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c_1 & c_2\\
-c_2^* & c_1^*
\end{bmatrix},
</math>
where * denotes [[complex conjugate]].
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-c_2^*&c_1^*&\frac{c_3}{\sqrt 2}\\
\frac{c_3^*}{\sqrt 2}&\frac{c_3^*}{\sqrt 2}&\frac{\left(-c_1-c_1^*+c_2-c_2*\right)}{2}\\
\frac{c_3^*}{\sqrt 2}&-\frac{c_3^*}{\sqrt 2}&\frac{\left(c_2+c_2^*+c_1-c_1^*\right)}{2}.
\end{bmatrix}
</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 — 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.
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\frac{c_3^*}{\sqrt 2}&\frac{c_3^*}{\sqrt 2}&\frac{\left(-c_1-c_1^*+c_2-c_2^*\right)}{2}&\frac{\left(-c_2-c_2^*+c_1-c_1^*\right)}{2}\\
\frac{c_3^*}{\sqrt 2}&-\frac{c_3^*}{\sqrt 2}&\frac{\left(c_2+c_2^*+c_1-c_1^*\right)}{2}&-\frac{\left(c_1+c_1^*+c_2-c_2^*\right)}{2}
\end{bmatrix}.
</math>
These codes achieve rate-1/2 and rate-3/4 respectively, as for their 3-antenna counterparts. <math>C_{4,3/4}</math> exhibits the same uneven power problems as <math>C_{3,3/4}</math>. An improved version of <math>C_{4,3/4}</math> is<ref>{{cite journal|author=G. Ganesan and P. Stoica|title=Space–time block codes: A maximum SNR approach|journal=IEEE Transactions on Information Theory|pages=1650–1656|volume=47|issue=4|month=May | year=2001|doi=10.1109/18.923754}}</ref>
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-c_3^*&0&c_1^*&-c_2\\
0&-c_3^*&c_2^*&c_1
\end{bmatrix},
</math>
which has equal power from all antennas in all time-slots.
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At time <math>t</math>, the signal <math>r_t^j</math> received at antenna <math>j</math> is:
:<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]]).
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:<math>R_i = \sum_{t=1}^{n_T}\sum_{j=1}^{n_R}r_t^j\alpha_{\epsilon_{t}(i)j}\delta_t(i)</math>
where <math>\delta_k(i)</math> is the sign of <math>s_i</math> in the <math>k</math><sup>th</sup> row of the coding matrix, <math>\epsilon_k(p)=q</math> denotes that <math>s_p</math> is (up to a sign difference), the <math>(k,q)</math> element of the coding matrix,
for <math>i=1,2
:<math>s_i = \arg{}\min_{s\in\mathcal{A}}\left(\left|R_i - s\right|^2 + \left(-1 + \sum_{k,l}^{}\left|\alpha_{kl}\right|^2\right)\left|s\right|^2\right),</math>
with <math>\mathcal{A}</math> the [[constellation diagram|constellation alphabet]]. Despite its appearance, this is a simple, linear decoding scheme that provides maximal diversity.
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It has been proven<ref name="COD">{{cite journal|author=Xue-Bin Liang|title=Orthogonal Designs With Maximum Rates|journal=IEEE Transactions on Information Theory|pages=2468–2503|volume=49|issue=10|month=October | year=2003|doi=10.1109/TIT.2003.817426}}</ref> that the highest rate any <math>n_T</math>-antenna code can achieve is
: <math>r_{\mathrm{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|author=Kejie Lu, Shengli Fu, and Xiang-Gen Xia|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|month=December | year=2005|doi=10.1109/TIT.2005.858943}}</ref>
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-c_3^* & -c_4^* & c_1^* & c_2^*\\
c_4 & -c_3 & -c_2 & c_1
\end{bmatrix}.
</math>
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 ratio]]s (SNRs). At high SNRs, though (above about 22dB 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.
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