From: Demus on
>> (By the same token, in my opinion the phase of a causal system can't
really
>> be said to be negative, at least not when the inputs and outputs are
>> non-stationary signals...)
>
>Negative phase slope indicates delay, which is perfectly reasonable to
>expect in a causal system. I think you're thinking of a positive phase
>slope, indicating lead. You can have lead in a real system, but only at
>the expense of a rising amplitude vs. frequency (think differentiation).
> The amplitude 'distortion' is the price you pay for the 'prediction'
>effect of differentiation.
>

Oh yeah, you're right... I had positive phases in mind. And yeah, of course
that makes sense.

Ok, but I'm not sure I understand this completely, even if dealing only
with ideal linear systems (and stable, minphase) and uncorrupted signals.
The answer is no, there shouldn't (ideally) be a pure time-delay in the
above mentioned setup?

>> Can someone please explain this situation to me, perhaps including how
>> phase relates to causality?
>
>The slope of the phase vs. frequency characteristic of a linear system
>transfer function indicates the effective delay of signals at that
>particular frequency. It can be positive, but as I mentioned, if the
>system is causal then the positive phase vs. frequency slope comes with
>a cost in the amplitude vs. frequency relationship.

So what does this mean in the time domain (still ideally - perfect
linearity, no noise)?

>
>There's some relationship, but I can't remember the details, even if I
>ever learned them. I know that if a system is minimum phase then its
>phase response is the Hilbert transform of its magnitude, but I couldn't
>even tell you if that's magnitude, magnitude squared, log magnitude,
>etc. But if you really want to know, you should be able to at least
>find a book reference with an internet search.

Isn't this Bode's Integral Theorem? I brought this up a couple of months
ago? I can verify (by the way) that the discrete time version is explicitly
stated in the first edition of Oppenheims' Digital Signal Processing.

Also, sorry, I posted my previous post before I saw your post.

Thanks!
From: Demus on
I think I understand the situation better now... I guess I tricked myself
before.

Thanks for the explanation!
From: Tim Wescott on
On 07/26/2010 09:17 AM, Demus wrote:
>>> (By the same token, in my opinion the phase of a causal system can't
> really
>>> be said to be negative, at least not when the inputs and outputs are
>>> non-stationary signals...)
>>
>> Negative phase slope indicates delay, which is perfectly reasonable to
>> expect in a causal system. I think you're thinking of a positive phase
>> slope, indicating lead. You can have lead in a real system, but only at
>> the expense of a rising amplitude vs. frequency (think differentiation).
>> The amplitude 'distortion' is the price you pay for the 'prediction'
>> effect of differentiation.
>>
>
> Oh yeah, you're right... I had positive phases in mind. And yeah, of course
> that makes sense.
>
> Ok, but I'm not sure I understand this completely, even if dealing only
> with ideal linear systems (and stable, minphase) and uncorrupted signals.
> The answer is no, there shouldn't (ideally) be a pure time-delay in the
> above mentioned setup?

Correct. In fact, for a linear, time-invariant, minimum-phase (no zeros
on the imaginary axis), purely stable system (no poles on the imaginary
axis) with a finite number of states, you can always find an inverse,
and the cascade of the system with its inverse will always have a
transfer function of one.

This should contradict your intuition, before you even get to Simulink.

>>> Can someone please explain this situation to me, perhaps including how
>>> phase relates to causality?
>>
>> The slope of the phase vs. frequency characteristic of a linear system
>> transfer function indicates the effective delay of signals at that
>> particular frequency. It can be positive, but as I mentioned, if the
>> system is causal then the positive phase vs. frequency slope comes with
>> a cost in the amplitude vs. frequency relationship.
>
> So what does this mean in the time domain (still ideally - perfect
> linearity, no noise)?

That you can't make a predictor that's not both inaccurate and noisy.

>> There's some relationship, but I can't remember the details, even if I
>> ever learned them. I know that if a system is minimum phase then its
>> phase response is the Hilbert transform of its magnitude, but I couldn't
>> even tell you if that's magnitude, magnitude squared, log magnitude,
>> etc. But if you really want to know, you should be able to at least
>> find a book reference with an internet search.
>
> Isn't this Bode's Integral Theorem?

No -- Bode's Integral Theorem is probably similar, but it has to do with
the sensitivity of a control system to disturbances and what you can do
with it. Although perhaps its a corollary -- or perhaps the one I'm
familiar with is the corollary.

> I brought this up a couple of months
> ago? I can verify (by the way) that the discrete time version is explicitly
> stated in the first edition of Oppenheims' Digital Signal Processing.

Oppenheim & Schafer? Copyright 1975? I can't find a reference to it in
the index or table of contents -- do you have a page number, or do I
gave the wrong book?

> Also, sorry, I posted my previous post before I saw your post.

No problem -- it's just a race condition.

--

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

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: Vladimir Vassilevsky on

H(z) x 1/H(z) === 1

This is scientific fact (except for the special case H(z) === 0).

Neither H(z) nor 1/H(z) have to be casual.
Neither H(z) nor 1/H(z) have to be stable.
Neither H(z) nor 1/H(z) have to be minimum phase.


Demus wrote:
> Hello,
>
> This might be a silly question, but I can't figure out what implications
> the phase of a transfer function has for causality.
>
> If I pass a signal to a filter and pass the output to the inverse filter...
> my intuition of causality tells me then that the output of the inverse
> filter can not be the same as what enters the first filter.

Intuition may be good at gambling.

> And by 'can not be the same' I mean to include it's place in time. It
> should have the same shape but not occur at the same time, which would then
> suggest that the result is a pure time delay.
>
> However, when I simulate this situation in simulink I get the opposite
> result, the two signals have the same time coordinates as well, so I guess
> my intuition is off.

"Matlab does all thinking for us" (TM)

> (By the same token, in my opinion the phase of a causal system can't really
> be said to be negative, at least not when the inputs and outputs are
> non-stationary signals...)
>
> Can someone please explain this situation to me, perhaps including how
> phase relates to causality?

There is no such relation.

VLV
From: Demus on
>No -- Bode's Integral Theorem is probably similar, but it has to do with
>the sensitivity of a control system to disturbances and what you can do
>with it. Although perhaps its a corollary -- or perhaps the one I'm
>familiar with is the corollary.

Yeah, the waterbed effect for "2-pole-excess" systems, right? They might
have the same name sometimes but I know that one at least as Bode's
Sensitivity Integral.

>> I brought this up a couple of months
>> ago? I can verify (by the way) that the discrete time version is
explicitly
>> stated in the first edition of Oppenheims' Digital Signal Processing.
>
>Oppenheim & Schafer? Copyright 1975? I can't find a reference to it in
>the index or table of contents -- do you have a page number, or do I
>gave the wrong book?

I don't have the book, I borrowed it from the library last time (and then I
only found it in the first edition, not the second edition). I might be way
off here but I'm pretty sure it was either theorem 5.21 or 7.21. Can't
remember.

Again, thanks for your replies. You've always been very helpful.
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