Spacetime block code (STBC)

Spacetime block code (STBC)[1–4]

Description

In a space-time block code, \(n\) transmitting antennas send symbols to \(m\) receiving antennas over \(T\) time slots. These symbols can be arranged in a \(n \times T\) matrix where the rows correspond to the channels, and the columns correspond to the time slots. The codewords \(\{X\}\) making up the code are thus \(n \times T\) matrices.

Protection

Provides protection against errors due to thermal noise and destructive interference arising from traversing an environment with scattering, reflection, and/or refraction.

Transmission occurs from \(n\) transmitting antennas to \(m\) receiving antennas over \(T\) time slots. The typical noise model, a fading channel [5], multiplies an incoming codeword by an \(m \times n\) fading matrix or damping matrix \(H\) and adds random (typically Gaussian) noise using an \(m \times T\) matrix \(Z\), \begin{align} X\to HX+Z = Y~. \tag*{(1)}\end{align} If \(H\) is known (unknown) to the receiver, then the receiver is called coherent (incoherent).

Decoding corresponds to choosing the codeword \(X\) that, when transformed under the channel, is closest to the corrupted output \(Y\) in the Frobenius norm, \begin{align} \min_{X\in C}\|Y-HX\|^{2}~. \tag*{(2)}\end{align}

Rate

If a codeword encodes \(k\) information symbols over \(T\) channel uses, then its rate is \(k/T\) symbols per channel use. A spacetime block code is said to be full rate when this equals the number \(n\) of transmit antennas, i.e., when \(k=nT\).

Realizations

High data-rate wireless communication, e.g., WiMAX (IEEE 802.16m) [6–8].

Member of code lists

Primary Hierarchy

Parents

Spacetime codes also use spatial and temporal diversity, but do not necessarily use blocks as codewords.

Spacetime block code (STBC)

Children

Linear STC

Orthogonal Spacetime Block Code (OSTBC)

STBCs whose codewords are orthogonal or unitary matrices are OSTBCs.

Multi-channel group-orbit code

References

[1]: B. L. Hughes, “Differential space-time modulation”, IEEE Transactions on Information Theory 46, 2567 (2000) DOI
[2]: B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading”, IEEE Transactions on Information Theory 46, 543 (2000) DOI
[3]: B. M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens, and R. Urbanke, “Systematic design of unitary space-time constellations”, IEEE Transactions on Information Theory 46, 1962 (2000) DOI
[4]: B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation”, IEEE Transactions on Communications 48, 2041 (2000) DOI
[5]: E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects”, IEEE Transactions on Information Theory 44, 2619 (1998) DOI
[6]: R. Vidhya Lavanya and M. Madheswaran, “Wimax (IEEE 802.16 m) system based on space time block code and discrete multiwavelet transform and implementation in FPGA”, Telecommunication Systems 56, 327 (2013) DOI
[7]: S. P. Alex and L. M. A. Jalloul, “Performance Evaluation of MIMO in IEEE802.16e/WiMAX”, IEEE Journal of Selected Topics in Signal Processing 2, 181 (2008) DOI
[8]: “Advances in smart antennas - MIMO-OFDM wireless systems: basics, perspectives, and challenges”, IEEE Wireless Communications 13, 31 (2006) DOI

Page edit log

Richard Barney (2022-04-05) — most recent

Cite as:

“Spacetime block code (STBC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/spacetime_block

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/spacetime/spacetime_block.yml.