Adaptive beamforming with linear constraints

Reference article:
[1]an alternative approach to linearly Constrained adaptive.1982 beamforming

Adaptive beamforming algorithm model with linear constraints

For the output of the M channel is XM (k) x_m (k), the output of the M channel is ideally: XM (k) =s (k) +nm (k). X_m (k) = S (k) + N_m (k).
Wherein, is S (k) s (k) is the desired signal, NM (k) N_m (k) represents the noise and interference of the first m microphone. the beam-formed signal y (k) y (k) is summed by the weighted delay of XM (k) x_m ( k): Y (k) =∑m=1m∑l=−kkam,lxm (k−l). Y (k) = \sum_{m=1}^m\sum_{l=-k}^ka_{m,l}x_m (L-K).
2k+1 is the number of sample points that participate in the calculation. Am,l A_{m,l} represents the weight of the first M-channel delay L. ( or understood as Am,l a_{m,l} is the tap factor of the FIR filter of the channel *i, 2k+1 2k+1 is the order of the FIR filter ). Representation of the Matrix: Y (k) =∑l=−kkat (L) X (k−l) y (k) =\sum_{l=-k }^k\boldsymbol{a}^t (L) \boldsymbol{x} (K-L)
, Al \boldsymbol{a}_l and X (k−l) \boldsymbol{x} (K-L) represent the filter coefficients and signal vectors at the first-time delay point [1].
The basic diagram of the input and output is as follows: