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### Pulse in white noise

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```What is the best filter for a pulse in white noise? I remember in the
depths of the past reading something about a filter with a reverse-
time impulse response of the pulse which you convolve it with, but
this is just an integrator.

Hardy
```
 0
Reply gyansorova (854) 6/7/2012 7:42:41 AM

```On Jun 7, 7:42=A0pm, HardySpicer <gyansor...@gmail.com> wrote:
> What is the best filter for a pulse in white noise? I remember in the
> depths of the past reading something about a filter with a reverse-
> time impulse response of the pulse which you convolve it with, but
> this is just an integrator.
>
> Hardy

ok it's a matched filter. How do you implement this - say in analogue?
I imagine just a leaky integrator?

Hardy
```
 0
Reply gyansorova (854) 6/7/2012 8:30:02 AM

```On Thu, 07 Jun 2012 01:30:02 -0700, HardySpicer wrote:

> On Jun 7, 7:42 pm, HardySpicer <gyansor...@gmail.com> wrote:
>> What is the best filter for a pulse in white noise? I remember in the
>> depths of the past reading something about a filter with a reverse-
>> time impulse response of the pulse which you convolve it with, but this
>> is just an integrator.
>>
>> Hardy
>
> ok it's a matched filter. How do you implement this - say in analogue?
> I imagine just a leaky integrator?
>
> Hardy

Does a leaky integrator's impulse response look like your pulse?  Betcha
could do better...
```
 0

```On Jun 7, 4:30=A0am, HardySpicer <gyansor...@gmail.com> wrote:
> On Jun 7, 7:42=A0pm, HardySpicer <gyansor...@gmail.com> wrote:
>
> > What is the best filter for a pulse in white noise? I remember in the
> > depths of the past reading something about a filter with a reverse-
> > time impulse response of the pulse which you convolve it with, but
> > this is just an integrator.
>
> > Hardy
>
> ok it's a matched filter. How do you implement this - say in analogue?
> I imagine just a leaky integrator?
>
> Hardy

Many matched filters are implemented as correlators,
that is, active circuits with switches that are variations
of integrate-and-dump circuits, and not as purely passive
(linear time-invariant) circuits, whether in the analog or
the digital domain.

Dilip Sarwate
```
 0
Reply dvsarwate (263) 6/7/2012 11:50:51 AM

```On 6/7/12 7:50 AM, dvsarwate wrote:
> On Jun 7, 4:30 am, HardySpicer<gyansor...@gmail.com>  wrote:
>> On Jun 7, 7:42 pm, HardySpicer<gyansor...@gmail.com>  wrote:
>>
>>> What is the best filter for a pulse in white noise? I remember in the
>>> depths of the past reading something about a filter with a reverse-
>>> time impulse response of the pulse which you convolve it with, but
>>> this is just an integrator.
>>
>> ok it's a matched filter. How do you implement this - say in analogue?
>> I imagine just a leaky integrator?
>
> Many matched filters are implemented as correlators,
> that is, active circuits with switches that are variations
> of integrate-and-dump circuits, and not as purely passive
> (linear time-invariant) circuits, whether in the analog or
> the digital domain.

first post that isn't answered by two words (and not "leaky integrator")

