# A Weighted Turbo Equalized Multi-Band Underwater Wireless Acoustic Communications
## 2 The Multi-Band Underwater Wireless Acoustic Sensor (UWAS) Communication
### Transmitter and Receiver Structure of the Multi-Band Underwater Communication  
Based on the packet shown in Figure 1, the multi-band transceiver structure is modeled, as shown in Figure 2. After \(K\) bits have passed through the channel coding at the transmission unit of Figure 2, \(N\) coded bits \(\{d_{0},d_{1},\cdots,d_{N-1}\}\) are created. Signals that have passed through an iterative encoder and an interleaver, to change a burst error to a random error. After the interleaver, \(n\) preamble bits, that is, \(\{p_{0},p_{1},\cdots,p_{n-1}\}\) are input for synchronous acquisition. The Data bit string \(\mathbf{D}\) with \(N_{T}=(n+N)\) bits, which is configured by the input of \(\{p_{0},p_{1},\cdots,p_{n-1}\}\), can be expressed by Equation (1).  
\[\mathbf{D}=\{p_{0},p_{1},\cdots,p_{n-1},d_{0},d_{1},\cdots,d_{N-1}\} \tag{1}\]  
Multi-band signal can be expressed by Equation (2).  
\[s(t)=\sum_{k=1}^{N_{b}}\mathbf{D}e^{i\mathbf{v}_{k}t} \tag{2}\]  
The transmission signal \(s(t)\) is propagated, after \(N_{b}\) different bands are multiplied by \(e^{i\mathbf{v}_{k}t}\), is modulated for each band and the modulated signals for \(N_{b}\) bands are added up.  
The received signal is expressed by Equation (3).  
\[r(t)=\sum_{l=1}^{L}\sum_{k=1}^{N_{b}}\mathbf{D}e^{i\mathbf{v}_{k}t}h_{l}(t)+\eta(t) \tag{3}\]  
\(l\) represents the \(l\)-th multi-path among the total \(L\) multi-paths. \(h_{l}(t)\) represents the channel response coefficient in the path \(l\), and \(\eta(t)\) represents the Gaussian noise. The receiver signal is demodulated for \(N_{b}\) different bands and the output signal of the \(k\)-th band, which had passed through the LPF and the equalizer, was referred to as \(y_{k}(t)\). Here, the input signal of the decoder can be expressed by Equation (4).  
\[z_{t}=\sum_{k=1}^{N_{b}}w_{k}y_{k}(t) \tag{4}\]  
As for the signal received, a matched filter is used to divide each frequency and acquire the corresponding information. Then, a DFE (Decision Feedback Equalizer) removes the multi-path interference from each band and a threshold value is determined, by which the performance of the  
Figure 2: Transceiver structure of multi-band communications.  
encoder is judged [16; 17]. After the threshold detection, weighting factor, \(w_{k}\), for the \(k\)-th band is assigned. In this manner, the data of a band, which comes close to the error decoding performance limit of the decoder, are decoded. The external input \(L(z_{t})\) is the extrinsic information of the decoder. The estimated extrinsic information of \(L(z_{t})\) at decoder output is given by  
\[L(z_{t})=log\frac{P\big{(}d_{j}=+1\big{)}}{P\big{(}d_{j}=-1\big{)}}. \tag{5}\]  
In the iterative turbo equalization method, equalizer and decoder are connected through the interleaving and de-interleaving functions that update each other's information, recursively. The interleaved output is canceled, a posteriori, from the proceeding received signal. The interleaving helps the receiver convergence. Its extrinsic value \(L(z_{t})\) calculates the post-probability which is error correction terms. The re-interleaving of the computed value, as \(L(z_{t}),\ L(\hat{z}_{t})\), is added to the received signal of each band, and input to the DFE. \(L(\hat{z}_{t})\) is the correction factor in order to compensate for the errors. As the number of iteration of turbo equalization increases, the updated error correction approaches the original signal which is to be transmitted, thereby, improving the BER (Bit Error Ratio) performance [18; 19].