This Letter proposes a low-complexity joint equalization and decoding reception scheme based on super-trellis per-survivor processing, making it possible to apply maximum likelihood sequence estimation in high-order underwater acoustic communications under fast time-varying channels. The technique combines trellis-coded modulation states and intersymbol interference states and uses per-survivor processing to track channel parameters. Furthermore, a general trellis configuration for arbitrary order quadrature amplitude modulation signal is provided when truncate the channel is used to describe the intersymbol interference state to 1. Sea trials results show that the performance of proposed method can be more than 1.4 dB superiority than conventional schemes.

As is widely recognized, the majority of underwater acoustic (UWA) channels exhibit notable time-varying characteristics.1 These variations in timing and carrier phase are caused by the Doppler effect resulting from the relative motion between the transmitter and receiver platforms, as well as the underwater dynamic environment. Additionally, the time-varying characteristic of the channel impulse response (CIR) is attributed to the changes in the medium/propagation.2 In recent years, there has been a growing interest in high-rate UWA communication (UAC), where high-order modulation techniques are commonly employed due to bandwidth limitations.

In the current literature of receivers for high-order modulation UCAs, the implicit channel-estimation-based decision feedback equalization (DFE) required the high signal-to-noise ratio (SNR) and the slower channel variation,3,4 which the practical applications scenarios of high-rate UACs are limited. The DFE based on symbol-by-symbol explicit channel estimation (CE) under the minimum mean square error (MMSE) criterion is considered the most effective reception scheme in rapidly changing channels.5 However, it introduces a high computational complexity due to matrix inversion.6 Recursive and iterative methods commonly employed to address this issue have been observed to degrade its performance by approximately 2 dB.7 Furthermore, as the modulation order increases, higher receiving SNR thresholds are required;8 thus, a reliable decision is demanded to provide by powerful coding techniques. However, DFE requires delay-free decoding feedback and unable to be directly integrated with a coding system. Alternatively, symbol-by-symbol linear equalization (LE) based on CE can be employed with a coding system which has short decoding delays, such as trellis-coded modulation (TCM).9 

Compared with MMSE criterion-based receiver, maximum-likelihood criterion-based receiver is optimal.10 The maximum likelihood sequence estimation (MLSE) strategy is commonly combined with per-survivor processing (PSP) to address time-varying channels.11 PSP can avoid the delay associated with the traceback length in CE using the least mean square or recursive least squares algorithms in MLSE, and can realize channel parameter tracking in fast time-varying channels. However, the calculation of MLSE increases exponentially with the channel memory length and the modulation order, it is mainly used in direct-sequence spread spectrum UAC.12,13 Assuming the complexity can be reduced, then undoubtedly, PSP-MLSE is the optimal reception scheme for high-order modulation signals in rapidly changing UWA channels. To reduce the intersymbol interference (ISI) trellis states, several methods have been proposed. These include the decision feedback sequence estimation algorithm with channel truncation14 and the reduced-state sequence estimation algorithm based on set partitioning.15 The trellis configuration of the reduced-state sequence estimation algorithm is complex, which varies with modulation order and different set partitioning ways, while decision feedback sequence estimation is more realizable. In addition, the ISI trellis can be merged with the TCM trellis to form a super-trellis (ST), enabling non-iterative joint equalization and decoding.9 This approach reduces the complexity of both equalization and decoding while achieving joint optimization.

In this Letter, we proposed a joint equalization and decoding reception scheme based on PSP with TCM-ISI ST (PSP: ST-JED) for high-order modulation signals in fast time-varying UAC channel. We truncate the channel length used to describe the finite state in decision feedback sequence estimation to 1 and combine it with 4-state TCM, propose a ST construction formula with the smallest number of states under arbitrary modulation orders. Moreover, the PSP framework is performed to track the CIR and the phase offsets. The effectiveness and superiority of this approach are validated through numerical simulation and sea trials, and achieving a good trade-off between complexity and performance of high-order modulation signal reception scheme in fast time-varying UWA channel.

