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 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= >>>>>>>>> >>>>>>>> _______________________________________________ >>>>>>>> dupswapdrop: music-dsp mailing list >>>>>>>> music-dsp@music.columbia.edu >>>>>>>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>>>>>>> <https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.columbia.edu_mailman_listinfo_music-2Ddsp&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=0Zfr9NX2z_qbqorZ4mvWlKWdhvCOnws4tZKFE3J0lxI&s=_0-DUAEnNzJ0nyrUgGHozY0ob4n_-0OWpipEf-p2Bps&e=> >>>>>>> >>>>>>> _______________________________________________ >>>>>>> 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=0Zfr9NX2z_qbqorZ4mvWlKWdhvCOnws4tZKFE3J0lxI&s=_0-DUAEnNzJ0nyrUgGHozY0ob4n_-0OWpipEf-p2Bps&e= >>>>>> >>>>>> _______________________________________________ >>>>>> dupswapdrop: music-dsp mailing list >>>>>> music-dsp@music.columbia.edu >>>>>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>>>>> <https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.columbia.edu_mailman_listinfo_music-2Ddsp&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=ggIGGD37NXAIrRak00WIRysmpvCxdGGCHkoma2TGgxc&s=2aCxaadCSRm8GtUxELE7DhnWmqkKUkkAymUl19tD-v4&e=> >>>>> >>>>> _______________________________________________ >>>>> 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=ggIGGD37NXAIrRak00WIRysmpvCxdGGCHkoma2TGgxc&s=2aCxaadCSRm8GtUxELE7DhnWmqkKUkkAymUl19tD-v4&e= >>>> >>>> _______________________________________________ >>>> dupswapdrop: music-dsp mailing list >>>> music-dsp@music.columbia.edu >>>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>>> <https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.columbia.edu_mailman_listinfo_music-2Ddsp&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=LVW8eOM2POVbM1MauwqppWYiBwmnAs5_i7qiMOEK0-o&s=Wiyf_pAPkjR4_Ox3pi0vTvCNZDjINUsf0bfxVKpiGW8&e=> >>> >>> _______________________________________________ >>> 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=LVW8eOM2POVbM1MauwqppWYiBwmnAs5_i7qiMOEK0-o&s=Wiyf_pAPkjR4_Ox3pi0vTvCNZDjINUsf0bfxVKpiGW8&e= >> >> _______________________________________________ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp >> <https://urldefense.proofpoint.com/v2/url?u=https-3A__lists.columbia.edu_mailman_listinfo_music-2Ddsp&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=w_CiiFx8eb9uUtrPcg7_DA&m=cdkdusrQSRQeoIdvFPBw-IHlrTvjeJt4YV8Nkf2Zfd4&s=WaORGHf6gkBiBL4HM9z9I3P-lBjNVIi5TkMPBpzVmfY&e=> > > _______________________________________________ > 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=cdkdusrQSRQeoIdvFPBw-IHlrTvjeJt4YV8Nkf2Zfd4&s=WaORGHf6gkBiBL4HM9z9I3P-lBjNVIi5TkMPBpzVmfY&e=
_______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