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (3908) 6/7/2012 5:04:35 PM

```robert bristow-johnson <rbj@audioimagination.com> writes:

> Hardy, is there something that i'm missing about your question in your
> first post that isn't answered by two words (and not "leaky
> integrator") in your second post?

"I imagine"?

> "Imagination is more important than knowledge."
--
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com
```
 0
Reply yates9428 (362) 6/7/2012 5:51:38 PM

```On Jun 8, 5:04=A0am, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On 6/7/12 7:50 AM, dvsarwate wrote:
>
>
>
>
>
>
>
>
>
> > On Jun 7, 4:30 am, HardySpicer<gyansor...@gmail.com> =A0wrote:
> >> On Jun 7, 7:42 pm, HardySpicer<gyansor...@gmail.com> =A0wrote:
>
> >>> What is the best filter for a pulse in white noise? I remember in the
> >>> depths of the past reading something about a filter with a reverse-
> >>> time impulse response of the pulse which you convolve it with, but
> >>> this is just an integrator.
>
> >> ok it's a matched filter. How do you implement this - say in analogue?
> >> I imagine just a leaky integrator?
>
> > Many matched filters are implemented as correlators,
> > that is, active circuits with switches that are variations
> > of integrate-and-dump circuits, and not as purely passive
> > (linear time-invariant) circuits, whether in the analog or
> > the digital domain.
>
> Hardy, is there something that i'm missing about your question in your
> first post that isn't answered by two words (and not "leaky integrator")
>
> --
>
> r b-j =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0r...@audioimagination.com
>
> "Imagination is more important than knowledge."

Sorry, I remembered after I had posted it what the name of the filter
was. I was unsure of the implementation. The literature
just has block diagrams. For instance, is it commonplace in the
digital domain to use FFTs for the convolution or just time-domain
methods. For example, to work out cross-correlation you can convolve
with the time-reversed impulse response which is what I need. To get
this I could computed cross spectral density (cross-periodogram) and
inverse FFT. I am wondering what the most common approach is since in
the books they only seem to cover the theory and not implmentation
issues.

Hardy
```
 0
Reply gyansorova (854) 6/7/2012 6:49:30 PM

```On 6/7/12 1:51 PM, Randy Yates wrote:
> robert bristow-johnson<rbj@audioimagination.com>  writes:
>
>> Hardy, is there something that i'm missing about your question in your
>> first post that isn't answered by two words (and not "leaky
>> integrator") in your second post?
>
> "I imagine"?
>
>> "Imagination is more important than knowledge."

LOL! (after initial confusion)

question in the second post about implementing a match filter in the
analog[ue] world.

i suppose, Hardy, to implement a matched filter in the analog world
would be to first, F.T. (DFT, whatever) the pulse you're trying to pick
out and design an analog filter with a frequency response that has about
the same magnitude response and a phase response that is the negative of
the phase component of the pulse spectrum with some necessary constant
delay or linear phase term added.

essentially, i never took a course in analog filters where we designed
directly for a *specific* impulse response.  sometimes given a prototype
impulse response that is the sum of various decaying exponentials and
damped sinusoids, i think we *have* designed for specific features in
the impulse response (like its integral, the step response having a
limited overshoot).

in the digital world, we just use an FIR with impulse response that is
proportional to the time-reversed copy of the pulse we're trying to
detect.  with simple additive white noise, that is pretty much the end
result of what a matched filter is in the digital domain.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (3908) 6/7/2012 6:57:27 PM

```our most recent posts "crossed in the mail".

On 6/7/12 2:49 PM, HardySpicer wrote:
>
> Sorry,

none needed, but if i am confused, i start asking basic questions.

> I remembered after I had posted it what the name of the filter
> was. I was unsure of the implementation. The literature
> just has block diagrams. For instance, is it commonplace in the
> digital domain to use FFTs for the convolution or just time-domain
> methods.

if it's a really, really long pulse, lotsa samples, convolution in the
frequency domain is the efficient way to do it.  but if the pulse (or
your approximation to it) is shorter than a few dozen samples, i think
the matched filter is likely to be a simple FIR in the time-domain.

one nifty use of truncated IIR filters (TIIR), which are a form of FIR
filter but implemented using recursive means (a moving-sum or
moving-average filter is maybe the simplest non-trivial example) is that
if the pulse you're trying to detect is some damped sinusoid in time
(which, seems to me, might be often found in nature or in physical
systems), you can approximated that damped sinusoid with a finite-length
replica and implement a finite-length time reversed copy (which has
exponentially increasing amplitude, at least for a while) efficiently
with a TIIR.

> For example, to work out cross-correlation you can convolve
> with the time-reversed impulse response which is what I need. To get
> this I could computed cross spectral density (cross-periodogram)

i think that's the F.T. of the cross-correlation, right?

> and inverse FFT.

then you get cross-correlation, then you look for peaks in that?

> I am wondering what the most common approach is since in
> the books they only seem to cover the theory and not implementation
> issues.

