Space–time block code

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 codes. Orthogonal STBCs can be shown to achieve the maximum diversity allowed by this criterion.

Diversity criterion

Call a codeword

${\displaystyle \mathbf{c}=c_1^1{c}_1^2\cdots c_1^{n_T}{c}_2^1{c}_2^2\cdots c_2^{n_T}\cdots c_T^1{c}_T^2\cdots c_T^{n_T}}$

and call an erroneously decoded received codeword

${\displaystyle \mathbf{e}=e_1^1{e}_1^2\cdots e_1^{n_T}{e}_2^1{e}_2^2\cdots e_2^{n_T}\cdots e_T^1{e}_T^2\cdots e_T^{n_T}.}$

Then the matrix

${\displaystyle \mathbf{B}(\mathbf{c},\mathbf{e})=\left[\begin{array}{cccc}{e}_1^1-c_1^1& e_2^1-c_2^1& \cdots & e_T^1-c_T^1\\ e_1^2-c_1^2& e_2^2-c_2^2& \cdots & e_T^2-c_T^2\\ ⋮& ⋮& \ddots & ⋮\\ e_1^{n_T}-c_1^{n_T}& e_2^{n_T}-c_2^{n_T}& \cdots & e_T^{n_T}-c_T^{n_T}\end{array}\right]}$

has to be full-rank for any pair of distinct codewords ${\displaystyle \mathbf{c}}$ and ${\displaystyle \mathbf{e}}$ to give the maximum possible diversity order of ${\displaystyle n_T{n}_R}$. If instead, ${\displaystyle \mathbf{B}(\mathbf{c},\mathbf{e})}$ has minimum rank ${\displaystyle b}$ over the set of pairs of distinct codewords, then the space–time code offers diversity order ${\displaystyle b{n}_R}$. 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 codes 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:

${\displaystyle C_2=\left[\begin{array}{cc}{c}_1& c_2\\ -c_2^{\ast }& c_1^{\ast }\end{array}\right],}$

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 ${\displaystyle 2n_R}$-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 ${\displaystyle n_R}$ 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 modulation symbols. Since almost all constellation diagrams 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.

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 assumption.

3 transmit antennas

Two straightforward codes for 3 transmit antennas are:

${\displaystyle C_{3,1/2}=\left[\begin{array}{ccc}{c}_1& c_2& c_3\\ -c_2& c_1& -c_4\\ -c_3& c_4& c_1\\ -c_4& -c_3& c_2\\ c_1^{\ast }& c_2^{\ast }& c_3^{\ast }\\ -c_2^{\ast }& c_1^{\ast }& -c_4^{\ast }\\ -c_3^{\ast }& c_4^{\ast }& c_1^{\ast }\\ -c_4^{\ast }& -c_3^{\ast }& c_2^{\ast }\end{array}\right]\text{and}{C}_{3,3/4}=\left[\begin{array}{ccc}{c}_1& c_2& \frac{c_3}{\sqrt{2}}\\ -c_2^{\ast }& c_1^{\ast }& \frac{c_3}{\sqrt{2}}\\ \frac{c_3^{\ast }}{\sqrt{2}}& \frac{c_3^{\ast }}{\sqrt{2}}& \frac{\left(-c_1-c_1^{\ast }+c_2-c_2\ast \right)}{2}\\ \frac{c_3^{\ast }}{\sqrt{2}}& -\frac{c_3^{\ast }}{\sqrt{2}}& \frac{\left(c_2+c_2^{\ast }+c_1-c_1^{\ast }\right)}{2}.\end{array}\right]}$

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 ${\displaystyle C_{3,3/4}}$ 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.

4 transmit antennas

Two straightforward codes for 4 transmit antennas are:

