Space–time block code

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.

[1] Seminal papers by Gerard J. Foschini and Michael J. Gans,[2] Foschini[3] and Emre Telatar[4] 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.

Proposed by Vahid Tarokh, Nambi Seshadri and Robert Calderbank, these space–time codes[5] (STCs) achieve significant error rate improvements over single-antenna systems.

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.

The code rate of an STBC measures how many symbols per time slot it transmits on average over the course of one block.

symbols, the code-rate is Only one standard STBC can achieve full-rate (rate 1) — Alamouti's code.

This means that the STBC is designed such that the vectors representing any pair of columns taken from the coding matrix is orthogonal.

Moreover, there exist quasi-orthogonal STBCs that achieve higher data rates at the cost of inter-symbol interference (ISI).

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.

An examination of the example STBCs shown below reveals that they all satisfy this criterion for maximum diversity.

There is no coding scheme included here — the redundancy purely provides diversity in space and time.

Siavash Alamouti invented the simplest of all the STBCs in 1998,[6] 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.

Using the optimal decoding scheme discussed below, the bit-error rate (BER) of this STBC is equivalent to

[5] That is to say that it is the only STBC that can achieve its full diversity gain without needing to sacrifice its data rate.

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.

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.

Subsequent generalizations of Alamouti's concept have led to a tremendous impact on the wireless communications industry.

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.

These two matrices give examples of why codes for more than two antennas must sacrifice rate — it is the only way to achieve orthogonality.

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.

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.

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,

[14] These codes exhibit partial orthogonality and provide only part of the diversity gain mentioned above.

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.