research article symbol estimation algorithm for mimo

Research ArticleSymbol Estimation Algorithm for MIMO Underwater AcousticCommunication System Based on Multiplicative Noise Model

Ling Zhang,1 Ming Li,1 Gangsheng Li,2 and Rui Wang1

1College of Engineering, Ocean University of China, Qingdao 266100, China2Department of Education, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Ming Li; [email protected]

Received 25 September 2014; Accepted 22 January 2015

Academic Editor: Hsuan-Ling Kao

Copyright © 2015 Ling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The stochastic and time-varying underwater acoustic (UWA) channels are usually affected by serious multipath delays, energyloss and distortion factors, thus making the modeling and estimation of the UWA channel challenging problems in the researchcommunity. Based on the analysis of the UWA channel, the system with multiplicative noise (SMN) model is established tocharacterize the complicated factors such as random time-variation, nonlinearity, and energy attenuation. As to the multiple-inputmultiple-output (MIMO) UWA communication, the complicated SMN model is established for MIMO UWA channels; basedon which, the transmitted symbols are estimated according to the optimal recursive filtering algorithm. The algorithm is derivedbased on the projection theorem, which is optimal in the sense of linear minimum variance, and can overcome the intersymbolinterference andnoise pollution efficiently.Theoptimal algorithm is computed recursively, which has the advantage of computation-efficiency and can track the random variation of the fast time-varying channel gain dynamically. Simulation results have validatedthe effectiveness of the algorithm. The model and the algorithm can be extended flexibly to certain practical problems, such as thejoint channel and symbol estimation in underwater acoustic communication systems.

1. Introduction

Underwater acoustic (UWA) communication usually worksin very complex transmission environment with seriouslynoisy background and very limited bandwidth. In addi-tion, the complex multipath propagation and time-frequencyselective fading are prone to causing serious distortion ofUWA communication signals, which set obstacles to thehigh-speed and reliable transmission of UWA communica-tion. Hence, how to characterize the underwater channel andestimate the transmitted symbol have become the focus ofcurrent research [1–4].

In recent years, there is an increasing demand for highdata rate underwater acoustic communication. Thus, someadvanced and sophisticated technologies from land wirelesscommunication are introduced into the field of underwatercoherent communication, such as multiple input multipleoutput (MIMO) technology [5–8] and orthogonal frequency

division multiplexing (OFDM) technology [9–11]. MIMOtechnology especially fully exploits space resources and trans-mits signals from various transmitting and receiving appara-tus occupying the same frequency band, without additionalreceiver transmitting power and spectrum resource; thereforeit can greatly increase channel capacity and reduce the biterror rate (BER) of system.

Underwater acoustic channel is an integral part of UWAcommunication system. Considering the various constraintsand the time-space-frequency variability of the channel,UWA channel is muchmore complex than the radio channel.Usually, there exist reflection and refraction in the propa-gation of acoustic wave in seawater as marine environmentis inhomogeneous, which results in a lot of acoustic sig-nals superimposed in the receiving end and the so-calledmultipath effect leads to intersymbol interference (ISI) andsignal amplitude fading. Currently, the adaptive equalizationtechnology is commonly used to overcome this interference

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 719025, 7 pageshttp://dx.doi.org/10.1155/2015/719025

2 Mathematical Problems in Engineering

[12–15]. But the equalization algorithm cannot perform wellunder complex and random time-varying communicationenvironment.

Recently, the research on themultiplicative noise in UWAcommunication has become a hot spot [16–19] because themultiplicative noise can describe many factors in practicalsystems such as nonlinearity, packet dropout, and frequencyoffset. Considering the various kinds of interference in UWAchannel, it can be characterized by a system with multi-plicative noise (SMN) model. In [20, 21], the UWA channelis seen as SMN with additive and multiplicative noises,and the underwater target is regarded as a special scatterer.Therefore, the problem of underwater target detection canbe converted into the estimation of the scattering coefficientsequence constituted by the target and the scatterer nearby.Conventionally, the SMN model is applied in single inputmultiple output (SIMO) UWA system and seldom considersthe MIMO system. This paper focuses on the modeling ofMIMO UWA channel and derives the optimal estimationalgorithm based on the new model, which can be applied formore complex UWA communication system.

