From a recent discussion here:

> >if i could generate some coefficents that had a 'negative' group delay for
> >a period of time, would you think that 'phase cloning' was new and
> >intersting??
>
> A time machine would be pretty revolutionary, yes.
>
> Negative group delay means that the output appears before the input
> arrives.

Fascinating concept, isn't it? I was curious enough to dig into the
topic for a while and write up what I found out. You can read about it
here:

http://www.dsprelated.com/showarticle/54.php

Regards,
Andor

On Mar 7, 5:18=A0pm, Andor <andor.bari...@gmail.com> wrote:
> From a recent discussion here:
>
> > >if i could generate some coefficents that had a 'negative' group delay =
for
> > >a period of time, would you think that 'phase cloning' was new and
> > >intersting??
>
> > A time machine would be pretty revolutionary, yes.
>
> > Negative group delay means that the output appears before the input
> > arrives.
>
> Fascinating concept, isn't it? I was curious enough to dig into the
> topic for a while and write up what I found out. You can read about it
> here:
>
> http://www.dsprelated.com/showarticle/54.php
>
> Regards,
> Andor

Hello Andor,

Very well written article. Hopefully this will put a lot of the
seeming paradoxes associated with group delay to bed. The previous
threads must have set a limit for their quantity.  Whole books have
been written on this subject as it is quite slippery. The D'enouement
makes itself apparent when you try to trick mother nature and the
predictability goes out the window.  You can have fun by feeding this
circuit's output back to its input and make a negative group delay
oscillator - wouldn't that be odd?

Good Job,

Clay

On Fri, 7 Mar 2008 14:18:06 -0800 (PST), Andor
<a...@gmail.com> wrote:

>From a recent discussion here:
>
>> >if i could generate some coefficents that had a 'negative' group delay for
>> >a period of time, would you think that 'phase cloning' was new and
>> >intersting??
>>
>> A time machine would be pretty revolutionary, yes.
>>
>> Negative group delay means that the output appears before the input
>> arrives.
>
>Fascinating concept, isn't it? I was curious enough to dig into the
>topic for a while and write up what I found out. You can read about it
>here:
>
>http://www.dsprelated.com/showarticle/54.php
>
>Regards,
>Andor

Nicely done!

Gotta admit, I learned some things there, and the article is good
enough that I do feel I understand the topic better.

Kudos for including the code, too!

But wouldn't it be impressive if I'd said that before the article was
published?

;)



Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

On Mar 7, 2:18 pm, Andor <andor.bari...@gmail.com> wrote:
> From a recent discussion here:
>
> > >if i could generate some coefficents that had a 'negative' group delay for
> > >a period of time, would you think that 'phase cloning' was new and
> > >intersting??
>
> > A time machine would be pretty revolutionary, yes.
>
> > Negative group delay means that the output appears before the input
> > arrives.
>
> Fascinating concept, isn't it? I was curious enough to dig into the
> topic for a while and write up what I found out. You can read about it
> here:
>
> http://www.dsprelated.com/showarticle/54.php
>
> Regards,
> Andor

Very nice write-up Andor.

Far from being an impossible time machine, it looks
like any pole(s) inside the unit circle and much nearer
than the canceling zero(s) could have a tiny frequency
band with negative group delay.

So what this says:

> > Negative group delay means that the output appears before the input
> > arrives.

might be be better stated at this:

A filter with negative group delay can produce the output
before the input if the input has been on and continues on
a highly predictable trajectory due to staying completely
within a certain constrained bandwidth.

No violation of causality becaues of the "if" clause.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

On Mar 7, 11:18=A0pm, Andor <andor.bari...@gmail.com> wrote:
> From a recent discussion here:
>
> > >if i could generate some coefficents that had a 'negative' group delay =
for
> > >a period of time, would you think that 'phase cloning' was new and
> > >intersting??
>
> > A time machine would be pretty revolutionary, yes.
>
> > Negative group delay means that the output appears before the input
> > arrives.
>
> Fascinating concept, isn't it? I was curious enough to dig into the
> topic for a while and write up what I found out. You can read about it
> here:
>
> http://www.dsprelated.com/showarticle/54.php

Interesting piece. Just a couple of questions:

- Why not use the BLT to transfor to discrete-time domain?
  I don't know Greg's stuff, I am familiar with the properties
  of the BLT.
- Why analyze in discrete-time domain at all? Your results
  would have been seriously interesting if you could demonstrate
  similar effects in continuos-time domain; here they are amusing.
- Why not use the impulse as test signal? You refer to 'some' who
  'claim' that system with negative group delays are noncausal, as
  if you contest (or at least not support) such a view. I have
  made such claims.

