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 <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 <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 < >> 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" <et...@polyspectral.com> >>> 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
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