## 4 Maximum Minimum Posterior SNR Criterion for the LDC-TAS  
As mentioned above, \(\mathbf{H}_{\mathrm{s}}\) is a more effective channel coefficient array chosen from that function \(\hbar\).  
\[\mathbf{H}_{\mathrm{s}}=\hbar(\mathbf{H}) \tag{6}\]  
For convenience of analysis, defining a linear transferring array:  
\[\chi:=[vec(M_{0}),\ vec(M_{1}),\ \ldots,\ vec(M_{N-1})] \tag{7}\]  
So the channel coefficient matrix can be expressed as  
\[\overline{H_{\mathrm{s}}}=I_{T}\otimes H_{\mathrm{s}} \tag{8}\]  
In the equation, \(vec(\cdot)\) is the vectorization operator, and \(\otimes\) is the Kronecker product, so \(Y\) can be re-written as below:  
\[y=\sqrt{E_{\mathrm{s}}H_{\mathrm{s}}}\chi s+v=\sqrt{E_{\mathrm{s}}\overline{H _{\mathrm{s}}}}+v. \tag{9}\]  
In it, \(y=vec(\mathbf{Y}),\ \mathbf{s}=[s_{0},\ s_{1},\ s_{2},\ldots s_{N-1}]^{T},\ \mathbf{v}=vec(\mathbf{V}),\ and\ \widehat{H_{\mathrm{s}}}:= \overline{H_{\mathrm{s}}}\chi\) being the \(M_{r}T\times N\) matrix of the equivalent channel coefficient.  
From the expression above, any decoding technology, which is suitable for BLAST can be applied for the LDC system decoding. Due to the complexity of the real practical system, only the linear receiver is taken into account in this article. When the \(N\times M_{r}T\) matrix equalizer \(\mathbf{G}\) is multiplied to \(\mathbf{y}\), an estimated \(\mathbf{s}\) can be achieved as below:  
Figure 1: Transmit antenna selection for linear dispersion codes.  
\[\hat{S}=Gy=\sqrt{E_{s}}G\widetilde{H_{s}}+Gv. \tag{10}\]  
The original information can be recovered from \(\hat{S}\). According to the zero-force (ZF) receiver:  
\[G=\widetilde{H}^{+}. \tag{11}\]  
However, for MMSE receivers, \(G\) can be written as:  
\[G={(\widetilde{H}^{H}\widetilde{H}+\frac{N_{0}}{E_{s}})}^{-1}\widetilde{H}^{H}. \tag{12}\]  
Maximum minimum posterior SNR criterion: Corresponding in ZF receiver, to \(N\) multiplexed sub streams, a posteriori SNR is:  
\[SNR_{n}^{\text{ZF}}(H_{k})=\frac{E_{s}}{NN_{0}{[\widetilde{H}_{k}^{H} \widetilde{H}_{k}]}_{nm}^{-1}}. \tag{13}\]  
For each element in the collection \(\mathbf{H}\), conjugate transposition of matrices is \({()}^{\text{H}}\), we should calculate the minimum SNR in the corresponding N multiplexed sub streams. As for each element of the set \(\hat{H}\), a minimum SNR of the corresponding N streams is calculated, so an antenna subset using the most maximum minimum SNR can be chosen.  
\[\text{H}_{s}=\text{argmax}\Big{[}\text{min}\Big{(}\text{SNR}_{\text{N}}^{ \text{ZF}}(\text{H}_{\text{k}})\Big{)}\Big{]}. \tag{14}\]  
The bit error rate (BER) and frame error rate (FER) profiles of these two systems are displayed in Figure 2. It is obviously seen that the LDC-TAS performance outperformed STBC-TAS, while the SNR was lower than 14 dB. While SNR was at a high level, because STBC could achieve full diversity gain, its performance was indeed better than LDC-TAS.  
BLAST-TAS used four transmitters and two receivers, QPSK modulation, two antennae for transmission; LDC-TAS used 16 QAM modulation, T = 3. It means that their spectral efficiency of  
Figure 2: Performance of linear dispersion code transmitting antenna selection (LDC-TAS) vs. space–time block code (STBC)-TAS.  
these two systems was exactly R = 3 bit/s/Hz. The FER and BER profiles of these two systems are displayed in Figure 3. LDC-TAS shows a significant diversity gain contrasted to BLAST-TAS under the equal spectral efficiency. It is worth mentioning that LDC-TAS approximately had a 6 dB performance gain of the system when BER reached \(10^{-2}\), moreover, with the increase of SNR, the performance gain was higher.