Your way of phrasing opens a whole new can of worms of semantic wars
etc - why not demonstarte once and for all that systems with negative
group delays exist and can be implemented in CT? (If that indeed is
your claim, of course; I could not find out from the article what
your stand on the issue is.)

I would suggest the following:

1) Make a clear statement of the 'usual' views on the issue,
   your own opinions, and exactly what this article aims
   to demonstrate.
2) Compute the phase response of the CT system function and
   demonstrate that there is a negative group delay there.
   If you have the tools, I would also suggest you simulate
   the impulse response of that circuit in CT domain.
3) Use standard techniques in DT domain (BLT, Kronecker delta)
   and repeat your analysis.

If all the results persist (system functions show negative
group delays, anticausal behaviour) after such a re-work, I
will consider to spend some time looking into your article
in more detail.

As the article stands, it only cretates confusion. A less
benevolent reviewer than me might sugest that the instability
caused by the truncated signal is caused by poles located
outside the unit circle, and thus suggest that there is a
fundamental flaw in the argument. As you know, if you can
repeat the results using standard analysis tools and from
multiple angles of attack, it will stop those sorts of
argument at the outset.

Rune

On Mar 8, 12:02 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On Mar 7, 11:18 pm, Andor <andor.bari...@gmail.com> wrote:
>
>
> Interesting piece. Just a couple of questions:
>
> - Why not use the BLT to transfor to discrete-time domain?
>   I don't know Greg's stuff, I am familiar with the properties
>   of the BLT.

I'm not sure it matters what the technique is?  Surely all that
matters is that a discrete-time filter is derived that has the key
properties of its CT counterpart (approximately flat -ve group delay
over a region with approximately flat magnitude response).  The fact
that the overall response is roughly the same is merely an aesthetic
bonus.


> - Why analyze in discrete-time domain at all? Your results
>   would have been seriously interesting if you could demonstrate
>   similar effects in continuos-time domain; here they are amusing.
> - Why not use the impulse as test signal? You refer to 'some' who
>   'claim' that system with negative group delays are noncausal, as
>   if you contest (or at least not support) such a view. I have
>   made such claims.

How would you suggest performing the experiment in CT, short of
actually building the circuit?  Even circuit analysis tools have to
operate in discrete time.


> Your way of phrasing opens a whole new can of worms of semantic wars
> etc - why not demonstarte once and for all that systems with negative
> group delays exist and can be implemented in CT? (If that indeed is
> your claim, of course; I could not find out from the article what
> your stand on the issue is.)

It would appear that he has.  He's presented a circuit whose transfer
function is easily derivable, and which clearly has a negative group
delay over some region (as the graph demonstrates).


> I would suggest the following:
>
> 1) Make a clear statement of the 'usual' views on the issue,
>    your own opinions, and exactly what this article aims
>    to demonstrate.
> 2) Compute the phase response of the CT system function and
>    demonstrate that there is a negative group delay there.
>    If you have the tools, I would also suggest you simulate
>    the impulse response of that circuit in CT domain.

I agree that graphs of the impulse responses would have been nice.
But on the other hand, they're kind of irrelevant to the argument, as
it is clear that both systems are causal!

> 3) Use standard techniques in DT domain (BLT, Kronecker delta)
>    and repeat your analysis.
>
> If all the results persist (system functions show negative
> group delays, anticausal behaviour) after such a re-work, I
> will consider to spend some time looking into your article
> in more detail.

I believe the point of the article was precisely to counter the belief
that -ve group delay equates to "anticausal behaviour".

