Matched Filters in Digital Communication Systems

Hi People,

I'm confused regarding matched filters in digital communication
systems.

In Lee and Messerschmitt, it is shown how a narrow receive filter
reduces the noise power at the slicer. This makes lots of intuitive
sense - a narrower filter lets through less noise. Of course the
natural inference is that less noise at the slicer will result in
fewer decision errors.

In Proakis (section 5.1.2) it is shown that the filter which optimizes
the SNR is the matched filter, i.e., the filter that matches the pulse
shape of the transmitted signal. I follow the math just fine - it
makes sense from a math point of view. Again it is inferred that
this is the best filter for the slicer since the SNR is max'ed.

However, these two perspectives seem to collide with one another when
the matched filter doesn't do a very good job of filtering out the
noise. That is, what if the transmit pulse shape has a wideband power
spectrum? Then it would be good to have a wideband filter since it
matches the pulse shape, but it would be good to have a narrow-band
filter since it limits the noise. Well which is it?

Any light on the subject would be greatly appreciated.

--Randy

Randy Yates wrote:
> Hi People,
>
> I'm confused regarding matched filters in digital communication
> systems.
>
> In Lee and Messerschmitt, it is shown how a narrow receive filter
> reduces the noise power at the slicer. This makes lots of intuitive
> sense - a narrower filter lets through less noise. Of course the
> natural inference is that less noise at the slicer will result in
> fewer decision errors.
>
> In Proakis (section 5.1.2) it is shown that the filter which optimizes
> the SNR is the matched filter, i.e., the filter that matches the pulse
> shape of the transmitted signal. I follow the math just fine - it
> makes sense from a math point of view. Again it is inferred that
> this is the best filter for the slicer since the SNR is max'ed.
>
> However, these two perspectives seem to collide with one another when
> the matched filter doesn't do a very good job of filtering out the
> noise. That is, what if the transmit pulse shape has a wideband power
> spectrum? Then it would be good to have a wideband filter since it
> matches the pulse shape, but it would be good to have a narrow-band
> filter since it limits the noise. Well which is it?
>
> Any light on the subject would be greatly appreciated.
>
> --Randy

A narrow receive filter will let in less noise, but it will also let in
less of the signal itself. If the signal is wideband and has
significant components outside the filter's bandwidth, using a
narrowband filter will result in signal power being filtered out. So,
it's a question of trading off how much of the signal power you want to
let in versus how much of noise you're willing to let in in the
process. The matched filter does the best job in this SNR tradeoff (for
Gaussian IID noise). It bandlimits the receive filter to the bandwidth
of the transmitted pulse so that most of the significant signal energy
comes through.

- Ravi Srikantiah

"Randy Yates" <yates@ieee.org> wrote in message 
news:1153460776.869435.258030@m73g2000cwd.googlegroups.com...
> Hi People,
>
> I'm confused regarding matched filters in digital communication
> systems.
>
> In Lee and Messerschmitt, it is shown how a narrow receive filter
> reduces the noise power at the slicer. This makes lots of intuitive
> sense - a narrower filter lets through less noise. Of course the
> natural inference is that less noise at the slicer will result in
> fewer decision errors.
>
> In Proakis (section 5.1.2) it is shown that the filter which optimizes
> the SNR is the matched filter, i.e., the filter that matches the pulse
> shape of the transmitted signal. I follow the math just fine - it
> makes sense from a math point of view. Again it is inferred that
> this is the best filter for the slicer since the SNR is max'ed.
>
> However, these two perspectives seem to collide with one another when
> the matched filter doesn't do a very good job of filtering out the
> noise. That is, what if the transmit pulse shape has a wideband power
> spectrum? Then it would be good to have a wideband filter since it
> matches the pulse shape, but it would be good to have a narrow-band
> filter since it limits the noise. Well which is it?
>
> Any light on the subject would be greatly appreciated.
>

Randy,

Maybe this will help:

You've already grasped the math of the matched filter so, in a way, that 
should tell the story.  But, it's not necessarily intuitive.

Consider that a matched filter and a correlation processor can be the same 
thing.  That is, the correlator's impulse response is the same as that of a 
matched filter.

Now, assume that the waveform is complicated so that correlation not only is 
done as a matched filter but will be required because the waveform is 
complicated - such as with broadband random sequences.  Does that help?

In a way, the perspective changes because one is in the frequency domain 
(filtering) and the other is in the time domain (correlation).

Here's a simple but different example:
The waveform is the sum of 3 sinusoids with particular temporal 
relationships and in pulses of particular width / thus bandwidth each.
So, a matched filter would look something like 3 bandpass filters in 
parallel.
The total bandwidth is "bigger" because of the 2 "added" sinusoids and their 
filters.  So, the noise received is higher.  But then, so is the signal 
received!  Thus the math.

Fred 

Randy Yates wrote:
> Hi People,
> 
> I'm confused regarding matched filters in digital communication
> systems.
> 
> In Lee and Messerschmitt, it is shown how a narrow receive filter
> reduces the noise power at the slicer. This makes lots of intuitive
> sense - a narrower filter lets through less noise. Of course the
> natural inference is that less noise at the slicer will result in
> fewer decision errors.
> 
> In Proakis (section 5.1.2) it is shown that the filter which optimizes
> the SNR is the matched filter, i.e., the filter that matches the pulse
> shape of the transmitted signal. I follow the math just fine - it
> makes sense from a math point of view. Again it is inferred that
> this is the best filter for the slicer since the SNR is max'ed.
> 
> However, these two perspectives seem to collide with one another when
> the matched filter doesn't do a very good job of filtering out the
> noise. That is, what if the transmit pulse shape has a wideband power
> spectrum? Then it would be good to have a wideband filter since it
> matches the pulse shape, but it would be good to have a narrow-band
> filter since it limits the noise. Well which is it?
> 
> Any light on the subject would be greatly appreciated.
> 
> --Randy
> 
As already stated, a filter that's narrower than the signal would cut 
off signal as well as noise, to the detriment of SNR.