The transmission system sends m information bits an per signaling interval Ts, and the TCM encoder produces m + 1 encoded bits bn, which are assigned to a symbol sn taken from a 2m+1-ary signal constellation according to the mapping rule of TCM.16 Then the transmitter produces a quadrature amplitude modulation (QAM) baseband signal of the form as
s ( t ) = n = 0 N 1 s n g ( t n T s ) , t [ 0 , T ] ,
(1)
where N represents the length of the symbol sequence, g(t) is a pulse shaped filter, and T = NTS denotes signal duration.
Assume that the considered fast time-varying UWA channel2 involves Np discrete path as
h ( t , τ ) = p = 1 N p A p ( t ) δ ( τ τ p ( t ) ) ,
(2)
where Ap(t) and τp(t) are the amplitude and delay of the pth path, respectively. Under the influence of additive noise w ̃ ( t ) and the UWA channel, the received passband signal at the receiving transducer can be represented as
r ̃ ( t ) = p = 1 N p A p ( t ) s ̃ ( t τ p ( t ) ) + w ̃ ( t ) ,
(3)
where s ̃ ( t ) is passband transmitted signal. After being sampled by a low-pass filter at a sampling rate of Ω T s, the discrete baseband input-output relationship can be represented as
r k = r ( t ) | t = k T s / Ω = n = 0 N 1 p = 1 N p A p , k s n g ( k T s Ω n T s τ p , k ) + w k , k [ 0 , Ω N 1 ] ,
(4)
where Ap,k and τp,k represent discrete sampling points for the amplitude and delay of the pth path, respectively. wk is the sampling point for baseband additive noise. In this scheme, a fractional interval receiver with a sampling rate of 2 T s is employed (Ω = 2).
The block diagram structure of PSP: ST-JED receiver is shown in Fig. 1. At the receiver front end, the Doppler factor is initially estimated using the training sequence through the ambiguity function method.17 Subsequently, in UAC, timescale interpolation can be performed using a well-known Farrow filter.18 The impact of residual Doppler-induced the phase offsets and the timing offsets are then addressed by PSP tracking based on a ST, as introduced in Sec. 3.4. The initial CIR also is estimated by the least squares algorithm using the training sequence. The estimated channel is transformed into a minimum-phase channel of length L + 1 through the utilization of a matched whitening filter based on linear prediction.19 The fractional interval CIR can be represented as
h ̂ = [ h ̂ 1 , 0 h ̂ 1 , 1 h ̂ 1 , L h ̂ 2 , 0 h ̂ 2 , 1 h ̂ 2 , L ] T ,
(5)
where (·)T represents the transpose.
Fig. 1.

The block diagram structure of PSP: ST-JED reception scheme.

Fig. 1.

The block diagram structure of PSP: ST-JED reception scheme.

Close modal
ST is combined the encoding state and the discrete channel ISI state. The state representation of the TCM-ISI ST state is as follows:
γ n = ( s n L , s n L 1 , , s n 1 ; σ n ) ,
(6)
where σn denotes the TCM encoding state. The symbol sequence { s n L , s n L 1 , , s n 1 } represents a path that transitions the TCM encoder from a previous state σn–1 to the present state σn, following the TCM coding rule. It can be observed that each encoding state corresponds to 2mL ISI states. Therefore, for an S-state TCM encoder, the number of states in the TCM-ISI ST is NS = S·2mL.

To reduce the computation and storage requirements of the TCM-ISI super trellis, the channel memory L + 1 is truncated to μ terms. Hence, the number of states in the ST are reduced to NS = S·2. For the sake of simplicity, this letter sets μ = 1. Figure 2 illustrates the 32-state TCM-ISI ST for the 4-state TCM encoder with a 16-QAM transmitted signal.

Fig. 2.

The 32-state TCM-ISI ST (4-state TCM, 16-QAM, μ = 1).

Fig. 2.

The 32-state TCM-ISI ST (4-state TCM, 16-QAM, μ = 1).

