Thank you. Yes it seems very theoretical and math heavy. In practice you will get this frequency response artifact no matter how small. It should factor into the math in some way, perhaps they are not looking at the laplacian
On Wed, Jun 24, 2020, 10:41 AM Corey K <corey...@gmail.com> wrote: > It's a classic paper. Google scholar shows it has been cited over 1000 > times. There's a link to it here here: > https://jontalle.web.engr.illinois.edu/uploads/537/Papers/Public/AllenRabiner77-ProcIEEE.pdf > <https://urldefense.proofpoint.com/v2/url?u=https-3A__jontalle.web.engr.illinois.edu_uploads_537_Papers_Public_AllenRabiner77-2DProcIEEE.pdf&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=EIWxvlh8qpdcU_coPNdiZxhQaqJEihTl-BcA8WN1eWw&s=e-NKtVrjHEINB5kEwTE3d4g9q2bZGxjnso8yyBhaUUM&e=> > > > On Wed, Jun 24, 2020 at 11:56 AM Zhiguang Eric Zhang <zez...@nyu.edu> > wrote: > >> unfortunately, i'm not familiar with that paper. could you please attach >> it or provide a link for reference? the Gibbs phenomenon is actually a >> very well-known and thoroughly characterized signal processing artifact >> that has been approached from a variety of angles as far as trying to find >> a solution. iit can be thought of as an unavoidable digital filter >> response of having to take X number of samples in one snapshot while >> capturing a finite instance in time (as you might know the Dirac delta is >> centered on DC) >> >> https://en.wikipedia.org/wiki/Ringing_artifacts >> <https://urldefense.proofpoint.com/v2/url?u=https-3A__en.wikipedia.org_wiki_Ringing-5Fartifacts&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=EIWxvlh8qpdcU_coPNdiZxhQaqJEihTl-BcA8WN1eWw&s=XkZObRCqtj2hzKnIyyVYQqrxJYhSfTrXnDzaDFzTsw4&e=> >> >> On Wed, Jun 24, 2020 at 10:12 AM Corey K <corey...@gmail.com> wrote: >> >>> I think you're mistaken, unfortunately. Block FFT convolution has been >>> around for 30+ years. In 1977 (43 years ago now), Jont Allen showed in his >>> paper "A Unified Approach to Short-Time Fourier Analysis" how you can >>> perform FIR filtering perfectly with the FFT, of COLA windows are used. See >>> equation 5.2.5 in that paper, and the analysis that precedes it. >>> >>> >>> >>> >>> >>> On Wed, Jun 24, 2020 at 11:16 AM Zhiguang Eric Zhang <zez...@nyu.edu> >>> wrote: >>> >>>> that's not true. with FFT/COLA you will necessarily have the Gibbs >>>> phenomenon / ringing / ripple artifacts. certain window types will >>>> minimize this but you will get this phenomenon nonetheless. >>>> >>>> On Wed, Jun 24, 2020 at 9:44 AM Corey K <corey...@gmail.com> wrote: >>>> >>>>> I see what you're getting at, I suppose. However, in the context of >>>>> FIR filtering I wouldn't refer to this as an artifact. Let's say you gave >>>>> me an FIR filter with N-taps and asked me to write a program to implement >>>>> that filter. I could implement this using a direct form structure (in the >>>>> time-domain), or with the FFT using OLA. Both would give the exact same >>>>> results down to numerical precision, with no "artifacts". That's why it >>>>> intrigued me when you said "of course it won't have the ripple artifacts >>>>> associated with FFT overlap windowing" when referring to software that >>>>> does >>>>> filtering. >>>>> >>>>> >>>>> On Wed, Jun 24, 2020 at 10:59 AM Zhiguang Eric Zhang <zez...@nyu.edu> >>>>> wrote: >>>>> >>>>>> ripple is just a known artifactual component of a windowing >>>>>> operation. it's also known as the Gibbs phenomenon >>>>>> >>>>>> http://matlab.izmiran.ru/help/toolbox/signal/filterd8.html >>>>>> <https://urldefense.proofpoint.com/v2/url?u=http-3A__matlab.izmiran.ru_help_toolbox_signal_filterd8.html&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=LVW8eOM2POVbM1MauwqppWYiBwmnAs5_i7qiMOEK0-o&s=XefFmTg_gx0qQrZnZTOJDTlaqMl3xt5WBzqxYAkoMKA&e=> >>>>>> >>>>>> i'm not referring to any equivalency between time/freq domain >>>>>> filtering >>>>>> >>>>>> >>>>>> On Wed, Jun 24, 2020 at 9:21 AM Corey K <corey...