### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

thanks for your offer, I can not really read Math lab code and always have a hard time even figuring out essentials of such code My phase vocoder already works kind of satisfactorily now as a demo in Native Instruments Reaktor, I do the forward FFT offline and the iFFT "just in time", that means 12 "butterflies" per sample, so you could bring down latency by speeding up the iFFT, though I am not sure what a reasonable latency is. I made a poll on some electronic musicians board and most people voted for 10 ms being just tolerable. I am half way content with the way it works now, for analysis I have twelve FFTs in parallel, one for each octave with window sizes based on ERB scale per octave, so it's not totally bad on transients but not good either. I assume there is still some room for improvements on the windows, but not very much. FFT size is 4096, and now I search for ways to improve it, mostly regarding transients. But I am not sure if that's possible with FFT cause I still have pre-ringing, and I cant see how to avoid that completely cause you can only shorten the windows on the low octaves so much. Maybe with an assymetric window? If you do the analysis with a IIR filter bank (or wavelets) you kind of have assymmetric windows, that is the filters integrate in a causal way with a decaying "window" they see, but I am not sure if this can be adapted somehow to an FFT. An other way that would reduce reverberation and shorten transient times somehwat would be using shorter FFTs for the resynthesis, this would also bring down CPU a bit and latency. So this is where I am at at the moment Am 09.11.2018 um 23:29 schrieb robert bristow-johnson: i don't wanna lead you astray. i would recommend staying with the phase vocoder as a framework for doing time-frequency manipulation. it **can** be used real-time for pitch shift, but when i have used the phase vocoder, it was for time-scaling and then we would simply resample the time-scaled output of the phase vocoder to bring the tempo back to the original and shift the pitch. that was easier to get it right than it was to *move* frequency components around in the phase vocoder. but i remember in the 90s, Jean Laroche doing that real time with a single PC. also a real-time phase vocoder (or any frequency-domain process, like sinusoidal modeling) is going to have delay in a real-time process. even if your processor is infinitely fast, you still have to fill up your FFT buffer with samples before invoking the FFT. if your buffer is 4096 samples and your sample rate is 48 kHz, that's almost 1/10 second. and that doesn't count processing time, just the buffering time. and, in reality, you will have to double buffer this process (buffer both input and output) and that will make the delay twice as much. so with 1/5 second delay, that's might be an issue. i offered this before (and someone sent me a request and i believe i replied, but i don't remember who), but if you want my 2001 MATLAB code that demonstrates a simple phase vocoder doing time scaling, i am happy to send it to you or anyone. it's old. you have to turn wavread() and wavwrite() into audioread() and audiowrite(), but otherwise, i think it will work. it has an additional function that time-scales each sinusoid *within* every frame, but i think that can be turned off and you can even delete that modification and what you have left is, in my opinion, the most basic phase vocoder implemented to do time scaling. lemme know if that might be helpful. L8r, r b-j Original Message ---------------- Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "gm" Date: Fri, November 9, 2018 5:02 pm To: music-dsp@music.columbia.edu -- > You get me intrigued with this > > I actually believe that wavelets are the way to go for such things, > but, besides that anything beyond a Haar wavelet is too complicated for me > (and I just grasp that Haar very superficially of course), > > I think one problem is the problem you mentioned - don't do anything > with the bands, > only then you have perfect reconstruction > > And what to do you do with the bands to make a pitch shift or to > preserve formants/do some vocoding? > > It's not so obvious (to me), my naive idea I mentioned earlier in this > thread was to > do short FFTs on the bands and manipulate the FFTs only > > But how? if you time stretch them, I believe the pitch goes down (thats > my intuition only, I am not sure) > and also, these bands alias, since the filters are not brickwall, > and the aliasing is only canceled on reconstruction I believe? > > So, yes, very interesting topic, that could lead me astray for another > couple o

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

� i don't wanna lead you astray.� i would recommend staying with the phase vocoder as a framework for doing time-frequency manipulation.� it **can** be used real-time for pitch shift, but when i have used the phase vocoder, it was for time-scaling and then we would simply resample the time-scaled output of the phase vocoder to bring the tempo back to the original and shift the pitch.� that was easier to get it right than it was to *move* frequency components around in the phase vocoder.� but i remember in the 90s, Jean Laroche doing that real time with a single PC.� also a real-time phase vocoder (or any frequency-domain process, like sinusoidal modeling) is going to have delay in a real-time process.� even if your processor is infinitely fast, you still have to fill up your FFT buffer with samples before invoking the FFT.� if your buffer is 4096 samples and your sample rate is 48 kHz, that's almost 1/10 second.� and that doesn't count processing time, just the buffering time.� and, in reality, you will have to double buffer this process (buffer both input and output) and that will make the delay twice as much.� so with 1/5 second delay, that's might be an issue. i offered this before (and someone sent me a request and i believe i replied, but i don't remember who), but if you want my 2001 MATLAB code that demonstrates a simple phase vocoder doing time scaling, i am happy to send it to you or anyone.� it's old.� you have to turn wavread() and wavwrite() into audioread() and audiowrite(), but otherwise, i think it will work.� it has an additional function that time-scales each sinusoid *within* every frame, but i think that can be turned off and you can even delete that modification and what you have left is, in my opinion, the most basic phase vocoder implemented to do time scaling.� lemme know if that might be helpful. L8r, r b-j � Original Message Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "gm" Date: Fri, November 9, 2018 5:02 pm To: music-dsp@music.columbia.edu -- > You get me intrigued with this > > I actually believe that wavelets are the way to go for such things, > but, besides that anything beyond a Haar wavelet is too complicated for me > (and I just grasp that Haar very superficially of course), > > I think one problem is the problem you mentioned - don't do anything > with the bands, > only then you have perfect reconstruction > > And what to do you do with the bands to make a pitch shift or to > preserve formants/do some vocoding? > > It's not so obvious (to me), my naive idea I mentioned earlier in this > thread was to > do short FFTs on the bands and manipulate the FFTs only > > But how? if you time stretch them, I believe the pitch goes down (thats > my intuition only, I am not sure) > and also, these bands alias, since the filters are not brickwall, > and the aliasing is only canceled on reconstruction I believe? > > So, yes, very interesting topic, that could lead me astray for another > couple of weeks but without any results I guess > > I think as long as I don't fully grasp all the properties of the FFT and > phase vocoder I shouldn't start anything new... > -- r b-j� � � � � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge." � � � � ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

