"Bob Cain" <arcane@arcanemethods.com> wrote in message news:c15lou0utf@enews1.newsguy.com...> This discussion has lead me to wonder what kind of > characteristics a "maximum phase" filter, with all its zeros > outside the unit circle, would be. Probably obvious to the > inhabitants of the complex plane but I'm only slowly taking > up residence there.It's not so much the zeros outside of the unit circle, as the unmatched poles inside the unit circle that make the difference in the z-plane. Phase-wise, overall, i.e., without regard to the localized effects that depend on the precise positions of the poles and zeros, a pole inside the circle is a single-sample delay, and a zero inside the unit circle is a single sample advance. Poles and zeros outside the unit circle do not have this effect. So when you move a zero outside the unit circle, the newly-unmatched pole adds an extra sample period to the filter's average group delay. The s-plane is a little different. There, a pole adds or subtracts to/from the integral of the filter's group delay over the whole spectrum, depending on whether it's on the left or right of the jw axis, and zeros have the opposite effect. When a zero is moved to the right of this axis, some delay is added, and some advancement removed.
FIR roots and frequency response
Started by ●February 13, 2004
Reply by ●February 22, 20042004-02-22
Reply by ●February 22, 20042004-02-22
Matt Timmermans wrote:> "Bob Cain" <arcane@arcanemethods.com> wrote in message > news:c15lou0utf@enews1.newsguy.com... > >>This discussion has lead me to wonder what kind of >>characteristics a "maximum phase" filter, with all its zeros >>outside the unit circle, would be. Probably obvious to the >>inhabitants of the complex plane but I'm only slowly taking >>up residence there. > > > It's not so much the zeros outside of the unit circle, as the unmatched > poles inside the unit circle that make the difference in the z-plane.In light of the fact that I'm only considering a FIR, where all the poles are at the origin I'm not sure how to interpret the remainder of your response. Imaging all the zeros that are outside to the inside makes it minimum phase and the reason for that I now understand. What I'm curious about is what kind of filter results from the opposite strategy, imaging all that are inside to the outside. Or what is the nature of the resulting FIR. Perhaps no such generalization exists. An ancilary question is what kind of calculation similar to the so-called root homomorphic spectrum would produce such a transformation. Is there an equivalent to the simple one I've attached below for the minimum phase transformation. If your answer should have explained that to me, I must plead density. :-( Thanks, Bob function y = minphase(x) %minphase - minimum phase from mixed phase impulse responses % minphase(x) returns minumum phase version of signal x % y = minphase(x) returns the minimum phase signal % derived from x. % References: % [1] A.V. Oppenheim and R.W. Schafer, Digital Signal % Processing, Prentice-Hall, 1975. % [2] Programs for Digital Signal Processing % IEEE Press % John Wiley & Sons, 1979, algorithm 7.2. % % Adapted from the Matlab rceps() function by Bob Cain % arcane@arcanemethods.com %Number of impulse responses m = size(x,2); %Size of each n = size(x,1); %The weighting function wn = [ones(1,m); 2*ones(n/2-1,m); ones(1,m); zeros(n/2-1,m)]; %The work y = real(ifft(exp(fft(wn.*real(ifft(log(abs(fft(x))))))))); -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 22, 20042004-02-22
Bob Cain wrote: ...> Imaging all the zeros that are outside to the inside makes > it minimum phase and the reason for that I now understand. > What I'm curious about is what kind of filter results from > the opposite strategy, imaging all that are inside to the > outside. Or what is the nature of the resulting FIR. > Perhaps no such generalization exists.When you make a transversal filter, the poles are all at the origin, whether or not the filter has symmetry. (A minimum-phase filter does not.) As zeros are moved outside the unit circle, the average delay increases. With one particular set of zeros outside, the coefficients become symmetric; we know about that. With all zeros inside, impulses cause post ringing. When all are outside, the dominant effect is pre ringing. When the phase is flat, ringing is symmetric and small. Just don't ask me to prove it. As you pointed out, an FIR's (a transversal filter's to be precise) poles are all at the origin. The angles of these poles are indeterminate. That makes them hard for me to think about. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 22, 20042004-02-22
Jerry Avins wrote:> Bob Cain wrote: > > ... > >> Imaging all the zeros that are outside to the inside makes >> it minimum phase and the reason for that I now understand. >> What I'm curious about is what kind of filter results from >> the opposite strategy, imaging all that are inside to the >> outside. Or what is the nature of the resulting FIR. >> Perhaps no such generalization exists. > > > When you make a transversal filter, the poles are all at the origin, > whether or not the filter has symmetry. (A minimum-phase filter does > not.) As zeros are moved outside the unit circle, the average delay > increases. With one particular set of zeros outside, the coefficients > become symmetric; we know about that. With all zeros inside, impulses > cause post ringing. When all are outside, the dominant effect is pre > ringing. When the phase is flat, ringing is symmetric and small. Just > don't ask me to prove it. > > As you pointed out, an FIR's (a transversal filter's to be precise) > poles are all at the origin. The angles of these poles are > indeterminate. That makes them hard for me to think about. > > ... > > JerryI need to think more about processes and less about math! To see what a maximum-phase FIR filter does to a signal, just filter the time-reversed signal and examine the result. Ringing becomes pre ringing, as I wrote above. Moreover, it's obvious when you look at it right that the impulse response is exactly the reverse of the minimum-phase case, so to make one, reverse the tap coefficients of the minimum-phase filter we know how to design. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 22, 20042004-02-22
