> 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
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