well, it's gonna depend on the shape and the length of the pulse you're
trying to detect that's buried in white noise.  probably, 90%+ of the
time it's done with a simple FIR.  i could be wrong about that percentage.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (3908) 6/7/2012 7:20:15 PM

```On Jun 8, 7:20=A0am, robert bristow-johnson <r...@audioimagination.com>
wrote:
> our most recent posts "crossed in the mail".
>
> On 6/7/12 2:49 PM, HardySpicer wrote:
>
>
>
> > Sorry,
>
> none needed, but if i am confused, i start asking basic questions.
>
> > I remembered after I had posted it what the name of the filter
> > was. I was unsure of the implementation. The literature
> > just has block diagrams. For instance, is it commonplace in the
> > digital domain to use FFTs for the convolution or just time-domain
> > methods.
>
> if it's a really, really long pulse, lotsa samples, convolution in the
> frequency domain is the efficient way to do it. =A0but if the pulse (or
> your approximation to it) is shorter than a few dozen samples, i think
> the matched filter is likely to be a simple FIR in the time-domain.
>
> one nifty use of truncated IIR filters (TIIR), which are a form of FIR
> filter but implemented using recursive means (a moving-sum or
> moving-average filter is maybe the simplest non-trivial example) is that
> if the pulse you're trying to detect is some damped sinusoid in time
> (which, seems to me, might be often found in nature or in physical
> systems), you can approximated that damped sinusoid with a finite-length
> replica and implement a finite-length time reversed copy (which has
> exponentially increasing amplitude, at least for a while) efficiently
> with a TIIR.
>
> > For example, to work out cross-correlation you can convolve
> > with the time-reversed impulse response which is what I need. To get
> > this I could computed cross spectral density (cross-periodogram)
>
> i think that's the F.T. of the cross-correlation, right?
>
> > and inverse FFT.
>
> then you get cross-correlation, then you look for peaks in that?
>
> > I am wondering what the most common approach is since in
> > the books they only seem to cover the theory and not implementation
> > issues.
>
> well, it's gonna depend on the shape and the length of the pulse you're
> trying to detect that's buried in white noise. =A0probably, 90%+ of the
> time it's done with a simple FIR. =A0i could be wrong about that percenta=
ge.
>
> --
>
> r b-j =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0r...@audioimagination.com
>
> "Imagination is more important than knowledge."

ok thanks for that
```
 0
Reply gyansorova (854) 6/7/2012 8:00:45 PM

```On Thu, 7 Jun 2012 01:30:02 -0700 (PDT), HardySpicer
<gyansorova@gmail.com> wrote:

>On Jun 7, 7:42=A0pm, HardySpicer <gyansor...@gmail.com> wrote:
>> What is the best filter for a pulse in white noise? I remember in the
>> depths of the past reading something about a filter with a reverse-
>> time impulse response of the pulse which you convolve it with, but
>> this is just an integrator.
>>
>> Hardy
>
>ok it's a matched filter. How do you implement this - say in analogue?
>I imagine just a leaky integrator?
>
>Hardy

An integrator, specifically an integrate-and-dump filter, is the
matched filter for receiving a rectangular pulse.    Is this what

If so, such an integrator is not that hard to build with analog
components.

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
 0
Reply eric.jacobsen (2385) 6/7/2012 8:25:41 PM

```On 6/7/2012 12:42 AM, HardySpicer wrote:
> What is the best filter for a pulse in white noise? I remember in the
> depths of the past reading something about a filter with a reverse-
> time impulse response of the pulse which you convolve it with, but
> this is just an integrator.
>
>
> Hardy

"Best" depends on the objectve:

a matched filter is best for peak detection / i.e. presence.

something else might be best for getting a clean eye pattern as in PAM -
particularly with respect to location of zero crossings of the filter
output.

Fred
```
 0

```On Thu, 07 Jun 2012 00:42:41 -0700, HardySpicer wrote:

> What is the best filter for a pulse in white noise? I remember in the
> depths of the past reading something about a filter with a reverse- time
> impulse response of the pulse which you convolve it with, but this is
> just an integrator.

If the duration of the pulse is known then you want a filter with an
impulse response is on _for that duration_.  If you don't know _when_ the
pulse is going to happen (i.e. radar), then the filter is hard to
implement and you need to think about spending lots of \$\$ (for fancy
things like delay lines) or you need to think about approximating the
response (with a 1st, 2nd, or 3rd order lowpass filter, probably).

If the _time_ of the pulse is known as well as it's duration, then an
integrate-and-dump will work, as mentioned.

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
```
 0
Reply tim177 (4404) 6/7/2012 11:32:35 PM

```On 6/7/12 4:25 PM, Eric Jacobsen wrote:
> On Thu, 7 Jun 2012 01:30:02 -0700 (PDT), HardySpicer
> <gyansorova@gmail.com>  wrote:
>
>> On Jun 7, 7:42=A0pm, HardySpicer<gyansor...@gmail.com>  wrote:
>>> What is the best filter for a pulse in white noise? I remember in the
>>> depths of the past reading something about a filter with a reverse-
>>> time impulse response of the pulse which you convolve it with, but
>>> this is just an integrator.
>>
>> ok it's a matched filter. How do you implement this - say in analogue?
>> I imagine just a leaky integrator?
>
> An integrator, specifically an integrate-and-dump filter, is the
> matched filter for receiving a rectangular pulse.    Is this what

Eric, is an integrate-and-dump filter one that has a rectangular pulse
for its impulse response?  just not sure about terminology.

