# Deep Learning-Based Spread-Spectrum FGSM for Underwater Communication
## 3 Proposed SS-FGSM Scheme
### SS-FGSM Scheme
 \tag{15}\]  
where \(B\) is the number of transmitted data blocks; \(\exists_{d_{I_{\backslash}}\backslash,k}\) and \(\exists_{d_{Q_{\backslash}}\backslash,k}\in\Xi\) are the consistent spreading sequence for the mapped bits of vector \(\mathbf{d}_{I_{\backslash}}^{T}\) and \(\mathbf{d}_{Q_{\backslash}}^{T}\), respectively. \(\backslash\) and \(\backslash\) are the two indices of selecting spreading codes out of \(\mathcal{N}\) orthogonal Walsh codes. \(p(t)\) is the pulse shaping filter, in this paper we use a rectangular pulse of unit amplitude in \([0,T_{c}]\) and \(f_{o}\) is the carrier frequency, without loss of generally we use one for time chip \(T_{c}\) in this paper. Table 2 shows an example for the proposed SS-FGSM, assuming a SE needed is 9 bits per channel use (bpcu). Figure 1: Proposed DLSS-FGSM transmitter and receiver structures. The proposed scheme achieves this SE by only using four transmitter/receiver (Tx/Rx) antennas. For example, consider the incoming block of bits: \(\mathbf{d}_{\text{\tiny{I}}}=\left[ \begin{array}{cc}\underbrace{10}_{data}& \underbrace{1011}_{code}&\underbrace{111}_{antenna}\\ \end{array} \right]\), the data modulated bits using any digital modulation (e.g., QAM) are represented in the first two bits \(\mathbf{d}_{M-ary_{\text{\tiny{I}}}}^{T}=\left[1\;0\right]\). The 1011 four bits represent the code bits, are partitioned into two sub-groups \(\mathbf{d}_{I_{\text{\tiny{I}}}}^{T}=\left[1\;0\right]\) and \(\mathbf{d}_{Q,\text{\tiny{I}}}^{T}=\left[1\;1\right]\), the first sub-group is used to choose the spreading code used in spreading real constellation symbols \(a_{\text{\tiny{I}}}\) and the other sub-group is used to choose the spreading code used for spreading the imaginary constellation symbols \(b_{\text{\tiny{I}}}\)). The resulted signal \(x_{\text{\tiny{q}}}(t)\) is mapped to be transmitted by one or two antennas based on the next three data bits \(\mathbf{d}_{(N_{\text{\tiny{I}}}-1),}^{T}=\left[1\;1\;1\right]\) as in the conventional way of the FGSM scheme [17]. Assuming a full knowledge of underwater acoustic channel matrix \(H\), therefore, the equivalent baseband received signal of the proposed SS-FGSM at the receiver antenna \(j\)-th can be expressed as:  
\[y_{j}(t)=\sum_{q=1}^{N_{\text{\tiny{N}}}}\sum_{j=1}^{B}\sum_{k=1}^{\mathcal{L} }\left\{ \begin{array}{cc}\mathbf{h}_{q,j,\mu}(t;\tau)a_{\text{\tiny{I}}}\exists d _{k_{\text{\tiny{I}}}\lambda}p(t-(\lambda\mathcal{L}+k)T_{\text{\tiny{c}}})\; \cos(2\pi f_{\text{\tiny{I}}}t)+\\ \mathbf{h}_{q,j,\mu}(t;\tau)b_{\text{\tiny{I}}}\exists d_{Q_{\text{\tiny{I}}}\lambda }p(t-(\lambda\mathcal{L}+k)T_{\text{\tiny{c}}})\;\sin(2\pi f_{\text{\tiny{I}}} t)+\tilde{n}_{j}(t)\end{array} \right. \tag{16}\]  
where \(\tilde{n}_{j}(t)\) is the term of additive white Gaussian noise after multiplied by the Hadamard-Walsh code. For the proposed SS-FGSM, the received signal is first pre-processed where the channel is estimated and equalized, and also the other effects such that Doppler effect and carrier frequency offset (CFO) are recovered before detecting the transmitted data. In our proposed scheme, if we use the conventional receiver of SMTs which use the ML decoder to estimate the active antennas indices \(\overline{q}\), spread data symbols \(s_{\text{\tiny{I}}}\)) and the index of the codes used in spreading these symbols, the ML decoder can be expressed in this case as:  
\[\left[\overline{q}\;\overline{d}_{\text{\tiny{I}}}\;\overline{d}_{Q}\; \overline{s}_{\text{\tiny{I}}}\right]=\arg\underbrace{min}_{q\
eq\mathcal{L}_{ \text{\tiny{I}}}d_{Q,\text{\tiny{I}}}s_{\text{\tiny{I}}}}\left\|y(t)-\;x_{ \text{\tiny{q}}}(t)h(t,\tau)\right\|^{2} \tag{17}\]  
where \(y(t)=\left[y_{1}^{T},y_{2}^{T},\;\cdots\cdots,\;y_{N_{\text{\tiny{I}}}}^{T} \right]^{T}\). The operation of the ML decoder is based on joined the detection of the transmit antenna index, spreading codes index of real and imaginary parts of the modulated signal and the modulated signal itself.