Space–time Block Code
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Space–time block coding is a technique used in
wireless communications Wireless communication (or just wireless, when the context allows) is the transfer of information (''telecommunication'') between two or more points without the use of an electrical conductor, optical fiber or other continuous guided mediu ...
to transmit multiple copies of a data stream across a number of antennas and to exploit the various received versions of the data to improve the reliability of data transfer. The fact that the transmitted signal must traverse a potentially difficult environment with
scattering In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
,
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, mirror-like reflection of waves from a surface *** Mirror image, a reflection in a mirror or in water ** Diffuse r ...
,
refraction In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commo ...
and so on and may then be further corrupted by
thermal noise A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
in the receiver means that some of the received copies of the data may be closer to the original signal than others. This redundancy results in a higher chance of being able to use one or more of the received copies to correctly decode the received signal. In fact,
space–time coding In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional continuum (measurement), continu ...
combines ''all'' the copies of the received signal in an optimal way to extract as much information from each of them as possible.


Introduction

Most work on wireless communications until the early 1990s had focused on having an antenna array at only one end of the wireless link — usually at the receiver. Seminal papers by Gerard J. Foschini and Michael J. Gans, Foschini and Emre Telatar enlarged the scope of wireless communication possibilities by showing that for the highly scattering environment, substantial capacity gains are enabled when antenna arrays are used at both ends of a link. An alternative approach to utilizing multiple antennas relies on having multiple transmit antennas and only optionally multiple receive antennas. Proposed by Vahid Tarokh, Nambi Seshadri and
Robert Calderbank Robert Calderbank (born 28 December 1954) is a professor of computer science, electrical engineering, and mathematics and director of the Information Initiative at Duke University. He received a BSc from Warwick University in 1975, an MSc f ...
, these space–time codes (STCs) achieve significant error rate improvements over single-antenna systems. Their original scheme was based on trellis codes but the simpler
block code In coding theory, block codes are a large and important family of Channel coding, error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. Th ...
s were utilised by Siavash Alamouti, and later Vahid Tarokh,
Hamid Jafarkhani Hamid Jafarkhani () (born 1966, in Tehran) is an Iranian-born American electrical engineer and professor. He serves as the Chancellor's Professor in electrical engineering and computer science in the Henry Samueli School of Engineering at the Uni ...
and
Robert Calderbank Robert Calderbank (born 28 December 1954) is a professor of computer science, electrical engineering, and mathematics and director of the Information Initiative at Duke University. He received a BSc from Warwick University in 1975, an MSc f ...
to develop space–time block-codes (STBCs). STC involves the transmission of multiple redundant copies of data to compensate for
fading In wireless communications, fading is the variation of signal attenuation over variables like time, geographical position, and radio frequency. Fading is often modeled as a random process. In wireless systems, fading may either be due to mul ...
and
thermal noise A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
in the hope that some of them may arrive at the receiver in a better state than others. In the case of STBC in particular, the data stream to be transmitted is encoded in blocks, which are distributed among spaced antennas and across time. While it is necessary to have multiple transmit antennas, it is not necessary to have multiple receive antennas, although to do so improves performance. This process of receiving diverse copies of the data is known as
diversity reception In telecommunications, a diversity scheme refers to a method for improving the reliability of a message signal by using two or more communication channels with different characteristics. Diversity is mainly used in radio communication and is a c ...
and is what was largely studied until Foschini's 1998 paper. An STBC is usually represented by a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
. Each row represents a time slot and each column represents one antenna's transmissions over time. : \text \begin \text\\ \left \downarrow \overrightarrow\right. \end Here, s_ is the
modulated Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...
symbol to be transmitted in time slot i from antenna j. There are to be T time slots and n_T transmit antennas as well as n_R receive antennas. This block is usually considered to be of 'length' T The
code rate In telecommunication and information theory, the code rate (or information rateHuffman, W. Cary, and Pless, Vera, ''Fundamentals of Error-Correcting Codes'', Cambridge, 2003.) of a forward error correction code is the proportion of the data-stre ...
of an STBC measures how many symbols per time slot it transmits on average over the course of one block. If a block encodes k symbols, the code-rate is : r = \frac. Only one standard STBC can achieve full-rate (rate 1) — Alamouti's code.


