Is there a method to speed up beam-forming if you only care about the total power in a given direction, or total power in frequency groups? I've googled this, and can't find anything. My intuition says that wouldn't make sense given that you need the power in each frequency to properly cancel out signals to create your beam.
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$\begingroup$ Direction Finding? $\endgroup$– user28715Commented Jul 17, 2019 at 20:13
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$\begingroup$ @StanleyPawlukiewicz does that mean DOA estimation? if so yeah. $\endgroup$– KrupipCommented Jul 17, 2019 at 20:15
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$\begingroup$ I get 63K hits for doa estimation in Google Scholar $\endgroup$– user28715Commented Jul 17, 2019 at 20:18
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$\begingroup$ @StanleyPawlukiewicz right... I thought you were just asking what I was doing, DOA != finding total energy in beam as an optimization for beamforming rather than considering individual frequencies. $\endgroup$– KrupipCommented Jul 17, 2019 at 20:22
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I am not sure regarding total power but you can certenly focuse on ccertain frequencies of interest.
Beamforming is a process in which different channels (a number of microphones for example) gather their information to cancel the uncorrelated component between them. Let $y(f)=[Y_1, Y_2, \ldots, Y_M]^T$ be signals received from $M$ channels S.T. $y(f)=x(f)+v(f)$, where $v(f)$ is noise. We construct:
$$Z(f)=\sum_{m=1}^{M}H^*_m(f)Y_m(f)=X_d(f)+V_r(f)$$
Where $X_d$ is our reconstructed desired signal and $V_r$ is the residual noise. Given a directional beam $d(f,cos\theta_d)=[1,e^{-j2\pi f\tau_0\cos\theta},\ldots,e^{-j2(M-1)\pi f\tau_0\cos\theta}]^T$, the white noise gain (WNG) of such a filter is given by: $$\mathcal{W}[h(f)]=\frac{|h^H(f)d(f,cos\theta_d)|^2}{h^H(f)h(f)}$$ Note that all of these are functions of $f$, meaning they can be applied in only a single or couple of specific frequencies of interest.
To find such a filter, simply solve for the maximum on $\mathcal{W}[h(f)]$ (the higher WNG is the better you reconstruction of the signal is) which yields the common delay and sum beamformer:
$$h_{DS}(f,cos\theta_d)=\frac{d(f,cos\theta_d)}{M}$$
There are many types of beamformers that can be produced such as the maximum directivity factor (DF) beamformer. I find this book is a good source. All of them can be applied as a function of $f$.