Close modal
For the case of μ = 1, we define the following state transition process in the ST. The index of previous state γn that the current state γn+1 may correspond to can be represented as
S S N s × N g = [ I N g × 1 [ 1 : N g / 2 N s / 2 + 1 : ( N s + N g ) / 2 ] I N g × 1 [ N g / 2 + 1 : N g ( N s + N g ) / 2 + 1 : N s / 2 + N g ] I N g × 1 [ N g + 1 : 3 N g / 2 N s / 2 + N g + 1 : N s / 2 + 3 N g / 2 ] I N g × 1 [ 3 N g / 2 + 1 : 2 N g N s / 2 + 3 N g / 2 + 1 : N s / 2 + 2 N g ] ] ,
(7)
where Ng = 2mμ, INg × 1 denotes an Ng × 1 dimensional identity matrix, and ⊗ is Kronecker product.
The input symbols (the symbol before encoding) from state γn to state γn+1 are written as
S I N s × N g = [ m [ 1 : 2 : 2 m + 1 ] I 1 × N g m [ 1 : 2 : 2 m + 1 ] I 1 × N g [ m [ N g + 1 : 2 : 2 m + 1 ] m [ 1 : 2 : N g ] ] I 1 × N g [ m [ N g + 1 : 2 : 2 m + 1 ] T m [ 1 : 2 : N g ] T ] I 1 × N g ] ,
(8)
where m 2 m + 1 × 1 = ( ( [ 0 : 2 m + 1 1 ] T ) 2 ) 10 . ( · ) 2 and ( · ) 10 , respectively, return the binary sequence from right to left and the decimal number obtained from left to right. For a vector x , x [ i : j ] returns the i-th to j-th elements of x, and x [ i : 2 : j ] represents selecting a value every two values starting from i.
The output symbols (the symbol after encoding) from state γn to state γn+1 are written as
S O N s × N g = [ m [ 1 : N g ] I 1 × N g m [ N g + 1 : 2 m + 1 ] I 1 × N g m [ N g / 2 + 1 : N g 1 : N g / 2 ] I 1 × N g m [ 3 N g / 2 + 1 : 2 m + 1 N g + 1 : 3 N g / 2 ] I 1 × N g ] .
(9)
The CIR corresponding to the state γn in the ST is given by
h ̂ ( γ n ) = [ h ̂ 1 , 0 ( γ n ) h ̂ 1 , 1 ( γ n ) h ̂ 1 , L ( γ n ) h ̂ 2 , 0 ( γ n ) h ̂ 2 , 1 ( γ n ) h ̂ 2 , L ( γ n ) ] T .
(10)
The state transition process mentioned in Sec. 3.2, at discrete time n, there are Ng × NS path transitions. For all possible transitions γn → γn+1, the branch metric for the soft-decision Viterbi algorithm is given by
b ( γ n γ n + 1 ) = k = Ω n Ω n + 1 | ( r k e j θ ̂ k ( γ n ) i = 0 μ h ̂ 1 , i ( γ n ) s n i ( γ n γ n + 1 ) i = μ + 1 L h ̂ 2 , i ( γ n ) s n i ( γ n γ n + 1 ) ) | 2 ,
(11)
where { s i ( γ n γ n + 1 ) } i = n L n is the string of the TCM coding sequence found in the survivor history, and arranging { s i ( γ n γ n + 1 ) } i = n L n into an (L + 1)-element vector s ( γ n γ n + 1 ). The phase sequence { θ ̂ k ( γ n ) } k = Ω n 1 Ω n denotes the PSP phase estimation at discrete time n of γn state.
The survivor branch metrics can be evaluated by
b min ( γ n γ n + 1 ) = min s n ( γ n γ n + 1 ) Q ( γ n γ n + 1 ) b ( γ n γ n + 1 ) ,
(12)
where Q ( γ n γ n + 1 ) denotes the symbol subset from the state transition γ n γ n + 1. For all successor states γ n + 1, the accumulated metrics M ( γ n + 1 ) can be determined by
M ( γ n + 1 ) = min γ n { M ( γ n ) + b min ( γ n γ n + 1 ) } .
(13)
At discrete time n, for all states survive the path transition γ n γ n + 1, the error between the received signal compensated for the phase offset θ ̂ ( γ n ) and the reconstructed received signal r ̂ ( γ n γ n + 1 ) is given by
e ( γ n γ n + 1 ) = r n ° e j θ ̂ ( γ n ) r ̂ ( γ n γ n + 1 ) = r n ° e j θ ̂ ( γ n ) h ̂ T ( γ n ) s ( γ n γ n + 1 ) ,
(14)
where r n = [ r Ω k r Ω k + 1 ] T, and ° represents Hadamard product.
The CIR estimation h ̂ ( γ n ) is then updated based on the low complexity recursive least squares algorithm20 
h ̂ ( γ n + 1 ) = h ̂ ( γ n ) + ( 1 η ) s * ( γ n γ n + 1 ) e T ( γ n γ n + 1 ) ,
(15)
where η denotes forgetting factor and (·)* represents conjugate.
The estimation of the phase offset θ ̂ ( γ n ) is updated according to
θ ̂ ( γ n + 1 ) = θ ̂ ( γ n ) + β { r ̂ ( γ n γ n + 1 ) ° r n * ° e j θ ̂ ( γ n ) } ,
(16)
where β is a suitable constant.

We validated the proposed scheme through numerical simulations and deep sea trials. This Letter primarily compares three reception schemes: the proposed joint equalization and decoding based on ST by PSP method (PSP: ST-JED), the ST-based joint equalization and decoding that utilizes the obtained tentative decoding result for symbol-by-symbol tracking of the channel and the phase offset (SBS: ST-JED), and the LE based on adaptive CE, which also utilizes the tentative decoding result for symbol-by-symbol tracking of the channel and the phase offset (SBS: CE-LE-TCM).

To validate the superiority of the proposed reception scheme, we extracted the CIR from a 900 m deep-sea vertical UAC trial, which was conducted in the LingShui waters of the South China Sea on August 4, 2021. Due to the slow time-varying nature of the channel in this trial, we resampled the received signals to simulate the time-varying scenarios. The center frequency was 10 kHz, with a bandwidth of 5 kHz, and the sampling frequency was 80 kHz. Each frame consisted of 2136 16-QAM symbols, with the first 200 symbols being training symbols, and the symbol rate is 5000 symbols/s. Combined Monte Carlo numerical simulations were conducted to evaluate the performance of the proposed scheme under the two cases presented in Fig. 3(a). In case 1, the wave has an amplitude of Aw = 1 m and a frequency of 0.1 Hz. In case 2, the relative radial acceleration between the transmitter and receiver is a = 0.1 m/s2, with an initial velocity of 1 m/s.

Fig. 3.

Numerical simulation results (BER = 10–5 denotes error-free). (a) CIR for two time-varying channels. (b) BER curves of 16-QAM with three schemes under two cases. (c) BER curves of 256-QAM with three schemes under two cases. (d) BER curves of the proposed scheme for 16-QAM to 256-QAM under case 1. (e) BER curves of the proposed scheme under different accelerations (16-QAM). (f) BER curves of the proposed scheme under varying wave amplitudes (16-QAM).

Fig. 3.

Numerical simulation results (BER = 10–5 denotes error-free). (a) CIR for two time-varying channels. (b) BER curves of 16-QAM with three schemes under two cases. (c) BER curves of 256-QAM with three schemes under two cases. (d) BER curves of the proposed scheme for 16-QAM to 256-QAM under case 1. (e) BER curves of the proposed scheme under different accelerations (16-QAM). (f) BER curves of the proposed scheme under varying wave amplitudes (16-QAM).

Close modal

Figures 3(b) and 3(c) illustrate the BER performance curves of 16-QAM and 256-QAM with the three reception schemes under two different cases. The proposed scheme PSP: ST-JED demonstrates the best performance, surpassing that of SBS: ST-JED by more than 1 dB and SBS: CE-LE-TCM by more than 2 dB. Furthermore, as the modulation order increases, the performance advantage becomes increasingly prominent. Figure 3(d) presents the performance of the proposed scheme for modulation schemes ranging from 16-QAM to 256-QAM. To further evaluate the performance of the proposed scheme in various time-varying channels, Fig. 3(e) conducts simulations with accelerations ranging from 0.2 to 1 m/s2. When the acceleration is 1 m/s2, the timing offset caused by the fast time-varying Doppler causes the performance degrades. Figure 3(f) illustrates the performance of the proposed scheme under wave amplitudes ranging from 1 to 9 m and the frequency of 0.2 Hz. The performance begins to degrade when the amplitude is 9 m. Moreover, Table 1 shows the number of multiplication operations in SBS: CE-LE-TCM (the length of LE Nf obtained from the MMSE algorithm is 64), PSP: MLSE-TCM (L = 5) without channel truncation, and PSP: ST-JED, and the number of TCM states is 4. The proposed scheme is the least complex.

Table 1.

The number of multiplication operations.

