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Statistical optimal beamformers: linearly constrained minimum variance beamformer

Beamformers are grouped into two general classes: data independent and statistically optimal beamformers. Data independent beamforming relies solely on spatial properties such as forming a beam at a specific direction of arrival whilst nulls are formed at other all other directions. In statistical optimal beamforming, the filter weights used are dependent on the statistics of the received data. The linearly constrained maximum beamformer (LCMV) is a used if when generally flexible constraints are used. In many applications of beamforming, the desired signal strength is unknown apriori. Also, the desired signal may be present for the entire temporal duration in question, as such it may be impractical to use the multiple sidelobe canceler which does because estimation of the signal to noise ratio and the noise covariance matrix becomes impractical from the received signals. In such cases, the LCMV beamformer is utilized.
Consider a far field source impinging N uniform linear array (ULA) microphones as shown in Figure 1 below:

N ULA microphones

Figure 1: N ULA microphones

Suppose the signal at each microphone i \in \{1, \cdots, N\} is given as

x_i(w) = s(w) W_{i}(w) +\sum\limits_{l=1}^L v_l(w) e^{\left(-jw (i-1)\frac{d}{c} \sin{\left(\psi_l \right)} \right)} +n_i(w)

where s(w) is the desired speech signal, \theta is the direction of arrival (DOA) of the speech signal with respect to the broadside,  v_l(w) is the correlated desired signal such that \mathbb{E}[v_l(w) v_l^*(w)] = \sigma_{v_l^2} and  \mathbb{E}[ s(w) e^{\left(-jw \frac{d}{c} \sin{\left((i-1)\psi - \theta\right)} \right)} v_l^*(w)] = 0, \forall i \in \{1, \cdots, N\}, \forall l \in \{1,\cdots,L\}. \psi_l is the direction of arrival (DOA) of the l^{th} correlated noise with respect to the broadside. Here,

W_{i}(w) = e^{\left(-jw (i-1)\frac{d}{c} \sin{\theta} \right)}

and

W(w) = [W_1(w), \cdots, W_N(w)]^T

The statistical optimal criterion for LCMV is to find weights  W(w) such that there is a fixed gain and phase in the direction of the desired signal such that the output signal variance  is minimized, thus:

\underset{W(w)}{\mathrm{argmin}} ~~\mathbb{E}\left [{W^H(w) R_{x}(w) W(w)}\right] ~ ~ s.t ~~ d(\theta, w)^H W(w) = f

where R_{x}(w) is the signal plus noise covariance matrices.

Notice that the directional signal \nu_l is not used in the estimate of the filter weights.

It can be shown that the optimum weights correspond to:

W(w) = R_x(w)^{-1} d(\theta, w) [d(\theta, w)^H(w) R_x^{-1}(w)d(\theta, w)]^{-1}f

The main drawback of this approach is the computational burden. Setting f = 1 results of the minimum variance distortionless response.

VOCAL Technologies offers custom designed solutions for beamforming with a robust voice activity detector, acoustic echo cancellation and noise suppression. Our custom implementations of such systems are meant to deliver optimum performance for your specific beamforming task. Contact us today to discuss your solution!

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