## 2 Asymmetric Turbo Code (ATC)
### Asymmetric Turbo-Coded Spatial Modulation (ATC-SM) Scheme with CMI  
The schematic of the ATC-SM scheme based on the CMI is illustrated in Figure 4. This scheme consists of \(\mathrm{N}_{\mathrm{T}}\) transmit antennas and \(\mathrm{N}_{\mathrm{R}}\) receive antennas. Figure 3: A weight-4 input sequence is mapped to another weight-4 input sequence after passing through the interleaver. The information bit sequence \(\mathbf{a}_{1}\) of length K is encoded into codewords \(\mathbf{v}\) of length 3K by an ATC encoder at the source node. The generated ATC codeword \(\mathbf{v}\) is divided into \(\mathbf{v}(\mathrm{j})\) with length \(\mathrm{d}\ =\log_{2}\mathrm{MN}_{\mathrm{T}}\) by the bit grouping block shown in Figure 4, where M denotes the modulation order and \(\mathrm{j}\in\{1,\ 2,\dots,\ 3K/\mathrm{d}\}\). Then, \(\mathbf{v}(\mathrm{j})\) is fed into the bit divider block. After the bit divider, the grouped bit sequence \(\mathbf{v}(\mathrm{j})\) is further partitioned into two parts, \(\mathbf{v}(\mathrm{j})_{\mathrm{ante}}\) and \(\mathbf{v}(\mathrm{j})_{\mathrm{modu}}\). The first bit train \(\mathbf{v}(\mathrm{j})_{\mathrm{ante}}\) with length \(\log_{2}\mathrm{N}_{\mathrm{T}}\) is sent to the antenna mapper to determine the active antenna index I(\(\mathrm{j})\), where I(\(\mathrm{j})\in\{1,\ 2,\dots,\ \mathrm{N}_{\mathrm{T}}\}\). Likewise, the M-PSK/QAM modulator obtains the remaining bit train \(\mathbf{v}(\mathrm{j})_{\mathrm{modu}}\) of length \(\log_{2}\mathrm{M}\) and modulates it into the M-PSK/QAM symbol \(\mathrm{v}_{\mathrm{s}}(\mathrm{j})\) with E[\(\mathrm{v}_{\mathrm{s}}(\mathrm{j})\)]\({}^{2}\)] = 1, where \(\mathrm{s}\in\{1,\ 2,\dots,\ \mathrm{M}\}\) and \(\mathrm{l}\) represents the Euclidean norm of a vector or matrix. Subsequently, the modulated symbol \(\mathrm{v}_{\mathrm{s}}(\mathrm{j})\) is assigned to the chosen active antenna index I(\(\mathrm{j})\). Thus, the coded SM mapper outputs the vector \(\mathbf{v}_{\mathrm{Is}}(\mathrm{j})\ =\ [0,\dots,\ 0,\ \mathrm{v}_{\mathrm{s}}( \mathrm{j}),\ 0,\dots,\ 0]^{\mathrm{T}}\), where \(\mathrm{v}_{\mathrm{s}}(\mathrm{j})\) represents the I(\(\mathrm{j}\))th non-zero component of the vector. Note that the components such as bit grouping, the bit divider, the antenna mapper, the M-PSK/QAM modulator and the coded SM mapper together constitute the main block called SM in this paper. Furthermore, the spectral efficiency of SM is evaluated as \(\mathrm{\eta}\ =\log_{2}\mathrm{MN}_{\mathrm{T}}\)[14]. The coded SM mapper outputs the transmission vector \(\mathbf{v}_{\mathrm{Is}}(\mathrm{j})\). Afterwards, the transmission vector \(\mathbf{v}_{\mathrm{Is}}(\mathrm{j})\) is propagated over the slow Rayleigh fading MIMO channel \(\mathbf{H}\ =\ [\mathbf{h}_{1},\ \mathbf{h}_{2},\dots,\ \mathbf{h}_{\mathrm{N}_{ \mathrm{T}}}]\), where \(\mathbf{h}_{\mathrm{i}}=\ [\mathbf{h}_{1,\mathrm{j}},\ \mathbf{h}_{2,\mathrm{j}},\dots,\ \mathbf{h}_{\mathrm{N}_{ \mathrm{T}},\mathrm{i}}]^{\mathrm{T}}\ (1\ \leq\ \mathrm{i}\ \leq\ \mathrm{N}_{\mathrm{T}})\). The entries of channel matrix \(\mathbf{H}\) are defined as independent and identically distributed (i.i.d.) complex Gaussian random variables with a mean of zero and a variance of 0.5 per dimension. At the destination, the received vector is modeled as:  
\[\mathbf{z}(\mathrm{j})=\ \mathbf{H}\mathbf{v}_{\mathrm{Is}}(\mathrm{j})+ \mathbf{n}(\mathrm{j})=\ \mathbf{h}^{\mathrm{I}(\mathrm{j})}\mathrm{v}_{\mathrm{s}}(\mathrm{j})+ \mathbf{n}(\mathrm{j}), \tag{13}\]  
where \(\mathbf{h}^{\mathrm{I}(\mathrm{j})}\) represents the I(\(\mathrm{j}\))th column of \(\mathbf{H}\), and \(\mathbf{n}(\mathrm{j})\ =\ [\mathbf{n}_{1}(\mathrm{j}),\ \mathbf{n}_{2}( \mathrm{j}),\dots,\ \mathbf{n}_{\mathrm{N}_{\mathrm{R}}}(\mathrm{j})]^{\mathrm{T}}\) is an i.i.d. complex additive Gaussian noise vector with each component \(\mathrm{n}_{\mathrm{k}}(\mathrm{j})\)-CN(\(0,\mathrm{N}_{0}\)) \((1\ \leq\ \mathrm{k}\ \leq\ \mathrm{N}_{\mathrm{R}})\), where \(\mathrm{N}_{0}\) is the noise power spectral density.