Re: Comparing matched and Wiener filters

emre wrote:
If you consider the outputs of the two for a binary communication
signal,
the output of the MF looks like triangles with peak amplitudes at the
sampling times; that is, the MF maximizes the SNR only at these
sampling
instances. On the other hand, the output of a Wiener filter, if you
really
wanted to use it, would look more like the transmitted binary
sequence;
this could not be the last stage for a detector, though.
I see what you're saying. However, isn't this just an artifact of the
"filter across all time and sample" viewpoint of matched-filtering, as
opposed to the "integrate-and-dump" viewpoint, where we only calculate
the output for the sampling points (i.e. correlation, equivalent to the
vector formation).

Not really. You are interested in comparing MF vs Wiener, and the outputs
of these two are significantly different. MF does not maximize SNR in the
same sense as Wiener. If you define the SNR like you define MSE, then
SNR_Wiener > SNR_MF.

Perhaps this is the essence of what I need to understand! My current
understanding is that:

MSE = E|x_est - x|^2
SNR = E|x'|^2 / E|v'|^2

where x' is the post-filter signal component, and v' is the post-filter
noise component of x_est (both scalar in my original example), i.e.
x_est = x' + v'.

If we normalise our matched filter (g = h*/||h||^2), then we can be sure
that x' = x (the original data). This leaves v' = x_est - x' = x_est -
x. Therefore:

SNR = S / E|x_est - x|^2

and equivalent to the MSE expression above (they're just
inversely-proportional).

How else would we define these quantities?


If you are thinking of appending an integrator
following the Wiener filter, that's another story.

Bad terminology on my part, I think. I was referring to the general
form of the colloquial "integrate-and-dump", i.e. weighted integration,
i.e. the inner-product with the filter kernel. No second
integration/filter implied!


What is your application? Perhaps the answer to this question would give
us some perspective as to why you are trying to compete these two against
each other.

I have no application in mind, I'm just trying to piece some more of the
puzzle together! Up until now, I've naively assumed that the matched
filter is just a special case of the Wiener/LMMSE formulation, but when
I thought about it and then tried to prove it, I got stuck.

Incidentally, it's possible we're talking at cross purposes here due to
the confusion between the two viewpoints of the operations; one as a
continuous process operating in the time domain, the other as a
"one-off" vector operation (as expressed in my OP). I contend that
these two viewpoints are equivalent if we only look at the time-domain
viewpoint at the sampling instants (i.e. the points of interest), and
treat all other points merely as an implementation artifact. If you
disagree, please correct me!
.

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