>
> As the article stands, it only cretates confusion. A less
> benevolent reviewer than me might sugest that the instability
> caused by the truncated signal is caused by poles located
> outside the unit circle, and thus suggest that there is a
> fundamental flaw in the argument. As you know, if you can
> repeat the results using standard analysis tools and from
> multiple angles of attack, it will stop those sorts of
> argument at the outset.

Even if there were poles outside the unit circle, that wouldn't allow
the system to become non-causal!  The article already states that both
CT and DT filters are minimum-phase.  A trivial analysis of the
transfer function polynomials demonstrates that they are stable.


--
Oli

On Mar 8, 2:38=A0pm, Oli Charlesworth <ca...@olifilth.co.uk> wrote:
> On Mar 8, 12:02 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > On Mar 7, 11:18 pm, Andor <andor.bari...@gmail.com> wrote:
>
> > Interesting piece. Just a couple of questions:
>
> > - Why not use the BLT to transfor to discrete-time domain?
> > =A0 I don't know Greg's stuff, I am familiar with the properties
> > =A0 of the BLT.
>
> I'm not sure it matters what the technique is?

It does to me. A result derived with a known, well-udnerstood
technique has a far greater impact than a unfamiliar, possibly
novel technique applied to a tricky question.

>=A0Surely all that
> matters is that a discrete-time filter is derived that has the key
> properties of its CT counterpart (approximately flat -ve group delay
> over a region with approximately flat magnitude response). =A0The fact
> that the overall response is roughly the same is merely an aesthetic
> bonus.

Wrong. If the claim applies to the CT cirquit, it is the CT
cirquit which must be analyzed.

> > - Why analyze in discrete-time domain at all? Your results
> > =A0 would have been seriously interesting if you could demonstrate
> > =A0 similar effects in continuos-time domain; here they are amusing.
> > - Why not use the impulse as test signal? You refer to 'some' who
> > =A0 'claim' that system with negative group delays are noncausal, as
> > =A0 if you contest (or at least not support) such a view. I have
> > =A0 made such claims.
>
> How would you suggest performing the experiment in CT, short of
> actually building the circuit? =A0Even circuit analysis tools have to
> operate in discrete time.

Derive and analyze the Laplace transform for the cirquit?
All analytical, should be easy.

> > Your way of phrasing opens a whole new can of worms of semantic wars
> > etc - why not demonstarte once and for all that systems with negative
> > group delays exist and can be implemented in CT? (If that indeed is
> > your claim, of course; I could not find out from the article what
> > your stand on the issue is.)
>
> It would appear that he has. =A0He's presented a circuit whose transfer
> function is easily derivable, and which clearly has a negative group
> delay over some region (as the graph demonstrates).

Well, that's *not* what the article says. The statement

"We will now proceed to find a discrete filter with
 comparable characteristics in order to be able to
 reproduce the experiment in Matlab world."

can only be interpreted as if the subsequent analysis
applies to the DT "comparable" system.

> > I would suggest the following:
>
> > 1) Make a clear statement of the 'usual' views on the issue,
> > =A0 =A0your own opinions, and exactly what this article aims
> > =A0 =A0to demonstrate.
> > 2) Compute the phase response of the CT system function and
> > =A0 =A0demonstrate that there is a negative group delay there.
> > =A0 =A0If you have the tools, I would also suggest you simulate
> > =A0 =A0the impulse response of that circuit in CT domain.
>
> I agree that graphs of the impulse responses would have been nice.
> But on the other hand, they're kind of irrelevant to the argument, as
> it is clear that both systems are causal!

That's what confuses me:

- I can't figure out what the claim in the article is; what is
  the 'usual' stand on the question and how is it contested?
- What is the premise for the debate? Are we talking about
  CT or DT systems? Online or offline in case of DT? Causal?
  Stable?
- What are the arguments? I don't know the Berchin method for
  CT->DT transform; if the claims are valid they should work
  just as well with the well-known BLT.
- What are the conclusions? I see some graphs but because
  I'm completely lost in earlier stages I don't understand
  what they signify or what the impact is.

