Hi, Sorry, late to the party and unable to read the backlog, but:

`The "FFT^-1" technique that Robert mentions is from a paper by Rodet and`

`Depalle that I can't find right now. It's widely cited in the literature`

`as "FFT^-1"`

`That paper only deals with steady-state sinusoids however. It won't`

`accurately deal with transients or glides.`

`There has been more recent work on spectral-domain synthesis and I'm`

`fairly sure that some techniques have found their way into some quite`

`famous commercial products.`

`Bonada, J.; Loscos, A.; Cano, P.; Serra, X.; Kenmochi, H. (2001).`

`"Spectral Approach to the Modeling of the Singing Voice". In Proc. of`

`the 111th AES Convention.`

> My goal is to resynthesize arbitary noises.

`In that case you need to think about how an FFT represents "arbitrary`

`noises".`

`One approach is to split the signal into sinusoids + noise (a.k.a.`

`spectral modeling synthesis).`

https://en.wikipedia.org/wiki/Spectral_modeling_synthesis

`It is worth reviewing Xavier Serra's PhD thesis for the basics (what was`

`already established in the late 1980s.)`

http://mtg.upf.edu/content/serra-PhD-thesis Here's the PDF: https://repositori.upf.edu/bitstream/handle/10230/34072/Serra_PhDthesis.pdf?sequence=1&isAllowed=y

`There was a bunch of in the early 90's on real-time additive synthesis`

`at CNMAT, e.g.`

https://quod.lib.umich.edu/i/icmc/bbp2372.1995.091/1/--bring-your-own-control-to-additive-synthesis?page=root;size=150;view=text

`Of course there is a ton of more recent work. You could do worse than`

`looking at the papers of Xavier Serra and Jordi Bonada:`

http://mtg.upf.edu/research/publications On 31/10/2018 1:35 PM, gm wrote:

But back to my question, I am serious, could you compress a spectrum byjust adding the bins that fall together?

`I'm not sure what "compress" means in this context, nor am I sure what`

`"fall together" means. But here's some points to note:`

`A steady state sine wave in the time domain will be transformed by a`

`short-time fourier transform into a spectral peak, convolved (in the`

`frequency domain) by the spectrum of the analysis envelope. If you know`

`that all of your inputs are sine waves, then you can perform "spectral`

`peak picking" (AKA MQ analysis) and reduce your signal to a list of sine`

`waves and their frequencies and phases -- this is the sinusoidal`

`component of Serra's SMS (explained in the pdf linked above).`

`Note that since a sinusoid ends up placing non-zero values in every FFT`

`bin, you'd need to account for that in your spectral estimation, which`

`basic MQ does not -- hence it does not perfectly estimate the sinusoids.`

`In any case, most signals are not sums of stationary sinusoids. And`

`since signals are typically buried in noise, or superimposed on top of`

`each other, so the problem is not well posed. For two very simple`

`examples: consider two stable sine waves at 440Hz and 441Hz -- you will`

`need a very long FFT to distinguish this from a single`

`amplitude-modulated sine wave? or consider a sine wave plus white noise`

`-- the accuracy of frequency and phase recovery will depend on how much`

`input you have to work with.`

`I think by "compression" you mean "represent sparsely" (i.e. with some`

`reduced representation.) The spectral modeling approach is to "model"`

`the signal by assuming it has some particular structure (e.g.`

`sinusoids+noise, or sinusoids+transients+noise) and then work out how to`

`extract this structure from the signal (or to reassemble it for synthesis).`

`An alternative (more mathematical) approach is to simply assume that the`

`signal is sparse in some (unknown) domain. It turns out that if your`

`signal is sparse, you can apply a constrained random dimensionality`

`reduction to the signal and not lose any information. This is the field`

`of compressed sensing. Note that in this case, you haven't recovered any`

`structure.`

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