# Matched Filter Question comes back

Started by December 1, 2006
```I was following the previous posts on Matched Filters. Very
interesting. Some questions keep haunting me. These may be stupid. I
dont know.

1. If the Transmitter Filter (i.e., Pulse Shaping Filter) G_T(t) is
symmetric, then the matched filter G_R(t) is same as the pulse shaping
filter. Hence for Raised Cosine (RC) pulse shaping filter, the matched
filter is also RC. For RRC pulse shaping, matched filter is also RRC.
But now, if RC is pulse shaping filter and RC is my matched filter,
then I destroy the zero-ISI rule.

So how is the priority determined? Maximize SNR or Zero ISI?  I am
looking at an implementation and noticing that zero ISI is given
importance. Hence, in the implementation I am looking at, if the pulse
shaping filter is RC, then the matched filter is taken to be an impulse
(in sampled time domain).

2. Is there a common matched filter used when the pulse shaping filter
is Gaussian?

I hope one day I will stop getting confused :-)

Sastry

```
```Hi Sastry,

Here's my take on things. I respond below.

> I was following the previous posts on Matched Filters. Very
> interesting. Some questions keep haunting me. These may be stupid. I
> dont know.
>
> 1. If the Transmitter Filter (i.e., Pulse Shaping Filter) G_T(t) is
> symmetric, then the matched filter G_R(t) is same as the pulse shaping
> filter. Hence for Raised Cosine (RC) pulse shaping filter, the matched
> filter is also RC. For RRC pulse shaping, matched filter is also RRC.
> But now, if RC is pulse shaping filter and RC is my matched filter,
> then I destroy the zero-ISI rule.

That is correct.

> So how is the priority determined? Maximize SNR or Zero ISI?

You do both. Don't use a RC for the transmit filter, for precisely the
reason you state. You want the cascade of the tranmit filter, channel
response, and receive filter to satisfy the Nyquist criteria (no
ISI). That's simply all there is to it.
--
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
%%% 919-577-9882                %  Verdi's always creepin' from her room."
%%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO
```
```Hi Randy,
Thats very true. But ideally.

In some practical scenarios like the following :

GMSK --> Transmit filter is Gaussian. So in such a case, I wonder if
different people have different implementations of receive filter.

TFM --> Tamed Frequency Modulated Pulse shaping. I wonder if there is a
standard implementation.

Sastry

Randy Yates wrote:
> Hi Sastry,
>
> Here's my take on things. I respond below.
>
>
> > I was following the previous posts on Matched Filters. Very
> > interesting. Some questions keep haunting me. These may be stupid. I
> > dont know.
> >
> > 1. If the Transmitter Filter (i.e., Pulse Shaping Filter) G_T(t) is
> > symmetric, then the matched filter G_R(t) is same as the pulse shaping
> > filter. Hence for Raised Cosine (RC) pulse shaping filter, the matched
> > filter is also RC. For RRC pulse shaping, matched filter is also RRC.
> > But now, if RC is pulse shaping filter and RC is my matched filter,
> > then I destroy the zero-ISI rule.
>
> That is correct.
>
> > So how is the priority determined? Maximize SNR or Zero ISI?
>
> You do both. Don't use a RC for the transmit filter, for precisely the
> reason you state. You want the cascade of the tranmit filter, channel
> response, and receive filter to satisfy the Nyquist criteria (no
> ISI). That's simply all there is to it.
> --
> %  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
> %% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
> %%% 919-577-9882                %  Verdi's always creepin' from her room."
> %%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO

```
```Sastry wrote:
> I was following the previous posts on Matched Filters. Very
> interesting. Some questions keep haunting me. These may be stupid. I
> dont know.
>
> 1. If the Transmitter Filter (i.e., Pulse Shaping Filter) G_T(t) is
> symmetric, then the matched filter G_R(t) is same as the pulse shaping
> filter. Hence for Raised Cosine (RC) pulse shaping filter, the matched
> filter is also RC. For RRC pulse shaping, matched filter is also RRC.
> But now, if RC is pulse shaping filter and RC is my matched filter,
> then I destroy the zero-ISI rule.

The overall response determines ISI. To get an overall raised-cosine
response, one uses RRC filters at both ends. Since the overall response
is the one-end response squared, overall becomes RC, and all is well.

> So how is the priority determined? Maximize SNR or Zero ISI?  I am
> looking at an implementation and noticing that zero ISI is given
> importance. Hence, in the implementation I am looking at, if the pulse
> shaping filter is RC, then the matched filter is taken to be an impulse
> (in sampled time domain).

RC pulse shaping is wrong for either end. Both ends cascaded should be RC.

> I am confused about priority.
>
> 2. Is there a common matched filter used when the pulse shaping filter
> is Gaussian?

Gaussian. That's what "matched" means.

> I hope one day I will stop getting confused :-)

Patience, my friend. It does come clear in the end.

Jerry
--
Engineering is the art of making what you want from things you can get.
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```
```Sastry wrote:
> Hi Randy,
> Thats very true. But ideally.
>
> In some practical scenarios like the following :
>
> GMSK --> Transmit filter is Gaussian. So in such a case, I wonder if
> different people have different implementations of receive filter.
>
> TFM --> Tamed Frequency Modulated Pulse shaping. I wonder if there is a
> standard implementation.
>
> Sastry
>

You are mixing things up:  the GMSK is a nonlinear modulation scheme
with memory.  So the receiver for a GMSK system is more than just
a matched filter.

