Target detection method for reverberant environments in continuous-wave
active sonar system based on group multichannel nonnegative matrix
factorization
Seokjin LEE(1)(1) Kyungpook National University, Republic of Korea, sjlee6@knu.ac.kr

Abstract
In the underwater environment, the active sonar system emits an acoustic wave and receives echo signals todetect a target. However, the echo signals consist of not only the echo from the target but also the reverberationfrom scatterers. If the target has a low speed, the detection problem is more difficult because the target echo
is interfered by the reverberation severely. In this paper, the target detection method for the reverberant environment is developed based on multichannel nonnegative matrix factorization. To extract the target echo signal
mixed with reverberations, received signal is transformed into bearing-time-frequency domain, and analyzed
into bearing, frequency, and temporal basis by the multichannel nonnegative matrix factorization. The simulation was performed to evaluate the proposed method, and the result shows that the proposed method has good
detection performance in the simulated reverberant environments.
Keywords: Active Sonar, Reverberation, Nonnegative Matrix Factorization, Target Detection

1INTRODUCTION

In the underwater acoustics systems, the active sonar emits a ping signal and analyzes the reflected signal
to detect the target. Because the active sonar system utilizes the emitted ping, the system has advantage toestimate the bearing and range of the target. However, the emitted ping signal also causes a problem: thereverberation from scatterers degrades the detection performance. The conventional target detection algorithm
including the matched filter is sensitive to the reverberation, so several researches have been conducted toimprove the detection performance of the active sonar system [1, 2, 3, 4, 5, 6, 7].
Because the target moves with a specific speed, we can classify the targets into two categories with respect
to the Doppler shift: the high-Doppler and the low-Doppler targets. Detections of the high-Doppler targets are
not severely affected by the reverberation because the beamformers can deal with the problem, but the reflectedecho from the low-Doppler targets are easily interfered by the reverberations [1], so additional algorithms isrequired to tackle the problem. Some researchers designed the transmission pulse to robust to the reverberation[2, 3]. While these approaches are effective, designed new pulses may be incompatible with the conventional
systems. Therefore, some signal processing algorithms have been developed to suppress the reverberation at
the receiver, including auto-regressive (AR) pre-whitening[4], principal component inverse (PCI)[5], and signal
subspace extraction (SSE)[6]. The AR pre-whitening algorithm shows good performances in the simulation, but
the performance is degraded if the model does not match the actual signal environment. The PCI and SSE
algorithms can be good alternatives, but the algorithms focus on the low-rank decomposition only and does not
utilize a signal characteristic.
Recently, a reverberation suppression algorithm based on the nonnegative matrix factorization (NMF), which
utilizes a time-frequency characteristic of the transmitted and received ping, has been developed [7]. The NMF
decomposes a nonnegative matrix into a multiplication of two nonnegative matrices [8], and the method candecompose a magnitude spectrogram of an acoustic signal into frequency basis and temporal basis matrices[9].
The NMF-based reverberation suppression algorithm has been developed to take into account the fact that theNMF algorithm can analyze the time and frequency characteristics. The algorithm shows a good performance
relative to the other signal processing algorithms, but the algorithm only utilizes the time and frequency char-5400

1+

2$
%

1−

2$%

Mainlobe
Sidelobe

OutsideFigure 1. Simulated reverberation of a frequency-bearing domain. The receiver is assumed to use beamformers.
The x-axis denotes a bearing and the y-axis denotes a relative frequency.
acteristics, and it does not use any inter-channel information e.g. bearing information.
In this paper, we develop a target detection method in the reverberant signal environment, especially for a lowDoppler target with transmitting a continuous-wave ping signal. The proposed algorithm is developed by using
the beamspace-domain multichannel NMF (BD-MC-NMF) [10], and dividing the basis vectors into two basis
groups – echo and reverberation basis groups.