${\displaystyle C_{4,1/2}=\left[\begin{array}{cccc}{c}_1& c_2& c_3& c_4\\ -c_2& c_1& -c_4& c_3\\ -c_3& c_4& c_1& -c_2\\ -c_4& -c_3& c_2& c_1\\ c_1^{\ast }& c_2^{\ast }& c_3^{\ast }& c_4^{\ast }\\ -c_2^{\ast }& c_1^{\ast }& -c_4^{\ast }& c_3^{\ast }\\ -c_3^{\ast }& c_4^{\ast }& c_1^{\ast }& -c_2^{\ast }\\ -c_4^{\ast }& -c_3^{\ast }& c_2^{\ast }& c_1^{\ast }\end{array}\right]\text{and}{C}_{4,3/4}=\left[\begin{array}{cccc}{c}_1& c_2& \frac{c_3}{\sqrt{2}}& \frac{c_3}{\sqrt{2}}\\ -c_2^{\ast }& c_1^{\ast }& \frac{c_3}{\sqrt{2}}& -\frac{c_3}{\sqrt{2}}\\ \frac{c_3^{\ast }}{\sqrt{2}}& \frac{c_3^{\ast }}{\sqrt{2}}& \frac{\left(-c_1-c_1^{\ast }+c_2-c_2^{\ast }\right)}{2}& \frac{\left(-c_2-c_2^{\ast }+c_1-c_1^{\ast }\right)}{2}\\ \frac{c_3^{\ast }}{\sqrt{2}}& -\frac{c_3^{\ast }}{\sqrt{2}}& \frac{\left(c_2+c_2^{\ast }+c_1-c_1^{\ast }\right)}{2}& -\frac{\left(c_1+c_1^{\ast }+c_2-c_2^{\ast }\right)}{2}\end{array}\right].}$

These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts. ${\displaystyle C_{4,3/4}}$ exhibits the same uneven power problems as ${\displaystyle C_{3,3/4}}$. An improved version of ${\displaystyle C_{4,3/4}}$ is

${\displaystyle C_{4,3/4}=\left[\begin{array}{cccc}{c}_1& c_2& c_3& 0\\ -c_2^{\ast }& c_1^{\ast }& 0& c_3\\ -c_3^{\ast }& 0& c_1^{\ast }& -c_2\\ 0& -c_3^{\ast }& c_2^{\ast }& c_1\end{array}\right],}$

which has equal power from all antennas in all time-slots.

Decoding

One particularly attractive feature of orthogonal STBCs is that maximum likelihood 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 ${\displaystyle t}$, the signal ${\displaystyle r_t^j}$ received at antenna ${\displaystyle j}$ is:

${\displaystyle r_t^j=\sum _{i=1}^{n_T}{\alpha }_{ij}{s}_t^i+n_t^j,}$

where ${\displaystyle {\alpha }_{ij}}$ is the path gain from transmit antenna ${\displaystyle i}$ to receive antenna ${\displaystyle j}$, ${\displaystyle s_t^i}$ is the signal transmitted by transmit antenna ${\displaystyle i}$ and ${\displaystyle n_t^j}$ is a sample of additive white Gaussian noise (AWGN).

The maximum-likelihood detection rule is to form the decision variables

${\displaystyle R_i=\sum _{t=1}^{n_T}\sum _{j=1}^{n_R}{r}_t^j{\alpha }_{{ϵ}_t(i)j}{\delta }_t(i)}$

where ${\displaystyle {\delta }_k(i)}$ is the sign of ${\displaystyle s_i}$ in the ${\displaystyle k}$^th row of the coding matrix, ${\displaystyle {ϵ}_k(p)=q}$ denotes that ${\displaystyle s_p}$ is (up to a sign difference), the ${\displaystyle (k,q)}$ element of the coding matrix, for ${\displaystyle i=1,2,\dots ,n_T}$ and then decide on constellation symbol ${\displaystyle s_i}$ that satisfies

${\displaystyle s_i=\arg \underset{s\in \mathcal{A}}{min}\left({\left|R_i-s\right|}^2+\left(-1+\sum _{k,l}^{}{\left|{\alpha }_{kl}\right|}^2\right){\left|s\right|}^2\right),}$

with ${\displaystyle \mathcal{A}}$ 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, ${\displaystyle T}$, 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 ${\displaystyle n_T}$-antenna code can achieve is

${\displaystyle r_{max}=\frac{n_0+1}{2n_0},}$

where ${\displaystyle n_T=2n_0}$ or ${\displaystyle n_T=2n_0-1}$, 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 ${\displaystyle 0,c_i,-c_i,c_i^{\ast },}$, or ${\displaystyle -c_i^{\ast }}$, which forces that any variable ${\displaystyle c_i}$ 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 above. An example reported by Hamid Jafarkhani is:

${\displaystyle C_{4,1}=\left[\begin{array}{cccc}{c}_1& c_2& c_3& c_4\\ -c_2^{\ast }& c_1^{\ast }& -c_4^{\ast }& c_3^{\ast }\\ -c_3^{\ast }& -c_4^{\ast }& c_1^{\ast }& c_2^{\ast }\\ c_4& -c_3& -c_2& c_1\end{array}\right].}$

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 ratios (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.