Orthogonality

STBCs as originally introduced, and as usually studied, are
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
. This means that the STBC is designed such that the
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s representing any pair of columns taken from the coding matrix is orthogonal. The result of this is simple,
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
,
optimal Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
decoding at the receiver. Its most serious disadvantage is that all but one of the codes that satisfy this criterion must sacrifice some proportion of their data rate (see Alamouti's code). Moreover, there exist quasi-orthogonal STBCs that achieve higher data rates at the cost of inter-symbol interference (ISI). Thus, their error-rate performance is lower bounded by the one of orthogonal rate 1 STBCs, that provide ISI free transmissions due to orthogonality.


Design of STBCs

The design of STBCs is based on the so-called diversity criterion derived by Tarokh et al. in their earlier paper on
space–time trellis code Space–time trellis codes (STTCs) are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a generalised TCM signal distributed over time and a number of antennas ('sp ...
s. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.


Diversity criterion

Call a codeword :\mathbf = c_1^1c_1^2\cdots c_1^c_2^1c_2^2\cdots c_2^\cdots c_T^1c_T^2\cdots c_T^ and call an erroneously decoded received codeword :\mathbf = e_1^1e_1^2\cdots e_1^e_2^1e_2^2\cdots e_2^\cdots e_T^1e_T^2\cdots e_T^. Then the matrix :\mathbf(\mathbf,\mathbf) = \begin e_1^1 - c_1^1 & e_2^1 - c_2^1 & \cdots & e_T^1 - c_T^1\\ e_1^2 - c_1^2 & e_2^2 - c_2^2 & \cdots & e_T^2 - c_T^2\\ \vdots & \vdots & \ddots & \vdots\\ e_1^ - c_1^ & e_2^ - c_2^ & \cdots & e_T^ - c_T^ \end has to be full-
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
for any pair of distinct codewords \mathbf and \mathbf to give the maximum possible diversity order of n_Tn_R. If instead, \mathbf(\mathbf,\mathbf) has minimum rank b over the set of pairs of distinct codewords, then the space–time code offers diversity order bn_R. An examination of the example STBCs shown
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...
reveals that they all satisfy this criterion for maximum diversity. STBCs offer only diversity gain (compared to single-antenna schemes) and not coding gain. There is no coding scheme included here — the redundancy purely provides diversity in space and time. This is contrast with
space–time trellis code Space–time trellis codes (STTCs) are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a generalised TCM signal distributed over time and a number of antennas ('sp ...
s which provide both diversity and coding gain since they spread a conventional trellis code over space and time.


Encoding


Alamouti's code

Siavash Alamouti invented the simplest of all the STBCs in 1998, although he did not coin the term "space–time block code" himself. It was designed for a two-transmit antenna system and has the coding matrix: :C_2 = \begin c_1 & c_2\\ -c_2^* & c_1^* \end, where * denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. It is readily apparent that this is a rate-1 code. It takes two time-slots to transmit two symbols. Using the optimal decoding scheme discussed below, the
bit-error rate In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors. The bit error rate ( ...
(BER) of this STBC is equivalent to 2n_R-branch maximal ratio combining (MRC). This is a result of the perfect orthogonality between the symbols after receive processing — there are two copies of each symbol transmitted and n_R copies received. This is a very special STBC. It is the only orthogonal STBC that achieves rate-1. That is to say that it is the only STBC that can achieve its full diversity gain without needing to sacrifice its data rate. Strictly, this is only true for
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
modulation symbols. Since almost all
constellation diagram A constellation diagram is a representation of a signal modulated by a digital modulation scheme such as quadrature amplitude modulation or phase-shift keying. It displays the signal as a two-dimensional ''xy''-plane scatter diagram in the co ...
s rely on complex numbers however, this property usually gives Alamouti's code a significant advantage over the higher-order STBCs even though they achieve a better error-rate performance. See ' Rate limits' for more detail. The significance of Alamouti's proposal in 1998 is that it was the first demonstration of a method of encoding which enables full diversity with ''linear'' processing at the receiver. Earlier proposals for
transmit diversity Transmit diversity is the use of multiple transmit antennas to achieve reliability in radio communication. It involves using signals that originate from two or more independent sources that have been modulated with identical information-bearing s ...
required processing schemes which scaled ''exponentially'' with the number of transmit antennas. Furthermore, it was the first
open-loop A control loop is the fundamental building block of control systems in general and industrial control systems in particular. It consists of the process sensor, the controller function, and the final control element (FCE) which controls the process ...
transmit diversity Transmit diversity is the use of multiple transmit antennas to achieve reliability in radio communication. It involves using signals that originate from two or more independent sources that have been modulated with identical information-bearing s ...
technique which had this capability. Subsequent generalizations of Alamouti's concept have led to a tremendous impact on the wireless communications industry.


