High Resolution Spherical Quantization of Sinusoids with Harmonically Related Frequencies

Sinusoidal coding is an essential tool in low-rate audio coding, and developing an efficient quantization scheme for the sinusoidal parameters is therefore crucial. In this work we derive optimal entropy constrained amplitude, phase and frequency quantizers for sinusoids whose frequencies are harmonically related, with respect to the l₂ distortion measure. This scheme exploits the harmonic structure of many speech and audio signals in the sense that besides amplitudes and phases, only fundamental frequencies need to be quantized, resulting in a significant decrease in the number of bits assigned to frequency parameters. The asymptotically optimal quantizers minimize a high-resolution approximation of the expected l₂ distortion while the corresponding quantization indices satisfy an entropy constraint. The quantizers turn out to be flexible and of low complexity, in the sense that they can be determined easily for varying bit rate requirements, without any sort of retraining or iterative procedures. In an objective rate-distortion comparison, the proposed scheme is shown to outperform two variants of a recently proposed scheme, in which all frequency parameters are quantized separately, either directly or differentially