In the last decade, there has been an explosive growth in different applications of wireless technology, due to users' increasing expectations for multi-media services. With the current trend, the present systems will not be able to handle the required data traffic. Lattice codes have attracted considerable attention in recent years, because they provide high data rate constellations. In this thesis, the applications of implementing lattice codes in different communication systems are investigated. The thesis is divided into two major parts. Focus of the first part is on constellation shaping and the problem of lattice labeling. The second part is devoted to the lattice decoding problem. In constellation shaping technique, conventional constellations are replaced by lattice codes that satisfy some geometrical properties. However, a simple algorithm, called lattice labeling, is required to map the input data to the lattice code points. In the first part of this thesis, the application of lattice codes for constellation shaping in Orthogonal Frequency Division Multiplexing (OFDM) and Multi-Input Multi-Output (MIMO) broadcast systems are considered. In an OFDM system a lattice code with low Peak to Average Power Ratio (PAPR) is desired. Here, a new lattice code with considerable PAPR reduction for OFDM systems is proposed. Due to the recursive structure of this lattice code, a simple lattice labeling method based on Smith normal decomposition of an integer matrix is obtained. A selective mapping method in conjunction with the proposed lattice code is also presented to further reduce the PAPR. MIMO broadcast systems are also considered in the thesis. In a multiple antenna broadcast system, the lattice labeling algorithm should be such that different users can decode their data independently. Moreover, the implemented lattice code should result in a low average transmit energy. Here, a selective mapping technique provides such a lattice code. Lattice decoding is the focus of the second part of the thesis, which concerns the operation of finding the closest point of the lattice code to any point in N-dimensional real space. In digital communication applications, this problem is known as the integer least-square problem, which can be seen in many areas, e.g. the detection of symbols transmitted over the multiple antenna wireless channel, the multiuser detection problem in Code Division Multiple Access (CDMA) systems, and the simultaneous detection of multiple users in a Digital Subscriber Line (DSL) system affected by crosstalk. Here, an efficient lattice decoding algorithm based on using Semi-Definite Programming (SDP) is introduced. The proposed algorithm is capable of handling any form of lattice constellation for an arbitrary labeling of points. In the proposed methods, the distance minimization problem is expressed in terms of a binary quadratic minimization problem, which is solved by introducing several matrix and vector lifting SDP relaxation models. The new SDP models provide a wealth of trade-off between the complexity and the performance of the decoding problem.

In the last decade, there has been an explosive growth in different applications of wireless technology, due to users' increasing expectations for multi-media services. With the current trend, the present systems will not be able to handle the required data traffic. Lattice codes have attracted considerable attention in recent years, because they provide high data rate constellations. In this thesis, the applications of implementing lattice codes in different communication systems are investigated. The thesis is divided into two major parts. Focus of the first part is on constellation shaping and the problem of lattice labeling. The second part is devoted to the lattice decoding problem. In constellation shaping technique, conventional constellations are replaced by lattice codes that satisfy some geometrical properties. However, a simple algorithm, called lattice labeling, is required to map the input data to the lattice code points. In the first part of this thesis, the application of lattice codes for constellation shaping in Orthogonal Frequency Division Multiplexing (OFDM) and Multi-Input Multi-Output (MIMO) broadcast systems are considered. In an OFDM system a lattice code with low Peak to Average Power Ratio (PAPR) is desired. Here, a new lattice code with considerable PAPR reduction for OFDM systems is proposed. Due to the recursive structure of this lattice code, a simple lattice labeling method based on Smith normal decomposition of an integer matrix is obtained. A selective mapping method in conjunction with the proposed lattice code is also presented to further reduce the PAPR. MIMO broadcast systems are also considered in the thesis. In a multiple antenna broadcast system, the lattice labeling algorithm should be such that different users can decode their data independently. Moreover, the implemented lattice code should result in a low average transmit energy. Here, a selective mapping technique provides such a lattice code. Lattice decoding is the focus of the second part of the thesis, which concerns the operation of finding the closest point of the lattice code to any point in N-dimensional real space. In digital communication applications, this problem is known as the integer least-square problem, which can be seen in many areas, e.g. the detection of symbols transmitted over the multiple antenna wireless channel, the multiuser detection problem in Code Division Multiple Access (CDMA) systems, and the simultaneous detection of multiple users in a Digital Subscriber Line (DSL) system affected by crosstalk. Here, an efficient lattice decoding algorithm based on using Semi-Definite Programming (SDP) is introduced. The proposed algorithm is capable of handling any form of lattice constellation for an arbitrary labeling of points. In the proposed methods, the distance minimization problem is expressed in terms of a binary quadratic minimization problem, which is solved by introducing several matrix and vector lifting SDP relaxation models. The new SDP models provide a wealth of trade-off between the complexity and the performance of the decoding problem.