Jerry Avins wrote: <trim>> Moreover, it's obvious when you look at it right that theimpulse> response is exactly the reverse of the minimum-phase case, so to make > one, reverse the tap coefficients of the minimum-phase filter we know > how to design.Things always seem to turn out much simpler than I imagine them. :-) Thanks, Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 22, 20042004-02-22
"Bob Cain" <arcane@arcanemethods.com> wrote in message news:c1b0hi14d0@enews3.newsguy.com...> Jerry Avins wrote: > > <trim> > > > Moreover, it's obvious when you look at it right that the > impulse > > response is exactly the reverse of the minimum-phase case, so to make > > one, reverse the tap coefficients of the minimum-phase filter we know > > how to design. > > Things always seem to turn out much simpler than I imagine > them. :-) >You actually have to put the poles outside the unit circle, as well as the zeros, to get that effect. I believe the term "maximum phase", though it isn't used much means with poles inside and zeros outside.
Reply by ●February 22, 20042004-02-22
"Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message news:Bie_b.2322$253.255196@news20.bellglobal.com...> You actually have to put the poles outside the unit circle, as well as the > zeros, to get that effect. I believe the term "maximum phase", though it > isn't used much means with poles inside and zeros outside.Oh, you mean FIRs... Nevermind.
Reply by ●February 22, 20042004-02-22
Matt Timmermans wrote:> "Bob Cain" <arcane@arcanemethods.com> wrote in message > news:c1b0hi14d0@enews3.newsguy.com... > >>Jerry Avins wrote: >> >><trim> >> >>> Moreover, it's obvious when you look at it right that the >> >>impulse >> >>>response is exactly the reverse of the minimum-phase case, so to make >>>one, reverse the tap coefficients of the minimum-phase filter we know >>>how to design. >> >>Things always seem to turn out much simpler than I imagine >>them. :-) >> > > > You actually have to put the poles outside the unit circle, as well as the > zeros, to get that effect. I believe the term "maximum phase", though it > isn't used much means with poles inside and zeros outside. >Curious how to do that if they are all at the origin. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 23, 20042004-02-23
In article f3WZb.25811$Cd6.1037211@news20.bellglobal.com, Matt Timmermans at mt0000@sympatico.nospam-remove.ca wrote on 02/21/2004 23:18: ...> What, exactly, a minimum phase filter minimizes is the average group delay > (discrete case) or integral of the group delay (continuous) over the entire > spectrum -- they're zero for minimum phase filters.i think that Minimum Phase Filters (whether they be FIR or IIR matters not) simply minimize the phase shift (or the negative of it). the minimizing of phase delay or group delay happens as a consequence of that.> Any minimum phase filter without zeros right on the frequency axis can be > reconstructed from its log-magnitude spectrum, by using the Hilbert > transform to derive its phase response. Exponentiate log-magnitude + > j*phase to get the frequency response, and transform that back into the time > domain to get the impulse response. The impulse response will be causal, > because log-magnitude + j*phase is an analytic signal, and exponentiation > preserves this property.i think to prove the association of MPF to the Hilbert Transform property of log-magnitude to phase in radians takes more than that. just because a complex function happens to be analytic (not to be confused with the use of the term "analytic" in "analytic signal") outside the unit circle does not necessarily mean that the log of it is analytic. zeros outside the unit circle don't hurt the analytic nature of the complex function but they *do* hurt the analyticality of the log of it.> From the above, we can deduce that, for minimum phase filters, the group > delay is the Hilbert transform of the derivative of the -log-magnitude > response, so I'm now quite confident that tiny changes in the log-magnitude > response won't move large portions of the energy significantly forward or > backward in time in the minimum phase response.that may be true. r b-j
Reply by ●February 23, 20042004-02-23
Bob Cain wrote:> Matt Timmermans wrote:...>> You actually have to put the poles outside the unit circle, as well as >> the >> zeros, to get that effect. I believe the term "maximum phase", though it >> isn't used much means with poles inside and zeros outside. >> > > Curious how to do that if they are all at the origin.They would all go to infinity, the only question being the angle. But it doesn't happen. I meant an FIR structure; how else could the tap coefficients be swapped end for end?, so the poles stay at the origin. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
