Coherent underwater acoustic communications are challenged by acoustic channels, which are often characterized as time-varying, dispersive, sparse, etc. [1]. Therefore, much of the recent research has been focused on the development of channel equalizers to remove intersymbol interference (ISI) in multipath environments, especially for high-rate coherent communications.

One receiver cannot avoid deep fading in time-varying channels, and thus the equalizers fail to remove ISI. With multiple sensors exploiting spatial diversity, Stojanovic et al. [2] has proposed a multichannel decision feedback equalizer (McDFE). The disadvantage of McDFE is its complexity due to the computational load, which increases with the time spread of underwater channels. Therefore, it is difficult to apply McDFE in underwater channels of long time spread, especially using a large number of receiving hydrophones.

Another novel method is the time reversal mirror (TRM), originally proposed by Fink [3], which was later applied for underwater communications. The focusing of TRM results in a significant reduction of ISI for underwater communications, which has been demonstrated by Edelman et al. [4,5]. Two vertical hydrophone arrays and two-way transmission are required by TRM to achieve the focusing at the transmitter. During the transmission, the underwater channel is required to be constant.

An alternative technique for underwater communications is proposed by Rouseff et al. [6] to take advantage of the focusing at the receiver, commonly referred as passive time reversal or passive-phase conjugation (PPC). This method requires only one receiving array and one-way transmission. ISI cannot be eliminated by the focusing, and thus a subsequent channel equalizer is used to remove residual ISI [6–9], where a single channel decision feedback equalizer (DFE) is used. It is referred to as PPC-DFE in this paper. Spatial diversity is used by the focusing to suppress ISI. In a real oceanic environment, it is difficult to predict time variant spatial coherence [10], when interchannel correlations impact spatial focusing.

Stojanovic [11] has discussed the upper bound performance of time reversal communications, but it is very difficult to predict real performance of time reversal communications, as spatial coherence is neglected in the model. By numerical simulations and experimental demonstrations, Yang [12] has demonstrated that McDFE achieves superior performance over that of PPC-DFE. This leads to a receiver structure which uses adaptive multichannel combining after PPC processing in each individual channel.

Zhang et al. [13] have presented a receiver structure—joint PPC and McDFE (PPC-McDFE). This receiver structure involves temporal focusing (pulse compression by PPC processing) for time delayed arrivals [14], and thus the computational load of a subsequent McDFE is much reduced. It is well known that temporal focusing degrades with time evolution in time-varying channels. To counter for this degradation, the block-based approach proposed by Song [15] can be used to extend PPC-McDFE in time-varying channels.

PPC processing requires information of the channel characteristics, which can be estimated using training symbols. Underwater channels are often sparse, especially at the high-frequency regime, where there are a few dominant arrivals. The dominant arrivals can be estimated using the matching pursuit (MP) algorithm [16]. Song [17] has shown that the MP algorithm exploits the channel sparseness to improve the performance of PPC-DFE. It is an open question whether the MP algorithm can improve the performance of PPC-McDFE, in comparison with the conventional channel estimation method—the least squares (LS) method.

The above brief introduction shows that different approaches have been proposed and have been tested in field experiments. However, the experiments were conducted under different conditions, and it is therefore difficult to compare the performance of different receiver structures. This has motivated the work of this paper. A recent field experiment was conducted to collect data over a range of 7.4 km, when three modulation schemes were used. Four data rates with a maximal data rate of 4 kilo-bits/s have been achieved. Using the same real data, we compare the performance of three receiver structures: McDFE, PPC-DFE, and PPC-McDFE. These structures are frequently discussed in the literature, and in the future we may extend the discussion to other structures and modulation schemes.

As required, information of the channel characteristics for PPC processing can be obtained by a channel probe signal or estimated using training symbols. For example, using a linear frequency modulation pulse (LFM) chirp as a channel probe signal, when the chirp is also used as a shaping pulse at the transmitter, the received LFM is immediately used for PPC processing. Alternatively, the channel is estimated using training symbols, when a root-raised-cosine pulse (RRC) is used as a shaping pulse. In this paper, we have also tested the scenario using the two shaping pulses.

The contributions of this paper include: (1) experimental assessment of the difference between two shaping pulses—LFM and RRC; (2) performance comparison of the McDFE, PPC-DFE, and PPC-McDFE structures; (3) evaluation of the block-based approach for PPC-McDFE; and (4) assessment of the MP algorithm for both PPC-DFE and PPC-McDFE, in which PPC processing is implemented in two modes—one block and multi-block.

This paper is organized as follows: Section 2 introduces the field experiment conducted in Trondheim harbor on 7 September 2011. Section 3 shows the receiver structures: (1) McDFE; (2) PPC-DFE; and (3) PPC-McDFE. Section 4 briefly introduces channel estimations for PPC processing, the LS method and the MP algorithm. In Section 5, the results are presented and discussed, and performance of the three structures is shown. Finally, Section 6 summarizes the work.