> > 3) Use standard techniques in DT domain (BLT, Kronecker delta)
> > =A0 =A0and repeat your analysis.
>
> > If all the results persist (system functions show negative
> > group delays, anticausal behaviour) after such a re-work, I
> > will consider to spend some time looking into your article
> > in more detail.
>
> I believe the point of the article was precisely to counter the belief
> that -ve group delay equates to "anticausal behaviour".

If so, the article would benefit greatly from a better structure
and presentation.

> > As the article stands, it only cretates confusion. A less
> > benevolent reviewer than me might sugest that the instability
> > caused by the truncated signal is caused by poles located
> > outside the unit circle, and thus suggest that there is a
> > fundamental flaw in the argument. As you know, if you can
> > repeat the results using standard analysis tools and from
> > multiple angles of attack, it will stop those sorts of
> > argument at the outset.
>
> Even if there were poles outside the unit circle, that wouldn't allow
> the system to become non-causal! =A0The article already states that both
> CT and DT filters are minimum-phase. =A0A trivial analysis of the
> transfer function polynomials demonstrates that they are stable.

Again, that may be the case but I am not able to follow
the chain of arguments to reach that conclusion.

Rune

On Mar 8, 2:01 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On Mar 8, 2:38 pm, Oli Charlesworth <ca...@olifilth.co.uk> wrote:
>
> > I'm not sure it matters what the technique is?
>
> It does to me. A result derived with a known, well-udnerstood
> technique has a far greater impact than a unfamiliar, possibly
> novel technique applied to a tricky question.

Perhaps.

> > Surely all that
> > matters is that a discrete-time filter is derived that has the key
> > properties of its CT counterpart (approximately flat -ve group delay
> > over a region with approximately flat magnitude response).  The fact
> > that the overall response is roughly the same is merely an aesthetic
> > bonus.
>
> Wrong. If the claim applies to the CT cirquit, it is the CT
> cirquit which must be analyzed.

Not necessarily.  See further down...

> > How would you suggest performing the experiment in CT, short of
> > actually building the circuit?  Even circuit analysis tools have to
> > operate in discrete time.
>
> Derive and analyze the Laplace transform for the cirquit?
> All analytical, should be easy.

Yes, we could analyse via the Laplace domain by putting in a signal
with a known equation, and examining the output signal's equation.
But given that the group delay and magnitude responses aren't
completely flat (over the bandlimited region of interest), there will
clearly be some distortion, so it's not going to be a case of y(t) =
x(t + T).  As far as I can see, the next logical step in analysing the
"delay" would be to graph both input and output signals.  But clearly,
the graphing process requires discretising the time axis.


> > It would appear that he has.  He's presented a circuit whose transfer
> > function is easily derivable, and which clearly has a negative group
> > delay over some region (as the graph demonstrates).
>
> Well, that's *not* what the article says. The statement
>
> "We will now proceed to find a discrete filter with
>  comparable characteristics in order to be able to
>  reproduce the experiment in Matlab world."
>
> can only be interpreted as if the subsequent analysis
> applies to the DT "comparable" system.

Figure 2 shows the group delay of the CT system (in blue); i.e. it has
already been demonstrated that the CT system has -ve group delay.


> > I agree that graphs of the impulse responses would have been nice.
> > But on the other hand, they're kind of irrelevant to the argument, as
> > it is clear that both systems are causal!
>
> That's what confuses me:
>
> - I can't figure out what the claim in the article is; what is
>   the 'usual' stand on the question and how is it contested?
> - What is the premise for the debate? Are we talking about
>   CT or DT systems? Online or offline in case of DT? Causal?
>   Stable?
> - What are the arguments? I don't know the Berchin method for
>   CT->DT transform; if the claims are valid they should work
>   just as well with the well-known BLT.
> - What are the conclusions? I see some graphs but because
>   I'm completely lost in earlier stages I don't understand
>   what they signify or what the impact is.

Andor would be able to answer these questions better than I, but here
is my guess.

I think it is clear that the article is discussing causal, stable
systems, given that the two systems presented are both causal and
stable, and that the typical confusion about -ve group delay is that
it somehow violates causality in practical (i.e. stable) systems.
Whether we're talking about online or offline processing is
irrelevant, as this doesn't affect the property of causality (at
least, not as far as analysing an impulse response is concerned).