2PROPOSED METHOD

2.1 Problem Description
If we assume the sonar receiver moves with the velocity of v and receives a reflected ping sT (t) which iscaused by the transmitted ping s(t) from the transmitter, the received signal can be expressed as [11]sT (t) = a |G (ψ − ψ0 )| s (t − td ) exp( j2π fd t),(1)where a is an attenuation factor, td is a time delay, fd is a Doppler frequency shift, ψ is an angle between thesignal incidence and the receiver moving direction, and G(ψ) is a directivity function of the beamformer. Thenthe spectrum ST ( f ) of the received signal is stated as [11]ST ( f ) = a |G (ψ − ψ0 )| S ( f − fd ) exp [− j2π ( f − fd )td ] ,(2)where S( f ) is a Fourier transform of s(t).
The reverberation can be modeled as a sum of several reflections from a large number of scatterers, so thespectrum SR ( f ) of reverberation can be expressed as [11]SR ( f ) = ∑ {an |G(ψn − ψ0 )| S ( f − fdn ) exp [− j2π ( f − fdn )tdn ]}(3)nwhere an , fdn , and tdn are the attenuation factor, Doppler frequency shift, and time delay of n-th scatterer,respectively. It can be assumed that the velocities of the scatterers are very small relative to the velocity of thesonar receiver, or the ship, so the Doppler shift from each scatterer is
fdn =2v cos ψnf0 ,c(4)5401

Figure 2. A block diagram of the proposed algorithm.
where f0 is a center frequency of the transmitted ping. Therefore, the incidance angle of the reverberation with
the Doppler frequency shift of fdn isc f dn
ψn = cos−1.
(5)
2v f0
Figure 1 shows that the magnitude of simulated reverberation in a frequency-bearing domain. It is clearly shownthat the reverberation can interfere the detection of low-Doppler targets because the energy of reverberation
is distributed around the zero-Doppler frequency. Considering −1 ≤ cos ψn ≤ 1, equation (4) shows that the2v
Doppler frequency of reverberation has the value from − 2v
c f 0 to c f 0 , and Figure 1 shows that the range of thereverberation. Therefore, the faster the receiver moves, the more difficult it is to detect the low-Doppler target.
2.2 Basis-Group Beamspace-Domain Multichannel Nonnegative Matrix Factorization
The proposed algorithm consists of NMF basis estimation and basis selection modules, as shown in Figure 2.
The basis groups consist of bearing, frequency, and time bases, and each basis is estimated by beamspacedomain multichannel NMF algorithm. If we gather the magnitude spectrogram of the output signals from thebeamformers, the element vm,kn of the 3-dimensional spectrogram tensor V ∈ RM×K×N is [10, 12]vm,kn = ∑ um j p j,kn + em,kn ,(6)jwhere um j is a gain for m-th beamformer output of j-th source, em,kn is a noise of (k, n)-th time-frequency binof a spectrogram of m-th beamformer output, and p j,kn is a (k, n)-th bin of a spectrogram P j of j-th source.
p j,kn can be expressed as
p j,kn = ∑ wkr hrn ,(7)r∈C jwhere wkr is a (k, r)-th element of a frequency basis matrix W, and hrn is a (r, n)-th element of a time basis
matrix H. If we divide the basis matrix to the echo and reverberation basis groups, equation (6) can be modified
as [12]
JRvm,kn = ∑ um j p j,kn +j=1

JR +JE∑ um j p j,kn + em,kn ,j=JR +1

where JR and JE are number of basis vectors in the reverberation and echo basis groups, respectively.

5402(8)

2.3 Estimation of the reverberation basis group
In order to develop the estimation algorithm of the NMF bases, the reconstructed spectrogram v̂m,kn is defined
asJR +JE

JRv̂m,kn = ∑ um j p j,kn +j=1∑ um j p j,kn .

(9)j=JR +1

The estimation algorithms of um j , wkr , and hrn have to minimize the distance between vm,kn and v̂m,kn . Therefore, we define the cost function C(Θ) with the distance measure asC(Θ) = d(vm,kn |v̂m,kn ),(10)where Θ = um j , wkr , hrn . The distance measure is defined by the Itakura-Saito divergence as
 xx
− 1.
d(x|y) = − logyy(11)In order to derive the update equation for um j that minimize the cost function (10), the cost function is differentiated with respect to um j as [10]!
K Np j,kn v j,kn p j,kn
∂C,(12)−=∑∑∂ um j k=1v̂2m,knn=1 v̂m,kn
According to Lee and Seung [13], um j can be updated using equation (12) based on the multiplicative update
rule as
11×K V̂−2m ⊗ Vm ⊗ (W j H j ) 1N×1

um j ← um j,(13)11×K V̂−1m ⊗ (W j H j ) 1N×1
−2
where V̂−1m and V̂m are element-wise inverse and squared inverse of V̂m , respectively, ⊗ is an element-wise
multiplication (Hadamard product), the fraction is an element-wise division, and 1a×b is a a × b matrix whoseelements are ones. In the same way, wkr and hrn can be updated as [12] T−2∑M
m=1 um j V̂m ⊗ Vm H jWj ← Wj
,
(14)
−1 T
∑M
m=1 um j V̂m H jandTHj ← Hj

∑M
m=1 (um j W j )V̂−2m ⊗ Vm−1∑M
m=1 (um j W j ) V̂mT
.