>
> If so, such an integrator is not that hard to build with analog
> components.

what do you use for an analog delay element?  some RC-ladder APF thingie
or CCD?

__

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (3908) 6/8/2012 12:40:07 AM

```On Jun 8, 8:25=A0am, eric.jacob...@ieee.org (Eric Jacobsen) wrote:
> On Thu, 7 Jun 2012 01:30:02 -0700 (PDT), HardySpicer
>
> <gyansor...@gmail.com> wrote:
> >On Jun 7, 7:42=3DA0pm, HardySpicer <gyansor...@gmail.com> wrote:
> >> What is the best filter for a pulse in white noise? I remember in the
> >> depths of the past reading something about a filter with a reverse-
> >> time impulse response of the pulse which you convolve it with, but
> >> this is just an integrator.
>
> >> Hardy
>
> >ok it's a matched filter. How do you implement this - say in analogue?
> >I imagine just a leaky integrator?
>
> >Hardy
>
> An integrator, specifically an integrate-and-dump filter, is the
> matched filter for receiving a rectangular pulse. =A0 =A0Is this what
>
> If so, such an integrator is not that hard to build with analog
> components.
>
> Eric Jacobsen
> Anchor Hill Communicationswww.anchorhill.com

aye. But an integrator tends to wander off at the slightest hint of
dc, hence the leaky integrator ie a big resistor across the capacitor.

Hardy
```
 0
Reply gyansorova (854) 6/8/2012 2:13:46 AM

```On Jun 7, 8:40=A0pm, robert bristow-johnson <r...@audioimagination.com>
wrote:

>
> Eric, is an integrate-and-dump filter one that has a rectangular pulse
> for its impulse response? =A0just not sure about terminology.

An integrate-and-dump (or better yet, an
integrate-sample-and.THEN.dump.,not.before)
"filter" is not a linear time-invariant
system but a time-varying system.  If it
integrates over T-second periods (e.g.
integrate over (0+,T-), sample at T, dump
at T+, and lather-rinse-repeat similarly
every T seconds, then it will respond to
an impulse at t =3D 0 with a rectangular pulse
lasting from 0++ to T. It will respond
to an impulse at t =3D 0.25T with a rectangular
pulse that lasts from t =3D 0.25T+ to T, etc.

Dilip Sarwate

```
 0
Reply dvsarwate (263) 6/8/2012 3:50:04 AM

```On Jun 7, 2:57=A0pm, robert bristow-johnson <r...@audioimagination.com>
wrote:

>
> in the digital world, we just use an FIR with impulse response that is
> proportional to the time-reversed copy of the pulse we're trying to
> detect. =A0with simple additive white noise, that is pretty much the end
> result of what a matched filter is in the digital domain.

So, if the pulse is rectangular, the FIR has impulse
response H(z) =3D 1 + z^-1 + z^-2 + ... z^-n ?

Dilip Sarwate
```
 0
Reply dvsarwate (263) 6/8/2012 3:53:11 AM

```On 6/7/12 11:53 PM, dvsarwate wrote:
> On Jun 7, 2:57 pm, robert bristow-johnson<r...@audioimagination.com>
> wrote:
>
>>
>> in the digital world, we just use an FIR with impulse response that is
>> proportional to the time-reversed copy of the pulse we're trying to
>> detect.  with simple additive white noise, that is pretty much the end
>> result of what a matched filter is in the digital domain.
>
>
> So, if the pulse is rectangular, the FIR has impulse
> response H(z) = 1 + z^-1 + z^-2 + ... z^-n ?

which is a moving-sum (a scaled moving-average) and can be implemented
with an integrator and delay (of n samples) and subtractor, instead of
adding up all those terms.  isn't this what Eric means by "integrate and
dump"?

On 6/7/12 11:50 PM, dvsarwate wrote:
> On Jun 7, 8:40 pm, robert bristow-johnson<r...@audioimagination.com>
> wrote:
>
>>
>> Eric, is an integrate-and-dump filter one that has a rectangular pulse
>> for its impulse response?  just not sure about terminology.
>
>
> An integrate-and-dump (or better yet, an
> integrate-sample-and.THEN.dump.,not.before)
> "filter" is not a linear time-invariant
> system but a time-varying system.

that's what i kinda thought.  but what triggers the dump?  just the
arrival of time nT (where n is an integer)?

what if the pulse we're trying to detect straddles the time nT?

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
 0
Reply rbj (3908) 6/8/2012 5:24:55 AM