In this paper, based on the characteristics of the UWAchannel, the multiplicative noise is introduced to model theMIMO UWA communication system in order to track thefast time-varying UWA channel dynamically; that is, a novelstate-space model with multiplicative noise is establishedto describe the channel variation and achieve effective esti-mations of the transmitted symbols. Based on the systemmodel, the optimal recursive filtering algorithm is derivedfor complicated multichannel system with multiplicativenoise in the form of general stochastic matrix, which cansolve the optimal estimation problem of systems undermore complicated environment. The filtering algorithm isoptimal under the linear minimum variance criterion and iscomputed recursively, which is suitable for high speed UWAcommunication. In the modeling process, the transmittedsymbols are included in the state vector.Therefore, accordingto this correspondence between the state vector in the systemmodel and the transmitted signals from various transmitters,the symbol detection problem can be converted into thefiltering estimation of the system state.

Thepaper is arranged as follows.The SMNmodel is estab-lished to characterize the MIMO UWA channel in Section 2.In Section 3, the symbol estimation algorithm is developedfor MIMO UWA system.The performance of the establishedmodel and the present algorithm is analyzed in Section 4.Simulation results are shown to validate the algorithm inSection 5. Finally, conclusions are drawn in Section 6.

2. MIMO UWA Channel ModelingBased on SMN

For an MIMO UWA communication system employing𝑁 transducer sources and 𝑀 hydrophone receivers,

the discrete-time baseband equivalent signal received at the𝑖th receiver is expressed by

𝑧𝑖 (𝑘) =

𝑁

∑

𝑗=1

𝐿−1

∑

𝑙=0

𝑠𝑗 (𝑘 − 𝑙) ℎ𝑗,𝑖 (𝑘, 𝑙) + V𝑖 (𝑘) , (1)

where 𝑠𝑗(𝑘) is the transmitted symbol from the 𝑗th transducer

at time instant 𝑘; ℎ𝑗,𝑖(𝑘, 𝑙) is the 𝑙th fading channel coefficient

between the 𝑗th transducer and the 𝑖th receiver at time 𝑘; 𝐿 isthe maximum length of channels; and V

𝑖(𝑘) is the zero-mean

additive white Gaussian noise on the 𝑖th receiver.If the UWA channel has ideal transmission characteris-

tics, the slowly varying and stable compositions will be thedominant channel characteristics [22]; thereby the channelparameters remain unchanged during a packet or even fewpackets of data; that is, ℎ

𝑗,𝑖(𝑘, 𝑙) ≈ ℎ

𝑗,𝑖(𝑙). Thus (1) can be

rewritten as

𝑧𝑖 (𝑘) =

𝑁

∑

𝑗=1

𝐿−1

∑

𝑙=0

𝑠𝑗 (𝑘 − 𝑙) ℎ𝑗,𝑖 (𝑙) + V𝑖 (𝑘) . (2)

Otherwise, for underwater remote communication sys-tems, UWA channel usually has more complex transmissioncharacteristics [16]. Signals from transmitters along differentpaths arrive at the receiving end at different times. There-fore, the received signal is the superposition of all thesesignals, causing waveform distortion. This effect of UWAchannel includes not only marine environment noise andtarget radiation noise as additive interferences, but also avariety of distortions and information energy attenuation asmultiplicative interferences due to the random time-varyingcharacteristics of UWA channel. Broadly speaking, those fac-tors that bring the received signal multiplicative interferencecan be characterized by themultiplicative noise.Therefore, inthis paper, based on the random characteristics of the UWAchannel, the multiplicative noise is applied to characterizethe complicated and time-varying UWA channel, and theconvolution model with multiplicative noise in the receivedsignal can be expressed as

𝑧𝑖 (𝑘) =

𝑁

∑

𝑗=1

𝐿−1

∑

𝑙=0

𝑠𝑗 (𝑘 − 𝑙)𝑚𝑖,𝑙+1 (𝑘) ℎ𝑗,𝑖 (𝑙) + V𝑖 (𝑘)

=

𝑁

∑

𝑗=1

𝑠𝑗 (𝑘)𝑚𝑖,1 (𝑘) ℎ𝑗,𝑖 (0) + 𝑠𝑗 (𝑘 − 1)𝑚𝑖,2 (𝑘) ℎ𝑗,𝑖 (1)

+ ⋅ ⋅ ⋅ + 𝑠𝑗 (𝑘 − 𝐿 + 1)𝑚𝑖,𝐿 (𝑘) ℎ𝑗,𝑖 (𝐿 − 1) + V𝑖 (𝑘) ,

(3)

where themultiplicative noise {𝑚𝑖𝑗(𝑘), 1 ≤ 𝑖 ≤ 𝑀, 1 ≤ 𝑗 ≤ 𝐿}

is of Gaussian distribution with the mathematic expectationE{𝑚𝑖𝑗(𝑘)} = 𝑚

𝑖𝑗(𝑘) and known covariance.

