From: vv on 5 Mar 2010 04:50 I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter Banks" on the topic of the monotonicity of the unwrapped phase response of an allpass digital filter. The proof is based on the observation that the group delay is always positive and hence slope of the phase response is always negative, ergo phi(w) is a decreasing function. The group delay > 0 is shown for a first order section, with the statement that the group delay for an Nth order system is the addition of N monotonically decreasing functions, and hence the result follows. Can someone suggest a reference (in DSP literature or math) to an alternative proof that is perhaps more rigorous? (without getting sidetracked by whether or not the the proof in PPV's text is rigorous enough :->). Thanks! All pass is, of course, H(z) = z^{-N} A(z^-1)/A(z), where A(z) is 1+ a1 z^{-1} + ... +aN z^{-N} and z \in C. -vv From: Rune Allnor on 5 Mar 2010 04:59 On 5 Mar, 10:50, vv wrote:> I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter > Banks" on the topic of the monotonicity of the unwrapped phase > response of an allpass digital filter.  The proof is based on the > observation that the group delay is always positive and hence slope of > the phase response is always negative, ergo phi(w) is a decreasing > function. There was a discussion here a few years ago where somebody (I can't remember who - RBJ? Andor?) demonstrated that a causal filter might in fact have negative group delay in parts of the frequency band. The effect showed up as a very short rise time in the impulse response of the filter. Rune From: Rune Allnor on 5 Mar 2010 05:42 On 5 Mar, 10:59, Rune Allnor wrote:> On 5 Mar, 10:50, vv wrote: > > > I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter > > Banks" on the topic of the monotonicity of the unwrapped phase > > response of an allpass digital filter.  The proof is based on the > > observation that the group delay is always positive and hence slope of > > the phase response is always negative, ergo phi(w) is a decreasing > > function. > > There was a discussion here a few years ago where somebody > (I can't remember who - RBJ? Andor?) demonstrated that a > causal filter might in fact have negative group delay in > parts of the frequency band. The effect showed up as a very > short rise time in the impulse response of the filter. > > Rune Found it: http://groups.google.no/group/comp.dsp/msg/c820aea7bdaf4cb2?hl=no It was Andor who saw the flaw in the argment that a causal system must necessarily have positive group delay everywhere. He found a filter fundtion that was *both* causal *and* had negative group delay over substantial parts of the frequency band. So if the proof in the book is based on the supposition that a causal filter response must have positive group delay everywhere, the proof is wrong. Rune From: VV on 5 Mar 2010 06:35 On Mar 5, 3:42 pm, Rune Allnor wrote:> So if the proof in the book is based on the supposition > that a causal filter response must have positive group > delay everywhere, the proof is wrong. The allpass is filter is causal and stable and its group delay is always positive. There is nothing wrong with the proof. Assume the pole is at r exp(j x), where r < 1. The group delay for a first order allpass is (1-r^2)/|1-r exp(j(x-w))|^2, which is always positive. If the filter is not allpass, causal and stable filters can give rise to an expression for group delay that goes negative for some w, which is also well-know, I guess. (On negative group delay, it may be of interest to some people to see Morgan Mitchell and Raymond Y. Chiao: Causality and Negative Group Delays in a simple band-pass amplifier, American Journal of Physics, Vol. 66 no. 1, January 1998). -vv From: Jerry Avins on 5 Mar 2010 09:54 Rune Allnor wrote:> On 5 Mar, 10:59, Rune Allnor wrote: >> On 5 Mar, 10:50, vv wrote: >> >>> I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter >>> Banks" on the topic of the monotonicity of the unwrapped phase >>> response of an allpass digital filter. The proof is based on the >>> observation that the group delay is always positive and hence slope of >>> the phase response is always negative, ergo phi(w) is a decreasing >>> function. >> There was a discussion here a few years ago where somebody >> (I can't remember who - RBJ? Andor?) demonstrated that a >> causal filter might in fact have negative group delay in >> parts of the frequency band. The effect showed up as a very >> short rise time in the impulse response of the filter. >> >> Rune > > Found it: > > http://groups.google.no/group/comp.dsp/msg/c820aea7bdaf4cb2?hl=no > > It was Andor who saw the flaw in the argument that a > causal system must necessarily have positive group delay > everywhere. He found a filter function that was *both* > causal *and* had negative group delay over substantial > parts of the frequency band. > > So if the proof in the book is based on the supposition > that a causal filter response must have positive group > delay everywhere, the proof is wrong. From what I read here, the proof in the book assumes that an allpass filter has positive group delay everywhere. I don't have the book, so I can't check that. Jerry -- Blaise Pascal: Men never do evil so completely and cheerfully as when they do it from religious conviction. ��������������������������������������������������������������