PSP: ST-JED ( S × ( 2 m μ ) 2)
SBS: CE-LE-TCM ( N f 3) PSP: MLSE-TCM ( S × ( 2 m L ) 2) μ = 1 μ = 2 μ = 3
16-QAM  262 144  4.3 × 109  256  16 384  1 048 576 
256-QAM  —  4.7 × 1021  65 536  1.1 × 109  1.8 × 1013 
PSP: ST-JED ( S × ( 2 m μ ) 2)
SBS: CE-LE-TCM ( N f 3) PSP: MLSE-TCM ( S × ( 2 m L ) 2) μ = 1 μ = 2 μ = 3
16-QAM  262 144  4.3 × 109  256  16 384  1 048 576 
256-QAM  —  4.7 × 1021  65 536  1.1 × 109  1.8 × 1013 

On September 24, 2023, we conducted UAC experiments at a depth of 3695 m in the South China Sea and collected a series of data to verify the performance of the proposed reception scheme. The transmitting transducer was located 3695 m below the sea surface, while the receiving transducer was vertically suspended at a depth of 20 m using a soft connection deployed from the mother ship. The transmitting transducer has a 3 dB beam width of 80°. The horizontal communication ranges from 961 to 3022 m. 16-QAM modulation was adopted in the experiment. The experimental parameter settings were consistent with those described in Sec. 4.1 of the simulation.

In the sea trails, the receiving transducer was suspended underwater using a soft connection, which was susceptible to surface wave fluctuations and underwater dynamic environments. As the CIR experienced rapidly changes, continuous updating of the CIR is required. Figures 4(a)–4(c) present the sea trial results at a horizontal communication distance of 961 m. Figures 4(a) and 4(b) provide the estimated CIR and Doppler spread function. The installation of baffles on the receiving transducer effectively suppressed the influence of sea surface reflections. The channel exhibited limited delay spread but significant Doppler spread, with a maximum Doppler spread of 8 Hz. Figure 4(c) compares the results of the three reception schemes, clearly demonstrating the advantages of the proposed scheme over SBS: ST-JED by 1 dB and SBS: CE-LE-TCM by 2.2 dB. Figures 4(d)–4(f) illustrate the sea trial results at a horizontal communication distance of 3022 m. Figures 4(d) and 4(e) present the estimated CIR and Doppler spread function. Compared to the 961 m sea trial, the Doppler spread of the 3022 m sea trial channel is smaller, around 1 Hz, and the measured SNR decreased by approximately 7 dB. Figure 4(f) compares the results of the three reception schemes, indicating performance degradation with increasing communication distance for all three schemes. However, PSP: ST-JED still exhibited a significant advantage.

Fig. 4.

Sea trial results at the horizontal communication distance of 961 and 3022 m. (a, e) CIR. (b, d) Delay-Doppler spread function. (c, f) BER curves for three schemes.

Fig. 4.

Sea trial results at the horizontal communication distance of 961 and 3022 m. (a, e) CIR. (b, d) Delay-Doppler spread function. (c, f) BER curves for three schemes.

Close modal

This Letter proposes a low-complexity ST-based joint equalization and decoding technique that utilizes PSP to track the CIR and the phase offset, and is effectively applied to the receiver of high-order modulated signals in fast time-varying UWA channels. Furthermore, a general trellis configuration method for arbitrary order QAM signal is provided when the channel used to describe ISI states is truncated to 1. Through numerical simulations and sea trials, a comparison is conducted with the symbol-by-symbol ST-based joint equalization and decoding technique and symbol-by-symbol LE based on CE. The results validate the performance advantage of the proposed scheme and achieve an excellent trade-off between complexity and performance.

This work was supported by National Key R&D Program of China Grant No. 2021YFC2800200, National Natural Science Foundation of China Grant No. 61971472, and Strategic Priority Research Program of the Chinese Academy of Sciences Grant No. XDA22030101.

