path: root/doc/stereo.lyx

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\begin_body

\begin_layout Title
Stereo Quantization Improvements in Opus/CELT
\end_layout

\begin_layout Author
Jean-Marc Valin
\end_layout

\begin_layout Section
Modifying stereo input vectors
\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\mathbf{x}$
\end_inset

 denote the normalized vector for a band of the left channel and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 denote the corresponding vector for the right channel.
 When quantizing stereo, the first step is to quantize the angle derived
 from the ratio of the magnitude of the mid to the magnitude of the side
\begin_inset Formula 
\[
\theta=\arctan\frac{\left\Vert \mathbf{M}\right\Vert }{\left\Vert \mathbf{S}\right\Vert }\,,
\]

\end_inset

where 
\begin_inset Formula $\mathbf{M}=\mathbf{x}+\mathbf{y}$
\end_inset

 and 
\begin_inset Formula $\mathbf{S}=\mathbf{x}-\mathbf{y}$
\end_inset

.
 
\end_layout

\begin_layout Standard
It can be shown that the angle is 
\begin_inset Formula $\theta$
\end_inset

 is related to the angle 
\begin_inset Formula $\phi$
\end_inset

 between 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 by 
\begin_inset Formula $\phi=2\theta$
\end_inset

, where
\begin_inset Formula 
\[
\cos\phi=\mathbf{x}^{T}\mathbf{y}\,.
\]

\end_inset


\end_layout

\begin_layout Standard
When 
\begin_inset Formula $\theta$
\end_inset

 is quantized to 
\begin_inset Formula $\hat{\theta}$
\end_inset

, it causes distortion to both channels.
 The distortion (sum of squared errors) for each channel is given by the
 law of cosines to be
\begin_inset Formula 
\[
D=2-2\cos\delta\,,
\]

\end_inset

where 
\begin_inset Formula $\delta$
\end_inset

 is the angle by which each of the vectors was 
\emph on
moved
\emph default
 by the quantization.
 Since both channels are affected by the same amount, 
\begin_inset Formula $\delta=\frac{\hat{\phi}-\phi}{2}=\hat{\theta}-\theta$
\end_inset

.
\end_layout

\begin_layout Standard
However, we may want to change that behaviour when the two channels differ
 in loudness.
 Let 
\begin_inset Formula $w_{x}$
\end_inset

 and 
\begin_inset Formula $w_{y}$
\end_inset

 be the weight we assign to each of the channels.
 The total weighted distortion then becomes
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
D=w_{x}\left(2-2\cos\delta_{x}\right)+w_{y}\left(2-2\cos\delta_{y}\right)\,.
\]

\end_inset


\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $S=\delta_{x}+\delta_{y}=\hat{\phi}-\phi$
\end_inset

 be a known value (from the quantization process).
 We can minimize the weighted distortion by substituting 
\begin_inset Formula $\delta_{y}=S-\delta_{x}$
\end_inset

 and solving:
\begin_inset Formula 
\begin{align*}
\frac{\partial D}{\partial\delta_{x}}=2w_{x}\sin\delta_{x}-2w_{y}\sin\left(S-\delta_{x}\right) & =0\\
2w_{x}\sin\delta_{x}-2w_{y}\left(\sin S\cos\delta_{x}-\cos S\sin\delta_{x}\right) & =0\\
w_{x}\sin\delta_{x}+w_{y}\cos S\sin\delta_{x} & =w_{y}\sin S\cos\delta_{y}\\
\sin\delta_{x}\cdot & \left(w_{x}+w_{y}\cos S\right)=w_{y}\sin S\cos\delta_{x}\\
\tan\delta_{x} & =\frac{w_{y}\sin S}{w_{x}+w_{y}\cos S}\,.
\end{align*}

\end_inset

Using a similar derivation, we can find
\begin_inset Formula 
\[
\tan\delta_{y}=\frac{w_{x}\sin S}{w_{y}+w_{x}\cos S}\,.
\]

\end_inset


\end_layout

\begin_layout Standard
Given these values, we want to compute 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 that will be quantized instead of 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

.
 Since quantizing 
\begin_inset Formula $\theta$
\end_inset

 keep 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 in the same plane, we also want 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 to lie on the same plane as 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

.
 We express them as linear combinations of 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 such that the angle between 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\mathbf{x}$
\end_inset

 is 
\begin_inset Formula $\delta_{x}$
\end_inset

 and the angle between 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 is 
\begin_inset Formula $\delta_{y}$
\end_inset

.
 To make the calcualtion easier, we are not yet concerned about the norm
 of 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

.
 Let us consider 
\begin_inset Formula $\tilde{\mathbf{x}}=\mathbf{x}+\alpha_{x}\mathbf{y}$
\end_inset

