# MIMO Underwater Acoustic Communications in Ports and Shallow Waters at Very High Frequency

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## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. Source Signals

_{i,h}(t) followed after a predefined time delay τ

_{msg}with the traditional Hermes message s

_{i,msg}(t), such that:

_{i,h}(t) consists of perfectly known pseudo noise (PN) sequences of length τ = 218.5 ms, up-sampled from the symbol frequency F

_{sym}to the sampling frequency and pulse-shaped using a raised-cosine filter. Figure 1 shows an example of source signal. Given a carrier frequency F

_{0}and a signal level SL, the transmitted signal is

_{sym}is the symbol period.

#### 2.2. Received Signals

#### 2.2.1. Channel Estimation

_{win}time windows, of time index t

_{win}and length L

_{win}. The received samples at receiver j can be written in matrix form [4,17]:

_{0}(t

_{win}) denotes the beginning of the time window. L

_{ij}is the length of the channel impulse response between transmitter i and receiver j, and . Finally, T

_{j}is the sum of the channel lengths over the total number of transmitters N

_{t}, so that .

**Figure 2.**Definition of the parameters of the time-window used to perform the channel estimation at receiver j.

_{j}is noted T

_{L}(this term is also used in the results section). The index j must be dropped, as the tunable parameter T

_{L}is assumed identical across every receiver. We study the influence of this parameter on channel estimation and deconvolution in the results section. The channel estimation is performed through minimization of the following quantity [18],

_{win},

^{H}represents the Hermitian operator. This operation requires a T

_{j}× T

_{j}matrix inversion.

#### 2.2.2. MIMO Deconvolution

_{L}; (2) the length T

_{k}of the pre-cursor and post-cursor of the linear equalization filter. We consider two equalization structures: conventional Linear Equalization (LE) and Interference Cancellation Linear Equalization (ICLE). ICLE provides a theoretical lower bound of the equalization process.

#### 2.2.2.1. Minimum Mean Squared Error Linear Equalization

_{ij},

_{1}and κ

_{2}(Figure 3) as the pre-cursor and post-cursors of the linear equalization filter. In this case, , and are defined as

**Figure 3.**Multiple-Input-Multiple-Output (MIMO) deconvolution process: pre-cursor and post-cursor definition.

_{k}. Next, we define the augmented matrix containing the matrices of channel impulse responses, denoted ,

_{k}= Hx

_{k}+ w

_{k}

**I**is the identity matrix of size (1 + κ

_{1}+ κ

_{2}) N

_{r}, is the variance of the noise and is the variance of the original sequence S

_{I,K}. denotes a column vector of size (L + κ

_{1}+ κ

_{2}) N

_{t}with 1 at index i and 0 at other positions. is the estimates of augmented channel matrix H or time window index t

_{win}Here, the estimated signal depends on length of the time window, which in turns depends on the measured channel response. If N

_{win}= 1, the channel is stationary over a message duration and

_{win}≥ 1, becomes a function of t

_{win}. In this case, the equalized signal for each sliding window is cropped and forms a section of the equalized output ,

_{win}, L

_{win}and O

_{R}represent the time window index, length and overlapping rate, respectively.

#### 2.2.2.2. MMSE Interference Cancellation Linear Equalizer

**y**

_{k}is defined in Equation (17).

**p**

_{i,k}and

**q**

_{i,k}respectively stand for the feed-forward and feed-back filters.

**v**

_{i,k}is defined as

_{ICLE}is the discrete delay index induced by the equalizer, such that 0 ≤ K

_{ICLE}≤ N

_{ICLE}+ L − 2, where N

_{ICLE}represents the discrete ICLE length. Under MMSE optimization, the feed-forward and feed-back equalization vectors become equal to [7]:

**e**

_{k}is a null column vector, with the exception of element N

_{t}(k − 1) + i equal to 1. represents the noise variance. The ICLE estimate depend on the number of time windows used to perform the channel estimation, thus Equations (23)–(25) also apply to .

## 3. Simulated and Experimental Results

#### 3.1. Experimental Setup

_{j}was calculated at each receiver j. The value of SNR

_{j}, averaged across every messages within a mission, is shown in Table 3. The observed SNR value varied between 27.1 dB and 34.3 dB from mission to mission. These variations are mostly due to the time varying characteristics of the channel, which dramatically impacts the average power of the received signals.

