Geometric piecewise uniform lattice vector quantization of the memoryless Gaussian source

The aim of this paper is to find a quantization technique that has low implementation complexity and asymptotic performance arbitrarily close to the optimum. More specifically, it is of interest to develop a new vector quantizer design procedure for a memoryless Gaussian source that yields vector quantizers with excellent performance and the structure required for fast quantization. To achieve this, we combined a fast lattice-encoding algorithm with a geometric approach to generate a model of a geometric piecewise-uniform lattice vector quantizer. Expressions for granular distortion and the optimal number of outputs points in each region were derived. Both exact and approximative asymptotic analyses were carried out. During this process, the constant probability density function of the input signal vector was kept inside the whole region. The analysis demonstrated the existence of piecewise-constant approximations to the input-vector probability density function, which is optimal for the proposed geometric piecewise-uniform vector quantizer. The considered quantization technique is near optimal for a memoryless Gaussian source. In other words, this paper proposes a method for a near-optimum, low-complex vector quantizer design based on probability density function discretization. The presented methodology gives a signal-to-quantization noise ratio that in some cases differs from the optimum by 0.1 dB or less. Improvements of the considered model in performance and complexity over some of the existing techniques are also demonstrated.