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 <et...@polyspectral.com
<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 <ethan.d...@gmail.com
<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
<et...@polyspectral.com <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
<r...@audioimagination.com
<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" <et...@polyspectral.com
<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."
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