[music-dsp] Re: Wavelet Convolution
Wen X
xue.wen at elec.qmul.ac.uk
Mon Oct 9 04:45:19 EDT 2006
This is very closely related to adaptive basis/frame selection, in which you
have a large family of bases to choose one from so that the signal is, in
some sense, optimally represented. The wavelet packets enable
frequency-dependent resolution and the local cosines enable time-dependent
resolution. There is also the block wavelet packets that first cut in time
and then find a wavelet packet for each time block so it does a
time-dependent frequency-dependent selection of basis (not time- and
frequency- dependent!). There are "fast" algorithms for finding the optimal
basis, based on dynamic programming, but much slower than just fft or dct or
dwt. However, if the signal property does not alter much across time and
frequency, a easier thing to do is to calcualte a bunch of fft's, each at a
different resolution, then choose one the optimizes some cost function.
----- Original Message -----
From: "Tony Robinson" <tony at tonyRobinson.com>
To: <music-dsp at music.columbia.edu>
Sent: Saturday, October 07, 2006 1:47 PM
Subject: [music-dsp] Re: Wavelet Convolution
>> Date: Sat, 7 Oct 2006 07:36:21 +0200
>> From: "Didier Dambrin" <didid at skynet.be>
>>
>> I think it would be helpful to see transients more accurately, while
>> having more definition in the lower freqs.
>
> The time resolution is inversely proportional to the frequency resolution,
> so you can set things up so that there are an equal number of frequency
> bins per octave, so giving you higher time resolution at higher
> frequencies. In ASCII art here are three octaves:
>
> +-+-+-+-+-+-+-+-+
> | | | | | | | | |
> | | | | | | | | |
> | | | | | | | | |
> +-+-+-+-+-+-+-+-+
> | | | | | | | | |
> | | | | | | | | |
> | | | | | | | | |
> +---+---+---+---+
> | | | | |
> +---+---+---+---+
> | | | | |
> +-------+-------+
> +-------+-------+
> +-------+-------+
>
> This is the standard way to be able to see transients more accuractly
> (through the time resolution at high frequencies), and you get better
> frequency definitition at lower frequencies.
>
> I'd like to see something different. If the signal is stationary I'd
> like to be able to see all the harmonics - e.g. via a long time period
> FFT, and if the signal isn't then I'd like to see the moment of transition
> (using short time period FFT).
>
> Anyone know of anything like this around? I guess the implementation
> hinges on what I mean by stationary, I don't have a formal definition just
> now, but say if the spectral distortion of two FFTs of length T is below
> some threshold then they are replaced by one FFT of length 2T.
>
>
> Tony
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