From (3), let

x (𝑘) = [𝑥1 (𝑘) ⋅ ⋅ ⋅ 𝑥𝐿 (𝑘) , 𝑥𝐿+1 (𝑘) ⋅ ⋅ ⋅ 𝑥2𝐿 (𝑘) , . . . ,

𝑥(𝑁−1)𝐿+1 (𝑘) ⋅ ⋅ ⋅ 𝑥𝑁𝐿 (𝑘)]

𝑇,

(4)

Mathematical Problems in Engineering 3

𝑠1 (𝑘) = 𝑥1 (𝑘) , . . . , 𝑠1 (𝑘 − 𝐿 + 1) = 𝑥𝐿 (𝑘) ,

𝑠2 (𝑘) = 𝑥𝐿+1 (𝑘) , . . . , 𝑠2 (𝑘 − 𝐿 + 1) = 𝑥2𝐿 (𝑘)

.

.

.

𝑠𝑁 (𝑘) = 𝑥(𝑁−1)𝐿+1 (𝑘) , . . . , 𝑠𝑁 (𝑘 − 𝐿 + 1) = 𝑥𝑁𝐿 (𝑘) .

(5)

For𝑁 ×𝑀 underwater acoustic communication system,define

z (𝑘) = [𝑧1 (𝑘) 𝑧2 (𝑘) ⋅ ⋅ ⋅ 𝑧𝑀 (𝑘)]𝑇,

k (𝑘) = [V1 (𝑘) V2 (𝑘) ⋅ ⋅ ⋅ V𝑀 (𝑘)]

𝑇,

w (𝑘) = [𝑤1 (𝑘) 𝑤2 (𝑘) ⋅ ⋅ ⋅ 𝑤𝑁 (𝑘)]𝑇,

with 𝑤𝑗 (𝑘) = 𝑠𝑗 (𝑘 + 1) , 𝑗 = 1, 2, . . . , 𝑁.

(6)

From (3)–(6), we have the following SMNmodel:

x (𝑘 + 1) = Ax (𝑘) + Bw (𝑘) ,

z (𝑘) = U (𝑘) x (𝑘) + k (𝑘) ,(7)

where

A =

[[[[[[[[

[

A1

0 ⋅ ⋅ ⋅ 0

0 A2

d...

.

.

. d d 00 ⋅ ⋅ ⋅ 0 A

𝑁

]]]]]]]]

]

, A𝑗=

[[[[[[

[

0 0 ⋅ ⋅ ⋅ 0

1 0 ⋅ ⋅ ⋅ 0

.

.

. d d...

0 ⋅ ⋅ ⋅ 1 0

]]]]]]

](𝐿×𝐿)

,

B =

[[[[[[[[

[

B1

0 ⋅ ⋅ ⋅ 0

0 B2

d...

.

.

. d d 00 ⋅ ⋅ ⋅ 0 B

𝑁

]]]]]]]]

]

, B𝑗=

[[[[[[

[

1

0

.

.

.

0

]]]]]]

](𝐿×1)

,

E {k (𝑘)} = 0, E {k (𝑘) kT (𝑙)} = R (𝑘) 𝛿𝑘𝑙, with 𝛿𝑘𝑙={

{

{

1 𝑘 = 𝑙

0 𝑘 = 𝑙,

U (𝑘) = [U1 (𝑘) U2 (𝑘) ⋅ ⋅ ⋅ U𝑁 (𝑘)] ,

U𝑗 (𝑘) =

[[[[[[[[

[

𝑚11 (𝑘) ℎ𝑗,1 (0) 𝑚

12 (𝑘) ℎ𝑗,1 (1) ⋅ ⋅ ⋅ 𝑚1𝐿 (𝑘) ℎ𝑗,1 (𝐿 − 1)

𝑚21 (𝑘) ℎ𝑗,2 (0) 𝑚

22 (𝑘) ℎ𝑗,2 (1) ⋅ ⋅ ⋅ 𝑚2𝐿 (𝑘) ℎ𝑗,2 (𝐿 − 1)

.

