V. Shashidhar

Full-diversity, high-rate space-time block codes from division algebras

We present some general techniques for constructing full-rank, minimal-delay, rate at least one space-time block codes (STBCs) over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras (field extensions) as well as using noncommutative division algebras of the rational field /spl Qopf/ embedded in matrix rings. The first half of the paper deals with constructions using field extensions of /spl Qopf/. Working with cyclotomic field extensions, we construct several families of STBCs over a wide range of signal sets that are of full rank, minimal delay, and rate at least one appropriate for any number of transmit antennas. We study the coding gain and capacity of these codes. Using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas. We also present a method of constructing STBCs using noncyclotomic field extensions. In the later half of the paper, we discuss two ways of embedding noncommutative division algebras into matrices: left regular representation, and representation over maximal cyclic subfields. The 4/spl times/4 real orthogonal design is obtained by the left regular representation of quaternions. Alamoutis (1998) code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it. Also, we discuss a general principle for constructing cyclic division algebras using the nth root of a transcendental element and study the capacity of the STBCs obtained from this construction. Another family of cyclic division algebras discovered by Brauer (1933) is discussed and several examples of STBCs derived from each of these constructions are presented.

Information-Lossless Space–Time Block Codes From Crossed-Product Algebras

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen proposed a design for two transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting space-time block code (STBC) when used with a signal set an information-lossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori specified signal set. Alamouti code and quasi-orthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossed-product algebras are information-lossless. We give some classes of crossed-product algebras, from which the STBCs obtained are information-lossless and also of full rank. We present some simulation results for two, three, and four transmit antennas to show that our STBCs perform better than some of the best known STBCs and also that these STBCs are approximately 1 dB away from the capacity of the channel with quadrature amplitude modulation (QAM) symbols as input

Low-complexity, full-diversity space-time-frequency block codes for MIMO-OFDM

We present a new class of space-time-frequency block codes (STFBC) for multiantenna orthogonal frequency division multiplexing (MIMO-OFDM) transmissions over frequency selective Rayleigh fading channels. We show that these codes admit symbol-by-symbol decoding (decoupled decoding) when the number of nonzero taps of the channel impulse response is equal to two and they admit reduced complexity (1/2 of that of known schemes) for more than two channel taps. We also present simulation results to show that our codes perform better than the known codes.

STBCs using capacity achieving designs from cyclic division algebras

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that only for the case of two transmit and one receive antennas. M.O. Damen et al. (see IEEE Trans. Inform. Theory, vol.48, no.3, p.753-60, 2002) gave a design for 2 transmit antennas, which achieves capacity for any number of receive antennas, calling it an information lossless STBC. We construct capacity achieving designs using cyclic division algebras for an arbitrary number of transmit and receive antennas. For the STBCs obtained using these designs, we present simulation results for those numbers of transmit and receive antennas for which Damen et al. also gave results, and show that our STBCs perform better than theirs.

Information-lossless STBCs from crossed-product algebras

This work presents the construction of STBCs, using crossed-product algebras, for arbitrary number of transmit antennas over an a priori specified signal set. It obtains a condition under which these STBCs from arbitrary crossed-product algebras are information-lossless.

High-rate, full-diversity STBCs from field extensions

A construction of high-rate codes is presented from field extensions that are full-rank also. Also, we discuss the coding gain and decoding of these codes.

STBCs using capacity achieving designs from crossed-product division algebras

We construct full-rank, rate-n space-time block codes (STBC), over any a priori specified signal set for n-transmit antennas using crossed-product division algebras and give a sufficient condition for these STBCs to be information lossless. A class of division algebras for which this sufficient condition is satisfied is identified. Simulation results are presented to show that STBCs constructed in this paper perform better than the best known codes, including those constructed from cyclic division algebras and also to show that they are very close to the capacity of the channel with QAM input.

Asymptotic-information-lossless designs and diversity-multiplexing tradeoff

It is known that neither the Alamouti nor the V-BLAST scheme achieves the Zheng-Tse diversity-multiplexing tradeoff (DMT) of the multiple-input multiple-output (MIMO) channel. With respect to the DMT curve, the Alamouti scheme achieves the point corresponding to maximum diversity gain only, whereas V-BLAST meets only the point corresponding to maximum multiplexing gain. It is also known that D-BLAST achieves the optimal DMT for n transmit and n receive antennas, but only under the assumption that the leading and trailing zeros are ignored. When these zeros are taken into account, D-BLAST achieves the point corresponding to zero multiplexing gain, but not the point corresponding to zero diversity gain. The first scheme to achieve the DMT is the coding scheme of Yao and Wornell for the case of two transmit and two receive antennas. In this paper, we introduce the notion of an asymptotic-information-lossless (AILL) design and obtain a necessary and sufficient condition under which a design is AILL. Analogous to the result that full-rank designs achieve the point corresponding to the zero multiplexing gain of the optimal DMT curve, we show AILL to be a necessary and sufficient condition for a design to achieve the point on the DMT curve corresponding to zero diversity gain. We also derive a lower bound on the tradeoff achieved by designs from field extensions and show that the tradeoff is very close to the optimal tradeoff in the case of a single receive antenna. A lower bound to the tradeoff achieved by designs from division algebras is presented which indicates that these designs achieve both extreme points (corresponding to zero diversity and zero multiplexing gain) of the optimal DMT curve. Finally, we present simulations results for n transmit and n receive antennas, for n=2,3,4, which suggest that designs from division algebras are likely to have the property of being DMT achieving.

Full-diversity group space-time-frequency (GSTF) codes from cyclic codes

It is known that multi-antenna transmissions over frequency-selective channels can provide a diversity gain that is product of the number of transmit antennas, the receive antennas and the length of the channel impulse response. Liu, Xin and Giannakis have studied multi-antenna orthogonal frequency division multiplexing (OFDM) through frequency-selective Rayleigh-fading channels and have introduced the concept of space-time frequency (STF) coding to enable maximum diversity and high coding gains. It is known that under some conditions, an n-length cyclic code C over Fqm (n|qm - 1, and m les n) can have fullrank i.e Rankq(C) = m. Designs for space-time codes suitable for both quasi-static fading channels and block-fading channels have been derived from n length cyclic codes over Fqm. In this paper, we present a simplified design of STF codes using designs derived from cyclic codes to obtain group space-time-frequency (GSTF) codes for frequency selective Rayleigh fading channels. These codes achieve maximum diversity gain

Full-diversity STBCs for block-fading channels from cyclic codes

Viewing an n-length vector over F(q/sup m/) (the finite field of q/sup m/ elements) as an m/spl times/n matrix over F/sub q/, by expanding each entry of the vector with respect to a basis of F(q/sup m/) over F/sub q/, the rank weight of the n-length vector over F(q/sup m/) is the rank of the corresponding m/spl times/n matrix over F/sub q/. Using the appropriate discrete Fourier transform (DFT), it is known that, under some conditions, n-length cyclic codes over F(q/sup m/), (n|q/sup m/-1 and m/spl les/n), have full-rank (=m). Using this result, we obtain designs for full-diversity space time block codes (STBCs) suitable for block-fading channels from n length cyclic codes over F(q/sup m/). These STBCs are suitable for m transmit antennas over signal sets matched to F/sub q/, where q=2 or q is a prime of the form 4k+1, (k=1, 2, ...). We also present simulation results which illustrate the performance of a few of these STBCs and show that our codes perform better than the well known codes for block-fading channels.