Space–time block code: Difference between revisions

Revision as of 11:25, 20 April 2013

Revision as of 17:00, 15 September 2013

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===Diversity criterion===

Call a codeword

:<math>\mathbf{c} = c_1^1c_1^2...\cdots c_1^{n_T}c_2^1c_2^2...\cdots c_2^{n_T}...\cdots c_T^1c_T^2...\cdots c_T^{n_T}</math>

and call an erroneously decoded received codeword

:<math>\mathbf{e} = e_1^1e_1^2...\cdots e_1^{n_T}e_2^1e_2^2...\cdots e_2^{n_T}...\cdots e_T^1e_T^2...\cdots e_T^{n_T}.</math>.

Then the matrix

:<math>\mathbf{B}(\mathbf{c},\mathbf{e}) =

\begin{bmatrix}

Line 51 ⟶ 56:

e_1^2 - c_1^2 & e_2^2 - c_2^2 & \cdots & e_T^2 - c_T^2\\

\vdots & \vdots & \ddots & \vdots\\

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{bmatrix}

</math>

has to be full-[[rank (linear algebra)|rank]] for any pair of distinct codewords <math>\mathbf{c}</math> and <math>\mathbf{e}</math> to give the maximum possible diversity order of <math>n_Tn_R</math>. If instead, <math>\mathbf{B}(\mathbf{c},\mathbf{e})</math> has minimum rank <math>b</math> over the set of pairs of distinct codewords, then the space–time code offers diversity order <math>bn_R</math>. An examination of the example STBCs shown [[#Encoding|below]] reveals that they all satisfy this criterion for maximum diversity.