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Sun Dec 2 21:22:33 CET 2007

## matched filter

differential BPSK data stream:
. .___. . .___.
\ / \ / \ /
X X X
./ \.___./ \./ \.___.
1 0 1 1 0
Using cosine crossfading as implemented in modulator.f is effectively
the same as using symbols 2 baud periods wide with a 1 + cos
envelope. this wavelet is the output filter which maps a binary +1,-1
PCM signal to the shaped BPSK signal.
This output filter needs to be matched in the receiver.
Now, about matched filters..
A matched filter in the presence of additive white gaussian noise is
just the time-reverse of the wavelet: one projects the observed signal
vector onto the 1D space spanned by the wavelet's vector in signal
space. This gets rid of all the disturbances orthogonal to the
wavelet's subspace.
When the noise is not white, the noise statistics are used to compute
an optimal subspace to project onto, such that most of the noise will
still vanish.
I don't have noise statistics, and I'm not going to use any online
estimation, which leaves me to plain and simple convolution with the
time-reversed signal.
I do wonder what all this talk is about 'designing' matched filters
for PSK31...

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