@gmail.com> wrote: >>>>>> >>>>>>> Not totally understanding you, unfortunately. But if what you are >>>>>>> describing is part of the normal filter response/ringing I guess I >>>>>>> wouldn't >>>>>>> refer to it as "artifacts"? FIR filtering can be performed equivalently >>>>>>> in >>>>>>> the time or frequency domain. Do you disagree with that statement? >>>>>>> >>>>>>> On Wed, Jun 24, 2020 at 10:02 AM Zhiguang Eric Zhang <zez...@nyu.edu> >>>>>>> wrote: >>>>>>> >>>>>>>> yes but any windowing operation is akin to taking a dirac delta >>>>>>>> function on X number of samples and thus you will get ringing/ripple >>>>>>>> artifacts as a necessary part of the filter response >>>>>>>> >>>>>>>> On Wed, Jun 24, 2020 at 6:30 AM Corey K <corey...@gmail.com> wrote: >>>>>>>> >>>>>>>>> >>>>>>>>> of course it won't have the ripple artifacts associated with FFT >>>>>>>>>> overlap windowing >>>>>>>>>> >>>>>>>>> >>>>>>>>> What is the ripple artifact you are talking about? When using >>>>>>>>> constant overlap add (COLA) windows the STFT is a perfect >>>>>>>>> reconstruction >>>>>>>>> filterbank. Likewise block FFT convolution can be used to implement >>>>>>>>> any FIR >>>>>>>>> filtering operation. >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>>> cheers, >>>>>>>>>> -ez >>>>>>>>>> >>>>>>>>>> On Mon, Apr 13, 2020 at 4:55 PM Andreas Gustafsson < >>>>>>>>>> g...@waxingwave.com> wrote: >>>>>>>>>> >>>>>>>>>>> Hello Spencer, >>>>>>>>>>> >>>>>>>>>>> You wrote: >>>>>>>>>>> > A while ago I read through some the literature [1] on >>>>>>>>>>> implementing >>>>>>>>>>> > an invertible CQT as a special case of the Nonstationary Gabor >>>>>>>>>>> > Transform. It's implemented by the essentia library [2] among >>>>>>>>>>> other >>>>>>>>>>> > places probably. >>>>>>>>>>> > >>>>>>>>>>> > The main idea is that you take the FFT of your whole signal, >>>>>>>>>>> then >>>>>>>>>>> > apply the filter bank in the frequency domain (just >>>>>>>>>>> > multiplication). Then you IFFT each filtered signal, which >>>>>>>>>>> gives you >>>>>>>>>>> > the time-domain samples for each band of the filter bank. Each >>>>>>>>>>> > frequency-domain filter has a different bandwidth, so your >>>>>>>>>>> IFFT is a >>>>>>>>>>> > different length for each one, which gives you the different >>>>>>>>>>> sample >>>>>>>>>>> > rates for each one. >>>>>>>>>>> >>>>>>>>>>> That's the basic idea, but the Gaborator rounds up each of the >>>>>>>>>>> per-band sample rates to the original sample rate divided by some >>>>>>>>>>> power of two. This means all the FFT sizes can be powers of two, >>>>>>>>>>> which tend to be faster than arbitrary sizes. It also results >>>>>>>>>>> in a >>>>>>>>>>> nicely regular time-frequency sampling grid where many of the >>>>>>>>>>> samples >>>>>>>>>>> coincide in time, as shown in the second plot on this page: >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__www.gaborator.com_gaborator-2D1.4_doc_overview.html&d=DwICAg&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=4rIFY1X4fS1G8-882xM72jF9DvsY6-Z2ckeHxjPPfTY&s=FG-ZGfFa09T-Y7nLajB8evbCy9WIADFrUqPwjz-LHow&e= >>>>>>>>>>> >>>>>>>>>>> Also, the Gaborator makes use of multirate processing where the >>>>>>>>>>> signal >>>>>>>>>>> is repeatedly decimated by 2 and the calculations for the lower >>>>>>>>>>> octaves run at successively lower sample rates. These >>>>>>>>>>> optimizations >>>>>>>>>>> help the Gaborator achieve a performance of millions of samples >>>>>>>>>>> per >>>>>>>>>>> second per CPU core. >>>>>>>>>>> >>>>>>>>>>> > They also give an "online" version where you do >>>>>>>>>>> > the processing in chunks, but really for this to work I think >>>>>>>>>>> you'd >>>>>>>>>>> > need large-ish chunks so the latency would be pretty bad. >>>>>>>>>>> >>>>>>>>>>> The Gaborator also works in chunks. A typical chunk size might >>>>>>>>>>> be >>>>>>>>>>> 8192 samples, but thanks to the multirate processing, in the >>>>>>>>>>> lowest >>>>>>>>>>> frequency bands, each of those 8192 samples may represent the >>>>>>>>>>> low-frequency