You get me intrigued with this I actually believe that wavelets are the way to go for such things, but, besides that anything beyond a Haar wavelet is too complicated for me (and I just grasp that Haar very superficially of course), I think one problem is the problem you mentioned - don't do anything with the bands, only then you have perfect reconstruction And what to do you do with the bands to make a pitch shift or to preserve formants/do some vocoding? It's not so obvious (to me), my naive idea I mentioned earlier in this thread was to do short FFTs on the bands and manipulate the FFTs only But how? if you time stretch them, I believe the pitch goes down (thats my intuition only, I am not sure) and also, these bands alias, since the filters are not brickwall, and the aliasing is only canceled on reconstruction I believe? So, yes, very interesting topic, that could lead me astray for another couple of weeks but without any results I guess I think as long as I don't fully graps all the properties of the FFT and phase vocoder I shouldn't start anything new... Am 09.11.2018 um 22:31 schrieb robert bristow-johnson: what you're discussing here appears to me to be about perfect reconstruction in the context of Wavelets and Filter Banks. there is a theorem that's pretty easy to prove that if you have complementary high and low filterbanks with a common cutoff at 1/2 Nyquist, you can downsample both high and low-pass filterbank outputs by a factor of 1/2 and later combine the two down-sampled streams of samples to get perfect reconstruction of the original. this result is not guaranteed if you **do** anything to either filter output in the filterbank. Original Message Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "gm" Date: Fri, November 9, 2018 4:19 pm To: music-dsp@music.columbia.edu -- > > hm, my application has also WOLA ... > > All I find is about up- and downsampling of time sequences and spectra > of the same length. > ... > > If anyone knows of an easy explanation of down- and up sampling spectra > it would be much appreciated. > > Am 09.11.2018 um 19:16 schrieb Ethan Duni: > .. >> The only applications I know of that tolerate time-domain aliasing in >> transforms are WOLA filter banks - which are explicitly designed to >> cancel these (severe!) artifacts in the surrounding time-domain >> processing. -- r b-j r...@audioimagination.com "Imagination is more important than knowledge." ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

what you're discussing here appears to me to be about perfect reconstruction in the context of Wavelets and Filter Banks. there is a theorem that's pretty easy to prove that if you have complementary high and low filterbanks with a common cutoff at 1/2 Nyquist, you can downsample both high and low-pass filterbank outputs by a factor of 1/2 and later combine the two down-sampled streams of samples to get perfect reconstruction of the original.� this result is not guaranteed if you **do** anything to either filter output in the filterbank. Original Message Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "gm" Date: Fri, November 9, 2018 4:19 pm To: music-dsp@music.columbia.edu -- > > hm, my application has also WOLA ... > > All I find is about up- and downsampling of time sequences and spectra > of the same length. > ... > > If anyone knows of an easy explanation of down- and up sampling spectra > it would be much appreciated. > > Am 09.11.2018 um 19:16 schrieb Ethan Duni: > .. >> The only applications I know of that tolerate time-domain aliasing in >> transforms are WOLA filter banks - which are explicitly designed to >> cancel these (severe!) artifacts in the surrounding time-domain >> processing. -- r b-j� � � � � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge." � � � � ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

s a string of stationary inputs it can fold them together into one big high-res DCT and code that instead. On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn mailto:et...@polyspectral.com>> wrote: I don't think that's correct -- DIF involves first doing a single stage of butterfly operations over the input, and then doing two smaller DFTs on that preprocessed data. I don't think there is any reasonable way to take two "consecutive" DFTs of the raw input data and combine them into a longer DFT. (And I don't know anything about the historical question!) -Ethan On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson mailto:r...@audioimagination.com>> wrote: Ethan, that's just the difference between Decimation-in-Frequency FFT and Decimation-in-Time FFT. i guess i am not entirely certainly of the history, but i credited both the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect historical impression. Original Message -------------------- Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "Ethan Fenn" mailto:et...@polyspectral.com>> Date: Mon, November 5, 2018 10:17 am To: music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> -- > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ > DFT's (that is, the DFT of the even-numbered samples and the DFT of the > odd-numbered samples) and combine them into one longer DFT. But here you're > talking about taking two _consecutive_ DFT's. I don't think there's any > cheap way to combine these to exactly recover an individual bin of the > longer DFT. > > Of course it's possible you'll be able to come up with a clever frequency > estimator using this information. I'm just saying it won't be exact in the > way Cooley-Tukey is. > > -Ethan > > -- r b-j r...@audioimagination.com <mailto:r...@audioimagination.com> "Imagination is more important than knowledge." ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