The authors declare no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
L.
Jing
,
T.
Zheng
,
C.
He
, and
H.
Yin
, “
Iterative adaptive frequency-domain equalization based on sliding window strategy over time-varying underwater acoustic channels
,”
JASA Express Lett.
1
(
7
),
076002
(
2021
).
2.
D.
Sun
,
J.
Wu
,
X.
Hong
,
C.
Liu
,
H.
Cui
, and
B.
Si
, “
Iterative double-differential direct-sequence spread spectrum reception in underwater acoustic channel with time-varying Doppler shifts
,”
J. Acoust. Soc. Am.
153
(
2
),
1027
1041
(
2023
).
3.
T.
Shimura
,
K. I. D. A.
Yukihiro
, and
M.
Deguchi
, “
High-rate acoustic communication at the data rate of 69 kbps over the range of 3,600 m developed for vertical uplink communication
,” in
OCEANS 2019-Marseille
(
IEEE
,
New York
,
2019
), pp.
1
6
.
4.
T.
Shimura
,
Y.
Kida
,
M.
Deguchi
,
Y.
Watanabe
, and
Y.
Maeda
, “
High-rate underwater acoustic communication at over 600 kbps × km for vertical uplink data transmission on a full-depth lander system
,” in
2021 Fifth Underwater Communications and Networking Conference
(
IEEE
,
New York
,
2021
), pp.
1
4
.
5.
M.
Rupp
, “
Robust design of adaptive equalizers
,”
IEEE Trans. Signal Process.
60
(
4
),
1612
1626
(
2012
).
6.
J.
Tao
,
Y.
Wu
,
Q.
Wu
, and
X.
Han
, “
Kalman filter based equalization for underwater acoustic communications
,” in
OCEANS 2019-Marseille
(
IEEE
,
New York
,
2019
), pp.
1
5
.
7.
T.
Chen
,
Y. V.
Zakharov
, and
C.
Liu
, “
Low-complexity channel-estimate based adaptive linear equalizer
,”
IEEE Signal Process. Lett.
18
(
7
),
427
430
(
2011
).
8.
S.
Jeong
and
J.
Moon
, “
Soft-in soft-out DFE and bi-directional DFE
,”
IEEE Trans. Commun.
59
(
10
),
2729
2741
(
2011
).
9.
P. R.
Chevillat
and
E.
Eleftheriou
, “
Decoding of trellis-encoded signals in the presence of intersymbol interference and noise
,”
IEEE Trans. Commun.
37
(
7
),
669
676
(
1989
).
10.
G.
Tabak
,
M. L.
Oelze
, and
A. C.
Singer
, “
Effects of acoustic nonlinearity on communication performance in soft tissues
,”
J. Acoust. Soc. Am.
152
(
6
),
3583
3594
(
2022
).
11.
R.
Raheli
,
A.
Polydoros
, and
C.-K.
Tzou
, “
Per-survivor processing: A general approach to MLSE in uncertain environments
,”
IEEE Trans. Commun.
43
(
2
),
354
364
(
1995
).
12.
X.
Xu
,
S.
Zhou
,
A. K.
Morozov
, and
J. C.
Preisig
, “
Per-survivor processing for underwater acoustic communications with direct-sequence spread spectrum
,”
J. Acoust. Soc. Am.
133
(
5
),
2746
2754
(
2013
).
13.
X.
Kuai
,
S.
Zhou
,
Z.
Wang
, and
E.
Cheng
, “
Receiver design for spread-spectrum communications with a small spread in underwater clustered multipath channels
,”
J. Acoust. Soc. Am.
141
(
3
),
1627
1642
(
2017
).
14.
A.
Duel-Hallen
and
C.
Heegard
, “
Delayed decision-feedback sequence estimation
,”
IEEE Trans. Commun.
37
(
5
),
428
436
(
1989
).
15.
M. V.
Eyuboglu
and
S. U. H.
Qureshi
, “
Reduced-state sequence estimation with set partitioning and decision feedback
,”
IEEE Trans. Commun.
36
(
1
),
13
20
(
1988
).
16.
G.
Ungerboeck
, “
Trellis-coded modulation with redundant signal sets. II: State of the art
,”
IEEE Commun. Mag.
25
(
2
),
12
21
(
1987
).
17.
M.
Johnson
,
L.
Freitag
, and
M.
Stojanovic
, “
Improved Doppler tracking and correction for underwater acoustic communications
,” in
1997 IEEE International Conference on Acoustics, Speech, and Signal Processing
(
IEEE
,
New York
,
1997
), Vol.
1
, pp.
575
578
.
18.
D.
Li
,
Y.
Wu
,
M.
Zhu
, and
J.
Tao
, “
An enhanced iterative receiver based on vector approximate message passing for deep-sea vertical underwater acoustic communications
,”
J. Acoust. Soc. Am.
149
(
3
),
1549
1558
(
2021
).
19.
W. H.
Gerstacker
,
F.
Obernosterer
,
R.
Meyer
, and
J. B.
Huber
, “
On prefilter computation for reduced-state equalization
,”
IEEE Trans. Wireless Commun.
1
(
4
),
793
800
(
2002
).
20.
M.
Stojanovic
,
L.
Freitag
, and
M.
Johnson
, “
Channel-estimation-based adaptive equalization of underwater acoustic signals
,” in
Oceans' 99 MTS/IEEE: Riding the Crest into the 21st Century. Conference and Exhibition. Conference Proceedings (IEEE Catalog No. 99CH37008)
(
IEEE
,
New York
,
1999
), Vol.
2
, pp.
985
990
.