One other reader of this list tells me he's a big fan of Gabel & Roberts, recently read it cover to cover, and is planning to use it for a course. It's still in print at a low price in Asian markets.In terms of the funny correction to impulse invariance for impulses with a step at t = 0 that Leland Jackson and Wolfgang Mecklenbraüker separately published in 2000, yes, it's in all three editions of Gabel & Roberts, going back to 1973; and not quite in any of Jim Kaiser's papers.More on gammatones in the Gabel & Roberts "Signals and Linear Systems" book:I got the 1980 second edition and 1973 first edition. Like the 1987 third, they have a table of z transforms of sampled impulse responses of gammatone-like filters (complex gammatones); the second and third go to order 4, but the first goes all the way to order 5! The fifth order has numerator [1, 11, 11, 1], which I haven't checked, but seems plausible; that makes zeros at z = -9.8990, -1.0000, -0.1010.
Speaking of Jim Kaiser, I sent him a copy of my book, which was delivered yesterday; his son says "He couldn't remember receiving the book yesterday. Our healthcare worker found the book this morning on his night stand. He had already started reading it but forgot that he had! He appreciated your inscription. He was able to recall your interactions at Bell in the early 70's. Long term memory still seems relatively good." Jim is 89, living in Chapel Hill NC.I repeat my question: does anybody know these guys (Robert A. Gabel and/or Richard A. Roberts)?DickOn Thu, Feb 1, 2018 at 7:19 PM, Richard F. Lyon <dicklyon@xxxxxxx> wrote:The correction term would not be needed, and the problem would never have existed, if the discrete impulse response at h[0] had been defined in terms of the continuous impulse response h(t) as (h(0-) + h(0+)/2; that is, as the average across the step discontinuity at 0 if there is one, as two different papers in 2000 pointed out. The examples in the older papers and the correction term in the 1966 paper make it clear that such a reasonable choice was not made at that time.The "correction" term in the 1966 version is not there in 1963, nor in 1964.I got a copy that Jim Beauchamp found in a library near him. Thanks, Jim.If anyone wants a copy, let me know. Or of the 1964 Golden & Kaiser BSTJ paper.This issue (but not its history) is covered in great detail in the 1987 book Signals and Linear Systems, third edition, by Gabel and Roberts (does anyone know these guys?). I haven't looked at earlier editions. They not only discuss the discontinuity in depth, but also address repeated poles, which are ignored in most treatments, and provide a table up to order 4 repeated poles, which agrees precisely with Volker Hohmann's derivation of zeros in the numerator of the impulse-invariance design of discrete-time complex gammatone filters: a numerator [1, 4, 1] independent of pole frequencies and dampings, yielding zeros at z = -3.7321 and z = -0.2679, which do just a little smoothing on top of the repeated-poles filter.DickOn Mon, Jan 29, 2018 at 8:15 PM, Richard F. Lyon <dicklyon@xxxxxxx> wrote:DickThe key observation is that using the naive impulse invariance method adds a constant (frequency independent) term to the frequency response of the digital filter proportional to the impulse response on the right side of time zero: T/2 * h(0+). He didn't go as far as the "corrections" which said to take the impulse response h[k] at k = 0 to be (h(0-) + h(0+))/2, though it's pretty obvious from there. It's funny that at some point he got as far as including that unwanted term yet didn't comment on the easy way to remove it. Maybe in the missing paper...Does anyone have the 1963 Proceedings of the First Allerton Conference on Circuit and System Theory? Or just "Design methods for sampled-data filters" by J. F. Kaiser?I'm trying to resolve a disconnect in derivations of the impulse-invariance method, which was "corrected" in several places over the years, though Kaiser had the key to the correction in his chapter "Digital Filters" in the 1966 "System Analysis by Digital Computer" book, in which he says his stuff on IIR design closely follows that missing paper as well as a BSTJ paper that does not have the key piece.