.

....

.

.

....

𝑚𝑀1 (𝑘) ℎ𝑗,𝑀 (0) 𝑚

𝑀2 (𝑘) ℎ𝑗,𝑀 (1) ⋅ ⋅ ⋅ 𝑚𝑀𝐿 (𝑘) ℎ𝑗,𝑀 (𝐿 − 1)

]]]]]]]]

](𝑀×𝐿)

.

(8)

3. Symbol Estimation Algorithm of MIMOUWA Communication System

During the process of SMN modeling, we can see that𝑤𝑗(𝑘 − 1) = 𝑠

𝑗(𝑘) = 𝑥

(𝑗−1)𝐿+1(𝑘), 𝑗 = 1, 2, . . . , 𝑁. For

binary phase shift keying (BPSK) modulation system, thetransmitted symbol is taken independently and equally from{1, −1}; hence, it is clear that the mean of 𝑤

𝑗(𝑘) is 0 and the

variance is 1. Based on the SMNmodel (7), we can obtain thestate estimation according to the optimal filtering algorithm.Because of the correspondence between the transmittedsignals of𝑁 transducer sources and the state vector x(𝑘), theestimation of the transmitted signals can be acquired easily.For convenience of understanding, the algorithm is listed asfollows.

TheOptimal Recursive Filtering Algorithm for SMN. Consider

x (𝑘/𝑘 − 1) = Ax (𝑘 − 1/𝑘 − 1) ,

x (𝑘/𝑘) = x (𝑘/𝑘 − 1)

+ K (𝑘) [z (𝑘) −M (𝑘) x (𝑘/𝑘 − 1)] ,

P (𝑘/𝑘 − 1) = AP (𝑘 − 1)AH+ BQBH

,

P (𝑘) = [I − K (𝑘)M (𝑘)]P (𝑘/𝑘 − 1) ,

K (𝑘) = P (𝑘/𝑘 − 1)MH(𝑘)R−1𝐿(𝑘) ,


S (𝑘) = AS (𝑘 − 1)AH+ BQBH

,

R𝐿 (𝑘) = F (𝑘) +M (𝑘)P (𝑘/𝑘 − 1)MH

(𝑘) + R (𝑘) ,

F (𝑘) = E {(U (𝑘) −M (𝑘)) x (𝑘) xH (𝑘) (U (𝑘) −M (𝑘))H} ,

𝑓𝑡𝑙 (𝑘) = E

{

{

{

𝑁

∑

𝑛=1

𝑁

∑

𝑗=1

𝐿

∑

𝑚=1

𝐿

∑

𝑖=1

Δ𝑡𝑖 (𝑘) Δ 𝑙𝑚 (𝑘)

⋅ 𝑎(𝑗−1)𝐿+𝑖,(𝑛−1)𝐿+𝑚 (𝑘)

⋅ ℎ𝑗,𝑡 (𝑖 − 1) ℎ𝑛,𝑙 (𝑚 − 1)

}

}

}

,

(9)

where (⋅)H denotes Hermitian transpose of a vector ormatrix,𝑓𝑡𝑙(𝑘) is the (𝑡, 𝑙)th element of matrix F(𝑘), and x(𝑘)xH(𝑘) =

{𝑎𝑖𝑗(𝑘)}𝑁𝐿×𝑁𝐿

;

M (𝑘) = [M1 (𝑘) M2 (𝑘) ⋅ ⋅ ⋅ M𝑁 (𝑘)] , with

M𝑗 (𝑘) =

[[[[[[[[

[

𝑚11 (𝑘) ℎ𝑗,1 (0) 𝑚

12 (𝑘) ℎ𝑗,1 (1) ⋅ ⋅ ⋅ 𝑚1𝐿 (𝑘) ℎ𝑗,1 (𝐿 − 1)

𝑚21 (𝑘) ℎ𝑗,2 (0) 𝑚

22 (𝑘) ℎ𝑗,2 (1) ⋅ ⋅ ⋅ 𝑚2𝐿 (𝑘) ℎ𝑗,2 (𝐿 − 1)

.