Consider a spread-spectrum application -- the signal is broadened, but 
has a specific (if obscure) correlation property.  A matched filter will 
minimize the noise received, even if the filter itself appears to be broad.

Consider also that all this math is done with the assumption of white 
noise -- if your interfering signal is a tone smack in the middle of 
your desired signal then the best filter is no longer a matched filter, 
it's some weird thing with a honking big notch on the tone.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Thank you Ravi, Fred, and Tim.

Is the matched filter, then, the Wiener filter when
the noise is white?

--Randy

>Thank you Ravi, Fred, and Tim.
>
>Is the matched filter, then, the Wiener filter when
>the noise is white?
>
>--Randy
>
>

No, the matched filter (MF) is a maximum likelihood scheme if the noise a
AWGN. The Wiener filter is a LMMSE scheme, for which you need to know the
noise variance at least. MF does not need this.

lanbaba wrote:
> >Thank you Ravi, Fred, and Tim.
> >
> >Is the matched filter, then, the Wiener filter when
> >the noise is white?
> >
> >--Randy
> >
> >
>
> No, the matched filter (MF) is a maximum likelihood scheme if the noise a
> AWGN.

In Proakis, the matched filter is one step along the way towards
Maximum Likelihood detection, but the MF isn't a "maximum likelihood
scheme" itself. It is simply a scheme that maximizes the output
SNR.

> The Wiener filter is a LMMSE scheme, for which you need to know the
> noise variance at least. MF does not need this.

Haykin shows (problem 7, chapter 3) that if the noise is Gauusian,
independent of the signal, zero-mean, and has a positive-definite
correlation matrix, then the minimum mean-squared error and
maximum SNR  yield the same filter.

--Randy

Randy Yates wondered:
.
>
> In Lee and Messerschmitt, it is shown how a narrow receive filter
> reduces the noise power at the slicer. This makes lots of intuitive
> sense - a narrower filter lets through less noise. Of course the
> natural inference is that less noise at the slicer will result in
> fewer decision errors.
>
> In Proakis (section 5.1.2) it is shown that the filter which optimizes
> the SNR is the matched filter, i.e., the filter that matches the pulse
> shape of the transmitted signal. I follow the math just fine - it
> makes sense from a math point of view. Again it is inferred that
> this is the best filter for the slicer since the SNR is max'ed.
>
> However, these two perspectives seem to collide with one another when
> the matched filter doesn't do a very good job of filtering out the
> noise. That is, what if the transmit pulse shape has a wideband power
> spectrum? Then it would be good to have a wideband filter since it
> matches the pulse shape, but it would be good to have a narrow-band
> filter since it limits the noise. Well which is it?

Let's take a simplistic example.  The received signal x(t) is a sinc
function with Fourier transform X(w) = A for |w| < W, and 0 otherwise.
The filter has transfer function H(w) = 1 for |w| < B, and 0 otherwise.

Then, if B = W, we have a matched filter for the signal; otherwise,
the filter is overly broad or too narrow.

The signal output is y(t) whose Fourier transform is X(w)H(w),
and the sampling instant is t = 0.  Then, the signal sample
value is y(0) = area under the curve X(w)H(w).  We thus
have y(0) = 2AB if B < W and y(0) = 2AW if B > W.

The noise at the filter input is white Gaussian with two-sided
power spectral density N.  Then, the sample at t = 0 includes
a Gaussian random variable of mean 0 and variance NB
regardless of whether B < W or B > W.

For BPSK, the error probability is
Q(y(0)/sqrt(NB)) = Q(2Asqrt(B/N)) if B < W and Q(2AW/sqrt(NB))
if B > W.  Thus, as B increases to W, the argument of Q
increases.  As B increases past W, the argument of Q decreases.
As noted by others in this discussion, the increasing bandwidth
does let more noise in but also grabs more of the signal.  The
latter overpowers the noise increase.  Once the filter has grabbed
all the signal, increasing the bandwidth further increases the
noise while the signal power remains the same;  not desirable
at all.

Hope this helps

--Dilip Sarwate

Randy,

I don't know if this helps, but this is how I explain a matched filter
to someone and also make sense of it to myself.

The matched filter is a replica of the ideal waveform.  This has the
effect of emphasising the bigger parts of the waveform while giving
less weight to the smaller parts.  Since one would expect larger parts
to have a higher SNR than the smaller parts (on average) this maximizes
the SNR at the filters output.  (Maximum ratio combining.)

I hope that this helps.

Phil

Phil wrote:

> The matched filter is a replica of the ideal waveform.  This has the
> effect of emphasising the bigger parts of the waveform while giving
> less weight to the smaller parts.  Since one would expect larger parts
> to have a higher SNR than the smaller parts (on average) this maximizes
> the SNR at the filters output.  (Maximum ratio combining.)


Yes indeed!  In fact, since x^2 > x for x > 1 and x^2 < x for x < 1,
the parts where the signal is large are emphasized while the parts
where the signal is small are de-emphasized.  This also works in
the frequency domain with the additional twist that the phase shift
is compensated for (the matched filter has the conjugate phase)
so that all the "sinusoids" comprising the signal are lined up to
simultaneously attain their maximum at the sampling instant.

--Dilip Sarwate