Multi-Terminal Source Coding (MTSC) addresses the problem of compressing correlated sources without communication links among them. In this thesis, the constructive approach of this problem is considered in an algebraic framework and a system design is provided that can be applicable in a variety of settings. Wyner-Ziv problem is first investigated: coding of an independent and identically distributed (i.i.d.) Gaussian source with side information available only at the decoder in the form of a noisy version of the source to be encoded. Theoretical models are first established and derived for calculating distortion-rate functions. Then a few novel practical code implementations are proposed by using the strategy of multi-dimensional nested lattice/trellis coding. By investigating various lattices in the dimensions considered, analysis is given on how lattice properties affect performance. Also proposed are methods on choosing good sublattices in multiple dimensions. By introducing scaling factors, the relationship between distortion and scaling factor is examined for various rates. The best high-dimensional lattice using our scale-rotate method can achieve a performance less than 1 dB at low rates from the Wyner-Ziv limit; and random nested ensembles can achieve a 1.87 dB gap with the limit. Moreover, the code design is extended to incorporate with distributed compressive sensing (DCS). Theoretical framework is proposed and practical design using nested lattice/trellis is presented for various scenarios. By using nested trellis, the simulation shows a 3.42 dB gap from our derived bound for the DCS plus Wyner-Ziv framework.

This thesis is concerned with decoding for wireless communications. In particular, computationally efficient lattice decoding algorithms are exploited to further improve the system performance. Based on this idea, five technical chapters are presented in this thesis. In Chapter 2, we propose novel lattice decoding based on segment Lenstra-Lenstra-Lovász (LLL) algorithm to further reduce the decoding complexity of coded Multiple-Input Multiple-Output (MIMO) cooperative channel. In particular, we extend the original segment LLL algorithm to the complex version, and prove that it can achieve the optimal diversity-multiplexing tradeoff (DMT). In Chapter 3, we present randomized lattice decoding based on Klein's sampling technique, which is a randomized version of Babai's nearest plane algorithm (i.e., successive interference cancellation). We analyze and optimize the performance of randomized lattice decoding resulting in reduced decoding complexity, and propose a very efficient implementation of random rounding. Chapter 4 is concerned with bounded distance decoding (BDD) based embedding technique. The embedding technique is used to reduce the γ-BDD problem to 1/(2γ)-unique shortest vector problem 1/(2γ)-uSVP). WE prove that the Lenstra, Lenstra and Lovász (LLL) algorithm can achieve 1/(2γ)-BDD for γ ≈ O(2n) for embedding decoding. We also prove that BDD of the regularized lattice is optimal in terms of the DMT. In Chapter 5, we present a detailed study of the soft output MIMO decoding. We show that the randomized decoding algorithm is an efficient way to compute soft output. In order to improve soft output quality, we propose variants of soft output decoding based on the sampling technique and embedding technique. Moreover, we derive a lower bound on the search radius for which list-based decoding can provide a near optimal solution to soft output. In Chapter 6, the performance limits of lattice reduction-aided precoding are investigated. The proximity factor is defined to measure the worst-case transmission power gap to sphere precoding. The second moment over precoding region is defined to measure the average-case transmission power loss. Afterward, low dimension lattice precoding is proposed to further reduce the transmission power.

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