Ensemble Properties of RVQ-Based Limited-Feedback Beamforming Codebooks

• Published in: IEEE transactions on information theory Vol. 59; no. 12; pp. 8224 - 8249
• Main Authors: Raghavan, Vasanthan, Veeravalli, Venugopal V.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive signaling; Applied sciences; Array signal processing; beamforming; Codes; Correlation; Correlation analysis; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; information rates; Information theory; Information, signal and communications theory; limited feedback; Matrix; Measurement; MIMO; multiantenna systems; precoding; Random variables; Sampling, quantization; Signal and communications theory; Signal to noise ratio; Signal, noise; spatial correlation; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Vectors; Singularity; Parameter estimation; Data compression; Spatial correlation; Information rate; Feedback regulation; limited feedback; Random vector; precoding; Beam forming; Information loss; Codebook; information rates; Signal quantization; Channel estimation; Adaptive signaling; MIMO; Pretreatment; Fading; MIMO system; Vector quantization; Signalling; Information transmission; multiantenna systems; Asymmetry; Transmitting antenna; Metric; beamforming; Mutual information; Signal to noise ratio
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2283719

The ensemble properties of random vector quantization (RVQ) codebooks for limited-feedback beamforming in multiinput multioutput (MIMO) systems are studied with the metrics of interest being the received \ssr SNR loss and mutual information loss, both relative to a perfect channel state information (CSI) benchmark. The simplest case of unskewed codebooks is first studied in the correlated MIMO setting and these loss metrics are computed as a function of the number of bits of feedback ( B), transmit antenna dimension ( Nt), and spatial correlation. In particular, it is established that: 1) the loss metrics are a product of two components-a quantization component and a channel-dependent component; 2) the quantization component, which is also common to analysis of channels with i.i.d. fading, decays as B increases at the rate 2 -B/(Nt-1) ; 3) the channel-dependent component reflects the condition number of the channel. Further, the precise connection between the received \ssr SNR loss and the squared singular values of the channel is shown to be a Schur-convex majorization relationship. Finally, the ensemble properties of skewed codebooks that are generated by skewing RVQ codebooks with an appropriately designed fixed skewing matrix are studied. Based on an estimate of the loss expression for skewed codebooks, it is established that the structure of a good skewing matrix is critically dependent on the condition numbers of the effective channel (product of the true channel and the skewing matrix) and the skewing matrix.