Wide band noise reduction is particularly challenging because of the difficulty in isolating one single spatial direction of arrival (DOA) of ambient noise in all frequency bands. It is typical for the DOAs to exhibit a persistent drift with a consequence of degrading the performance of nullforming algorithms. Consider the microphone array topology illustrated in Figure 1 below:
Figure 1: 2 microphone array
where 
is the DOA of the desired signal. Sub band differential signals can be synthesized as shown in figure 2 below:
Figure 2: sub-band adaptive nullforming
The delay unit is a function of the separation distance 
and the sampling frequency. The outputs of the filter bank could be down sampled depending on the particular implementation. It can be shown that the optimal weights will satisfy:
where ![\Phi_{x_i(w),x_j(w)} = \mathbb{E}[x_i(w) x^*_j(w)]](https://s0.wp.com/latex.php?latex=%5CPhi_%7Bx_i%28w%29%2Cx_j%28w%29%7D+%3D+%5Cmathbb%7BE%7D%5Bx_i%28w%29+x%5E%2A_j%28w%29%5D&bg=ffffff&fg=000000&s=0 "\Phi_{x_i(w),x_j(w)} = \mathbb{E}[x_i(w) x^*_j(w)]")
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