, the angle between 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\mathbf{x}$
\end_inset

 is given by
\begin_inset Formula 
\[
\delta_{x}=\arctan\frac{\alpha_{x}\sin\phi}{1+\alpha_{x}cos\phi}\,,
\]

\end_inset

where again 
\begin_inset Formula $\phi$
\end_inset

 is the angle between 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

.
 Solving for 
\begin_inset Formula $\alpha_{x}$
\end_inset

, we get
\begin_inset Formula 
\begin{align*}
\tan\delta_{x}\left(1+\alpha_{x}\cos\phi\right) & =\alpha_{x}\sin\phi\\
\tan\delta_{x} & =\alpha_{x}\sin\phi-\alpha_{x}\cos\phi\tan\delta_{x}\\
\alpha_{x} & =\frac{\tan\delta_{x}}{\sin\phi-\cos\phi\tan\delta_{x}}\,.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
Since we are not concerned with scaling, we can avoid the division by simply
 defining a denormalized 
\begin_inset Formula 
\[
\tilde{\mathbf{x}}_{d}=g_{xx}\mathbf{x}+g_{xy}\mathbf{y}\,,
\]

\end_inset

with
\begin_inset Formula 
\begin{align*}
g_{xx} & =\sin\phi-\cos\phi\tan\delta_{x}\\
g_{xy} & =\tan\delta_{x}\,.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
Using the law of cosines, the magnitude of 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 is given by
\begin_inset Formula 
\begin{align*}
\left\Vert \tilde{\mathbf{x}}_{d}\right\Vert  & =\tan^{2}\delta_{x}+\left(\sin\phi-\cos\phi\tan\delta_{x}\right)^{2}+2\cos\phi\tan\delta_{x}\left(\sin\phi-\cos\phi\tan\delta_{x}\right)\\
 & =\tan^{2}\delta_{x}+\sin^{2}\phi+\cos^{2}\phi\tan^{2}\delta_{x}-2\sin\phi\cos\phi\tan\delta_{x}+2\cos\phi\tan\delta_{x}\sin\phi-2\cos^{2}\phi\tan^{2}\delta_{x}\\
 & =\tan^{2}\delta_{x}+\sin^{2}\phi-\cos^{2}\phi\tan^{2}\delta_{x}\\
 & =\left(1-\cos^{2}\phi\right)\tan^{2}\delta_{x}+\sin^{2}\phi\\
 & =\sin^{2}\phi\left(1+\tan^{2}\delta_{x}\right)\\
 & =\frac{\sin^{2}\phi}{\cos^{2}\delta_{x}}\,.
\end{align*}

\end_inset

Knowing this, we can compute a normalized 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 as 
\begin_inset Formula 
\[
\tilde{\mathbf{x}}=\frac{\cos\delta_{x}}{\sin\phi}\tilde{\mathbf{x}}_{d}\,.
\]

\end_inset


\end_layout

\begin_layout Standard
We can then compute 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 similarly.
 Replacing 
\begin_inset Formula $\mathbf{x}$
\end_inset

 and 
\begin_inset Formula $\mathbf{y}$
\end_inset

 with 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 in the quantization process, we can give more weight to one channel or
 the other.
 When trying multiple values of 
\begin_inset Formula $\hat{\theta}$
\end_inset

, we will derive a different value of 
\begin_inset Formula $\tilde{\mathbf{x}}$
\end_inset

 and 
\begin_inset Formula $\tilde{\mathbf{y}}$
\end_inset

 and each 
\begin_inset Formula $\hat{\theta}$
\end_inset

.
 
\end_layout

\begin_layout Section
Stereo bit allocation
\end_layout

\begin_layout Standard
By dumping quantization data from the encoder and looking at the normalized
 distortion as a function of the angle 
\begin_inset Formula $\phi$
\end_inset

 and the rate, we have come up with the following approximation that best
 fits the data with a simple enough function:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
D=3\left(4^{-r}\sin\phi+4^{-2r}\left(1-\sin\phi\right)\right)\,,
\]

\end_inset

where 
\begin_inset Formula $r$
\end_inset

 is the bit depth
\begin_inset Formula 
\[
r=\frac{b}{2N-1}\,.
\]

\end_inset


\end_layout

\begin_layout Standard
Solving for 
\begin_inset Formula $r$
\end_inset

, we get
\begin_inset Formula 
\[
R=\frac{-3\sin\phi+\sqrt{9\sin^{2}\phi+12D\left(1-\sin\phi\right)}}{6\left(1-\sin\phi\right)}\,,
\]

\end_inset

with 
\begin_inset Formula $r=-\log_{4}R$
\end_inset

.
 
\end_layout

\end_body
\end_document