## 3 Key Modules  
### OSDM Theory  
First, the data X, in the 1 \(\times\) MN dimension vector, is grouped into N groups of 1 \(\times\) M dimension vector as follows:  
Figure 1: System structure of pseudo-noise sequence-based underwater acoustic orthogonal signal division multiplex (OSDM) communication system (UWAOSDM/PN).  
\[\begin{array}{l}X_{0}=(x_{00},x_{01},\ldots,x_{0(M-1)})\\ X_{1}=(x_{10},x_{11},\ldots,x_{1(M-1)})\\ \cdot\\ \cdot\\ \cdot\\ X_{N-1}=(x_{(N-1)0},x_{(N-1)1},\ldots,x_{(N-1)(M-1)})\end{array} \tag{1}\]  
As we all know, inverse discrete Fourier transform (IDFT) formula is:  
\[x(n)=\frac{1}{N}\sum_{k=0}^{N-1}X(k)e^{j\frac{2\pi}{N}nk}=\frac{1}{N}\sum_{k=0} ^{N-1}X(k)W_{N}^{-nk} \tag{2}\]  
IDFT matrix is:  
\[F_{N}^{-1}=\left\{\begin{array}{c}f_{0}\\ f_{1}\\ \cdot\\ \cdot\\ f_{k}\\ \cdot\\ f_{N-1}\end{array}\right\} \tag{3}\]  
where,  
\[f_{k}=\frac{1}{\sqrt{N}}(W_{N}^{0},W_{N}^{k},W_{N}^{2k},\ldots,W_{N}^{(N-1)k}) \tag{4}\]  
Using the Kronecker product between formulae (1) and (4), we can get:  
\[S_{k} =f_{k}\otimes X_{k} \tag{5}\]  
Formula (5) can be expanded to  
\[\begin{array}{c}W_{N}^{0}(x_{00},x_{01},\ldots,x_{0(M-1)}),W_{N}^{0}(x_{00}, x_{01},\ldots,x_{0(M-1)}),\ldots,W_{N}^{0}(x_{00},x_{01},\ldots,x_{0(M-1)})\\ W_{N}^{0}(x_{10},x_{11},\ldots,x_{1(M-1)}),W_{N}^{1}(x_{10},x_{11},\ldots,x_{1 (M-1)}),\ldots,W_{N}^{1*(N-1)}(x_{10},x_{11},\ldots,x_{1(M-1)})\\ \vdots\\ W_{N}^{0}(x_{(N-1)0},x_{(N-1)1},\ldots,x_{(N-1)(M-1)}),W_{N}^{k}(x_{(N-1)0},x _{(N-1)1},\ldots,x_{(N-1)(M-1)}),\ldots,\\ W_{N}^{k*(N-1)}(x_{(N-1)0},x_{(N-1)1},\ldots,x_{(N-1)(M-1)})\end{array} \tag{6}\]  
The orthogonality between any two \(S_{k}\) can be demonstrated [11]. By summing \(S_{k}\), we can get the modulated signal \(s\) which is the baseband OSDM signal. This is a good mathematical expression for the principle of OSDM modulation.  
Signal modulated by Formulae (1-6) also can be modulated by grouping and inverse fast Fourier transform (IFFT). As shown in Figure 2, first, all data are divided into M groups in a matrix denoted as \(d\). Then, M times of N-point IFFT are calculated. The obtained data are stored in a data table by column and the data are taken out by row, where, \(x\) is the original data and \(s\) is IFFT results of \(x\). In this way, MN data are modulated on N subcarriers, each subcarrier carrying M data.  
We define permutation matrix:  
\[P_{N,M}=\left[\begin{array}{c}I_{N}\otimes e_{M}^{T}(0)\\ I_{N}\otimes e_{M}^{T}(1)\\ \vdots\\ I_{N}\otimes e_{M}^{T}(M-1)\end{array}\right] \tag{7}\]  
To do OSDM modulation, the above modulation process can be expressed as:  
\[s=P_{N,M}^{H}(I_{M}\otimes P_{N}^{H})P_{N,M}d \tag{8}\]  
where, matrix \(P_{N,M}^{H}\), \(I_{M}\otimes P_{N}^{H}\) and \(P_{N,M}\) is row-wise write operations, N-point IDFT and row-wise read operations, respectively, and \(d\) is the left matrix in Figure 2.  
As shown in Figure 3, the data of the same group of IFFT are dispersed on the time scale. It is obvious that the OSDM modulation separates the original IFFT-transformed symbols into the time domain by (M-1) bits information in the same vector. In this way, not only the advantage of orthogonal carrier in the frequency domain is utilized, but also the time-scale diversity in the time domain is used. It can resist the influence of multipath of a certain delay on the data in the same IFFT group. Thus, OSDM can be considered as a hybrid modulation method in the time-frequency domain.  
At the receiver, in the OSDM demodulation, only the inverse process of the modulation process needs to be performed, and then the modulated signal can be demodulated. The demodulation can be expressed as:  
\[\mathrm{x}= P_{N,M}^{H}(I_{M}\otimes F_{N})P_{N,M}r \tag{9}\]  
where, \(r\) means the data after channel equalization.