gm wrote: >This is brining up my previous question again, how do you decimate a spectrum >by an integer factor properly, can you just add the bins? To decimate by N, you just take every Nth bin. >the orginal spectrum represents a longer signal so I assume folding >of the waveform occurs? Yeah, you will get time-domain aliasing unless your DFT is oversampled (i.e., zero-padded in time domain) by a factor of (at least) N to begin with. For critically sampled signals the result is severe distortion (i.e., SNR ~= 0dB). >but maybe this doesn't matter in practice for some applications? The only applications I know of that tolerate time-domain aliasing in transforms are WOLA filter banks - which are explicitly designed to cancel these (severe!) artifacts in the surrounding time-domain processing. Ethan D On Fri, Nov 9, 2018 at 6:39 AM gm wrote: > This is brining up my previous question again, how do you decimate a > spectrum > by an integer factor properly, can you just add the bins? > > the orginal spectrum represents a longer signal so I assume folding > of the waveform occurs? but maybe this doesn't matter in practice for some > applications? > > The background is still that I want to use a higher resolution for > ananlysis and > a lower resolution for synthesis in a phase vocoder. > > Am 08.11.2018 um 21:45 schrieb Ethan Duni: > > Not sure can get the odd bins *easily*, but it is certainly possible. > Conceptually, you can take the (short) IFFT of each block, then do the > (long) FFT of the combined blocks. The even coefficients simplify out as > you observed, the odd ones will be messier. Not sure quite how messy - I've > only looked at the details for DCT cases. > > Probably the clearest way to think about it is in the frequency domain. > Conceptually, the two consecutive short DFTs are the same as if we had > taken two zero-padded long DFTs, and then downsampled each by half. So the > way to combine them is to reverse that process: upsample them by 2, and > then add them together (with appropriate compensation for the > zero-padding/boxcar window). > > Ethan D > > On Thu, Nov 8, 2018 at 8:12 AM Ethan Fenn wrote: > >> I'd really like to understand how combining consecutive DFT's can work. >> Let's say our input is x0,x1,...x7 and the DFT we want to compute is >> X0,X1,...X7 >> >> We start by doing two half-size DFT's: >> >> Y0 = x0 + x1 + x2 + x3 >> Y1 = x0 - i*x1 - x2 + i*x3 >> Y2 = x0 - x1 + x2 - x3 >> Y3 = x0 + i*x1 - x2 - i*x3 >> >> Z0 = x4 + x5 + x6 + x7 >> Z1 = x4 - i*x5 - x6 + i*x7 >> Z2 = x4 - x5 + x6 - x7 >> Z3 = x4 + i*x5 - x6 - i*x7 >> >> Now I agree because of periodicity we can compute all the even-numbered >> bins easily: X0=Y0+Z0, X2=Y1+Z1, and so on. >> >> But I don't see how we can get the odd bins easily from the Y's and Z's. >> For instance we should have: >> >> X1 = x0 + (r - r*i)*x1 - i*x2 + (-r - r*i)*x3 - x4 + (-r + r*i)*x5 + i*x6 >> + (r + r*i)*x7 >> >> where r=sqrt(1/2) >> >> Is it actually possible? It seems like the phase of the coefficients in >> the Y's and Z's advance too quickly to be of any use. >> >> -Ethan >> >> >> >> On Mon, Nov 5, 2018 at 3:40 PM, Ethan Duni wrote: >> >>> You can combine consecutive DFTs. Intuitively, the basis functions are >>> periodic on the transform length. But it won't be as efficient as having >>> done the big FFT (as you say, the decimation in time approach interleaves >>> the inputs, so you gotta pay the piper to unwind that). Note that this is >>> for naked transforms of successive blocks of inputs, not a WOLA filter >>> bank. >>> >>> There are Dolby codecs that do similar with a suitable flavor of DCT >>> (type II I think?) - you have your encoder going along at the usual frame >>> rate, but if it detects a string of stationary inputs it can fold them >>> together into one big high-res DCT and code that instead. >>> >>> On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn >>> wrote: >>> >>>> I don't think that's correct -- DIF involves first doing a single stage >>>> of butterfly operations over the input, and then doing two smaller DFTs on >>>> that preprocessed data. I don't think there is any reasonable way to take >>>> two "consecutive" DFTs of the raw input data and combine them into a longer >>>> DFT. >>>> >>>> (And I don't know anything about the historical question!) >>>> >>>> -Ethan >>>> >>>> >>>> >>>> On Mon, Nov 5,

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

This is brining up my previous question again, how do you decimate a spectrum by an integer factor properly, can you just add the bins? the orginal spectrum represents a longer signal so I assume folding of the waveform occurs? but maybe this doesn't matter in practice for some applications? The background is still that I want to use a higher resolution for ananlysis and a lower resolution for synthesis in a phase vocoder. Am 08.11.2018 um 21:45 schrieb Ethan Duni: Not sure can get the odd bins *easily*, but it is certainly possible. Conceptually, you can take the (short) IFFT of each block, then do the (long) FFT of the combined blocks. The even coefficients simplify out as you observed, the odd ones will be messier. Not sure quite how messy - I've only looked at the details for DCT cases. Probably the clearest way to think about it is in the frequency domain. Conceptually, the two consecutive short DFTs are the same as if we had taken two zero-padded long DFTs, and then downsampled each by half. So the way to combine them is to reverse that process: upsample them by 2, and then add them together (with appropriate compensation for the zero-padding/boxcar window). Ethan D On Thu, Nov 8, 2018 at 8:12 AM Ethan Fenn <mailto:et...@polyspectral.com>> wrote: I'd really like to understand how combining consecutive DFT's can work. Let's say our input is x0,x1,...x7 and the DFT we want to compute is X0,X1,...X7 We start by doing two half-size DFT's: Y0 = x0 + x1 + x2 + x3 Y1 = x0 - i*x1 - x2 + i*x3 Y2 = x0 - x1 + x2 - x3 Y3 = x0 + i*x1 - x2 - i*x3 Z0 = x4 + x5 + x6 + x7 Z1 = x4 - i*x5 - x6 + i*x7 Z2 = x4 - x5 + x6 - x7 Z3 = x4 + i*x5 - x6 - i*x7 Now I agree because of periodicity we can compute all the even-numbered bins easily: X0=Y0+Z0, X2=Y1+Z1, and so on. But I don't see how we can get the odd bins easily from the Y's and Z's. For instance we should have: X1 = x0 + (r - r*i)*x1 - i*x2 + (-r - r*i)*x3 - x4 + (-r + r*i)*x5 + i*x6 + (r + r*i)*x7 where r=sqrt(1/2) Is it actually possible? It seems like the phase of the coefficients in the Y's and Z's advance too quickly to be of any use. -Ethan On Mon, Nov 5, 2018 at 3:40 PM, Ethan Duni mailto:ethan.d...@gmail.com>> wrote: You can combine consecutive DFTs. Intuitively, the basis functions are periodic on the transform length. But it won't be as efficient as having done the big FFT (as you say, the decimation in time approach interleaves the inputs, so you gotta pay the piper to unwind that). Note that this is for naked transforms of successive blocks of inputs, not a WOLA filter bank. There are Dolby codecs that do similar with a suitable flavor of DCT (type II I think?) - you have your encoder going along at the usual frame rate, but if it detects a string of stationary inputs it can fold them together into one big high-res DCT and code that instead. On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn mailto:et...@polyspectral.com>> wrote: I don't think that's correct -- DIF involves first doing a single stage of butterfly operations over the input, and then doing two smaller DFTs on that preprocessed data. I don't think there is any reasonable way to take two "consecutive" DFTs of the raw input data and combine them into a longer DFT. (And I don't know anything about the historical question!) -Ethan On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson mailto:r...@audioimagination.com>> wrote: Ethan, that's just the difference between Decimation-in-Frequency FFT and Decimation-in-Time FFT. i guess i am not entirely certainly of the history, but i credited both the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect historical impression. Original Message -------- Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "Ethan Fenn" mailto:et...@polyspectral.com>> Date: Mon, November 5, 2018 10:17 am To: music-dsp@music.columbia.edu <mailto:music-dsp@music.columbia.edu> -- > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ > DFT's (that is, the DFT of the even-numbered samples and the DFT of the > odd-numbered samples) a