Higher order STBCs

Tarokh et al. discovered a set of STBCs that are particularly straightforward, and coined the scheme's name. They also proved that no code for more than 2 transmit antennas could achieve full-rate. Their codes have since been improved upon (both by the original authors and by many others). Nevertheless, they serve as clear examples of why the rate cannot reach 1, and what other problems must be solved to produce 'good' STBCs. They also demonstrated the simple, linear decoding scheme that goes with their codes under perfect
channel state information In wireless communications, channel state information (CSI) is the known channel properties of a communication link. This information describes how a signal propagates from the transmitter to the receiver and represents the combined effect of, for ...
assumption.


3 transmit antennas

Two straightforward codes for 3 transmit antennas are: : C_= \begin c_1 & c_2 & c_3\\ -c_2 &c_1& -c_4\\ -c_3&c_4&c_1\\ -c_4&-c_3&c_2\\ c_1^* & c_2^*&c_3^*\\ -c_2^* &c_1^*& -c_4^*\\ -c_3^*&c_4^*&c_1^*\\ -c_4^*&-c_3^*&c_2^* \end \quad\text\quad C_= \begin c_1&c_2&\frac\\ -c_2^*&c_1^*&\frac\\ \frac&\frac&\frac\\ \frac&-\frac&\frac. \end These codes achieve rate-1/2 and rate-3/4 respectively. These two matrices give examples of why codes for more than two antennas must sacrifice rate — it is the only way to achieve orthogonality. One particular problem with C_ is that it has uneven power among the symbols it transmits. This means that the signal does not have a
constant envelope Constant envelope is achieved when a sinusoidal waveform reaches equilibrium in a specific system. This happens when negative feedback in a control system, such as in radio automatic gain control or when an amplifier reaches steady state. Ste ...
and that the power each antenna must transmit has to vary, both of which are undesirable. Modified versions of this code that overcome this problem have since been designed.


4 transmit antennas

Two straightforward codes for 4 transmit antennas are: : C_= \begin c_1 & c_2 & c_3&c_4\\ -c_2 &c_1&-c_4&c_3\\ -c_3&c_4&c_1&-c_2\\ -c_4&-c_3&c_2&c_1\\ c_1^* & c_2^*&c_3^*&c_4^*\\ -c_2^* &c_1^*&-c_4^*&c_3^*\\ -c_3^*&c_4^*&c_1^*&-c_2^*\\ -c_4^*&-c_3^*&c_2^*&c_1^* \end \quad\text\quad C_= \begin c_1&c_2&\frac&\frac\\ -c_2^*&c_1^*&\frac&-\frac\\ \frac&\frac&\frac&\frac\\ \frac&-\frac&\frac&-\frac \end. These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts. C_ exhibits the same uneven power problems as C_. An improved version of C_ is : C_= \begin c_1&c_2&c_3&0\\ -c_2^*&c_1^*&0&c_3\\ -c_3^*&0&c_1^*&-c_2\\ 0&-c_3^*&c_2^*&c_1 \end, which has equal power from all antennas in all time-slots.


