Experimental Allocation of Shape and Gain Rates

Let $R$ be the transmission rate of the wrapped SVQ and let the shape code rate $R_s$ and gain code rate $R_g$ satisfy $R_s + R_g = R$. The rate $R_s$ determined by (7) can be altered by rescaling $\Lambda$ so that more or fewer points are contained in ${\rm W}_\Lambda$ . We numerically determine the allocation of rate $R$ between $R_s$ and $R_g$ that minimizes the distortion of the wrapped SVQ, using the design algorithm given in Table III-A. In the next section, we provide an analytical solution for large rates. Since the gain codebook size is an integer, the values of $R_g$ are restricted to a finite set and the optimal value of $R_g$ can be found exactly. (This is in contrast to optimizations over an infinite set, in which an iterative algorithm may not converge to precisely the optimal value in bounded time.)

For a given pair $(R_g,R_s)$, the gain codebook is optimized using the Lloyd-Max algorithm with $R_g$ bits. Since each Voronoi cell corresponds to one lattice point, the number of shape quantizer codevectors is closely approximated by the $(k-1)$-dimensional content of the sphere $\Omega _k$ divided by the volume of one Voronoi cell (recall, $V(\Lambda) \approx V(h_i(\Lambda))$ [30]). That is, $2^{kR_s} \approx S_k / V(\Lambda)$ and it was shown in [30] that

$$\lim_{R_s \to\infty} V(\Lambda) 2^{kR_s} = S_k .$$ (21)

Thus, for a given shape rate $R_s$, we scale $\Lambda$ before the shape codebook is constructed such that the volume of the Voronoi cell satisfies $V(\Lambda) = S_k 2^{-kR_s}$. After optimization is complete, the actual number of codevectors is computed by evaluating the theta function. This more time-consuming step is avoided during the optimization step, which only uses estimates of the codebook sizes.