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Not sure can get the odd bins *easily*, but it is certainly possible. Conceptually, you can take the (short) IFFT of each block, then do the (long) FFT of the combined blocks. The even coefficients simplify out as you observed, the odd ones will be messier. Not sure quite how messy - I've only looked at the details for DCT cases. Probably the clearest way to think about it is in the frequency domain. Conceptually, the two consecutive short DFTs are the same as if we had taken two zero-padded long DFTs, and then downsampled each by half. So the way to combine them is to reverse that process: upsample them by 2, and then add them together (with appropriate compensation for the zero-padding/boxcar window). Ethan D On Thu, Nov 8, 2018 at 8:12 AM Ethan Fenn wrote: > I'd really like to understand how combining consecutive DFT's can work. > Let's say our input is x0,x1,...x7 and the DFT we want to compute is > X0,X1,...X7 > > We start by doing two half-size DFT's: > > Y0 = x0 + x1 + x2 + x3 > Y1 = x0 - i*x1 - x2 + i*x3 > Y2 = x0 - x1 + x2 - x3 > Y3 = x0 + i*x1 - x2 - i*x3 > > Z0 = x4 + x5 + x6 + x7 > Z1 = x4 - i*x5 - x6 + i*x7 > Z2 = x4 - x5 + x6 - x7 > Z3 = x4 + i*x5 - x6 - i*x7 > > Now I agree because of periodicity we can compute all the even-numbered > bins easily: X0=Y0+Z0, X2=Y1+Z1, and so on. > > But I don't see how we can get the odd bins easily from the Y's and Z's. > For instance we should have: > > X1 = x0 + (r - r*i)*x1 - i*x2 + (-r - r*i)*x3 - x4 + (-r + r*i)*x5 + i*x6 > + (r + r*i)*x7 > > where r=sqrt(1/2) > > Is it actually possible? It seems like the phase of the coefficients in > the Y's and Z's advance too quickly to be of any use. > > -Ethan > > > > On Mon, Nov 5, 2018 at 3:40 PM, Ethan Duni wrote: > >> You can combine consecutive DFTs. Intuitively, the basis functions are >> periodic on the transform length. But it won't be as efficient as having >> done the big FFT (as you say, the decimation in time approach interleaves >> the inputs, so you gotta pay the piper to unwind that). Note that this is >> for naked transforms of successive blocks of inputs, not a WOLA filter >> bank. >> >> There are Dolby codecs that do similar with a suitable flavor of DCT >> (type II I think?) - you have your encoder going along at the usual frame >> rate, but if it detects a string of stationary inputs it can fold them >> together into one big high-res DCT and code that instead. >> >> On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn >> wrote: >> >>> I don't think that's correct -- DIF involves first doing a single stage >>> of butterfly operations over the input, and then doing two smaller DFTs on >>> that preprocessed data. I don't think there is any reasonable way to take >>> two "consecutive" DFTs of the raw input data and combine them into a longer >>> DFT. >>> >>> (And I don't know anything about the historical question!) >>> >>> -Ethan >>> >>> >>> >>> On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson < >>> r...@audioimagination.com> wrote: >>> >>>> >>>> >>>> Ethan, that's just the difference between Decimation-in-Frequency FFT >>>> and Decimation-in-Time FFT. >>>> >>>> i guess i am not entirely certainly of the history, but i credited both >>>> the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect >>>> historical impression. >>>> >>>> >>>> >>>> Original Message >>>> >>>> Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for >>>> realtime synthesis? >>>> From: "Ethan Fenn" >>>> Date: Mon, November 5, 2018 10:17 am >>>> To: music-dsp@music.columbia.edu >>>> >>>> -- >>>> >>>> > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two >>>> _interleaved_ >>>> > DFT's (that is, the DFT of the even-numbered samples and the DFT of >>>> the >>>> > odd-numbered samples) and combine them into one longer DFT. But here >>>> you're >>>> > talking about taking two _consecutive_ DFT's. I don't think there's >>>> any >>>> > cheap way to combine these to exactly recover an individual bin of the >>>> > longer DFT. >>>> > >>>> > Of course it's

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

No matter how you go about this, the the Fast Fourier will in almost every case act as some sort of ensemble measurement over it's length, and maybe do some filtering between consecutive transform steps. Maybe you even continuously average in the frequency domain, using per sample sliding FFT frames, even then you're measuring which "bins" of the FFT respond to the samples in you signal, not more and not less. Usually if the math/algebra doesn't proof anything conclusive, no high spun expectations about the mathematical relevance of the result is in order... T.V. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

