The zero forcing number is a graph parameter first introduced as a tool for solving the minimum rank problem, which is: Given a simple, undirected graph G, and a field F, let S(F,G) denote the set of all symmetric matrices A=[a_{ij}] with entries in F such that a_{ij} doess not equal 0 if and only if ij is an edge in G. Find the minimum possible rank of a matrix in S(F,G). It is known that the zero forcing number Z(G) provides an upper bound for the maximum nullity of a graph. I investigate properties of the zero forcing number, including its behavior under various graph operations.

Massive MU-MIMO (Multiuser-Multiple Input and Multple Output) is a promising technology for 5G wireless communications because of its spectrum and energy efficiency. To combat the distortion from multipath fading channel, the acquisition of channel state information is essential, which generally requires the training signal that lowers the data rate. In addition, coarse quantization can reduce the high computational energy and cost, yet results in the loss of information. In this thesis, an iterative decision feedback receiver, including iterative Channel Estimation (CE) and equalization, is constructed for a Massive MU-MIMO uplink system. The impact of multipath distortion and coarse quantization can be gradually reduced due to the iterative structure that exploits extrinsic feedback to improve the CE and data detection, so that the data rate is improved by reducing training signals for CE and by using low precision quantization. To observe and evaluate the convergence behaviour, an Extrinsic Information Transfer (EXIT) chart method is utilized to visualize the performance of the iterative receiver. / Graduate

Let G be a simple graph on n vertices, and let S(G) be the class of all real-valued symmetric nxn matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The smallest rank achieved by a matrix in S(G) is called the minimum rank of G, denoted mr(G). The maximum nullity achieved by a matrix in S(G) is denoted M(G). For each graph G, there is an associated minimum rank class, MR(G) consisting of all matrices A in S(G) with rank A = mr(G). Although no restrictions are applied to the diagonal entries of matrices in S(G), sometimes diagonal entries corresponding to specific vertices of G must be zero for all matrices in MR(G). These vertices are known as nil vertices (see [6]). In this paper I discuss some basic results about nil vertices in general and nil vertices in cographs and prove that cographs with a nil vertex of a particular form contain two other nil vertices symmetric to the first. I discuss several open questions relating to these results and a counterexample. I prove that for all cographs G without an induced complete tripartite graph with independent sets all of size 3, the zero-forcing number Z(G), a graph theoretic parameter, is equal to M(G). In fact this result holds for a slightly larger class of cographs and in particular holds for all threshold graphs. Lastly, I prove that the maximum of the minimum ranks of all cographs on n vertices is the floor of 2n/3.

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<p>In this thesis, we consider a coherent MIMO system, emphasizing on the simplicity of implementation at both the code generator and the receiver. Specifically, we consider the transmission of a space-time block code (STBC) that is a linear combination of coding matrices weighted by the information symbols through a receiver-correlated flat-fading channel and received by a linear ZF detector. Our target is the design of a code which, while maintaining full data-transmission rate, minimizes the asymptotic average (over all the random channel coefficients) bit error probability of an ZF detector. To this end, we first ensure that the full data rate of symbols is maintained, and then, based on the BER for 4-QAM signals, we derive the conditions for optimal codes and establish a code structure that minimizes the asymptotic average bit error probability. We also prove that the diversity gain of our M × N MIMO system is N − M + 1. The resulting optimum code structure requires the individual coding matrices to be mutually orthogonal when vectorized and is related to covariance matrix of correlated channel. The first optimum structural characteristics of the coding matrices is described as trace-orthogonal. A new approach to express expected value of random correlated channel has been proposed as well. From simulation results we can see that advantage of optimum code over uncoded system is more apparent as channel correlation is higher.</p> / Master of Applied Science (MASc)

No / In this work we present the work on the equalization algorithms to be used in future orthogonally multiplexed wavelets based multi signaling communication systems. The performance of ZF and MMSE algorithms has been analyzed using SISO and MIMO communication models. The transmitted electromagnetic waves were subjected through Rayleigh multipath fading channel with AWGN. The results showed that the performance of both of the above mentioned algorithms is the same in SISO channel but in MIMO environment MMSE has better performance.

