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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.

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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.

Note that this convention is not universal in the FIR filtering literature; I have seen papers where the convolution sum for a system with complex-valued impulse response does not include the conjugate operator. However, I prefer to use the form that you showed in your question.

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  • $\begingroup$ could you please provide a good reference where convolution operator is defined as an inner product? $\endgroup$ – user4673 Jul 21 '14 at 14:19
  • $\begingroup$ @user4673: One text that I know uses this notation is "Adaptive Filter Theory" by Haykin. $\endgroup$ – Jason R Jul 22 '14 at 2:52

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