Space–time block code
Encyclopedia
Space–time block coding is a technique used in wireless communications
to transmit multiple copies of a data stream across a number of antenna
s 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
, reflection
, refraction
and so on and may then be further corrupted by thermal noise in the receiver
means that some of the received copies of the data will be 'better' 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
combines all the copies of the received signal in an optimal way to extract as much information from each of them as possible.
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
, 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
s were utilised by Siavash Alamouti
, and later Vahid Tarokh
, Hamid Jafarkhani
and Robert Calderbank
to develop space–time block-codes (STBCs). STC involves the transmission of multiple redundant copies of data to compensate for fading
and thermal noise 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 and is what was largely studied until Foschini's 1998 paper.
An STBC is usually represented by a matrix
. Each row represents a time slot and each column represents one antenna's transmissions over time.
Here,
is the modulated
symbol to be transmitted in time slot
from antenna 
. There are to be 
time slots and 
transmit antennas as well as 
receive antennas. This block is usually considered to be of 'length' 
The code rate
of an STBC measures how many symbols per time slot it transmits on average over the course of one block. If a block encodes
symbols, the code-rate is
.
Only one standard STBC can achieve full-rate (rate 1) — Alamouti's code.
, optimal
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.
s. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.
and call an erroneously decoded received codeword
.
Then the matrix
has to be full-rank
for any pair of distinct codewords
and 
to give the maximum possible diversity order of 
. If instead, 
has minimum rank 
over the set of pairs of distinct codewords, then the space–time code offers diversity order 
. An examination of the example STBCs shown below 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
s which provide both diversity and coding gain since they spread a conventional trellis code over space and time.
,
where * denotes complex conjugate
.
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 (BER) of this STBC is equivalent to
-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 
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
modulation symbols. Since almost all constellation diagram
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
required processing schemes which scaled exponentially with the number of transmit antennas. Furthermore, it was the first open-loop
transmit diversity
technique which had this capability. Subsequent generalizations of Alamouti's concept have led to a tremendous impact on the wireless communications industry.
assumption.
.
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
is that it has uneven power among the symbols it transmits. This means that the signal does not have a constant envelope
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.
.
These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts.
exhibits the same uneven power problems as 
. An improved version of 
is
,
which has equal power from all antennas in all time-slots.
decoding can be achieved at the receiver with only linear
processing. In order to consider a decoding method, a model of the wireless communications system is needed.
At time
, the signal 
received at antenna 
is:
,
where
is the path gain from transmit antenna 
to receive antenna 
, 
is the signal transmitted by transmit antenna 
and 
is a sample of additive
white
Gaussian noise
(AWGN).
The maximum-likelihood detection rule is to form the decision variables
where
is the sign of 
in the 
th row of the coding matrix, 
denotes that 
is (up to a sign difference), the 
element of the coding matrix,
for
...
and then decide on constellation symbol
that satisfies
,
with
the constellation alphabet
. Despite its appearance, this is a simple, linear decoding scheme that provides maximal diversity.
, 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
-antenna code can achieve is
,
where
or 
, if no linear processing is allowed in the code matrix. This rate limit is conjectured to be held even when linear processing is allowed. Closed-form recursive designs have been found.
is:
.
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
s (SNRs). At high SNRs, though (above about 22dB 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.
Wireless
Wireless telecommunications is the transfer of information between two or more points that are not physically connected. Distances can be short, such as a few meters for television remote control, or as far as thousands or even millions of kilometers for deep-space radio communications...
to transmit multiple copies of a data stream across a number of antenna
Antenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
s 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
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
, reflection
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...
, refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
and so on and may then be further corrupted by thermal noise in the receiver
Receiver (radio)
A radio receiver converts signals from a radio antenna to a usable form. It uses electronic filters to separate a wanted radio frequency signal from all other signals, the electronic amplifier increases the level suitable for further processing, and finally recovers the desired information through...
means that some of the received copies of the data will be 'better' 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
Space–time code
A space–time code is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas...
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 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
Vahid Tarokh
Vahid Tarokh is an electrical engineer with contributions to telecommunication, specifically to signal processing for wireless communications.-Life:...
, Nambi Seshadri and Robert Calderbank
Robert Calderbank
A. Robert Calderbank is the dean of Natural Sciences and professor of Computer Science, Electrical Engineering, and Mathematics at Duke University, and a professor of Electrical Engineering, Mathematics and Applied and Computational Mathematics at Princeton University...
, these space–time codes(STCs) achieve significant error rate improvements over single-antenna systems. Their original scheme was based on trellis codes
Convolutional code
In telecommunication, a convolutional code is a type of error-correcting code in which* each m-bit information symbol to be encoded is transformed into an n-bit symbol, where m/n is the code rate and...
but the simpler block code
Block code
In coding theory, block codes refers to the large and important family of 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...
s were utilised by Siavash Alamouti
Siavash Alamouti
Siavash Alamouti is an Iranian-American engineer who is best known for the invention of the so-called Alamouti space–time block code, filed in 1997 and patented jointly with Vahid Tarokh. Alamouti's code is a 2 transmit antenna space-time block code and has been adopted in various global standards...
, and later Vahid Tarokh
Vahid Tarokh
Vahid Tarokh is an electrical engineer with contributions to telecommunication, specifically to signal processing for wireless communications.-Life:...
