# use of complex conjugate of weight vectors in beamforming literature

in beamforming literature, people generally quote the following as a starting point:

$y(k) = \sum_{i} w_i^*\cdot x_i(k)$

Here:

i varies from 1 to N where N is the number of sensors

$w_i^*$ is the complex conjugate of the ith weight vector

$x_i(k)$ is the kth sample received at the ith sensor

y(k) is the output of the beamformer for kth sample

The question is why is the complex conjugate of the weight vector $w_i^*$ used(instead of the complex weight vector $w_i$ iteslf). The literature seems to suggest that it simplifies notation but I can't see how.

This comes from the fact that the convolution operator for discrete-time linear systems is typically defined as the inner product between the input signal $x_i(k)$ in your example) and the system's impulse response ($w_i$). The definition of the inner product for complex vectors includes the conjugation:
$$a \cdot b = \sum a_i b_i^*$$
For the above definition, note that there doesn't seem to be consistent agreement over which of the arguments to the dot operator gets conjugated, so be careful when reading the literature. For the application of discrete-time FIR filters (which your example is), the typical convention is to conjugate the vector of filter taps, hence the $w_i^*$ term in the sum.