Generalized Complex Orthogonal Designs Research Articles

On the criteria for designing complex orthogonal space-time block codes

A complex orthogonal design (COD) used in space-time block codes is a kind of combinatorial design. It has been well studied because it has a fast maximum-likelihood decoding algorithm and achieves full diversity. When designing CODs, there are seven characteristics that should be considered, which include code rate, decoding delay, transceiver signal linearization, peak-to-average power ratio, etc. In this paper, we study the relationships among these design criteria. We prove that the maximum rate of generalized CODs (GCODs) that meet the last five criteria is 1/2. Moreover, we present a new method to construct GCODs based on CODs. Using this method on balanced complex orthogonal designs (BCODs), we obtain a novel class of GCODs that performs almost perfectly with respect to all the seven properties.

The Final Case of the Decoding Delay Problem for Maximum Rate Complex Orthogonal Designs

Complex orthogonal space-time block codes (COSTBCs) based on generalized complex orthogonal designs (CODs) have been successfully implemented in wireless systems with multiple transmit antennas and single or multiple receive antennas. It has been shown that for a maximum rate COD with 2m-1 or 2m columns, a lower bound on decoding delay is ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> ) and this delay is achievable when the number of columns is congruent to 0, 1 , or 3 modulo 4. In this paper, the final case is addressed, and it is shown that when the number of columns is congruent to 2 modulo 4, the lower bound on decoding delay cannot be achieved. In this case, the shortest decoding delay a maximum rate COD can achieve is twice the lower bound. New techniques for analyzing CODs are introduced with connections to binary vector spaces.

Quaternion orthogonal designs from complex companion designs

The success of applying generalized complex orthogonal designs as space–time block codes recently motivated the definition of quaternion orthogonal designs as potential building blocks for space–time-polarization block codes. This paper offers techniques for constructing quaternion orthogonal designs via combinations of specially chosen complex orthogonal designs. One technique is used to build quaternion orthogonal designs on complex variables for any even number of columns. A second related technique is applied to maximum rate complex orthogonal designs to generate an infinite family of quaternion orthogonal designs on complex variables such that the resulting designs have no zero entries. This second technique is also used to generate an infinite family of quaternion orthogonal designs defined over quaternion variables that display a regular redundancy. The proposed constructions are theoretically important because they provide the first known direct techniques for building infinite families of orthogonal designs over quaternion variables for any number of columns.

The Minimum Decoding Delay of Maximum Rate Complex Orthogonal Space–Time Block Codes

The growing demand for efficient wireless transmissions over fading channels motivated the development of space-time block codes. Space-time block codes built from generalized complex orthogonal designs are particularly attractive because the orthogonality permits a simple decoupled maximum-likelihood decoding algorithm while achieving full transmit diversity. The two main research problems for these complex orthogonal space-time block codes (COSTBCs) have been to determine for any number of antennas the maximum rate and the minimum decoding delay for a maximum rate code. The maximum rate for COSTBCs was determined by Liang in 2003. This paper addresses the second fundamental problem by providing a tight lower bound on the decoding delay for maximum rate codes. It is shown that for a maximum rate COSTBC for 2m - 1 or 2m antennas, a tight lower bound on decoding delay is r = ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> ) . This lower bound on decoding delay is achievable when the number of antennas is congruent to 0, 1, or 3 modulo 4. This paper also derives a tight lower bound on the number of variables required to construct a maximum rate COSTBC for any given number of antennas. Furthermore, it is shown that if a maximum rate COSTBC has a decoding delay of r where r < r les 2r, then r=2r. This is used to provide evidence that when the number of antennas is congruent to 2 modulo 4, the best achievable decoding delay is 2( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m_</sup> ).

Performance of Orthogonal Space-Time Block Codes With Generalized Complex Orthogonal Design for Arbitrary Rectangular QAM Over Correlative Fading Channels

We derive the symbol error probability (SEP) expressions for orthogonal space-time block codes with any form of generalized complex orthogonal design (GCOD) employing arbitrary rectangular quadratic-amplitude modulation signaling over correlative fading channels. By first decorrelating the physical branches into uncorrelated virtual branches, a general expression for the moment generating function of the received signal-to noise ratio (SNR) is obtained, from which the SEP is derived. The result can be applied to correlative fading channels with any form of Gaussian fading gains. Moreover, channels having joint fading models, as well as mixed channel powers, are considered. We also discover that, for certain GCOD codes, different information symbols may exhibit different SEPs. The above features are demonstrated by two GCOD examples used for simulations. Theoretical performance curves are compared with Monte Carlo simulated results in excellent agreement.

A construction technique for generalized complex orthogonal designs and applications to wireless communications

We introduce a construction technique for generalized complex linear processing orthogonal designs, which are p × n matrices X satisfying X H X = fI, where f is a complex quadratic form, I is the identity matrix, and X has complex entries. These matrices generalize the familiar notions of orthogonal designs and generalized complex orthogonal designs. We explain the application of these matrices to space–time block coding for multiple-antenna wireless communications. In particular, we discuss the practical strengths of the space–time block codes constructed via our proposed technique.

On the nonexistence of rate-one generalized complex orthogonal designs

Orthogonal space-time block coding proposed recently by Alamouti (1998) and Tarokh et al. (1999) is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh et al. extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space-time block codes (STBC) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBC constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation.

On Space-Time Block Codes from Complex Orthogonal Designs

Space-time block codes from orthogonal designs recently proposed by Alamouti, and Tarokh-Jafarkhani-Calderbank have attracted considerable attention due to the fast maximum-likelihood (ML) decoding and the full diversity. There are two classes of space-time block codes from orthogonal designs. One class consists of those from real orthogonal designs for real signal constellations which have been well developed in the mathematics literature. The other class consists of those from complex orthogonal designs for complex constellations for high data rates, which are not well developed as the real orthogonal designs. Since orthogonal designs can be traced back to decades, if not centuries, ago and have recently invoked considerable interests in multi-antenna wireless communications, one of the goals of this paper is to provide a tutorial on both historical and most recent results on complex orthogonal designs. For space-time block codes from both real and (generalized) complex orthogonal designs (GCODs) with or without linear processing, Tarokh, Jafarkhani and Calderbank showed that their rates cannot be greater than 1. While the maximum rate 1 can be reached for real orthogonal designs for any number of transmit antennas from the Hurwitz–Radon constructive theory, Liang and Xia recently showed that rate 1 for the GCODs (square or non-square size) with linear processing is not reachable for more than two transmit antennas. For GCODs of square size, the designs with the maximum rates have been known, which are related to the Hurwitz theorem. In this paper, We briefly review these results and give a simple and intuitive interpretation of the realization. For GCODs without linear processing (square or non-square size), we prove that the rates cannot be greater than 3/4 for more than two transmit antennas.

Two generalized complex orthogonal space-time block codes of rates 7/11 and 3/5 for 5 and 6 transmit antennas

Space-time block codes from orthogonal designs have two advantages, namely, fast maximum-likelihood (ML) decoding and full diversity. Rate 1 real (pulse amplitude modulation-PAM) space-time codes (real orthogonal designs) for multiple transmit antennas have been constructed from the real Hurwitz-Radon families, which also provides the rate 1/2 complex (quadrature amplitude modulation-QAM) space-time codes (complex orthogonal designs) for any number of transmit antennas. Rate 3/4 complex orthogonal designs (space-time codes) for three and four transmit antennas have existed in the literature but no high rate (>1/2) complex orthogonal designs for other numbers of transmit antennas exist. We present rate 7/11 and rate 3/5 generalized complex orthogonal designs for five and six transmit antennas, respectively.