Space-Time block codes from orthogonal designs
We introduce space-time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. Maximum-likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum-likelihood decoding algorithm which is based only on linear processing at the receiver. Space-time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm. The classical mathematical framework of orthogonal designs is applied to construct space-time block codes. It is shown that space-time block codes constructed in this way only exist for few sporadic values of n. Subsequently, a generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas. These codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as PAM. For an arbitrary complex constellation such as PSK and QAM, space-time block codes are designed that achieve 1/2 of the maximum possible transmission rate for any number of transmit antennas. For the specific cases of two, three, and four transmit antennas, space-time block codes are designed that achieve, respectively, all, 3/4, and 3/4 of maximum possible transmission rate using arbitrary complex constellations. The best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well.
Space-time block coding for wireless communications: performance results
We document the performance of space-time block codes, which provide a new paradigm for transmission over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code, and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. Maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver. We review the encoding and decoding algorithms for various codes and provide simulation results demonstrating their performance. It is shown that using multiple transmit antennas and space-time block coding provides remarkable performance at the expense of almost no extra processing.
A quasi-orthogonal space-time block code
It has been shown that a complex orthogonal design that provides full diversity and full transmission rate for a space-time block code is not possible for more than two antennas. Previous attempts have been concentrated in generalizing orthogonal designs which provide space-time block codes with full diversity and a high transmission rate. We design rate one codes which are quasi-orthogonal and provide partial diversity. The decoder of the proposed codes works with pairs of transmitted symbols instead of single symbols.
Minimal non-orthogonality rate 1 space-time block code for 3+ Tx antennas
We propose a full rate space-time block code for 3+ Tx antennas. The code is chosen to minimize the non-orthonormality that arises from increasing the rate above the maximum allowed by orthogonality. A linear decoding based on iterative interference cancellation between parts of the code approaches the maximal likelihood decoding performance.
Space-Time block codes: A maximum SNR approach
In Tarokh et al. (1999) space-time block codes were introduced to obtain coded diversity for a multiple-antenna communication system, in this work, we cast space-time codes in an optimal signal-to-noise ratio (SNR) framework and show that they achieve the maximum SNR and, in fact, they correspond to a generalized maximal ratio combiner. The maximum SNR framework also helps in calculating the distribution of the SNR and in deriving explicit expressions for bit error rates. We bring out the connection between the theory of amicable orthogonal designs and space-time codes. Based on this, we give a much simpler proof to one of the main theorems on space-time codes for complex symbols. We present a rate 1/2 code for complex symbols which has a smaller delay than the code already known. We also present another rate 3/4 code which is simpler than the one already known, in the sense it does not involve additions or multiplications. We also point out the connection between generalized real designs and generalized orthogonal designs.
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. Alamouti's (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.
A quasi-orthogonal space-time block code
It has been shown that a complex orthogonal design which provides full diversity and full transmission rate for a space-time block code is not possible for more than two antennas. Previous attempts have been concentrated in generalizing orthogonal designs which provide space-time block codes with full diversity and a high transmission rate. In this work, we design rate one codes which are quasi-orthogonal and provide partial diversity. The decoder of the proposed codes works with pairs of transmitted symbols instead of single symbols.
Space-time block codes: a capacity perspective
Space-time block codes are a remarkable modulation scheme discovered recently for the multiple antenna wireless channel. They have an elegant mathematical solution for providing full diversity over the coherent, flat-fading channel. In addition, they require extremely simple encoding and decoding. Although these codes provide full diversity at low computational costs, we show that they incur a loss in capacity because they convert the matrix channel into a scalar AWGN channel whose capacity is smaller than the true channel capacity. In this letter the loss in capacity is quantified as a function of channel rank, code rate, and number of receive antennas.
Square-matrix embeddable space-time block codes for complex signal constellations
Space-time block codes for providing transmit diversity in wireless communication systems are considered. Based on the principles of linearity and unitarity, a complete classification of linear codes is given in the case when the symbol constellations are complex, and the code is based on a square matrix or restriction of such by deleting columns (antennas). Maximal rate delay optimal codes are constructed within this category. The maximal rates allowed by linearity and unitarity fall off exponentially with the number of transmit antennas.