.

....

.

.

....

𝑚𝑀1 (𝑘) ℎ𝑗,𝑀 (0) 𝑚

𝑀2 (𝑘) ℎ𝑗,𝑀 (1) ⋅ ⋅ ⋅ 𝑚𝑀𝐿 (𝑘) ℎ𝑗,𝑀 (𝐿 − 1)

]]]]]]]]

](𝑀×𝐿)

;

E {𝑚𝑖𝑗 (𝑘)} = 𝑚𝑖𝑗 (𝑘) ; Δ

𝑖𝑗 (𝑘) = 𝑚𝑖𝑗 (𝑘) − 𝑚𝑖𝑗 (𝑘) .

(10)

4. Performance Analysis

4.1. Optimality. The recursive filtering algorithm proposed inthis paper is based on the well-known projection theorem[23], which is optimal in the sense of linear minimumvariance. Therefore, the theoretical performance of the esti-mation algorithm is guaranteed such as the optimality and theconvergence rate, which will be illustrated by the simulationexamples in the next section. Furthermore, the model andthe corresponding algorithm can be more flexibly extendedto solve some practical systems under complex environmentinterfered by multiplicative noises, such as the symbol esti-mation problem of the MIMO UWA communication systemwith packet dropout and the estimation of Doppler frequencyoffset in MIMO OFDM systems.

4.2. Recursive Computation. The algorithm is computedrecursively with computational complexity 𝑂(𝑁2), where 𝑁is the number of the transducers, that is, the dimension of thestate vector. Since the algorithm has low computation cost, itcan track the fast time-varying UWA channel dynamically,which allows the UWA channel gain changing even over onesymbol interval, while the equalization algorithm cannot.Theconvergence rate of the proposed algorithm will be verifiedby the simulation in the next section. The above advantagesof the new algorithm will better adapt to the demand ofthe future high-speed underwater acoustic communicationsystem.

5. Simulation and Comparison

It is known that UWA channels change randomly overtime, so we have some simulation results for the optimalrecursive filtering algorithm when the multiplicative noise

U(𝑘) is a general stochastic matrix with every componentrelated to each other at the same time index. In addition,based on the characteristics of UWA channel, we assumethat it obeys Rayleigh distribution. Since the multiplicativenoise can describe the fading, nonlinearity, and distortion ofthe channel, it is reasonable to model the Rayleigh fadingchannel. The received signal obtained from the multipathRayleigh fading channel is estimated by the proposed optimalfiltering algorithm in this paper and the adaptive equalizationalgorithm from [12], respectively, and the BER performanceof these two algorithms is compared under the same signal tonoise ratio (SNR) in Figures 4, 5, and 6.

5.1. Simulation of the Proposed Optimal Filtering Algorithm.For 2 × 2 UWA communication system with 2 transmittersand 2 receivers, the least square (LS) estimation of thechannel impulse response can be obtained by the methodin [24]. Set 𝐿 = 5. Based on the characteristics of UWAchannel, we presume that the mathematic expectation ofthe multiplicative noise 𝑚

𝑖𝑗(𝑘) is 0.9; the variance and the

covariance of each component in U𝑗(𝑘) are 0.02.

Figure 1 depicts the plot of BER versus SNR by BPSKmodulation using the proposed algorithm to estimate trans-mitted signals from the two transducer sources. Figure 2shows BER versus SNR by quadrature phase shift keying(QPSK) modulation from the two transducer sources usingthe present algorithm.

From Figures 1 and 2, we can see that BER is about10−3 when SNR is 16 dB; therefore, the simulation resultsshow that the proposed optimal filtering algorithm of SMNcan effectively estimate the transmitted characters, and it issuitable for solving the symbol detection problem of MIMOUWA communication system.


0 2 4 6 8 10 12 14 16BPSK SNR (dB)

BER

BER for transducer 1BER for transducer 2

100

10−1

10−2

10−3

Figure 1: BER versus SNR of two transducer sources for BPSKmodulation.

0 2 4 6 8 10 12 14 16QPSK SNR (dB)

BER

BER for transducer 1BER for transducer 2

100

10−1

10−2

10−3

Figure 2: BER versus SNR of two transducer sources for QPSKmodulation.

