## 2 System Model  
### The Transmitter  
The block diagram of the proposed transmitter is shown in Figure 1. First, the bit stream of a frame is divided into three parts, where \(\mathbf{b_{I}}=\left[b_{I_{1}},b_{I_{2}},\cdots,b_{I_{M_{p}}}\right]\) and \(\mathbf{b_{Q}}=\left[b_{Q_{1}},b_{Q_{2}},\cdots,b_{Q_{M_{p}}}\right]\) represent the index bits used for index modulation of branches I and Q, respectively, and \(\mathbf{b}=\left[b_{1},b_{2},\cdots,b_{n_{t}}\right]\) represents the modulated bits used for MPSK constellation mapping.  
Bit stream \(\mathbf{b}\) is divided into \(J\) groups and mapped into \(J\) constellation symbols \(\left\{S_{1}^{i},\cdots,S_{J}^{i}\right\}\) in the \(i\)th block. \(M_{c}\) is the number of bits contained in an \(M\)-ary constellation symbol, so the length of \(\mathbf{b}\) satisfies the equation \(n_{t}=JM_{c}\). Bit streams \(\mathbf{b_{I}}\) and \(\mathbf{b_{Q}}\) are used for index modulation to obtain the cyclic shift index vectors \(I_{seq}^{l}=\left\{k_{I,j}\right\}_{j=1}^{J}\) and \(J_{seq}^{Q}=\left\{k_{Q,j}\right\}_{j=1}^{J}\), then the original spreading sequence is cyclically shifted according to the index vectors to obtain two groups of \(J\) different sequences. To convert index bits to the index vectors, the natural numbers to \(k\)-combinations mapping algorithm was adopted [19]. As shown in Figure 1, the shift length is calculated by \(k_{j}(L+1)\), where \(L+1\) is the cyclic shift step size and \(L\) is usually equal to or greater than the number of channel multipaths. Two groups of spreading sequences carry the real and imaginary parts of constellation symbols, respectively. Then, all sequences with a pilot sequence, which is used for channel estimation, are superimposed. Moreover, the number of index bits \(M_{p}\) is calculated as \(n_{t}=\left|\log_{2}{\binom{I_{max}-1}{J}}\right|\), where \(\binom{I_{max}-1}{J}=\frac{(I_{max}-1)!}{J!(I_{max}-1-J)!}\) is the combination number, and \(I_{max}=\left|\frac{M}{L+1}\right|\) is the maximum number of available spreading sequences that are superimposed, where \(M\) is the length of the spreading sequences. The transmitted signal \(\mathbf{x}(i)\) is represented as  
\[\mathbf{x}(i)=\sum_{j=0}^{J}\left(s_{j,R}^{i}\mathbf{T}^{k_{I,j}(L+1)}\mathbf{c}+ \sqrt{-1}s_{j,I}^{i}\mathbf{T}^{k_{Qj}(L+1)}\mathbf{c}\right), \tag{1}\]  
where \(s_{j,R}^{i}\) and \(s_{j,I}^{i}\) represent the real part and imaginary part of the \(j\)th transmitted constellation symbols during the \(i\)th block, respectively. The circular shift matrix \(\mathbf{T}\) is defined as  
\[\mathbf{T}=\left[\begin{array}{cc}\mathbf{0}_{1\times(M-1)}&1\\ \mathbf{I}_{M-1}&\mathbf{0}_{(M-1)\times 1}\end{array}\right], \tag{2}\]  
where \(\mathbf{I}\) denotes the identity matrix and \(\mathbf{0}\) is the all zero matrix. It is worth pointing out that \(s_{0,R}^{i}\), \(s_{0,I}^{i}\) denote the real part and imaginary part of the pilot signal when \(j=0\), \(k_{I,0}=k_{Q,0}=0\). In this design, the pilot signal and the information symbols go through the same channel after being spread and superimposed for a spreading block, thus it can estimate a more accurate channel.  
Figure 1: The block diagram of the MSSS-I/Q-IM transmitter.