## 3 The Receiver Structures  
Generally, the receiver recovers distorted information by baseband signal processing, where multipath channels are often modeled as finite filters of multiple taps. In digitized form, the received signal at \(k\)th hydrophone can be written as:  
\[V_{k}^{n}=\sum_{l=0}^{L-1}H_{k}^{l}I_{n-l}+W_{k}^{n},\,k=1,\ldots,K \tag{1}\]  
where \(H_{k}^{l}\) denotes the \(l\)th tap of channel impulse response (CIR) \(H_{k}\) which spans \(L\) symbol interval, \(I_{n}\) is the \(n\)th symbol of a sequence \(\left\{I_{n}\right\},\) and \(W_{k}^{n}\) represents a bandwidth limited noise. In a multipath channel, where \(L\
eq 1\), ISI caused by \(H_{k}\) results in errors. The objective of a channel equalizer is to remove the ISI. Figure 5 shows the block diagram of McDFE [2]. The tap coefficients of the \(K\)-channel feed-forward filters plus one channel feedback filter are jointly updated by the recursive least squares (RLS) algorithm for its fast rate of convergence [19]. The technique of a second order digital phase-locked loop (DPLL) is implemented for the carrier-phase tracking. The DPLLs \(\left\{e^{-i\hat{\theta}_{k}^{n}}\right\}\) operate on a symbol-by-symbol basis to remove phase changes caused by the carrier frequency shift. In order to deconvolve \(H_{k}\), the number of taps for the McDFE is determined by the time spread \(L\), and it is usually chosen in an _ad hoc_ manner. The computational load increases with \(L\), and it may become  
Figure 5: Block diagram of McDFE using the RLS algorithm. There are \(K\) receiving channels. \(\hat{\theta}_{k}^{n}\) denotes the estimate of phase offset \(\theta_{k}^{n}\) at the \(k\)th receiving channel, \(\hat{\mathbf{l}}_{n}\) presents the soft estimate of \(I_{n}\), and \(\tilde{I}_{n}\) is the decided symbol which best matches \(\hat{I}_{n}\). prohibitive, when a large number of hydrophones are used. Moreover, under the same channel conditions, the number of taps increases with the symbol rate. Figure 6 shows the receiver structure of passive time reversal--PPC-DFE. Following the focusing, only one channel DFE is required to remove residual ISI [8], when one DPLL is implemented for carrier-phase tracking. The focusing mitigates ISI, the number of taps for the one channel DFE is much reduced, and thus the complexity of PPC-DFE is much lower than that of McDFE. Note that the focusing degrades with time in time-varying channels. As suggested by [PERSON] [15], a block-based approach extends PPC-DFE to be implemented in time-varying channels. The idea is that channel estimations \(\left\{\hat{H}_{k}^{\ast}\right\}\) are updated on a block-by-block basis, when the channel is assumed time-invariant within each block of a short time interval. The channel estimations \(\left\{\hat{H}_{k}^{\ast}\right\}\) are subsequently updated using detected symbols in the previous block. This block-based approach does not change the basic principle of TR focusing, which obtains spatial diversity by to mitigate ISI. [PERSON] _et al_. [13] has discussed the impact of the time variant interchannel correlations on the performance of PPC-DFE. The receiver structure PPC-McDFE is shown of Figure 7. Here, pulse compression is achieved by PPC processing in each individual channel, and then a subsequent McDFE is implemented to remove residual ISI by adaptive multichannel combining. The RLS algorithm updates the tap coefficients to minimize output mean square error (MSE). Pulse compression is achieved in the same way for single receiver [20,21]. Thus it is used by PPC-McDFE to reduce the complexity of the subsequent McDFE which obtains spatial gain. As discussed by [PERSON] [7], the peak-to-sidelobe ratio of pulse compression is determined by the channel response, while the pulse compression acts as a rake receiver recombining time delayed arrivals. In time-varying channels, PPC-McDFE can be extended using the block-based approach [15]. Figure 6: Block diagram of passive time reversal receiver structure PPC-DFE.