, Hamid Jafarkhani
Hamid Jafarkhani
Hamid Jafarkhani , born in 1966 in Tehran, is a Chancellor's Professor in electrical engineering and computer science at the University of California, Irvine's Henry Samueli School of Engineering...
and Robert Calderbank
Robert Calderbank
A. Robert Calderbank is the dean of Natural Sciences and professor of Computer Science, Electrical Engineering, and Mathematics at Duke University, and a professor of Electrical Engineering, Mathematics and Applied and Computational Mathematics at Princeton University...
to develop space–time block-codes (STBCs). STC involves the transmission of multiple redundant copies of data to compensate for fading
Fading
In wireless communications, fading is deviation of the attenuation that a carrier-modulated telecommunication signal experiences over certain propagation media. The fading may vary with time, geographical position and/or radio frequency, and is often modelled as a random process. A fading channel...
and thermal noise 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
Block code
In coding theory, block codes refers to the large and important family of 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...
, 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 and is what was largely studied until Foschini's 1998 paper.
An STBC is usually represented by a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. Each row represents a time slot and each column represents one antenna's transmissions over time.
Here,
is the modulatedModulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
symbol to be transmitted in time slot
from antenna . There are to be time slots and transmit antennas as well as receive antennas. This block is usually considered to be of 'length' The code rate
Code rate
In telecommunication and information theory, the code rate of a forward error correction code is the proportion of the data-stream that is useful...
of an STBC measures how many symbols per time slot it transmits on average over the course of one block. If a block encodes
symbols, the code-rate is.Only one standard STBC can achieve full-rate (rate 1) — Alamouti's code.
Orthogonality
STBCs as originally introduced, and as usually studied, are orthogonal. This means that the STBC is designed such that the vectors representing any pair of columns taken from the coding matrix is orthogonal. The result of this is simple, linearLinear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
, optimal
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
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 codeSpace–time trellis code
Space–time trellis codes are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a trellis code distributed over time and a number of antennas . These multiple, 'diverse' copies of the data are used by the receiver to...
s. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.
Diversity criterion
Call a codewordand call an erroneously decoded received codeword
.Then the matrix
has to be full-rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
for any pair of distinct codewords
and to give the maximum possible diversity order of . If instead, has minimum rank over the set of pairs of distinct codewords, then the space–time code offers diversity order . An examination of the example STBCs shown below 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 code
Space–time trellis codes are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a trellis code distributed over time and a number of antennas . These multiple, 'diverse' copies of the data are used by the receiver to...
s which provide both diversity and coding gain since they spread a conventional trellis code over space and time.
Alamouti's code
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:,where * denotes complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
.
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 (BER) of this STBC is equivalent to
-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 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 number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
modulation symbols. Since almost all constellation diagram
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 scatter diagram in the complex plane at symbol sampling instants...
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
Transmit diversity is radio communication using signals that originate from two or more independent sources that have been modulated with identical information-bearing signals and that may vary in their transmission characteristics at any given instant....
required processing schemes which scaled exponentially with the number of transmit antennas. Furthermore, it was the first open-loop
Open-loop
Open-loop may refer to:*Open-loop controller of a dynamical system*Open-loop model in game theory*Open loop rhetorical device...
transmit diversity
Transmit diversity
Transmit diversity is radio communication using signals that originate from two or more independent sources that have been modulated with identical information-bearing signals and that may vary in their transmission characteristics at any given instant....
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 informationChannel state information
In wireless communications, channel state information refers to 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 example, scattering, fading, and power decay with...
assumption.
3 transmit antennas
Two straightforward codes for 3 transmit antennas are:.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
is that it has uneven power among the symbols it transmits. This means that the signal does not have a constant envelopeEnvelope detector
An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the "envelope" of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls...
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:.These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts.
exhibits the same uneven power problems as . An improved version of is,which has equal power from all antennas in all time-slots.
Decoding
One particularly attractive feature of orthogonal STBCs is that maximum likelihoodMaximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
decoding can be achieved at the receiver with only linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
processing. In order to consider a decoding method, a model of the wireless communications system is needed.
At time
, the signal received at antenna is:,where
is the path gain from transmit antenna to receive antenna , is the signal transmitted by transmit antenna and is a sample of additiveAdditive white Gaussian noise
Additive white Gaussian noise is a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density and a Gaussian distribution of amplitude. The model does not account for fading, frequency selectivity, interference,...
white
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
Gaussian noise
Gaussian noise
Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...
(AWGN).
The maximum-likelihood detection rule is to form the decision variables
where
is the sign of in the th row of the coding matrix, denotes that is (up to a sign difference), the element of the coding matrix,for
... and then decide on constellation symbolConstellation 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 scatter diagram in the complex plane at symbol sampling instants...
that satisfies,with
the constellation alphabetConstellation 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 scatter diagram in the complex plane at symbol sampling instants...
. 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 three 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,, 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
-antenna code can achieve is,where
or , if no linear processing is allowed in the code matrix. This rate limit is conjectured to be held even when linear processing is allowed. Closed-form recursive designs have been found.Quasi-orthogonal STBCs
These codes exhibit partial orthogonality and provide only part of the diversity gain mentioned above. An example reported by Hamid JafarkhaniHamid Jafarkhani
Hamid Jafarkhani , born in 1966 in Tehran, is a Chancellor's Professor in electrical engineering and computer science at the University of California, Irvine's Henry Samueli School of Engineering...
is:
.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
Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
s (SNRs). At high SNRs, though (above about 22dB 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 (MIMO)
- Space-time block coding based transmit diversitySpace-time block coding based transmit diversitySpace-time block coding based transmit diversity 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...
(STTD) - Space–time codeSpace–time codeA space–time code is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas...
- Space–time trellis codeSpace–time trellis codeSpace–time trellis codes are a type of space–time code used in multiple-antenna wireless communications. This scheme transmits multiple, redundant copies of a trellis code distributed over time and a number of antennas . These multiple, 'diverse' copies of the data are used by the receiver to...
- Differential space–time codeDifferential space–time codeDifferential 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 usually based on space–time block codes, and transmit...