# Spatial moving average in array antenna processing

In array processing the theoretical spatial response of the array is calculated by doing DFT on sensor locations for some assumed wave number (1/wavelength). The figures below shows the sensor locations of an array (example is from seismology) and the array responses for different wavelength hitting the array. The mainlobe, the blob in the middle of response plots, says something about the resolution while the sidelobes tells of aliasing effects this array configuration gives for assumed wavelength.

In Fiber optic sensing, although every point in the fiber can be considered for measurement, in reality the strain on fiber (because of incoming wave field) is measured and averaged (moving average) along some length of the fiber, so called gauge length. In general, the averaging is equivalent to multiplying the array response with a low pass filter response in spatial frequency domain (wavenumber domain). The filter response for example in 2D is like: $$\frac{\sin(2{\pi}{k_x L}) \sin(2{\pi}{k_y L})}{L\sin({\pi}{k_x})L\sin({\pi}{k_y})}$$

where $$L$$ is the length of the averaging filter and $$k_x, k_y$$ are the wavenumber components (spatial frequency components). But the averaging examples above assumes regular sampling in either 1D or 2D, whereas the averaging in fiber optics is done along the path of sensor locations. So, in the sensor plot above, the averaging is 1D while the spatial sampling is obviously 2D (in $$k_x$$ and $$k_y$$, i.e. $$x$$ and $$y$$ components of wavenumber).

I made some synthetic data and performed beamforming to see how the averaging effect is for different wavelengths. But, I need to do this in spatial domain, i.e. find the theoretical array response and multiply it with the response of a moving average filter, where averaging is done along sensor path.

So, my question is: how to calculate the response of the moving average?