Decomposition into Shape and Gain Distortions

The distortion of the proposed Gaussian quantizer decomposes into gain and shape distortions in much the same way as for Sakrison's spherical vector quantizer in (5). The gain distortion can be evaluated using numerical integration. The shape distortion can be closely approximated and verified to be accurate by simulations.

The MSE per dimension of ${\rm W}_\Lambda$ can be decomposed as

$$D = {1 \over k} E[\Vert X - \hat g \hat S\Vert^2]$$

$$= {1 \over k} E[\Vert X - \hat{g}S\Vert^2]$$

$$+ {2 \over k} E[(X-\hat{g}S)^T(\hat{g}S-\hat g \hat S)]$$

$$+ {1 \over k} E[\Vert\hat{g}S - \hat g \hat S\Vert^2]$$ (8)

$$\equiv D_g + D_c + D_s$$

where $D_g$, $D_c$, and $D_s$ denote the first, second, and third terms of (8), respectively. Thus,

$$D_g = {1 \over k} E\left[\left\Vert\left(1-{\hat{g} \over g}\right)X\right\Vert^2\right]$$

$$= {1 \over k} E[(g-\hat{g})^2]$$ (9)

which is the per-dimension distortion due to the gain quantizer. If $\hat{g}S$ is known, then $\hat{g}$, $S$, and $\hat{S}$ are each also known, so that

$$D_c = {2 \over k} E[E[(X-\hat{g}S)^T(\hat{g}S-\hat g \hat S) \vert \hat{g}S]]$$

$$= {2 \over k} E[E[(X-\hat{g}S)^T\vert \hat{g}S](\hat{g}S-\hat g \hat S)]$$

$$= {2 \over k} E[E[(g-\hat{g})\vert \hat{g}]S^T(\hat{g}S-\hat g \hat S)]$$ (10)

$$=$$ 0 (11)

where (10) follows by the independence of $g$ and $\hat{g}$ from $S$, and (11) follows from the centroid condition of the gain quantizer. Finally,

$$D_s = {1 \over k} E[\hat{g}^2] E[\Vert S-\hat{S}\Vert^2],$$ (12)

$$= {1 \over k} (E[g^2] - E[(g-\hat{g})^2])E[\Vert S-\hat{S}\Vert^2]$$ (13)

$$\approx {1 \over k} E[g^2] E[\Vert S-\hat{S}\Vert^2]$$ (14)

$$= \sigma^2 E[\Vert S-\hat{S}\Vert^2]$$ (15)

where (12) follows from the independence of $S$ and $\hat{S}$ from $g$, (13) follows from the centroid condition of $\hat{g}$, and (15) follows from (3). The approximation in (14) is accurate for high signal-to-noise ratios (SNR) for the gain quantizer, which we will assume. It can be made more exact by estimating the error term via high resolution analysis using Bennett's integral, but we will not need to do so here. Hence, $D_s$ acts as a ``shape distortion'' (multiplied by the constant $E[g^2]$).

In summary, the distortion of ${\rm W}_\Lambda$ is

$$D \approx {1 \over k} E[(g-\hat{g})^2] + \sigma^2 E[\Vert S-\hat{S}\Vert^2]$$ (16)

which partitions the distortion of ${\rm W}_\Lambda$ into shape and gain components, as in [25]. The decomposition of $D$ allows us to optimize ${\rm W}_\Lambda$ by separately optimizing the shape and gain components.

The gain distortion is given by

$$D_g = {1 \over k} E[(g-\hat g)^2] = {1 \over k} \int_0^\infty (r-\hat g(r))^2 f_g(r)\, dr$$ (17)

where $\hat g(r)$ is the gain quantization of $r$ and where $f_g(r)$ is given in (1). This integral can be numerically evaluated once the gain quantizer has been designed.

We estimate the shape distortion $D_s$, use it in the design algorithm, and validate its accuracy by the observed shape distortion in the simulations for ${\rm W}_\Lambda$ . In all cases reported, the approximate computations of distortion agree with the simulated results within 0.1dB.

It follows from [30, Lemma 4.2] that if $U, V \in A_i$ and $\Vert h_i(U)-h_i(V)\Vert = O(d_\Lambda)$, then

$$1-O(\sqrt{d}) \le \frac{\Vert h_i(U)-h_i(V)\Vert^2}{\Vert U-V\Vert^2} \le 1,$$ (18)

i.e., the mapping used in ${\rm W}_\Lambda$ nearly preserves distances. Thus, for asymptotically high $R_s$, the distortion, $E[\Vert S-\hat S\Vert^2]$, of ${\rm W}_\Lambda$ , for $S$ uniformly distributed on $\Omega _k$, is equal to the distortion of the underlying lattice quantizer with codebook $\Lambda$ for a uniform source in ${\mathbb{R}}^{k-1}$. Let $\Pi$ be a Voronoi region of a $(k-1)$-dimensional lattice $\Lambda$ such that $0\in \Pi$, and let $V(\Lambda)$ denote the volume of $\Pi$. The normalized second moment of $\Pi$ (or of the lattice $\Lambda$) is

$$G(\Lambda) = {{1 \over k-1} \int_\Pi \Vert t\Vert^2\, dt \over V(\Lambda)^{1+{2 \over k-1}}}$$

and the $(k-1)$-dimensional vector mean-squared error when $\Lambda$ is used to quantize a uniform source, neglecting overload distortion, is the mean-squared error in any Voronoi region, given by

$$\frac{1}{V(\Lambda)} \int_\Pi \Vert t\Vert^2 \, dt = (k-1) G(\Lambda) V(\Lambda)^{2 \over k-1}.$$

Thus, for finite $R_s$ the shape distortion is approximated by

$$D_s \approx \sigma^2 E[\Vert S - \hat S\Vert^2] \approx (k-1)\sigma^2 G(\Lambda) V(\Lambda)^{2 \over k-1}.$$ (19)

For asymptotically large $R_s$ and $R_g$, the first approximation in (19) becomes tight by (13) because $E[(g-\hat g)^2]\rightarrow 0$, and the second becomes tight because $d\rightarrow 0$ in (18). Thus,

$$\lim_{R_s\rightarrow\infty} D_s V(\Lambda)^{-2 \over k-1} = \sigma^2 \lim_{R_s\rightarrow\infty} E[\Vert S - \hat S\Vert^2] V(\Lambda)^{-2 \over k-1}$$

$$= (k-1)\sigma^2 G(\Lambda)$$ (20)

The values (or close approximations) of $G(\Lambda)$ are given for the best known lattices for the uniform source in [37, pg. 61]. The shape distortion is affected by scaling $\Lambda$. For example, doubling the minimum distance of $\Lambda$ increases $V(\Lambda)$ by a factor of $2^{k-1}$, while $G(\Lambda)$ is invariant to scaling, and the shape distortion therefore increases by a factor of four. The total distortion $D = D_g + D_s$ is estimated using (17) and (19).

Table II: Optimization algorithm for construction of ${\rm W}_\Lambda$ at rate $R$.