Decoding

One particularly attractive feature of orthogonal STBCs is that
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
decoding can be achieved at the receiver with only
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
processing. In order to consider a decoding method, a model of the wireless communications system is needed. At time t, the signal r_t^j received at antenna j is: :r_t^j = \sum_^\alpha_s_t^i + n_t^j, where \alpha_ is the path gain from transmit antenna i to receive antenna j, s_t^i is the signal transmitted by transmit antenna i and n_t^j is a sample of
additive white Gaussian noise Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: * ''Additive'' because it is added to any nois ...
(
AWGN Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: * ''Additive'' because it is added to any nois ...
). The maximum-likelihood detection rule is to form the decision variables :R_i = \sum_^\sum_^r_t^j\alpha_\delta_t(i) where \delta_k(i) is the sign of s_i in the kth row of the coding matrix, \epsilon_k(p)=q denotes that s_p is (up to a sign difference), the (k,q) element of the coding matrix, for i=1,2,\ldots,n_T and then decide on constellation symbol s_i that satisfies :s_i = \arg\min_\left(\left, R_i - s\^2 + \left(-1 + \sum_^\left, \alpha_\^2\right)\left, s\^2\right), with \mathcal the constellation alphabet. Despite its appearance, this is a simple, linear decoding scheme that provides maximal diversity.


Rate limits

Apart from there being no full-rate, complex, orthogonal STBC for more than 2 antennas, it has been further shown that, for more than two antennas, the maximum possible rate is 3/4. Codes have been designed which achieve a good proportion of this, but they have very long block-length. This makes them unsuitable for practical use, because decoding cannot proceed until ''all'' transmissions in a block have been received, and so a longer block-length, T, results in a longer decoding delay. One particular example, for 16 transmit antennas, has rate-9/16 and a block length of 22 880 time-slots! It has been proven that the highest rate any n_T-antenna code can achieve is : r_\max = \frac, where n_T = 2n_0 or n_T = 2n_0 - 1, if no linear processing is allowed in the code matrix (the above maximal rate proved in only applies to the original definition of orthogonal designs, i.e., any entry in the matrix is 0, c_i, -c_i, c_i^*,, or -c_i^*, which forces that any variable c_i can not be repeated in any column of the matrix). This rate limit is conjectured to hold for any complex orthogonal space–time block codes even when any linear processing is allowed among the complex variables. Closed-form recursive designs have been found.


Quasi-orthogonal STBCs

These codes exhibit partial orthogonality and provide only part of the diversity gain mentioned
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. An example reported by
Hamid Jafarkhani Hamid Jafarkhani () (born 1966, in Tehran) is an Iranian-born American electrical engineer and professor. He serves as the Chancellor's Professor in electrical engineering and computer science in the Henry Samueli School of Engineering at the Uni ...
is: :C_ = \begin c_1 & c_2 & c_3 & c_4\\ -c_2^* & c_1^* & -c_4^* & c_3^*\\ -c_3^* & -c_4^* & c_1^* & c_2^*\\ c_4 & -c_3 & -c_2 & c_1 \end. The orthogonality criterion only holds for columns (1 and 2), (1 and 3), (2 and 4) and (3 and 4). Crucially, however, the code is full-rate and still only requires linear processing at the receiver, although decoding is slightly more complex than for orthogonal STBCs. Results show that this Q-STBC outperforms (in a bit-error rate sense) the fully orthogonal 4-antenna STBC over a good range of
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
s (SNRs). At high SNRs, though (above about 22 dB in this particular case), the increased diversity offered by orthogonal STBCs yields a better BER. Beyond this point, the relative merits of the schemes have to be considered in terms of useful data throughput. Q-STBCs have also been developed considerably from the basic example shown.


See also

*
Multiple-input and multiple-output In radio, multiple-input and multiple-output (MIMO) () is a method for multiplying the capacity of a radio link using multiple transmission and receiving antenna (radio), antennas to exploit multipath propagation. MIMO has become an essential e ...
(MIMO) *
Space–time block coding based transmit diversity Space–time block coding based transmit diversity (STTD) is a method of transmit diversity used in UMTS third-generation cellular systems. STTD is optional in the UTRAN air interface but mandatory for user equipment ( UE). STTD utilizes spaceâ ...
(STTD) *
Space–time code A space–time code (STC) is a method employed to improve the reliability of data transmission (telecommunications), transmission in wireless, wireless communication systems using multiple transmit antenna (radio), antennas. STCs rely on trans ...
*
Space–time trellis code Space–time trellis codes (STTCs) are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a generalised TCM signal distributed over time and a number of antennas ('sp ...
*
Differential space–time code Differential space–time codes are ways of transmitting data in wireless communications. They are forms of space–time code that do not need to know the channel impairments at the receiver in order to be able to decode the signal. They are ...


References

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