Diagonal algebraic space-time block codes
We construct a new family of linear space-time (ST) block codes by the combination of rotated constellations and the Hadamard transform, and we prove them to achieve the full transmit diversity over a quasi-static or fast fading channels. The proposed codes transmit at a normalized rate of 1 symbol/s. When the number of transmit antennas n=1, 2, or n is a multiple of four, we spread a rotated version of the information symbol vector by the Hadamard transform and send it over n transmit antennas and n time periods; for other values of n, we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n /spl times/ n ST code matrix. The codes maintain their rate, diversity, and coding gains for all real and complex constellations carved from the complex integers ring Z [i], and they outperform the codes from orthogonal design when using complex constellations for n > 2. The maximum-likelihood (ML) decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity. It is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity.
Applications of space-time block codes and interference suppression for high capacity and high data rate wireless systems
This paper presents a combined interference suppression and ML decoding scheme for space-time block codes that can effectively suppress interference from other co-channel users while providing each user with a diversity benefit. We consider a multiuser environment with K synchronous co-channel users, each is equipped with N transmit antennas and uses the space-time block coding. By exploiting the temporal and spatial structure of these codes, we develop a minimum mean-squared error (MMSE) interference suppression technique. Assuming that the receiver uses M/spl ges/K receive antennas, these technique will perfectly suppress the interference from the K-1 co-channel space-time users and provide a diversity order of N/spl times/(M-K+1) to each of the K users. Moreover, this MMSE solution tends itself to an adaptive implementation and does not require any explicit knowledge about the interference. In conjunction with this interference suppression technique, we show how space-time block codes can be used to increasing the capacity and/or data rate of wireless communication systems.
Limited feedback unitary precoding for orthogonal space-time block codes
Orthogonal space-time block codes (OSTBCs) are a class of easily decoded space-time codes that achieve full diversity order in Rayleigh fading channels. OSTBCs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information. When channel state information is available at the transmitter, though, precoding the space-time codeword can be used to support different numbers of transmit antennas and to improve array gain. Unfortunately, transmitters in many wireless systems have no knowledge about current channel conditions. This motivates limited feedback precoding methods such as channel quantization or antenna subset selection. This paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver. The receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error-free, zero-delay feedback channel. A criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system. Low average distortion codebooks are derived based on the optimal codeword selection criterion. The resulting design is found to relate to the famous applied mathematics problem of subspace packing in the Grassmann manifold. Codebooks designed by this method are proven to provide full diversity order in Rayleigh fading channels. Monte Carlo simulations show that limited feedback precoding performs better than antenna subset selection.
Upper bounds of rates of complex orthogonal space-time block code
We derive some upper bounds of the rates of (generalized) complex orthogonal space-time block codes. We first present some new properties of complex orthogonal designs and then show that the rates of complex orthogonal space-time block codes for more than two transmit antennas are upper-bounded by 3/4. We show that the rates of generalized complex orthogonal space-time block codes for more than two transmit antennas are upper-bounded by 4/5, where the norms of column vectors may not be necessarily the same. We also present another upper bound under a certain condition. For a (generalized) complex orthogonal design, its variables are not restricted to any alphabet sets but are on the whole complex plane. A (generalized) complex orthogonal design with variables over some alphabet sets on the complex plane is also considered. We obtain a condition on the alphabet sets such that a (generalized) complex orthogonal design with variables over these alphabet sets is also a conventional (generalized) complex orthogonal design and, therefore, the above upper bounds on its rate also hold. We show that commonly used quadrature amplitude modulation (QAM) constellations of sizes above 4 satisfy this condition.
Performance analysis of space-time block codes over keyhole Nakagami-m fading channels
In multiple-input-multiple-output (MIMO) fading environments, degenerate channel phenomena, called keyholes or pinholes, may exist under the realistic assumption that the spatial fading is uncorrelated at the transmitter and the receiver, but the channel has a rank-deficient transfer matrix. In this paper, we analyze the exact average symbol error rate (SER) of orthogonal space-time block codes (STBCs) with M-PSK and M-QAM constellations over Nakagami-m fading channels in the presence of the keyhole. We derive the moment generating function (MGF) of instantaneous signal-to-noise ratio (SNR) after space-time block decoding (signal combining) in such channels. Using a well-known MGF-based analysis approach, we express the average SER of the STBC in the form of single finite-range integrals whose integrand contains only the derived MGF. Numerical results show that the keyhole significantly degrades the SER performance of the STBC from idealistic behaviors in independent identically distributed MIMO channels.
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