I'd really like to understand how combining consecutive DFT's can work. Let's say our input is x0,x1,...x7 and the DFT we want to compute is X0,X1,...X7 We start by doing two half-size DFT's: Y0 = x0 + x1 + x2 + x3 Y1 = x0 - i*x1 - x2 + i*x3 Y2 = x0 - x1 + x2 - x3 Y3 = x0 + i*x1 - x2 - i*x3 Z0 = x4 + x5 + x6 + x7 Z1 = x4 - i*x5 - x6 + i*x7 Z2 = x4 - x5 + x6 - x7 Z3 = x4 + i*x5 - x6 - i*x7 Now I agree because of periodicity we can compute all the even-numbered bins easily: X0=Y0+Z0, X2=Y1+Z1, and so on. But I don't see how we can get the odd bins easily from the Y's and Z's. For instance we should have: X1 = x0 + (r - r*i)*x1 - i*x2 + (-r - r*i)*x3 - x4 + (-r + r*i)*x5 + i*x6 + (r + r*i)*x7 where r=sqrt(1/2) Is it actually possible? It seems like the phase of the coefficients in the Y's and Z's advance too quickly to be of any use. -Ethan On Mon, Nov 5, 2018 at 3:40 PM, Ethan Duni wrote: > You can combine consecutive DFTs. Intuitively, the basis functions are > periodic on the transform length. But it won't be as efficient as having > done the big FFT (as you say, the decimation in time approach interleaves > the inputs, so you gotta pay the piper to unwind that). Note that this is > for naked transforms of successive blocks of inputs, not a WOLA filter > bank. > > There are Dolby codecs that do similar with a suitable flavor of DCT (type > II I think?) - you have your encoder going along at the usual frame rate, > but if it detects a string of stationary inputs it can fold them together > into one big high-res DCT and code that instead. > > On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn wrote: > >> I don't think that's correct -- DIF involves first doing a single stage >> of butterfly operations over the input, and then doing two smaller DFTs on >> that preprocessed data. I don't think there is any reasonable way to take >> two "consecutive" DFTs of the raw input data and combine them into a longer >> DFT. >> >> (And I don't know anything about the historical question!) >> >> -Ethan >> >> >> >> On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson < >> r...@audioimagination.com> wrote: >> >>> >>> >>> Ethan, that's just the difference between Decimation-in-Frequency FFT >>> and Decimation-in-Time FFT. >>> >>> i guess i am not entirely certainly of the history, but i credited both >>> the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect >>> historical impression. >>> >>> >>> >>> Original Message >>> >>> Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for >>> realtime synthesis? >>> From: "Ethan Fenn" >>> Date: Mon, November 5, 2018 10:17 am >>> To: music-dsp@music.columbia.edu >>> >>> -- >>> >>> > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two >>> _interleaved_ >>> > DFT's (that is, the DFT of the even-numbered samples and the DFT of the >>> > odd-numbered samples) and combine them into one longer DFT. But here >>> you're >>> > talking about taking two _consecutive_ DFT's. I don't think there's any >>> > cheap way to combine these to exactly recover an individual bin of the >>> > longer DFT. >>> > >>> > Of course it's possible you'll be able to come up with a clever >>> frequency >>> > estimator using this information. I'm just saying it won't be exact in >>> the >>> > way Cooley-Tukey is. >>> > >>> > -Ethan >>> > >>> > >>> >>> >>> >>> -- >>> >>> r b-j r...@audioimagination.com >>> >>> "Imagination is more important than knowledge." >>> >>> >>> >>> >>> >>> >>> >>> >>> ___ >>> dupswapdrop: music-dsp mailing list >>> music-dsp@music.columbia.edu >>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> >> ___ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp > > > ___ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp > ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

I think I figured it out. I use 2^octave * SR/FFTsize -> toERBscale -> * log2(FFTsize)/42 as a scaling factor for the windows. Means the window of the top octave is about 367 samples at 44100 SR - does that seem right? Sounds better but not so different, still pretty blurry and somewhat reverberant. I used the lower frequency limit of the octaves for the window sizes and Hann windows cause I don't want the windows to be too small. Do you think using Gaussian windows and the center of the octave will make a big difference? Or do I just need more overlaps in resynthesis now? ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Further tests let me assume that you can do it on a log2 scale but that appropriate window sizes are crucial. But how to derive these optmal window sizes I am not sure. I could calculate the bandwitdh of the octave band (or an octave/N band) in ERB for instance but then what? How do I derive a window length from that for that band? I understand that bandwitdh is inversly proportional to window length. So it seems very easy actually but I am stuck here... Am 06.11.2018 um 16:13 schrieb gm: At the moment I am using decreasing window sizes on a log 2 scale. It's still pretty blurred, and I don't know if I just don't have the right window parameters, and if a log 2 scale is too coarse and differs too much from an auditory scale, or if if I don't have enough overlaps in resynthesis (I have four). Or if it's all together. The problem is the lowest octave or the lowest two octaves, where I need a long window for frequency estimation and partial tracking, it just soundded bad when the window was smaller in this range because the frequencies are blurred too much I assume. Unfortunately I am not sure what quality can be achieved and where the limits are with this approach. Am 06.11.2018 um 14:20 schrieb Ross Bencina: On 7/11/2018 12:03 AM, gm wrote: A similar idea would be to do some basic wavelet transfrom in octaves for instance and then do smaller FFTs on the bands to stretch and shift them but I have no idea if you can do that - if you shift them you exceed their bandlimit I assume? and if you stretch them I am not sure what happens, you shift their frequency content down I assume? Its a little bit fuzzy to me what the waveform in a such a band represents and what happens when you manipulate it, or how you do that. Look into constant-Q and bounded-Q transforms. Ross. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

