[opus] Antw: [EXT] Re: Coarse energy predictor confusion

[Ulrich Windl]

>>> Jake Taylor <yupferris at gmail.com> schrieb am 03.07.2021 um 13:51 in Nachricht
<CALkcPkLpPT1kG5KLpQ4ZHoTYRHJukNH58=ThYCJF-m7kZRV9Xw at mail.gmail.com>:
> Hi again all,
> 
> I've solved it (in fact the clues showed up in earlier notes, but I didn't
> put it together entirely until now)! I was making a crucial conceptual
> mistake. I assumed the output of the predictor was the *prediction*,
> however, this is incorrect - instead it outputs the *residual* directly.
> Under that interpretation, everything falls into place. So I no longer need
> help with this, and sorry for the noise while I figured it out!

Personally I think your thoughts could be valuable to people trying to understand the code.
Maybe it could be "weaved in" or added to the resources at the web site.

> 
> For the curious, once again, notes in pdf/lyx format are attached, as well
> as on my dropbox:
> pdf:
> https://www.dropbox.com/s/rzm08055hytjy6c/predictor-confusion-5.pdf?dl=0 
> <https://www.dropbox.com/s/l4tzyvq1mne8jan/predictor-confusion-4.pdf?dl=0>
> lyx:
> https://www.dropbox.com/s/vj1s3pbm8q914vf/predictor-confusion-5.lyx?dl=0 
> <https://www.dropbox.com/s/map3slkbnhyctui/predictor-confusion-4.lyx?dl=0>
> and plaintext below.
> 
> Thanks again,
> Jake
> 
> ---
> 
> Opus Coarse Energy Predictor Notes (Round 5)
> 
> Jake Taylor (yupferris at gmail.com), 3 July 2021
> 
> I've solved it - I was simply mistaken about what the output of
> the filters should be. Instead of the filters outputting the
> prediction, they output the residual directly. Under this
> interpretation, everything falls into place.
> 
> According to the paper[0]/RFC[1], the 2D z-transform should be:
> 
> A(z_{\ell},z_{b})=(1-\alpha z_{\ell}^{-1})\cdot\frac{1-z_{b}^{-1}}{1-\beta
> z_{b}^{-1}}
> 
> I've simplified the source code (in unquant_coarse_energy in
> quant_bands.c in libopus 1.3.1[2]) to the following C (pseudo)code:
> 
> void unquant_coarse_energy(float *e, int bands) {
>   float alpha = /* ... */;
>   float beta = /* ... */;
>   float prev = 0.0f;
>   for (int b = 0; b < bands; b++) {
>     float r = /* read from bitstream */;
>     e[i] = alpha * e[i] + prev + r;
>     prev = prev + (1 - beta) * r;
>   }
> }
> 
> The goal is simple: extract difference equations from the code,
> derive the corresponding z-transforms, and then solve the final z
> -transform which should hopefully match what the paper/RFC list.
> 
> The code is governed by the following difference equations:
> 
> e[\ell,b]=\alpha e[\ell-1,b]+prev[\ell,b-1]+r[\ell,b]
> 
> prev[\ell,b]=prev[\ell,b-1]+(1-\beta)r[\ell,b]
> 
> The corresponding z-transforms are:
> 
> E(z_{\ell},z_{b})=\alpha
> z_{\ell}^{-1}E(z_{\ell},z_{b})+z_{b}^{-1}Prev(z_{\ell},z_{b})+R(z_{\ell},z_{
> b})
> 
> Prev(z_{\ell},z_{b})=z_{b}^{-1}Prev(z_{\ell},z_{b})+(1-\beta)R(z_{\ell},z_{b\par >