Mission Number and Date | Source 1 Position | Source 2 Position | Receiver 1 Position | Receiver 2 Position | Number of Messages Retained |
---|---|---|---|---|---|

1-07/27/2011 | Pos1 | Pos7 | Rx1 | Rx3 | 50 |

2-07/27/2011 | Pos7 | Pos1 | Rx1 | Rx3 | 50 |

3-08/29/2011 | Pos7 | Pos1 | Rx1 | Rx3 | 100 |

4-08/29/2011 | Pos1 | Pos7 | Rx1 | Rx3 | 100 |

Distance on Figure 4 | Distance (m) |
---|---|

Rx1-Pos1 | 24 |

Rx1-Pos7 | 27 |

Rx3-Pos1 | 23.3 |

Rx3-Pos7 | 25.8 |

Pos1-Pos7 | 6.15 |

Rx1-Rx3 | 2.68 |

Mission Number and Date | SNR_{1}(dB) | SNR_{2}(dB) |
---|---|---|

1-07/27/2011 | 27.1 | 27.3 |

2-07/27/2011 | 28.8 | 27.8 |

3-08/29/2011 | 30.9 | 34.3 |

4-08/29/2011 | 29.2 | 33.2 |

#### 3.2. Simulation Parameters

_{0}between the signals measured at every receiver. In this paper, we only consider full overlap between received messages. The results for partial overlap are presented in [17]. The channel model considered here includes both the specular reflection from the sea bottom and scattering from the sea surface and bottom.

Name | Symbol | Value (units) | Name | Symbol | Value (units) |
---|---|---|---|---|---|

Sources Depth [4,17] | D_{Si} | 1 m | Source Level | SL | 179 dB re 1 µPa @ 1 m |

Receivers Depth [4,17] | D_{Rj} | 1.5 m | Noise Level | NL | 83.7 dB re 1 µPa |

Water Depth [4,17] | D_{W} | 3 m | Sampling Frequency | F_{S} | 150 kHz in base band 750 kHz in pass band |

Water Sound Speed [4,17] | C | 1,500 m/s | Symbol Rate | D_{sym} | 75 kHz |

Water Density [4,17] | ρ | 1,023 kg/m^{3} | Carrier Frequency | f_{0} | 0 kHz in base band 300 kHz in pass band |

Sandy Sediment Sound Speed [4,17] | C_{b} | 1,800 m/s | MIMO Sequence Duration | τ_{h} | 218.5 ms |

Sandy Sediment Density [4,17] | ρ_{b} | 1,800 kg/m^{3} | Dead-Time Duration | τ_{msg} | 300 ms |

Sea Bottom Loss [4,17] | L_{SB} | 5 dB | Correlation Threshold Parameter | K_{thr} | 20 |

Beginning of Time Window [4,17] | k_{0} | 0 | Time Window Length | τ_{win} | 518.5 ms |

Number of Transmitters [4,17] | N_{t} | 2 | Number of Receivers | N_{r} | 2 |

Distance Source 1 Receiver 1 [4,17] | R_{11} | 23.3 m | Distance Source 1 Receiver 2 [4,17] | R_{12} | 24 m |

Distance Source 2 Receiver 1 [4,17] | R_{21} | 25.8 m | Distance Source 2 Receiver 2 [4,17] | R_{22} | 27 m |

#### 3.3. Performance Metrics

_{MIMO_LE}is compared to another relative RMSE between emitted and raw received signals. This second metric is calculated by comparing the received MIMO header signal (prior to any interference removal) and the corresponding source signal [17]:

#### 3.4. MIMO Channel Estimation Results

_{CE}as a function of T

_{L}for both experimental and simulated data. The maximum value of T

_{L}is 5.33 ms, as higher values of T

_{L}lead to singularities and does not produce accurate results. As a reminder, RMSE

_{CE}measures the error between the received signals and the emitted sequences convolved with the estimated channels. RMSE

_{CE}is averaged across every receiver, so that the results presented in Table 5 translate the accuracy of the channel estimation across every sub-channel.