At the moment I am using decreasing window sizes on a log 2 scale. It's still pretty blurred, and I don't know if I just don't have the right window parameters, and if a log 2 scale is too coarse and differs too much from an auditory scale, or if if I don't have enough overlaps in resynthesis (I have four). Or if it's all together. The problem is the lowest octave or the lowest two octaves, where I need a long window for frequency estimation and partial tracking, it just soundded bad when the window was smaller in this range because the frequencies are blurred too much I assume. Unfortunately I am not sure what quality can be achieved and where the limits are with this approach. Am 06.11.2018 um 14:20 schrieb Ross Bencina: On 7/11/2018 12:03 AM, gm wrote: A similar idea would be to do some basic wavelet transfrom in octaves for instance and then do smaller FFTs on the bands to stretch and shift them but I have no idea if you can do that - if you shift them you exceed their bandlimit I assume? and if you stretch them I am not sure what happens, you shift their frequency content down I assume? Its a little bit fuzzy to me what the waveform in a such a band represents and what happens when you manipulate it, or how you do that. Look into constant-Q and bounded-Q transforms. Ross. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

On 7/11/2018 12:03 AM, gm wrote: A similar idea would be to do some basic wavelet transfrom in octaves for instance and then do smaller FFTs on the bands to stretch and shift them but I have no idea if you can do that - if you shift them you exceed their bandlimit I assume? and if you stretch them I am not sure what happens, you shift their frequency content down I assume? Its a little bit fuzzy to me what the waveform in a such a band represents and what happens when you manipulate it, or how you do that. Look into constant-Q and bounded-Q transforms. Ross. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

I'm definitely not the most mathy person on the list, but I think there's something about the complex exponentials, real transforms and the 2-point case. For all real DFTs you should get a real-valued sample at DC and Nyquist, which indeed you do get with your matrix. However, there should be some complex numbers in a matrix for a 4-point DFT, which you won't get no matter how many matrices of that form you multiply together. My guess is that yours is a special case of a DFT Matrix for 2 bins. I suspect if you took a 4-point DFT Matrix and tried the same it might work out better? https://en.wikipedia.org/wiki/DFT_matrix Stefan On Mon, Nov 5, 2018, 12:40 Ethan Duni You can combine consecutive DFTs. Intuitively, the basis functions are > periodic on the transform length. But it won't be as efficient as having > done the big FFT (as you say, the decimation in time approach interleaves > the inputs, so you gotta pay the piper to unwind that). Note that this is > for naked transforms of successive blocks of inputs, not a WOLA filter > bank. > > There are Dolby codecs that do similar with a suitable flavor of DCT (type > II I think?) - you have your encoder going along at the usual frame rate, > but if it detects a string of stationary inputs it can fold them together > into one big high-res DCT and code that instead. > > On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn wrote: > >> I don't think that's correct -- DIF involves first doing a single stage >> of butterfly operations over the input, and then doing two smaller DFTs on >> that preprocessed data. I don't think there is any reasonable way to take >> two "consecutive" DFTs of the raw input data and combine them into a longer >> DFT. >> >> (And I don't know anything about the historical question!) >> >> -Ethan >> >> >> >> On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson < >> r...@audioimagination.com> wrote: >> >>> >>> >>> Ethan, that's just the difference between Decimation-in-Frequency FFT >>> and Decimation-in-Time FFT. >>> >>> i guess i am not entirely certainly of the history, but i credited both >>> the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect >>> historical impression. >>> >>> >>> >>> Original Message >>> >>> Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for >>> realtime synthesis? >>> From: "Ethan Fenn" >>> Date: Mon, November 5, 2018 10:17 am >>> To: music-dsp@music.columbia.edu >>> >>> -- >>> >>> > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two >>> _interleaved_ >>> > DFT's (that is, the DFT of the even-numbered samples and the DFT of the >>> > odd-numbered samples) and combine them into one longer DFT. But here >>> you're >>> > talking about taking two _consecutive_ DFT's. I don't think there's any >>> > cheap way to combine these to exactly recover an individual bin of the >>> > longer DFT. >>> > >>> > Of course it's possible you'll be able to come up with a clever >>> frequency >>> > estimator using this information. I'm just saying it won't be exact in >>> the >>> > way Cooley-Tukey is. >>> > >>> > -Ethan >>> > >>> > >>> >>> >>> >>> -- >>> >>> r b-j r...@audioimagination.com >>> >>> "Imagination is more important than knowledge." >>> >>> >>> >>> >>> >>> >>> >>> >>> ___ >>> dupswapdrop: music-dsp mailing list >>> music-dsp@music.columbia.edu >>> https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> >> ___ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp > > ___ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

