Final Project of EE359

Project Proposal of EE359, Winter 2009

Final Report (.pdf format)

Project Proposal:

Space-Time Block Codes with Symbol-by-Symbol Maximum Likelihood Detections

Ming-Yang Chen

Winter, 2009

Coding for multiple-input multiple-output (MIMO) channels can significantly increase the transmit diversity of communication systems in fading environments [1]. Among various MIMO transmission and receiving techniques, the rate-1 orthogonal space-time block codes constitute a powerful scheme to achieve full diversity and decouple the received signals into parallel channels [2]. The optimal decoders can therefore be realized through symbol-by-symbol maximum likelihood (ML) detections. Nevertheless, Tarokh, Jafarkhani, and Calderbank [2] proved that the well-known Alamouti scheme for two transmit antennas [3] is the only existing rate-1 complex orthogonal space-time block code, wherefore quasi-orthogonal space-time block codes (QOSTBCs) are proposed to have better spectral efficiency with four transmit antennas [4, 5, 6]. A complete list of QOSTBCs that can be identified as rings is given jointly in [7] and [8]. Including those of [4, 5, 6], each of the QOSTBCs in [7] and [8] can achieve full diversity and the optimal coding gains with possibly having to rotate half of the transmitted symbols' constellations, as quadrature amplitude modulations (QAMs) [9] or phase-shift keying (PSK) modulations [10] are utilized. A common character for those QOSTBCs is that the corresponding optimal decoders can only be implemented via complex symbol pairwise ML detections. Motivated by this, Yuen, Guan, and Tjhung [11] re-stacked the Alamouti schemes with different input symbols into another QOSTBC that is symbol-by-symbol ML detectable. This QOSTBC is different from the traditional ones merely in that the transmitted symbols are precoded by coordinate-interleavers before transmissions occur. In [12], Wang, Wang, and Xia further used two coordinate-interleavers involving two real symbols together with other two involving four real symbols to achieve the maximum coding gains over general QAM constellations. A QOSTBC with coordinate-interleavers is no longer linear over the field of all complex numbers with respect to the input symbols so rotating all constellations with the same angle usually varies the corresponding normalized diversity product. An optimal rotation that provides the maximum normalized diversity product is then always embedded with the code. Subsequently, both [11] and [12] claim by numerical simulation that their performance of bit error rates (BERs) is identical with the rate-1 coordinate-interleaved orthogonal design (CIOD) of [13] over the 4-QAM constellation. By applying matrix representations of Clifford algebras, Karmakar and Rajan [14] presented codes with the similar properties.

The ultimate goals of this project are summarized as follows.

(i). Try to find an analytic proof for the conclusions observed via numerical simulation in [11] and [14], i.e., the QOSTBCs with symbol-by-symbol ML detections (abbreviated by SSD-QOSTBCs) of [11, 14] achieve full diversity and the same coding gains as the rate-1 CIOD [13] by individually using their optimal constellation rotations over the 4-QAM constellation.

(ii). Merited by the proof found in (i), try to generalize its conclusion from the 4-QAM constellation to an arbitrary QAM or even PSK constellation mathematically.

(iii). Try to provide insightful connections between the SSD-QOSTBCs of [11, 14] and QOSTBCs in [7, 8] that are capable of creating SSD-QOSTBCs systematically from each QOSTBC in [7] and [8] for four transmit antennas over quasi-static Rayleigh fading channels. The philosophy kept in mind here is to generate new designs that can still preserve the truthfulness substantiated in (ii).

(iv). With the aid of MATLAB simulation, try to characterize the coding loss in exchange for simplifying the optimal decoders of QOSTBCs from complex symbol pairwise to symbol-by-symbol ML detections (over the constellation schemes obtained in (ii)).