Index Assignment

In order to implement the shape-gain spherical quantizer, the $M = 2^{kR_s}$ quantizer codevectors must be uniquely identified by binary strings of length $kR$ which are transmitted across the channel. The assignment is accomplished in a similar manner as in [14] for the pyramid vector quantizer for the Laplacian source. First, the number of codevectors in each annulus of the shape codebook is counted using the theta function. We report on specific results using the Leech lattice $\Lambda_{24}$, for which the ${\rm W}_{\Lambda _{24}}$ codes need a one-time computation of the first few hundred coefficients of the theta function of the Leech lattice, which are stored and used as needed.

It is assumed that there is an efficient method for assigning indices to the underlying lattice. This is the case with many lattices, including the Leech lattice $\Lambda_{24}$ (e.g., see [40]).

The codevectors of the wrapped spherical code are assigned to integers according to their quantized gain, annulus, and order within their annulus, as follows. Let $N$ represent the number of annuli of the shape codebook. Let $P_j$ be the number of points in the $j$th annulus of a shell, and let $P$ be the total number of points in the shape codebook. Assuming all indices start at 0, the $l$th point within the $j$th annulus of the $i$th gain shell is assigned to the number

$$iP + \sum_{a=0}^{j-1} P_a + l.$$ (44)

Both the encoder and decoder must compute this summation. This can be made efficient by storing in memory the partial summations $\sum_{a=0}^{j-1} P_a$, for $j = 0, 1, \ldots, N-1$. The memory required for this is equal to the total number of annuli in the codebook, which is generally not large. For example, in the codebook ${\rm W}_{\Lambda _{24}}$of rate 4, there are 36 total annuli.