular-frequency bins that sample $[-\pi ,\pi ]$.

For a bandpass filter at a center frequency of $f_c$, let $H_{f_c}(p)$ represent the filter transfer function’s digital Fourier transform at the pth tap, i.e., at a digital angular frequency of $2 \pi \frac{p}{P}$. Then, define the $M \times P$ matrix:

$$\begin{aligned} \mathbf{H}_{f_c}= & {} \mathbf{1} \; \left[ H_{f_c}(1), H_{f_c}(2), \ldots , H_{f_c}(P)\right] , \end{aligned}$$

(16)

where $\mathbf{1}$ symbolizes an M-entry column vector of all ones.

The bandpass-filtered output has a DFT-domain expression of

$$\begin{aligned} {\tilde{\mathbf{Y}}}\left( f_c\right)= & {} {\tilde{\mathbf{X}}}_w \odot \mathbf{H}_{f_c}, \end{aligned}$$

(17)

where $\odot$ denotes the element-wise product. Its corresponding time-domain expression equals

$$\begin{aligned} \mathbf{X}\left( f_c\right)= & {} \mathrm{IDFT}\left\{ {\tilde{\mathbf{Y}}}(f_c)\right\} , \end{aligned}$$

(18)

where $\mathrm{IDFT}\left\{ \cdot \right\}$ applies the discrete Fourier transform separately on each row of the matrix inside the curly brackets.

Thus, $\mathbf{X}(f_c)$ represents the narrowband space-time data at a center frequency of $f_c$. As the array manifold generally varies with frequency, denote the array manifold by $\mathbf{a}(\theta , \phi , f_c)$, and construct the “null direction” matrix $\mathbf{A}_\mathrm{null}(f_c)$ similarly as in (1).

Substituting $\mathbf{A}_\mathrm{null}$ by $\mathbf{A}_\mathrm{null}(f_c)$ and $\mathbf{X}$ by $\mathbf{X}(f_c)$ in (S1) to (S4) of Sect. 2.2, a narrowband NCMP beamformer $\mathbf{w}_o (f_c)$ may be realized at $f_c$.

Lastly, applying the bank of bandpass filters at various center frequencies, wideband NCMP beamforming is realized by combining the bank of beamformers $\mathbf{w}_o(f_c)$.

Appendix B: The “Linearly Constrained Minimum-Variance” (LCMV) Beamformer

A competitor to the proposed beamformer is the linearly constrained minimum-variance (LCMV) beamformer, which minimizes the beamformer output power (i.e., variance) subject to $I \le L-1$ number of linear constraints $\{ \mathbf{c}_1, \mathbf{c}_2, \cdots , \mathbf{c}_{I} \}$ to either pass the desired signals incident from preset “look directions” and/or to reject interference impinging from pre-determined directions. Mathematically, the LCMV beamforming weight vector $(\mathbf{w}_\mathrm{LCMV})$ is obtained by solving this optimization problem [10]:

$$\begin{aligned} \mathbf{w}_\mathrm{LCMV}:= & {} \mathop { \min }_{\mathbf{w}} \; \mathbf{w}^H \mathbf{R} \mathbf{w}, \end{aligned}$$

(19)

subject to

$$\begin{aligned} \mathbf{w}_\mathrm{LCMV}^H \overbrace{\left[ \mathbf{c}_1, \mathbf{c}_2, \cdots , \mathbf{c}_I \right] }^{ \mathbf{C} :=}= & {} \overbrace{\left[ \begin{array}{c} g_1 \\ g_2 \\ \vdots \\ g_{I} \end{array}\right] }^{ \mathbf{g} :=}. \end{aligned}$$

(20)

Here, the $I \times 1$ vector $\mathbf{g}_i$ specifies the desired gain for the ith constraint, $\mathbf{c}_i$.⁶