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 kth hydrophone can be written as: (1) V k n = ∑ l = 0 L − 1 H k l I n − l + W k n ,   k = 1 , … , Kwhere H k l denotes the lth tap of channel impulse response (CIR) H_k which spans L symbol interval, I_n is the nth symbol of a sequence {I_n}, and W k n represents a bandwidth limited noise. In a multipath channel, where L ≠ 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 { e − j θ ^ k n } 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 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 Song [15], a block-based approach extends PPC-DFE to be implemented in time-varying channels. The idea is that channel estimations { H ^ k n } 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 { H ^ k n } 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. Zhang 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 Yang [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].

This section briefly introduces two channel estimation methods. Using training symbols, the channel estimations for PPC processing can be obtained using both the LS method [22] and the MP algorithm [16].

By combining M observed symbols, Equation (1) is rewritten as: (2) [ V k n − M + 1 V k n − M ⋮ V k n ] = [ I n − M + 1 ⋯ I n − M − L + 2 ⋮ ⋮ I n … I n − L + 1 ]   [ H k 0 H k 1 ⋮ H k L − 1 ] + [ W k n − M + 1 W k n − M ⋮ W k n ]which is simplified to: (3) V k = IH k + W k

In the channel estimation problem, the information matrix I is known as training symbols. An estimation of Ĥ_k can be obtained by solving LS problem: (4) H ^ k = arg   min H ˜ k { ‖ V k − I H ˜ k ‖ 2 }which gives the solution: (5) H ^ k = [ I H   I ] − 1 I H   V k

In practice, the LS method is sensitive to noise. When a channel is sparse, the CIR consists of a large number of zeros among several dominant taps, and the LS method will suffer from the noise between dominant taps. Besides, the LS method involves matrix inversion, and it sometimes suffers ill-conditioned problem of a matrix of large eigenvalue spread.

To exploit the sparse property of channels, the channel estimation problem can be reconsidered as an approximation problem. It is assumed that the received signal vector is approximated by: (6) ( V ^ k ) M = ∑ i = 0 M − 1 H ^ k p i ( I ) p iwhere (I)_pi is the p_i th column of information matrix I. Finding the approximation of (V̂_k)_M that minimizes ||(V̂_k)_M – V_k|| is an non-deterministic polynomial-time hard problem [23,24], which means there is unknown polynomial time algorithm that can solve this problem. MP [16] is a greedy algorithm that achieves non-optimal yet computational efficient approximation of V_k.

The MP algorithm selects one column in matrix I which is best aligned with residual signal r_p−1, where r₀ = V_k at initial step. In practice at the pth step, the selected l_p th column of I is determined by: (7) l p = arg   max l { ‖ ( I ) l H r p − 1 ‖ / ‖ ( I ) l ‖ }

Correspondingly, the tap value H ^ k l p is estimated by: (8) H ^ k l p = ( I ) l p H   r p − 1 ‖ ( I ) l p ‖ 2and r_p is updated by: (9) r p = r p − 1 − ( I ) l p H   r p − 1 ‖ ( I ) l p ‖ 2 ( I ) l p

This iteration is terminated until the preset P taps have been estimated. In practice, one column in I is probably selected more than once. To deal with this problem, we can either exclude previously selected columns in the search press shown in Equation (7), or the tap value calculated in Equation (8) can be added to the value found in previous steps [25]. In this paper, we use the former method.

Recorded signals of 15 periods are processed with parameters given in Table 1, in which some are chosen in an ad hoc manner. For instance, the number of taps N¹_ff, and N¹_fb. As suggested by Stojanovic [26], the integral tracking constant K₂ is chosen as 10 time smaller than the proportional tracking constant K₁. In subsections of 5.2 and 5.3, the performance of McDFE is selected as a benchmark.

Three receiver structures have been assessed by processing data collected in a recent experiment conducted over a range of 7.4 km. In this high frequency (10–14 kHz) experiment, coherent underwater communications of different symbol rates were achieved, e.g., 1 to 2 kilo-symbol/s. In a large time scale, in terms of period of 202.044 s, the time-variant characteristics of underwater channel are observed by the communication results in terms of output SNR.

As shaping pulses, it is shown that the difference between LFM and RRC is minimal. The LFM shaping pulse provides a simple method for PPC processing, where the received channel probe signal of LFM is immediately used. Using a RRC shaping pulse, it is flexible to select a channel estimation method for PPC processing, e.g., the MP algorithm. In addition, the block-based approach can be implemented in time-varying channels.

As evident, PPC-DFE achieves the worst performance in the assessment, and the performance of PPC-McDFE approximates that of McDFE. Time-variant reverberations result in unpredictable spatial coherence, which may impact on the performance of PPC-DFE. Therefore, it is preferable that the adaptive multichannel combining obtains much spatial gain, especially in the scenarios of a small number of receivers. For instance, it is preferable to use PPC-McDFE instead of PPC-DFE in a channel of long time spread.

In the sparse channel, the MP algorithm has been assessed in two modes. One is the conventional single block approach, and the other is the multi-block approach. The multi-block approach assumes that the channel is constant within each block of a short time interval, and then PPC processing is extended to time-varying channels. Comparing with PPC-DFE, PPC-McDFE is less sensitive to the channel variations. It has been demonstrated that the MP algorithm improves the performance of communications using PPC processing, and thus the MP algorithm is suggested in sparse channels.