5.2. Comparison with Other Related Algorithms. The lit-erature [25] describes the time-varying characteristics ofRayleigh fading channel from the perspective of UWAcommunication and utilizes a simplified Rayleigh statisticalmodel to simulate the random time-varying and fadingUWAchannel, which is a generally accepted UWA channel modelcompared with the actual underwater channel.

In this paper, UWA channel is taken as a Rayleigh fadingchannel with three paths; transmitted signals are modulated

0 1000 2000 3000 4000 5000−4

−3

−2

−1

0

1

2

3

4

Number of symbols

Am

plitu

de

Figure 3: Scatter plot of received baseband signals from Rayleighfading channel with three paths.

by BPSK; the carrier frequency is 5 kHz; the sampling fre-quency is 20 kHz; the symbol rate is 1 k baud; the multipathdelay vector is [0, 1, 2]ms; and the multipath power vectoris [0, −1, −9] dB. Set the channel parameters change every200 symbol intervals. The received signals obtained fromthis multipath Rayleigh fading channel are processed by theadaptive equalization algorithm from [12] and the proposedoptimal filtering algorithm in this paper, respectively. Therelevant parameters of the adaptive equalization algorithmcan be obtained from [12]. Since the equalization algorithm[12] is suitable for the single input single output (SISO) UWAcommunication system, set𝑁 = 1 and𝑀 = 1 in the proposedoptimal filtering algorithm. The following simulation resultscompare the performance of the optimal filtering algorithmand the equalization algorithm in terms of the optimality,convergence rate, and so forth.

Figure 3 is the scatter plot of received baseband signalsobtained from the Rayleigh fading channel model with threepaths.

Figures 4 and 5 are the scatter plots of signals processedby the adaptive equalization algorithm [12] and the proposedoptimal filtering algorithm, respectively.

Figure 6 depicts BER versus SNR of the two algorithmsfor comparison of the estimation performance.

From Figures 4 and 5, we can see that the severely aliasingreceived signals can be clearly separated by the adaptiveequalization algorithm [12] and the proposed optimal filter-ing algorithm, but the convergence rate of the proposed algo-rithm is faster than that of the equalization algorithm whichconverges after about 500 symbols as shown in Figure 4. FromFigure 6, we can see that BER of the proposed algorithmis lower than that of the equalization algorithm [12], whichconfirms the optimality of the proposed algorithm.Therefore,the effectiveness of the proposed optimal filtering algorithmfor the UWA communication system is validated.


0 1000 2000 3000 4000 5000−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Number of symbols

Am

plitu

de

Figure 4: Scatter plot of signals processed by the adaptive equaliza-tion algorithm.

0 1000 2000 3000 4000 5000−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Number of symbols

Am

plitu

de

Figure 5: Scatter plot of signals processed by the proposed optimalfiltering algorithm.

6. Conclusions

In this paper, the complicated and time-varying MIMOunderwater acoustic channel is characterized by an SMNmodel with a random general multiplicative noise matrix.Based on the system model, the estimation of UWA symbolsis derived through the optimal recursive filtering algorithmof SMN. The algorithm is optimal in the sense of linearminimum variance with recursive calculation process, whichprovides faster convergence rate and lower computationalcomplexity compared with the typically used equalizationalgorithm in underwater acoustic communication. The pro-posed recursive optimal filtering algorithm can track therandom time-varying UWA channels dynamically; therebythe SMN channel model in association with the optimalfiltering algorithm couldmeet the requirements of high speedUWA communication system in complex environment. Sim-ulation results show that the proposedmethod can effectively

0 2 4 6 8 10 12SNR (dB)

BER

Optimal filtering algorithmAdaptive equalization algorithm

100

10−1

10−2

10−3

Figure 6: BER versus SNR of the equalization algorithm [12] andthe optimal filtering algorithm.

estimate the transmitted symbols corrupted by the stochasticmultiplicative noise in complex communication environmentwith random time-varying UWA channels. Furthermore,the algorithm can be flexibly extended for solving the jointchannel and symbol estimation problem of MIMO UWAcommunication system contaminated by the multiplicativeinterference. Overall the SMN model has higher practicalmerits and wider range of applications.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This project is supported by the National Natural ScienceFoundation of China (Grants nos. 61132005, 51279185, and61401111) and ShandongNatural Science Foundation of China(Grant no. ZR2010DQ003).

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