In my opinion, the abstract states what the article is trying to
address (i.e. its "arguments").  Namely, the surpise/confusion that
many people exhibit when confronted with the topic of -ve group delay.

I believe the purpose of the article was to demonstrate that -ve group
delay is possible; the experiments in the DT domain clearly
demonstrate this, along with the apparently non-intuitive illusion of
non-causality.  The article concludes with an explanation for this
illusion.  Although there is no proof, surely it is logical to
conclude that if the apparent paradox has been cleared up in the DT
domain, it also serves to clear up any apparent paradox in the CT
domain?  Therefore, I'm not sure that it matters what the specific
discretisation technique was.

> > I believe the point of the article was precisely to counter the belief
> > that -ve group delay equates to "anticausal behaviour".
>
> If so, the article would benefit greatly from a better structure
> and presentation.

Perhaps.

> > Even if there were poles outside the unit circle, that wouldn't allow
> > the system to become non-causal!  The article already states that both
> > CT and DT filters are minimum-phase.  A trivial analysis of the
> > transfer function polynomials demonstrates that they are stable.
>
> Again, that may be the case but I am not able to follow
> the chain of arguments to reach that conclusion.

I'm not sure the chain of arguments affects a numerical analysis of
the transfer function polynomials...


--
Oli

On Mar 8, 2:43 pm, Oli Charlesworth <ca...@olifilth.co.uk> wrote:
> On Mar 8, 2:01 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > > How would you suggest performing the experiment in CT, short of
> > > actually building the circuit?  Even circuit analysis tools have to
> > > operate in discrete time.
>
> > Derive and analyze the Laplace transform for the cirquit?
> > All analytical, should be easy.
>
> Yes, we could analyse via the Laplace domain by putting in a signal
> with a known equation, and examining the output signal's equation.
> But given that the group delay and magnitude responses aren't
> completely flat (over the bandlimited region of interest), there will
> clearly be some distortion, so it's not going to be a case of y(t) =
> x(t + T).  As far as I can see, the next logical step in analysing the
> "delay" would be to graph both input and output signals.  But clearly,
> the graphing process requires discretising the time axis.
>

Just realised that this is irrelevant, and should be ignored!  You are
right, a CT analysis would be tractable.


--
Oli

On Mar 8, 3:43=A0pm, Oli Charlesworth <ca...@olifilth.co.uk> wrote:
> On Mar 8, 2:01 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > On Mar 8, 2:38 pm, Oli Charlesworth <ca...@olifilth.co.uk> wrote:
>
> > > I'm not sure it matters what the technique is?
>
> > It does to me. A result derived with a known, well-udnerstood
> > technique has a far greater impact than a unfamiliar, possibly
> > novel technique applied to a tricky question.
>
> Perhaps.

Certainly. If a conclusion depends on using a novel method
to do a standard computation, the first awkward questions
will inevitably concern the soundness of this new method.
It will only strengthen the conclusion if it can be demonstrated
that the conclusion is independent of numerical methods used.


> > > Surely all that
> > > matters is that a discrete-time filter is derived that has the key
> > > properties of its CT counterpart (approximately flat -ve group delay
> > > over a region with approximately flat magnitude response). =A0The fact=

> > > that the overall response is roughly the same is merely an aesthetic
> > > bonus.
>
> > Wrong. If the claim applies to the CT cirquit, it is the CT
> > cirquit which must be analyzed.
>
> Not necessarily. =A0See further down...
>
> > > How would you suggest performing the experiment in CT, short of
> > > actually building the circuit? =A0Even circuit analysis tools have to
> > > operate in discrete time.
>
> > Derive and analyze the Laplace transform for the cirquit?
> > All analytical, should be easy.
>
> Yes, we could analyse via the Laplace domain by putting in a signal
> with a known equation, and examining the output signal's equation.

No. The LT describes the linear system regardless of
input and output signals. If the LP is invalid, then
the system is nonlinear and thus the concept of group
delay is undefined.

Thanks for pointing out another item for my list of points
that ought to be clarified.

Rune