The optimum ordering strategies for the coded V-BLAST system with capacity achieving temporal codes on each stream are studied in this thesis. Mathematical representations of the optimum detection ordering strategies for the coded V-BLAST under instantaneous rate allocation (IRA), uniform power/rate allocation (URA), instantaneous power allocation(IPA) and instantaneous power/rate allocation (IPRA) are derived. For two transmit antennas, it is shown that the optimum detection strategies are based on the per-stream before-processing channel gains. Based on approximations of the per-stream capacity equation, closed-form expressions of the optimal ordering strategy under the IRA at low and high signal to noise ratio (SNR) are derived. Necessary optimality conditions under the IRA are given. Thresholds for the low, intermediate and high SNR regimes in the 2-Tx-antenna system under the IPRA are determined, and the SNR gain of the ordering is studied for each regime. Performances of simple suboptimal ordering strategies are analysed, some of which perform very close to the optimum one.

This thesis examines various signal processing techniques that are required for establishing efficient (near optimal) communications in multiuser multiple-input multiple-output (MIMO) environments. The central part of this thesis is dedicated to acquisition of information about the MIMO channel state - at both the receiver and the transmitter. This information is required to organize a communication set up which utilizes all the available channel resources. Realistic channel model, i.e., the spatial channel model (SCM), has been used in this study, together with modern long-term evolution (LTE) standard. The work consists of three major themes: (a) estimation of the channel at the receiver, also known as tracking; (b) quantization of the channel information and its feedback from receiver to the transmitter (feedback quantization); and (c) reconstruction of the channel knowledge at the transmitter, and its use for data precoding during communication transmission. / Communications

The optimum ordering strategies for the coded V-BLAST system with capacity achieving temporal codes on each stream are studied in this thesis. Mathematical representations of the optimum detection ordering strategies for the coded V-BLAST under instantaneous rate allocation (IRA), uniform power/rate allocation (URA), instantaneous power allocation(IPA) and instantaneous power/rate allocation (IPRA) are derived. For two transmit antennas, it is shown that the optimum detection strategies are based on the per-stream before-processing channel gains. Based on approximations of the per-stream capacity equation, closed-form expressions of the optimal ordering strategy under the IRA at low and high signal to noise ratio (SNR) are derived. Necessary optimality conditions under the IRA are given. Thresholds for the low, intermediate and high SNR regimes in the 2-Tx-antenna system under the IPRA are determined, and the SNR gain of the ordering is studied for each regime. Performances of simple suboptimal ordering strategies are analysed, some of which perform very close to the optimum one.

Previous studies have demonstrated that many existing communication systems can be formulated within a unified multirate filterbank transceiver model. A redundant block transmission system implemented via this unified multirate filterbank transceiver model is usually known as a multiple-input-multiple-output (MIMO) system in literature. This dissertation devises an optimum linear block-based precoder and the corresponding equalizer for MIMO systems over perfect reconstruction (PR) channels by exploiting the proposed oblique projection framework. Particularly, two main criteria of interest in a digital communication link with limited transmission power are investigated, namely, average bit error rate (BER) minimization and mutual information rate maximization. The study framework is developed as follows. For a block-based precoder, a received signal model is formulated for the two redundancy schemes, viz., trailing-zeros (TZ) and cyclic-prefix (CP). By exploiting the property of oblique projection, a cascaded equalizer for block transmission systems (i.e., MIMO systems) is proposed and implemented with a scheme, in which the inter-block interference (IBI) is completely eliminated by the oblique projection and followed by a matrix degree of freedom for inter-symbol interference (ISI) equalization. With the available channel state information at the transmitter side, the matrix for ISI equalization of the cascaded equalizer is utilized to design an optimum linear block-based precoder, such that the BER is minimized (or the mutual information rate is maximized), subject to the ISI-free and the transmission power constraints. Accordingly, the cascaded equalizer with the ISI-free constraint yields a cascaded ZF equalizer. Theoretical derivations and simulation results confirm that the proposed framework not only retains identical BER and information rate performances to previous works for cases with sufficient redundancy, but also allows their results to be extended to the cases of insufficient redundancy.