T_{L}(ms) | Simulated RMSE_{CE} (dB) | Experimental RMSE_{CE} (dB) |
---|---|---|

0.667 | −9.5 | −0.4 |

1.333 | −12.5 | −0.9 |

2.0 | −18.8 | −4.1 |

2.667 | −34.7 | −4.9 |

3.333 | −34.7 | −6.2 |

4.0 | −34.7 | −9.1 |

4.667 | −34.8 | −11.4 |

5.33 | −34.8 | −25.7 |

_{L}increases, the accuracy of the channel estimation improves. However, while RMSE

_{CE}reaches a sweet-spot at T

_{L}= 2.667 ms using simulated data ( ), the experimental results on RMSE

_{CE}differ. Indeed, the minimum RMSE

_{CE}is obtained for T

_{L}= 5.33 ms (RMSE

_{CE}= −25.7 dB) as shown in Table 5 and Figure 5. If, as explained earlier on, values higher than T

_{L}= 5.33 ms cannot be considered, the minimum value of RMSE

_{CE}using experimental data is of the same order of magnitude as RMSE

_{CE}in the simulation framework. The channel estimation can therefore be considered as very accurate.

_{L}on the channel estimation accuracy in the specific case of mission 4. Figure 6 shows that RMSE

_{CE}drops and the confidence interval [−σ

_{CE};σ

_{CE}] gets narrower as T

_{L}increases.

_{CE}at the output of the channel estimator (Equation (28)) is the inverse of the SNR. In this ideal case, the relationship can be rewritten in dB,

_{CE}(dB)

_{CE}(dB) (Equation (28)) obtained for each of the corresponding four missions and for each receiver. In theory, when the best possible estimation of the channel impulse response is calculated, Equation (32) applies, as shown using a solid line in Figure 6. The experimental results, labeled as individual points in Figure 6, remain very close to this theoretical limit, which indicates that the channel estimation algorithm works very well indeed. For example, the channel estimation in the data set recorded during in mission 2 at receiver 2 results in a value of RMSE

_{CE}(dB) that is almost exactly the opposite of SNR(dB). Some discrepancies are also observed, as in mission 3 at receiver 2. In this case, the channel estimator produces significant amounts of additive noise.

#### 3.5. MIMO Deconvolution Results

_{L}on the performance metrics is clearly observed. The results are shown for , which lead to the lowest RMSE values [17]. Both simulations and experimental results (averaged over the total number of missions carried out) are presented in Table 6. Clearly, RMSE

_{MIMO}_

_{LE}and RMSE

_{MIM}

_{O}_

_{ICLE}decrease as T

_{L}increases using either simulated or field data. Note that the equalizer length is limited to T

_{L }= 5.33 ms, as higher values of T

_{L}led to singularities.

_{MIMO}_

_{LE}, computed with both simulated and experimental data, is shown in Figure 7. Figure 7 shows the influence of the pre-cursor and post-cursor length: if T

_{L}is sufficiently large to provide accurate channel estimation, RMSE

_{MIMO_LE}drops as T

_{k}increasing. Therefore, showing the influence of T

_{L}on RMSE

_{MIMO}_

_{LE}is not of great interest: this is why we chose to represent RMSE

_{MIMO}_

_{LE}as a function of T

_{k}only (T

_{L}= 5.33 ms). For example, for an equalizer length T

_{L }= 5.33 ms, RMSE

_{MIMO_LE}= −6.2 dB with T

_{k}= 0.7 ms and drops to RMSE

_{MIMO_LE}= −20.5 dB with T

_{k}= 6.0 ms. As it has been shown in the channel estimation process results, the confidence interval also narrows as the process gets more and more reliable.

T_{L} (ms) | RMSE
_{MIMO_Raw} (dB) | |
---|---|---|

Simulations | Experiments | |

0.667 ms, 1.333 ms, 2 ms, 2.667 ms, 3.333 ms, 4 ms, 4.667 ms, 5.33 ms | 0.04 | 3 |

**Table 7.**RMSE between emitted and received Signals after LE processing, RMSE

_{MIMO_LE}and after ICLE processing, RMSE

_{MIMO_ICLE}as functions of T

_{L}.

