Coefficient Quantization

The coefficients in a band are all quantized with the same step size $Q$. To store $Q$ in the bit stream, it is indexed as

$$Q_i = {\rm round}(10\log(Q)+35),\ {\rm (min\ 0,\ max\ 50)},$$ (34)

and thereafter Huffman encoded. The $Q$ controls a nonuniform coefficient quantizer, with the following characteristic.

$$x_Q = {\rm sign}(x){\rm round}\left ( \left \vert \frac{x}{Q} \right \vert ^{1/c} \right ),$$ (35)

where a $c=1.3$ has proved to work well. A nonuniform quantizer does not follow the masking threshold theories in section 3.4, but works very well -- it is used in e.g MPEG-2 AAC [7].

The coefficients of any orthogonal linear transform can be quantized as described above, since

$$Qerr({\bf X}) = ({\bf X}-{\bf X_Q})^T({\bf X}-{\bf X_Q}) = (U^T({\bf x}-{\bf x_Q}))^T(U^T({\bf x}-{\bf x_Q})) =$$

$$= (({\bf x}-{\bf x_Q}))^T(({\bf x}-{\bf x_Q})) = Qerr({\bf x})$$ (36)

and so the quantizer makes no difference between the coefficients from the different modes in section 4.2.2.