Space–time block code: Difference between revisions

Revision as of 15:16, 13 December 2019

Revision as of 09:43, 17 August 2020

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</math>

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 &mdash; it is the only way to achieve orthogonality. One particular problem with <math>C_{3,3/4}</math> is that it has uneven power among the symbols it transmits. This means that the signal does not have a [[constant envelope|constant]] [[envelope detector|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====

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:<math>r_t^j = \sum_{i=1}^{n_T}\alpha_{ij}s_t^i + n_t^j,</math>

where <math>\alpha_{ij}</math> is the path gain from transmit antenna <math>i</math> to receive antenna <math>j</math>, <math>s_t^i</math> is the signal transmitted by transmit antenna <math>i</math> and <math>n_t^j</math> is a sample of [[additive white Gaussian noise|additive]] [[white noise|white]] [[Gaussian noise]] ([[AWGN]]).

The maximum-likelihood detection rule<ref name="perform" /> is to form the decision variables

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: <math>r_\max = \frac{n_0 + 1}{2n_0},</math>

where <math>n_T = 2n_0</math> or <math>n_T = 2n_0 - 1</math>, if no linear processing is allowed in the code matrix (the above maximal rate proved in <ref name="COD"/> only applies to the original definition of orthogonal designs, i.e., any entry in the matrix is <math>0, c_i, -c_i, c_i^*,</math>, or <math>-c_i^*</math>, which forces that any variable <math>c_i</math> 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.<ref name="bounds" /> Closed-form recursive designs have been found.<ref>{{cite journal|author1=Kejie Lu |author2=Shengli Fu |author3=Xiang-Gen Xia |last-author-amp=yes |title=Closed-Form Designs of Complex Orthogonal Space-Time Block Codes of Rates (k+1)/(2k) for 2k-1 or 2k Transmit Antennas|journal=IEEE Transactions on Information Theory|pages=4340–4347|volume=51|issue=12|date=December 2005|doi=10.1109/TIT.2005.858943}}</ref>

==Quasi-orthogonal STBCs==