```On Fri, 08 Jun 2012 01:24:55 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On 6/7/12 11:53 PM, dvsarwate wrote:
>> On Jun 7, 2:57 pm, robert bristow-johnson<r...@audioimagination.com>
>> wrote:
>>
>>>
>>> in the digital world, we just use an FIR with impulse response that is
>>> proportional to the time-reversed copy of the pulse we're trying to
>>> detect.  with simple additive white noise, that is pretty much the end
>>> result of what a matched filter is in the digital domain.
>>
>>
>> So, if the pulse is rectangular, the FIR has impulse
>> response H(z) = 1 + z^-1 + z^-2 + ... z^-n ?
>
>which is a moving-sum (a scaled moving-average) and can be implemented
>with an integrator and delay (of n samples) and subtractor, instead of
>adding up all those terms.  isn't this what Eric means by "integrate and
>dump"?

No, usually an I&D filter in a comm system is implemented strictly as
an integrator that accumulates for the symbol period, is sampled, and
then reset to integrate again for the next symbol period.

This requires synchronization with the symbol period.   If the time of
arrival of the pulse is unknown for a single-pulse detection system,
then you can subtract off old samples as you suggest or run N I&D
filters in parallel with different delays.

>On 6/7/12 11:50 PM, dvsarwate wrote:
>> On Jun 7, 8:40 pm, robert bristow-johnson<r...@audioimagination.com>
>> wrote:
>>
>>>
>>> Eric, is an integrate-and-dump filter one that has a rectangular pulse
>>> for its impulse response?  just not sure about terminology.
>>
>>
>> An integrate-and-dump (or better yet, an
>> integrate-sample-and.THEN.dump.,not.before)
>> "filter" is not a linear time-invariant
>> system but a time-varying system.
>
>that's what i kinda thought.  but what triggers the dump?  just the
>arrival of time nT (where n is an integer)?

In a comm system the timing recovery loop steers the integrator period
to align with the symbol period.

>what if the pulse we're trying to detect straddles the time nT?

An I&D is only matched to a pulse with a duration matching the
integration time.  So, as mentioned, you can either subtract off old
samples as the window slides along (for a digital implementation) or
run N filters in parallel with time offsets small enough to get
whatever time resolution is required.

It's not all that elegant, but the point is that an I&D filter is the
match (in a matched filter sense) to a rectangular pulse.   It sounded
like that was appropriate for the problem at hand but I'm not sure.

Hardy mentioned that DC offset is a concern, which it may be if the
pulse is very long.

>--
>
>r b-j                  rbj@audioimagination.com
>
>"Imagination is more important than knowledge."
>
>

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
 0
Reply eric.jacobsen (2385) 6/8/2012 3:12:43 PM

```On Thu, 07 Jun 2012 20:25:41 +0000, Eric Jacobsen wrote:

> On Thu, 7 Jun 2012 01:30:02 -0700 (PDT), HardySpicer
> <gyansorova@gmail.com> wrote:
>
>>On Jun 7, 7:42=A0pm, HardySpicer <gyansor...@gmail.com> wrote:
>>> What is the best filter for a pulse in white noise? I remember in the
>>> depths of the past reading something about a filter with a reverse-
>>> time impulse response of the pulse which you convolve it with, but
>>> this is just an integrator.
>>>
>>> Hardy
>>
>>ok it's a matched filter. How do you implement this - say in analogue? I
>>imagine just a leaky integrator?
>>
>>Hardy
>
> An integrator, specifically an integrate-and-dump filter, is the matched
> filter for receiving a rectangular pulse.    Is this what you're asking
>
> If so, such an integrator is not that hard to build with analog
> components.
>
>
> Eric Jacobsen
> Anchor Hill Communications
> www.anchorhill.com

I assumed - perhaps Hardy could fill us in - that we don't have the
synchronization needed for this; and that he was looking for a cheap
approximation, maybe something a bit better than a  leaky integrator (one
reactive component) but not necessarily approaching something in some
sense "optimal".

A quite-low-Q bandpass filter (two reactive components) would be the next
step up in approaching a rectangular-looking impulse response.  Just take
the FT of the time domain response, do a Pade approximation on the
transform (carefully choosing how many terms you want) and design your
circuit from there.  Others have already mentioned methods for generating
more DSP-ish methods.

I've used a similar (analog) technique in designing an ECG waveform
discriminator, though that was years ago.  In that system the various
signal distortion mechanisms were wrapped into the theoretical ECG
waveform before the approximation step.  The final result was a
considerable improvement in rejecting motion artifacts and other noise
contributions, and sufficiently simple and low-power for the application.
```
 0

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