content of something like 1024 samples of the >>>>>>>>>>> original >>>>>>>>>>> signal. This gives an effective chunk size of some 8 million >>>>>>>>>>> samples >>>>>>>>>>> without actually having to perform any FFTs that large. >>>>>>>>>>> >>>>>>>>>>> Latency is certainly high, but I would not say it is a >>>>>>>>>>> consequence of >>>>>>>>>>> the chunk size as such. Rather, both the high latency and the >>>>>>>>>>> need >>>>>>>>>>> for a large (effective) chunk size are consequences of the >>>>>>>>>>> lengths of >>>>>>>>>>> the band filter impulse responses, which get exponentially >>>>>>>>>>> larger as >>>>>>>>>>> the constant-Q bands get narrower towards lower frequencies. >>>>>>>>>>> >>>>>>>>>>> Latency in the Gaborator is discussed in more detail here: >>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__www.gaborator.com_gaborator-2D1.4_doc_realtime.html&d=DwICAg&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=4rIFY1X4fS1G8-882xM72jF9DvsY6-Z2ckeHxjPPfTY&s=uuRzi0taGcXI9Sq63G_xTTrCjaz9cu3ewu8jfzIUcVc&e= >>>>>>>>>>> >>>>>>>>>>> > The whole process is in some ways dual to the usual STFT >>>>>>>>>>> process, >>>>>>>>>>> > where we first window and then FFT. in the NSGT you first FFT >>>>>>>>>>> and >>>>>>>>>>> > then window, and then IFFT each band to get a Time-Frequency >>>>>>>>>>> > representation. >>>>>>>>>>> >>>>>>>>>>> Yes. >>>>>>>>>>> >>>>>>>>>>> > For resynthesis you end up with a similar window overlap >>>>>>>>>>> constraint >>>>>>>>>>> > as in STFT, except now the windows are in the frequency >>>>>>>>>>> domain. It's >>>>>>>>>>> > a little more complicated because the window centers aren't >>>>>>>>>>> > evenly-spaced, so creating COLA windows is complicated. There >>>>>>>>>>> are >>>>>>>>>>> > some fancier approaches to designing a set of synthesis >>>>>>>>>>> windows that >>>>>>>>>>> > are complementary (inverse) of the analysis windows, which is >>>>>>>>>>> what >>>>>>>>>>> > the frame-theory folks like that Austrian group seem to like >>>>>>>>>>> to use. >>>>>>>>>>> >>>>>>>>>>> The Gaborator was inspired by the papers from that Austrian >>>>>>>>>>> group and >>>>>>>>>>> uses complementary resynthesis windows, or "duals" as frame >>>>>>>>>>> theorists >>>>>>>>>>> like to call them. The analysis windows are Gaussian, and the >>>>>>>>>>> dual >>>>>>>>>>> windows used for resynthesis end up being slightly distorted >>>>>>>>>>> Gaussians. >>>>>>>>>>> >>>>>>>>>>> > One of the nice things about the NSGT is it lets you be really >>>>>>>>>>> > flexible in your filterbank design while still giving you >>>>>>>>>>> > invertibility. >>>>>>>>>>> >>>>>>>>>>> Agreed. >>>>>>>>>>> >>>>>>>>>>> In a later message, you wrote: >>>>>>>>>>> > Whoops, just clicked through to the documentation and it looks >>>>>>>>>>> like >>>>>>>>>>> > this is the track you're on also. I'm curious if you have any >>>>>>>>>>> > insight into the window-selection for the analysis and >>>>>>>>>>> synthesis >>>>>>>>>>> > process. It seems like the NSGT framework forces you to be a >>>>>>>>>>> bit >>>>>>>>>>> > smarter with windows than just sticking to COLA, but the dual >>>>>>>>>>> frame >>>>>>>>>>> > techniques should apply for regular STFT processing, right? >>>>>>>>>>> >>>>>>>>>>> I'm actually not that familiar with traditional STFTs and COLA, >>>>>>>>>>> but as >>>>>>>>>>> far as I can tell, the STFT is a special case of the NSGT and >>>>>>>>>>> the same >>>>>>>>>>> dual frame techniques should apply. >>>>>>>>>>> -- >>>>>>>>>>> Andreas Gustafsson, g...@waxingwave.com >>>>>>>>>>> _______________________________________________ >>>>>>>>>>> dupswapdrop: music-dsp mailing list >>>>>>>>>>> music-dsp@music.columbia.edu >>>>>>>>>>> >>>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.columbia.edu_mailman_listinfo_music-2Ddsp&d=DwICAg&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=4rIFY1X4fS1G8-882xM72jF9DvsY6-Z2ckeHxjPPfTY&s=br6gIADk3PB9_kF8YoA7aZdcf5McFvCCOlyYso5D2BI&e= >>>>>>>>>>> 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