Filter design with 'freqsamp' method?
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From: Fred Marshall on 5 Jul 2010 16:30 robert bristow-johnson wrote: > On Jul 3, 6:21 pm, Fred Marshall <fmarshallx(a)remove_the_xacm.org> > wrote: >> robert bristow-johnson wrote: >>> On Jul 1, 10:59 am, Fred Marshall <fmarshallx(a)remove_the_xacm.org> >>> wrote: >>>> r b-j, >>>> Well, maybe I've had it wrong all these years but I'd say that the >>>> windowing method starts with N frequency samples where N is the length >>>> of the filter you want. >>> so then, since the DFT and iDFT are bijective (i love using fancy- >>> pants words), why window? if h[n] has N samples and N degrees of >>> freedom, so does H[k]. you specify your N frequency samples and you >>> can hit it perfectly with no windowing. >>> so, that seems curious to me. >>> r b-j >> You window because the frequency response between those points can be >> nasty. Either you don't window and convolve those samples with a >> matching sinc er.. Dirichlet or you window and convolve those samples >> with something else. > > you're always windowing it (unless your impulse response is an > infinitely-repeating periodic function. you draw your N (or more) > discrete points in the frequency domain, you iDFT it and what you have > there is still a periodic sequence. if you use just one cycle of that > (and zero the rest, as you would for an FIR), then you've applied the > rectangular window. > >> The trade is that the Dirichlet matches all the samples exactly. Other >> windows don't. Some match them all except the adjacent two as in the >> 1/2 1 1/2 sum of adjacent Dirichlets. It still has zeros in the sum >> at all the other sample points - so the convolution doesn't perturb >> their values. And, the values between points are better behaved. > > the values between points exhibit less "high frequency" behavior or > are less smooth. that's normally thought of as "better". and the > reason is that in the other domain (where the window is applied), > you've reduced the amplitude of the points further from h[0] (the > "high frequency" points, but they're really the high time-displacement > points that fuel the "high frequency" wiggling in the frequency > domain. > >> And making N larger to begin with doesn't affect any of this in my way >> of looking at it. > > it allows you to draw the frequency response more densely. so you can > explicitly specify how "wiggly" you want it between the sparser points > you have with your smaller "N". but to do so, the impulse response is > very large, too large. so you have to shorten it and that is > windowing. then, starting with the ideal (but too long) impulse > response, you can try shortening it in a variety of different ways > (using different windows) and see how well you do. you can also try > shortening it to different lengths (using whatever window that looks > best) to make a tradeoff decision on FIR length and its performance. > > it's very similar in philosophy to what we do with the windowed-sinc > LPF design. start with the ideal (which is too long) and see what > happens when you settle for something shorter. > > r b-j > > r b-j, OK. That interesting - the approaches have the same underlying ingredients and are only different by virtue of how the starting points are viewed. In the "infinite-N" case you would be seeking N based on responses and, I guess would decide on a non-rectangular window (or windows) to use in the process. As soon as you do the first test case it's the same as the "fixed-N" approach. It rather assumes that you don't know N very well, right? In the "fixed-N" case you would be seeking acceptable response by choosing non-rectangular window functions of length N. And then, if you aren't satisfied, iterate N. Just as the "infinite-N" case does in the end. I think of both as the "windowing method". How you approach it depends on how closely you figure you know N and your own preference. Either way, just like using P-M, you have to choose an N at some point. My way of roughly choosing N is to look at the minimum transition band width W and spec N so that the temporal length of the filter is approx 1.5/W. And, for even rougher estimates just 1/W which is what I usually use when discussing the "physics". You get to pick based on where you put the break points in the transition - so that choice usually gets mixed up with band ripple - particularly with minimax criterion. The lower the ripple, the wider the transition and the longer the filter. But, it's sure that the filter can't be any *shorter*!! :-) And, then, if you like you can use one of those length-estimating formulas. Fred |