# Matched a phase only correlation filters

## Matched Filter Results

The easiest way to analyses the correlator is with a computer
simulation. If our input signal is the the letters ‘A’ and ‘B’ and the target filter is the letter ‘A’ The intensity of the cross correlation (matched filter) is then

Note that the left hand peak is the correlation of the two A’s
and the right is the cross correlation of ‘A’ and the ‘B’. There is some
similarity between the ‘A’ and ‘B’ so the peak is not zero in height. This example
shows how the matched filter is not very good at discriminating between
objects.

## Phase Only Filter

One of the short comings of most realistic optical filters is
that they can only modulate one parameter. Typically this is either phase or
amplitude. For a practical application the filter is usually dynamically updatable
so we use an SLM (Spatial Light Modulator). Liquid crystal SLMs can modulate
the phase of light and these are commonly used as filters for correlators.
The just after the filter the signal is then

$$F(u,v)exp(-i*\theta(-u,-v))$$

where $$\theta$$ is the phase of $$G$$, the filter function. Now when the two signals match the phase components cancel to zero and the result is the Fourier transform of the amplitude of $latex F$. This usually a much smaller width peak than the matched filter result.

Here is an example result with the same inputs as before.

This filter is much more discriminating than the matched
filter. The correlation of the ‘A’ and ‘B’ is now smaller and the peak for the ‘A’ correlation is sharp making it much easier to locate. However, the phase only filter is perhaps too sensitive. If we replace the input with the following:

The second ‘A’ has been rotated by 5 degrees and the resultant correlation is

The small change in rotation has resulted in the peak collapsing.

The reason for this is missing amplitude modulation in the filter. The amplitude component acts to attenuate the higher frequencies, without it they are over emphasised resulting in the filter been extremely sensitive to any small change in the signal. If the input is also represented by a phase only SLM, the result is a phase-phase correlator which is even more sensitive.

If we add some Gaussian noise to one of the ‘A’s and remove the rotation so the input is now

The phase only filter result is

So even with the small amount of noise, the peak drop is noticeable.  On the other hand if we use a matched filter on the same input we get