RMSE
_{MIMO_Raw} (dB) | RMSE
_{MIMO_ICLE} (dB) | ||||
---|---|---|---|---|---|

Simulations | Experiments | Simulations | Experiments | ||

T_{L }(ms) | 0.667 | 0.4 | 19.7 | −4.5 | 15.8 |

1.333 | −3.4 | 14.4 | −6.7 | 12.1 | |

2.0 | −10.3 | 8.3 | −13 | 6.4 | |

2.667 | −20.5 | 5.5 | −33.4 | 3.1 | |

3.333 | −20.5 | 2.2 | −33.4 | 0.7 | |

4.0 | −20.5 | −1.9 | −33.3 | −3 | |

4.667 | −20.5 | −3.9 | −33.3 | −8.8 | |

5.33 | −20.5 | −3.3 | −33.2 | −26.9 |

_{k}= 6.0 ms varies dramatically when computed with simulation and experimental data. In the best case scenario, we use T

_{L }= 5.33 ms and T

_{k}= 6 ms. Table 7 shows that RMSE

_{MIMO}_

_{LE}= −3.9 dB using real data vs. RMSE

_{MIMO}_

_{LE}= −20.5 dB using simulated data. Nevertheless, the LE process on experimental data dramatically improves the quality of the received signal: Table 6 shows that RMSE

_{MIMO}_

_{Raw}= 3 dB whereas Table 7 shows that RMSE

_{MIMO}_

_{LE}= −3.3 dB when T

_{L }= 5.33 ms.

_{MIM}

_{O}_

_{ICLE}as a function of T

_{L}: if the MIMO sequence is known, the accuracy of the process improves as T

_{L}increases. On average, RMSE

_{MIM}

_{O}_

_{ICLE}= −26.9 dB when T

_{L }= 5.33 ms using experimental data vs. −33.2 dB using simulated data.

_{MIM}

_{O}_

_{ICLE}and RMSE

_{MIMO}_

_{LE}calculated using simulated and experimental data reveals that LE process does not reach the lower bound provided by ICLE structure. Using simulation data, we find that RMSE

_{MIM}

_{O}_

_{ICLE}= −33.2 dB while RMSE

_{MIMO}_

_{LE}= −20.5 dB. This phenomenon is especially pronounced in the case of experimental data, where RMSE

_{MIM}

_{O}_

_{ICLE}= −26.9 dB and RMSE

_{MIMO}_

_{LE}= −3.3 dB for T

_{L}= 5.33 ms. One can conclude that LE alone is not totally sufficient to remove the whole interference terms provided by the frequency selective channel and multi-antenna architecture. Non-linear approaches like iterative processing strategy or decision feedback equalization appear thus necessary.

## 4. Conclusions

_{L}. Computer simulations showed that for T

_{L}≥ 2.667 ms, the relative root mean-square error used to measure the accuracy of the estimation reached a plateau. In this configuration, the RMSE (labeled RMSE

_{CE}(dB)) between the received MIMO header and the MIMO header convolved with the estimated channel (averaged across every message and every receiver) was equal to −34.8 dB. This performance was compared to experimental data: in this case the same metric was equal to −25.7 dB for T

_{L}= 5.33 ms, indicating that the proposed technique evaluated fairly accurately the acoustic channel between every source and receiver.

_{L}= 5.33 ms and T

_{k}= 6 ms, the RMSE was estimated at 0.03 dB before equalization (labeled RMSE

_{MIMO_Raw}(dB)), −20.5 dB after LE (labeled RMSE

_{MIMO_LE}(dB)) and −33.3 dB after ICLE (labeled RMSE

_{MIMO_ICLE}(dB)). For experimental data (with the same values for T

_{L}and T

_{k}), the RMSE was estimated at −3 dB before equalization, −3.3 dB after LE and to −26.9 dB after ICLE.

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Real, G.; Beaujean, P.-P.; Bouvet, P.-J. MIMO Underwater Acoustic Communications in Ports and Shallow Waters at Very High Frequency. *J. Sens. Actuator Netw.* **2013**, *2*, 700-716.
https://doi.org/10.3390/jsan2040700

**AMA Style**

Real G, Beaujean P-P, Bouvet P-J. MIMO Underwater Acoustic Communications in Ports and Shallow Waters at Very High Frequency. *Journal of Sensor and Actuator Networks*. 2013; 2(4):700-716.
https://doi.org/10.3390/jsan2040700

**Chicago/Turabian Style**

Real, Gaultier, Pierre-Philippe Beaujean, and Pierre-Jean Bouvet. 2013. "MIMO Underwater Acoustic Communications in Ports and Shallow Waters at Very High Frequency" *Journal of Sensor and Actuator Networks* 2, no. 4: 700-716.
https://doi.org/10.3390/jsan2040700