Sajid Ali

A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.

The use of quaternion orthogonal designs (QODs) to describe point-to-point communication among dual-polarized antennas has the potential to provide higher rate orthogonal and quasi-orthogonal complex designs exploiting polarization diversity among space and time diversities. Furthermore, it is essential to have a space time block code (STBC) which offers a linear and decoupled decoder which quasi-orthogonal designs fail to attain. In this paper, we show how the realm of quaternions unexpectedly offers us a possible solution and codes obtained from quaternion designs mostly achieve both linear and decoupled decoders. This motivated us to perform an indispensable search for QODs such that the code rate is bounded below by 1/2 and does not sharply decrease as the number of transmit antennas increases. It is shown that three famous recursive techniques do not satisfy this criteria and their code rates decrease rather rapidly. Therefore, we propose another method of constructing quaternion designs suitable for any number of transmit antennas and verify that these attain linear and decoupled decoders with the system model based on quaternionic channel. It is shown that such designs outperform others in terms of transmit diversity, code rates and the optimality of the proposed decoder is validated through simulation results.

A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the computer algebra system {\sf GAP}4.

Receiver complexity is an important performance metric in space time block coding (STBC). After comprehensive study and analysis of STBCs over complex domain, quaternion orthogonal designs (QODs) have been explored that can provide less decoder complexity. However, some researchers argue that existing maximum likelihood (ML) decoding rule for QODs does not yield optimal performance for quasi-orthogonal STBC-based QODs and, therefore, the design of optimal decoder for such designs remains a big challenge of this field. To counter this observation, we modify the system model slightly. In addition to this, we present a new generalized QOD construction that can generate QODs for any number of transmit antennas. These codes are quasi-orthogonal in complex domain and provide code rates higher than their respective complex orthogonal designs (CODs). We show that the proposed ML decoder provides optimal linear decoupled decoding solution for these codes because of their distinctive construction. Simulation results verify the optimal performance of the ML decoder for the proposed codes and also show that the diversity of the proposed quaternion-based codes is same as CODs, however, they provide better code rates.

Over the past few decades, wireless communication aims to support high data rates and reliability and thus several techniques exploiting space, time and polarization diversities have been used to achieve large diversity gains. Orthogonal space time block codes (OSTBC) in combination with polarization diversity promise optimal diversity gains. For dual-polarized antennas, we propose a new system model based on quater- nionic structure of the channel, which offers a way to exploit polarization diversity quite independently of other forms of diversities. Moreover, such orthogonal space time polarization block codes (OSTPBC) achieve better throughput and provide a linear decoupled decoding solution at the receiver, which significantly reduces computational complexity.