You can combine consecutive DFTs. Intuitively, the basis functions are periodic on the transform length. But it won't be as efficient as having done the big FFT (as you say, the decimation in time approach interleaves the inputs, so you gotta pay the piper to unwind that). Note that this is for naked transforms of successive blocks of inputs, not a WOLA filter bank. There are Dolby codecs that do similar with a suitable flavor of DCT (type II I think?) - you have your encoder going along at the usual frame rate, but if it detects a string of stationary inputs it can fold them together into one big high-res DCT and code that instead. On Mon, Nov 5, 2018 at 11:34 AM Ethan Fenn wrote: > I don't think that's correct -- DIF involves first doing a single stage of > butterfly operations over the input, and then doing two smaller DFTs on > that preprocessed data. I don't think there is any reasonable way to take > two "consecutive" DFTs of the raw input data and combine them into a longer > DFT. > > (And I don't know anything about the historical question!) > > -Ethan > > > > On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson < > r...@audioimagination.com> wrote: > >> >> >> Ethan, that's just the difference between Decimation-in-Frequency FFT and >> Decimation-in-Time FFT. >> >> i guess i am not entirely certainly of the history, but i credited both >> the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect >> historical impression. >> >> >> >> ------------ Original Message >> Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for >> realtime synthesis? >> From: "Ethan Fenn" >> Date: Mon, November 5, 2018 10:17 am >> To: music-dsp@music.columbia.edu >> -- >> >> > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two >> _interleaved_ >> > DFT's (that is, the DFT of the even-numbered samples and the DFT of the >> > odd-numbered samples) and combine them into one longer DFT. But here >> you're >> > talking about taking two _consecutive_ DFT's. I don't think there's any >> > cheap way to combine these to exactly recover an individual bin of the >> > longer DFT. >> > >> > Of course it's possible you'll be able to come up with a clever >> frequency >> > estimator using this information. I'm just saying it won't be exact in >> the >> > way Cooley-Tukey is. >> > >> > -Ethan >> > >> > >> >> >> >> -- >> >> r b-j r...@audioimagination.com >> >> "Imagination is more important than knowledge." >> >> >> >> >> >> >> >> >> ___ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp >> > > ___ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Well, I dunno shit about the history also. I just ascribed all of the radix-2 FFT to Cooley and Tukey.But I think you're mistaken about the technical claim. If you have or can get Oppenheim and Schafer and go to the FFT chapter of whatever revision you have, and there are several different 8 point FFTs that they illustrate.--r b-j r...@audioimagination.com"Imagination is more important than knowledge." Original message From: Ethan Fenn Date: 11/5/2018 11:34 AM (GMT-08:00) To: robert bristow-johnson , music-dsp@music.columbia.edu Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? I don't think that's correct -- DIF involves first doing a single stage of butterfly operations over the input, and then doing two smaller DFTs on that preprocessed data. I don't think there is any reasonable way to take two "consecutive" DFTs of the raw input data and combine them into a longer DFT.(And I don't know anything about the historical question!)-EthanOn Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson wrote: Ethan, that's just the difference between Decimation-in-Frequency FFT and Decimation-in-Time FFT.i guess i am not entirely certainly of the history, but i credited both the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect historical impression. Original Message -------- Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "Ethan Fenn" Date: Mon, November 5, 2018 10:17 am To: music-dsp@music.columbia.edu -- > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ > DFT's (that is, the DFT of the even-numbered samples and the DFT of the > odd-numbered samples) and combine them into one longer DFT. But here you're > talking about taking two _consecutive_ DFT's. I don't think there's any > cheap way to combine these to exactly recover an individual bin of the > longer DFT. > > Of course it's possible you'll be able to come up with a clever frequency > estimator using this information. I'm just saying it won't be exact in the > way Cooley-Tukey is. > > -Ethan > > -- r b-j r...@audioimagination.com "Imagination is more important than knowledge." ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

I don't think that's correct -- DIF involves first doing a single stage of butterfly operations over the input, and then doing two smaller DFTs on that preprocessed data. I don't think there is any reasonable way to take two "consecutive" DFTs of the raw input data and combine them into a longer DFT. (And I don't know anything about the historical question!) -Ethan On Mon, Nov 5, 2018 at 2:18 PM, robert bristow-johnson < r...@audioimagination.com> wrote: > > > Ethan, that's just the difference between Decimation-in-Frequency FFT and > Decimation-in-Time FFT. > > i guess i am not entirely certainly of the history, but i credited both > the DIT and DIF FFT to Cooley and Tukey. that might be an incorrect > historical impression. > > > > Original Message ---------------- > Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for > realtime synthesis? > From: "Ethan Fenn" > Date: Mon, November 5, 2018 10:17 am > To: music-dsp@music.columbia.edu > -- > > > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ > > DFT's (that is, the DFT of the even-numbered samples and the DFT of the > > odd-numbered samples) and combine them into one longer DFT. But here > you're > > talking about taking two _consecutive_ DFT's. I don't think there's any > > cheap way to combine these to exactly recover an individual bin of the > > longer DFT. > > > > Of course it's possible you'll be able to come up with a clever frequency > > estimator using this information. I'm just saying it won't be exact in > the > > way Cooley-Tukey is. > > > > -Ethan > > > > > > > > -- > > r b-j r...@audioimagination.com > > "Imagination is more important than knowledge." > > > > > > > > > ___ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp > ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

� Ethan, that's just the difference between Decimation-in-Frequency FFT and Decimation-in-Time FFT. i guess i am not entirely certainly of the history, but i credited both the DIT and DIF FFT to Cooley and Tukey.� that might be an incorrect historical impression. Original Message Subject: Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? From: "Ethan Fenn" Date: Mon, November 5, 2018 10:17 am To: music-dsp@music.columbia.edu -- > It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ > DFT's (that is, the DFT of the even-numbered samples and the DFT of the > odd-numbered samples) and combine them into one longer DFT. But here you're > talking about taking two _consecutive_ DFT's. I don't think there's any > cheap way to combine these to exactly recover an individual bin of the > longer DFT. > > Of course it's possible you'll be able to come up with a clever frequency > estimator using this information. I'm just saying it won't be exact in the > way Cooley-Tukey is. > > -Ethan > > � -- r b-j� � � � � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge." � � � � ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Am 05.11.2018 um 16:17 schrieb Ethan Fenn: Of course it's possible you'll be able to come up with a clever frequency estimator using this information. I'm just saying it won't be exact in the way Cooley-Tukey is. Maybe, but not the way I laid it out. Also it seems wiser to interpolate spektral peaks, as had been suggested to me before. But it doesn's sound much better then to get the frequency from the phase step so the bad sound for frequency shifts at an FFT size of 2048 has other reasons than just a bad phase estimate. Maybe it's just stupid to find a solution for this FFT size and a frequency domain shift. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

