Bode's Amplitude/Phase relation for discrete time systems
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From: Demus on 22 May 2010 05:03 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?
From: Jerry Avins on 22 May 2010 08:33 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. �����������������������������������������������������������������������
From: Greg Berchin on 22 May 2010 08:56 On Sat, 22 May 2010 04:03:24 -0500, "Demus" <sodemus(a)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 �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
From: Demus on 22 May 2010 11:30 >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!
From: Tim Wescott on 22 May 2010 13:10
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 |