I do not know what TFM is.

Now, for the big picture on matched filtering and systems with zero
ISI.
ISI is inter-symbol interference.  This concept is properly applied to
modulation
systems without memory, so that the decoding of one symbol is
independent
of other symbols.  If your modulation is linear, and the channel is
AWGN, then
the optimal receiver is the matched filter.

Now, a great many set of pulse shapes can satisfy the zero-ISI
condition.
For example, the sinc function, and many other interpolating functions.
The
question is, what function do we have such that the convolution of this
function by itself also has zero ISI?  And what function's power
spectrum decay
quickly enough in the frequency and time domain?  These are the reasons
that
the square-root raised cosine is chosen.

```
```Hi Julius,

Even in GMSK, on the transmitter side, I can say,  Step 1 ---> (pulse
shape all the symbols with Gaussian filter) and Step 2 --> Accumulate
all the symbols as phases.

So you are right that we have memory now.

But on the receive side, I can do "Phase Difference" between adjacent
phases and after that I have individual symbols on whom I can apply
match filter. So the question is what can I choose as a match filter ?
One that is Gaussian (Time reversed of transmitter filter) ? or a
filter (such that convolution with transmitter filter is near nyquist)
?

I am having a feeling that I am confusing you guys more than clearly
framing the discussion. Let me know if I am confusing.

Sastry

julius wrote:
> Sastry wrote:
> > Hi Randy,
> > Thats very true. But ideally.
> >
> > In some practical scenarios like the following :
> >
> > GMSK --> Transmit filter is Gaussian. So in such a case, I wonder if
> > different people have different implementations of receive filter.
> >
> > TFM --> Tamed Frequency Modulated Pulse shaping. I wonder if there is a
> > standard implementation.
> >
> > Sastry
> >
>
> You are mixing things up:  the GMSK is a nonlinear modulation scheme
> with memory.  So the receiver for a GMSK system is more than just
> a matched filter.
>
> I do not know what TFM is.
>
> Now, for the big picture on matched filtering and systems with zero
> ISI.
> ISI is inter-symbol interference.  This concept is properly applied to
> modulation
> systems without memory, so that the decoding of one symbol is
> independent
> of other symbols.  If your modulation is linear, and the channel is
> AWGN, then
> the optimal receiver is the matched filter.
>
> Now, a great many set of pulse shapes can satisfy the zero-ISI
> condition.
> For example, the sinc function, and many other interpolating functions.
>  The
> question is, what function do we have such that the convolution of this
> function by itself also has zero ISI?  And what function's power
> spectrum decay
> quickly enough in the frequency and time domain?  These are the reasons
> that
> the square-root raised cosine is chosen.

```
```Sastry wrote:
> Hi Julius,
>
> Even in GMSK, on the transmitter side, I can say,  Step 1 ---> (pulse
> shape all the symbols with Gaussian filter) and Step 2 --> Accumulate
> all the symbols as phases.
>
> So you are right that we have memory now.
>
> But on the receive side, I can do "Phase Difference" between adjacent
> phases and after that I have individual symbols on whom I can apply
> match filter. So the question is what can I choose as a match filter ?
> One that is Gaussian (Time reversed of transmitter filter) ? or a
> filter (such that convolution with transmitter filter is near nyquist)
> ?
>
> I am having a feeling that I am confusing you guys more than clearly
> framing the discussion. Let me know if I am confusing.
>
> Sastry
>

OK, let me see if I can understand where you are stuck.

The role of the Gaussian filter for the phase information is to smooth
out
the phase *trajectory* of the transmitted signal.  So in the strict
sense
if you can get the phase information back at the receiver, assuming
perfect synchronization, the optimal demodulator is a sequence
estimator.

So I disagree with what you say above that you can get individual
symbols.
What are you getting at, I think, is a suboptimal scheme.  You can
compute or estimate how the trajectory of the phase will change as you
apply a Gaussian filter at the output of the phase estimator at the
and then do a sequence estimation procedure based on this new
trajectory.

Think about it this way:  a Gaussian filter induces ISI, and so does a
Gaussian filter convolved with another Gaussian filter.  Given that you
know
that you have ISI, you can always do sequence estimation if you can
derive
a trellis model of your entire system.

I would say that your question is the same as if somebody were to ask
how
to demodulate a signal where there is ISI, except that you work
entirely in
the phase domain.  And like I said, this is fine if you assume perfect
synchronization between the transmitter and the receiver.  What is the