It's not exactly Cooley-Tukey. In Cooley-Tukey you take two _interleaved_ DFT's (that is, the DFT of the even-numbered samples and the DFT of the odd-numbered samples) and combine them into one longer DFT. But here you're talking about taking two _consecutive_ DFT's. I don't think there's any cheap way to combine these to exactly recover an individual bin of the longer DFT. Of course it's possible you'll be able to come up with a clever frequency estimator using this information. I'm just saying it won't be exact in the way Cooley-Tukey is. -Ethan On Mon, Nov 5, 2018 at 12:28 AM, gm wrote: > > > Am 05.11.2018 um 01:56 schrieb gm: > >> so you do the "radix 2 algorithm" if you will on a subband, and now what? >> the bandlimits are what? the neighbouring upper and lower bands? >> >> how do I get a frequency estimate "in between" out of these two real >> values that describe the upper and lower limit of the band but have no >> further information? >> >> thank you. >> > The way I see it: > > If you do that 2 point transform on a band you get 2 data points instead > of one (or rather instead of two sucsessive ones of course), representing > the upper and lower bandwith limit of the band, but not very well seperated. > But if you take the result of the previous frame also into account you now > get 4 points representing the corner of a bin > of the original spectrum so to say, however in bewteen spectra, and you > now can do bilinear interpolation between these 4 points. > > But in the end this is just crude averaging between two sucessive spectra, > and I am not sure if it sounded better > or worse. It's hard to tell a difference, it works quite well on a sine > sweep though. > > But there must be a better way to make use of these 2 extra data points. > > In the end you now have the same amount of information as with a spectrum > of double size. > So you should be able to obtain the same quality from that. > That was my way of thinking, however flawed that is, I'd like to know. > > > > ___ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp > > ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Am 05.11.2018 um 01:56 schrieb gm: so you do the "radix 2 algorithm" if you will on a subband, and now what? the bandlimits are what? the neighbouring upper and lower bands? how do I get a frequency estimate "in between" out of these two real values that describe the upper and lower limit of the band but have no further information? thank you. The way I see it: If you do that 2 point transform on a band you get 2 data points instead of one (or rather instead of two sucsessive ones of course), representing the upper and lower bandwith limit of the band, but not very well seperated. But if you take the result of the previous frame also into account you now get 4 points representing the corner of a bin of the original spectrum so to say, however in bewteen spectra, and you now can do bilinear interpolation between these 4 points. But in the end this is just crude averaging between two sucessive spectra, and I am not sure if it sounded better or worse. It's hard to tell a difference, it works quite well on a sine sweep though. But there must be a better way to make use of these 2 extra data points. In the end you now have the same amount of information as with a spectrum of double size. So you should be able to obtain the same quality from that. That was my way of thinking, however flawed that is, I'd like to know. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

Am 05.11.2018 um 01:39 schrieb robert bristow-johnson: mr. g, I think what you're describing is the Cooley-Tukey Radix-2 FFT algorithm. yes that seems kind of right, though I am not describing something but posting a question actually and the "other thing" was an answer to a question maybe my post was too long, boring and naive for you, so I am sorry for that. I am reusing the same subject lien to help people to ignore this, if they want so you do the "radix 2 algorithm" if you will on a subband, and now what? the bandlimits are what? the neighbouring upper and lower bands? how do I get a frequency estimate "in between" out of these two real values that describe the upper and lower limit of the band but have no further information? thank you. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

and the other thing you're describing is what they usually call "sinusoidal modeling.".--r b-j r...@audioimagination.com"Imagination is more important than knowledge." Original message From: gm Date: 11/4/2018 4:14 PM (GMT-08:00) To: music-dsp@music.columbia.edu Subject: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? bear with me, I am a math illiterate.I understand you can do a Discrete Fourier Transform in matrix form,and for 2-point case it is simply[ 1, 1 1,-1]like the Haar transform, average and difference.My idea is, to use two successive DFT frames, and to transform resepctive bins of two successive frames like this.To obtain a better frequency estimate (subbands) from two smaller DFTs instead of an DFT double the size.This should be possible? and the information obtained, time and frequency resolution wise, identical.Except that you can overlap the two DTFs.I basically want to find the dominant frequency in the FFT bin, and sepreate it and discard the rest.And a subband resolution of 2 seems to be a sufficient increase in resolution.But how do I get that from this when there is no phase other then 0?I can see which of the two bands has more energy, but how do I know "the true frequency" of Nyquist and DC?There is not enough information.The problem persists for me if I resort to a 4-point transform, what to do with the highest/lowest subband.(and also to understand how to calculate the simple 4 point matrix, cause I am uneducated..)Or do I need the 4-point case and discard "DC" and Nyquist subbands?Or is the idea totally nonsens?Or is it justified to pick the subband that has more energy, and then, what?___dupswapdrop: music-dsp mailing listmusic-dsp@music.columbia.eduhttps://lists.columbia.edu/mailman/listinfo/music-dsp___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp

### Re: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis?

mr. g,I think what you're describing is the Cooley-Tukey Radix-2 FFT algorithm.--r b-j r...@audioimagination.com"Imagination is more important than knowledge." Original message From: gm Date: 11/4/2018 4:14 PM (GMT-08:00) To: music-dsp@music.columbia.edu Subject: [music-dsp] 2-point DFT Matrix for subbands Re: FFT for realtime synthesis? bear with me, I am a math illiterate.I understand you can do a Discrete Fourier Transform in matrix form,and for 2-point case it is simply[ 1, 1 1,-1]like the Haar transform, average and difference.My idea is, to use two successive DFT frames, and to transform resepctive bins of two successive frames like this.To obtain a better frequency estimate (subbands) from two smaller DFTs instead of an DFT double the size.This should be possible? and the information obtained, time and frequency resolution wise, identical.Except that you can overlap the two DTFs.I basically want to find the dominant frequency in the FFT bin, and sepreate it and discard the rest.And a subband resolution of 2 seems to be a sufficient increase in resolution.But how do I get that from this when there is no phase other then 0?I can see which of the two bands has more energy, but how do I know "the true frequency" of Nyquist and DC?There is not enough information.The problem persists for me if I resort to a 4-point transform, what to do with the highest/lowest subband.(and also to understand how to calculate the simple 4 point matrix, cause I am uneducated..)Or do I need the 4-point case and discard "DC" and Nyquist subbands?Or is the idea totally nonsens?Or is it justified to pick the subband that has more energy, and then, what?___dupswapdrop: music-dsp mailing listmusic-dsp@music.columbia.eduhttps://lists.columbia.edu/mailman/listinfo/music-dsp___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp