Bode's Amplitude/Phase relation for discrete time systems

Could someone please show me what the discrete-time version of Bode's
amplitude/phase relation looks like?

On May 22, 12:00&#4294967295;pm, "Demus" <sodemus@n_o_s_p_a_m.hotmail.com> wrote:
> Could someone please show me what the discrete-time version of Bode's
> amplitude/phase relation looks like?

Discrete-time is an approximation to analogue so Bodes Theorem is the
same. There are no quick short-cuts for plotting Bode-plots however
such as asmyptotic approximations ie straight line approximations with
slopes multiples of +/- 20ndB/decade where n is 1,2,3etc

Hardy

On 05/21/2010 08:03 PM, HardySpicer wrote:
> On May 22, 12:00 pm, "Demus"<sodemus@n_o_s_p_a_m.hotmail.com>  wrote:
>> Could someone please show me what the discrete-time version of Bode's
>> amplitude/phase relation looks like?
>
> Discrete-time is an approximation to analogue so Bodes Theorem is the
> same.

Eh?  Aside from being completely wrong that's a very sensible and wise 
statement.

Discrete time is discrete time, and within discrete time the z transform 
is exact.  The Bode theorem is exact within discrete time control 
because the z transform is exact.

There's absolutely no reason at all that one has to make any 
approximations when doing discrete time control of a continuous-time 
plant (ask me how before you start your flame).  Sometimes the 
approximations are convenient, but they are not necessary.

> There are no quick short-cuts for plotting Bode-plots however
> such as asmyptotic approximations ie straight line approximations with
> slopes multiples of +/- 20ndB/decade where n is 1,2,3etc

Too true, at least for traditional "on paper" short cuts.  Fortunately 
Scilab/Matlab/Octave/etc. make _really good_ short cuts, and even Excel 
can be coerced to do the job.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On 05/21/2010 05:00 PM, Demus wrote:
> Could someone please show me what the discrete-time version of Bode's
> amplitude/phase relation looks like?

A heck of a lot like the continuous-time version.  The interpretation of 
a Bode plot of a sampled-time control system is exactly the same as the 
interpretation of one for a continuous-time system -- only the fact that 
the thing repeats at Fs/2 and beyond is different (and handy -- you're 
never left worrying if you've ignored something important).

Here's some examples:

http://www.wescottdesign.com/articles/zTransform/z-transforms.html

That paper got cut up and scattered around in two or three chapters of 
my book, which has a much more complete treatment of just about 
everything the paper touches:

http://www.wescottdesign.com/actfes/actfes.html,
http://www.elsevier.com/wps/find/bookdescription.cws_home/707797/description#description.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On May 22, 4:41&#4294967295;pm, Tim Wescott <t...@seemywebsite.now> wrote:
> On 05/21/2010 08:03 PM, HardySpicer wrote:
>
> > On May 22, 12:00 pm, "Demus"<sodemus@n_o_s_p_a_m.hotmail.com> &#4294967295;wrote:
> >> Could someone please show me what the discrete-time version of Bode's
> >> amplitude/phase relation looks like?
>
> > Discrete-time is an approximation to analogue so Bodes Theorem is the
> > same.
>
> Eh? &#4294967295;Aside from being completely wrong that's a very sensible and wise
> statement.
>
> Discrete time is discrete time, and within discrete time the z transform
> is exact. &#4294967295;The Bode theorem is exact within discrete time control
> because the z transform is exact.
>
> There's absolutely no reason at all that one has to make any
> approximations when doing discrete time control of a continuous-time
> plant (ask me how before you start your flame). &#4294967295;Sometimes the
> approximations are convenient, but they are not necessary.
>
> > There are no quick short-cuts for plotting Bode-plots however
> > such as asmyptotic approximations ie straight line approximations with
> > slopes multiples of +/- 20ndB/decade where n is 1,2,3etc
>
> Too true, at least for traditional "on paper" short cuts. &#4294967295;Fortunately
> Scilab/Matlab/Octave/etc. make _really good_ short cuts, and even Excel
> can be coerced to do the job.
>
> --
> Tim Wescott
> Control system and signal processing consultingwww.wescottdesign.com

Actually no. There is nothing special because you sample a system or a
signal.
If you sample fast enought you get back to analogue. Digital is just
an approximation to analogue, nothing more.
For example, take an integrator K/s. Use the Bilinear transform and
sample high enough and it will have a slope of -20dB/decade just as
the analogue case and a phase of -90 degrees.. However, if you are
thinking of special contrived digital systems eg FIR then I might
agree.

Hardy

Well, I see your points. In the limit the discrete time and the continuous
time systems approximate each other arbitrarily well but that exact methods
to analyze their respective behaviour exist separately, right (and that
discrete and continuous systems of course also exist separately)?

I mean, as for a sampled cont. sys., one could integrate the cont. state
transition matrix to describe the behavior between samples, but why bother
when you can't change the input (nor observe the system) between the
samples anyway (assuming zero order hold)?

My question might have been a little unclear though... What I was looking
for is the discrete time version of:

Phase(freq) = 2*freq/pi * "integral from 0 to inf w.r.t. dummy_var" { [
log(Amplitude(dummy_var)) - log(Amplitude(freq)) ] / (dummy_var^2-freq^2)
}

..assuming a stable minimum phase systems.

This is what I want! :) Is it trivial?

On 5/21/2010 8:00 PM, Demus wrote:
> Could someone please show me what the discrete-time version of Bode's
> amplitude/phase relation looks like?

I don't understand. A Bode plot consists of two graphs of a transfer 
function. Both use a horizontal axis of log(frequency). One of them is 
of log(amplitude), the other of phase. I suppose that it is necessary 
that the transfer function be continuous, but that is not a severe 
constraint.

The properties of Bode plots that make them easy to draw and make it 
possible to infer the phase plot from the amplitude plot apply to 
minimum-phase systems only. Most transfer functions of systems of 
discrete components are minimum phase. Most digital transfer functions 
are not.

Jerry
-- 
"I view the progress of science as ... the slow erosion of the tendency
  to dichotomize." --Barbara Smuts, U. Mich.
�����������������������������������������������������������������������

On Sat, 22 May 2010 04:03:24 -0500, "Demus" <sodemus@n_o_s_p_a_m.hotmail.com>
wrote:

>What I was looking
>for is the discrete time version of:
>
>Phase(freq) = 2*freq/pi * "integral from 0 to inf w.r.t. dummy_var" { [
>log(Amplitude(dummy_var)) - log(Amplitude(freq)) ] / (dummy_var^2-freq^2)
>}
>
>..assuming a stable minimum phase systems.
>
>This is what I want! :) Is it trivial?

I don't see any reason that a similar expression should not apply to discrete
time systems.  Even though they are discrete in time, they are continuous in
frequency.

Thumbing through Oppenheim & Shafer (1975 version), I find magnitude and phase
of discrete time minimum phase filters expressed as Cauchy Principal Value
integrals in equations (7.21) and (7.22):

(aw, they're too complicated to try to render in ASCII; look them up)

A notable difference is that the discrete time limits of integration are &#4294967295;PI,
which makes sense given the periodic nature of the spectrum.

The Cauchy expression for phase is not identical to the one that you presented,
but it does show that integral relationships for magnitude and phase still apply
in the discrete time case.

Greg

>I don't see any reason that a similar expression should not apply to
discrete
>time systems.  Even though they are discrete in time, they are continuous
in
>frequency.
>
>Thumbing through Oppenheim & Shafer (1975 version), I find magnitude and
phase
>of discrete time minimum phase filters expressed as Cauchy Principal
Value
>integrals in equations (7.21) and (7.22):
>
>(aw, they're too complicated to try to render in ASCII; look them up)
>

Yes, this was what I was looking for! Sorry for the earlier confusion...
Thanks for the reference.
>A notable difference is that the discrete time limits of integration are
�PI,
>which makes sense given the periodic nature of the spectrum.
>
Right, that I figured... though I really didn't think the integrand would
be that much different... 

>The Cauchy expression for phase is not identical to the one that you
presented,
>but it does show that integral relationships for magnitude and phase still
apply
>in the discrete time case.
>
>Greg
I'll look into this book when I get the chance (I'll have to borrow it from
the library, though one SHOULD probably own it...)
Again, thanks!

On 05/22/2010 02:03 AM, Demus wrote:
> Well, I see your points. In the limit the discrete time and the continuous
> time systems approximate each other arbitrarily well but that exact methods
> to analyze their respective behaviour exist separately, right (and that
> discrete and continuous systems of course also exist separately)?

Correct (and ignore the back and forth between Hardy and I -- the 
difference is philosophical, and makes little difference to you at this 
stage in your education).

> I mean, as for a sampled cont. sys., one could integrate the cont. state
> transition matrix to describe the behavior between samples,

Yes.

> but why bother
> when you can't change the input (nor observe the system) between the
> samples anyway (assuming zero order hold)?

For analyzing stability there is no reason.  For a system that is being 
sampled slowly compared to the speed at which the plant reacts, you may 
need to know if your plant behavior is acceptable between samples.

> My question might have been a little unclear though... What I was looking
> for is the discrete time version of:
>
> Phase(freq) = 2*freq/pi * "integral from 0 to inf w.r.t. dummy_var" { [
> log(Amplitude(dummy_var)) - log(Amplitude(freq)) ] / (dummy_var^2-freq^2)
> }
>
> ..assuming a stable minimum phase systems.

Ah.  I misunderstood.  You were speaking of the Bode integral, not the 
Bode plot (which also has to do with Bode expressing amplitude and phase 
relationships).

There is a transform, something like w = 0.5 * (z - 1) / (z + 1).  Note 
that it's _a_ bilinear transform, but it's not _the_ bilinear transform 
that folks use to approximate s in the z domain.  At any rate, the w 
transform maps the z plane onto the w plane, and maps the unit circle on 
the z plane onto the imaginary axis of the w plane (if the above 
equation I give doesn't have this property then I got it wrong -- I'm 
away from my books right now, and frankly I'm too lazy to go look).

The Bode integral should work on the w plane, and should lead to exact 
predictions for the disturbance rejection of a sampled time system.  It 
_will_ fall down in predicting disturbance rejection for a 
continuous-time system that is sampled and controlled, as nothing done 
in the sampled-time domain takes into account the plant behavior between 
samples.  But it should get you close.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com