Unfortunately, to understand Matched Filters, a basic review of Linear Systems is necessary:
Review of Convolution:
For a system response function g(t) and an input signal f(t), the following integral calculates the output signal shape in the time domain. This is the standard convolution integral:
Please note the following:
1.) In the time domain, the above convolution integral may be difficult to calculate. The calculation consists of calculating the overlap of the input pulse and the system response.
2.) In the frequency domain, the output response is much easier to calculate. In fact, the output pulse shape is determined by simply multiplying the transforms of the input and response, and then inverse transforming.
3. Network analyzer operate in the frequency domain and engineers tend to think of filter functions in the frequency domain.
4.) Time domain analysis is equivalent, but not as intuitive.
The following plot shows a familiar situation, an RC filter, but the convolution is done in the time domain.
|
The red curve is the impulse response,in the time domain, of the RC filter.
The green signal is the square wave input.
The yellow is the overlap(integral) and the value of
the overlap is plotted instant by instant on the bottom curve.
(the time axis is the same in the top and bottom plots.)
The bottom curve is the overlap integral, but it is also
the signal shape for a square pulse propagating through a simple RC Filter!
|
|
|
The plot shows a square pulse sent into a system with a simple RC system response. The lower plot shows graphically that using the time domain convolution we can calculate the familiar exponential rise and fall for the output.
Matched Filters
What is a matched filter and why does anyone care? In order to answer this we need to establish a few facts:
1. There must exist an optimum filter shape for maximizing SNR and thus sensitivity.
2. Output pulse shapes can be evaluated in the time or frequency domain - both are equivalent.
3. The optimum filter shape can be calculated for a particular input pulse shape and is different for each type of pulse.(Obviously fast pulses need a higher BW filter than slow pulses.
4. For obvious reasons, the optimum filter is called a Matched Filter.
5. The identification of the matched filter response for each input pulse is difficult, however the results are simple:
In the time domain, the impulse response of the filter should exactly match the shape of the incoming data.
Shape Arguments - skip math
There is a simple intuitive explanation that makes the result reasonable:
1. For a trial guess at a matched filter, pick a filter impulse response that exactly matches the input signal shape(red) and time reverse it(blue) and then for another trial guess, pick a square impulse response(dotted line).
2. Clearly this (blue) impulse response, which is a mirror image of the signal shape, maximizes the convolution integral(remember the convolution integral is a measure of the overlap) as well as any other shape because it captures the full area; therefore it maximizes the signal output. To be fair, our other trial h(t), the dotted line 'flat' impulse response, also captures the full area of the pulse and also maximizes the signal output, so each filter shape brings in the maximum signal.
3. While the 'dotted line' trial guess does capture all the signal, it also brings in too much noise....
|
|
In the 'flat' response case(dotted line), as the overlap integral is calculated,
some of the time the overlap is just adding noise and no signal
to the output and thus degrading the signal to noise ratio |
4. Choosing a system response that is the mirror image of the input pulse response is optimum because, at overlap it brings in maximum signal and NO extra noise.
|
|
Any other shape would bring in either extra noise or would
miss some signal.
In addition, this shape underweights parts of the signal with
poor SNR and overweights parts of the signal with good SNR.
|
NRZ pulses and Matched Filters
What is the optimum filter for NRZ pulses typically found in a fiberoptic communication system?
1.Since NRZ optical pulses are roughly square in time , the theoretical optimum matched filter for NRZ signals has a square impulse response.
2. What would the output shape look like for an ideal NRZ input pulse convolved with a mirror image, matched filter?
|
|
Red is the impulse response, in the time domain,
of an appropriate matched filter for an NRZ signal.
Green is the NRZ input pulse, also in the time domain.
As before, the yellow area is the convolution or overlap.
The bottom plot is the instant by instant value of the
overlap and thus gives the output pulse shape. |
The output of a Matched Filter for NRZ data is a Triangular Pulse !
Conclusions:
A matched filter yields the optimum SNR for signals with white Gaussian noise.
A matched filter for NRZ data (in the time domain), has a square, mirror image impulse response.
The matched filter output for NRZ data(ie convolution of 'square on square') is a triangular output pulse.
The best decision point is at the point of maximum overlap, top of triangle.
| 
