Abstract
We propose a novel recursive multi-stage approach to Grassmannian quantization. Compared to the commonly employed single-stage quantization, our method has the advantage of significantly decreasing the number of codebook searches required for quantization and, thus, reducing the complexity. On the downside, the multi-stage approach causes a slight rate-distortion degradation compared to single-stage quantization. We analyze the rate-distortion performance of the proposed recursive quantization approach, considering random vector quantization within the individual stages. We furthermore propose a bit-allocation optimization amongst the stages of the quantizer, given a constraint on the total number of quantization bits.
Highlights
G RASSMANNIAN quantization deals with quantization of points on a Grassmann manifold G(n, m), i.e., with quantization of m-dimensional subspaces of an n-dimensional realor complex-valued Euclidean space
Contribution: In this paper, we propose a general approach for recursive decomposition of a large Grassmannian quantization problem into a series of smaller Grassmannian quantization problems, which can be solved with reduced computational complexity
We consider quantization of points that are uniformly distributed on the complex-valued Grassmann manifold G(n, m)
Summary
G RASSMANNIAN quantization deals with quantization of points on a Grassmann manifold G(n, m), i.e., with quantization of m-dimensional subspaces of an n-dimensional realor complex-valued Euclidean space. It is well known that maximally spaced subspace packings are optimal for memoryless quantization of uniformly distributed points on the Grassmannian [3] Such packings are difficult to obtain in general; yet, suboptimal codebooks with good performance can efficiently be constructed [4]. Cube-split is essentially a scalar quantizer that transforms the quantization variables on-the-fly to allow for scalar quantization on the unit interval, thereby avoiding storage of quantization codebooks It requires storage of a look-up-table for the cumulative distribution function (CDF) of Gaussian random variables, with a resolution that grows exponentially with the number of quantization bits; the claimed “zero storage requirement” is only partially true.
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