(15)

2.4 Estimation of the echo basis group
The bearing and time bases, um j and hrn , can be updated by the same method as the reverberation basis groupcase. However, the frequency bases can be contructed in advance if we know the frequency structure of thetransmitted ping signal, because the received ping can be assumed as a Doppler-shifted replica of the transmitted
ping, as shown in equation (2). Therefore, the frequency bases are constructed in the proposed method as [12]W j = w p,↑{ j−JR −1−(JE −1)/2} ,JR + 1 ≤ j ≤ jE ,(16)where w j is a frequency spectrum of transmitted ping signal, and w p,↑∆k is a vector with w p shifted upward by∆k bins.

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2.5 Own-Doppler compensation
The NMF bases can be updated iterativly using equations (13)-(16). However, the estimated bearing basis tends
to have large value at the own-Doppler reverberation (which is denoted by "Mainlobe" in Figure 1). Fromequations (3) and (5), the beamformer gain of the own-Doppler reverberation isc ∆k j ∆ f−1
GOD (m, j) = Gm cos,(17)2v f0where ∆k j is a number of bins for the Doppler shift and ∆ f is a frequency resolution between the adjacentbins. Considering the own-Doppler gains, equation (13) for the echo basis group is modified asGOD (m, j)
um j ← αum j + (1 − α) 1 −,(18)maxm, j GOD (m, j)
where α is a tuning constant for the own-Doppler compensation whose value is between 0.0 and 1.0.
2.6 Basis selection
To detect the bearing, frequency, and the range of the target from the estimated NMF bases, we choose a properbasis which can represent the reflected echo from the target. As shown in the previous research [7], the energyof the reflected echo signal is concentrated in a short time period. Therefore, the target basis is selected by theenergy of continuous short duration as [12]r ∈ {CJR +1 , . . . ,CJR +JE } ,rtarget = arg max h̄(r),r(19)

where,
ih̄(r) = maxi

∑

hrn .

(20)

n=max(i−n p +1,1)

The target bearing number mtarget , target frequency bin number ktarget , and target time frames ntarget is calculatedasmtarget = arg max um j kC j 3rtarget ,(21)ktarget = arg max wkrtarget ,kntarget = nkhrtarget n > η ,

(22)

m(23)

where η is a pre-defined threshold to determine that target echo exists.

3SIMULATION RESULTS

To evaluate the proposed algorithm, a simulation was performed with a simulated reverberant signal environment. The target echo was simulated by a attenuated and delayed replica of transmitted ping signal, and thereceived reverberation signal was synthesized by the reverberation model of Abraham and Lyons[14]. The transmitted ping was a 500 ms continuous-wave signal with a Doppler frequency ratio ( f / f0 ) of 1.015. The system
was assumed to have 37 beamformers between 0◦ and 180◦ , and the null-to-null beamwidth of each beamformer
was set to 10◦ . The output signal of each beamformer was short-time-Fourier-transformed, as shown in Figure
2, by 133 ms Hamming window with 25% overlap.
The number of sources in the reverberation group JR was set to 10, and the number of basis per source in thereverberation group was set to 4. The number of sources in the echo group JE was set to 27. Because eachsource of the echo group has one basis vector, the total number of the basis vectors was 67. The own-Doppler
compensation constant α was set to 0.5, and the threshold for activation of target echo η was set asη = 0.2 max hrtarget n .
n5404

(24)

Figure 3. Results of estimated (a) bearing, (b) time, and (c) frequency basis matrices. Red doted rectangles
denotes the reverberation and echo basis groups, and the ground truth to be estimated. Green dashed line
denotes the selected basis rtarget by the proposed algorithm.

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Figure 4. Selected (a) bearing, (b) time, and (c) frequency basis vectors. Green dashed line denotes the groundtruth.
Figure 3 show results of the simulation. Figure 3(a) shows that the echo basis group includes the groundtruth of bearing, and the selected basis has a large value at the ground truth. Figure 3(b) shows that the echobasis matrix have distinctive large values in the ground truth frames, and proper time basis is selected. Figure3(c) shows that the echo basis group consists of the frequency-shifted replica of the transmitted ping, and theselected frequency basis indicates the ground truth well. Figure 4 show the selected basis vectors, and theresults show that the large values of each selected basis vector are matched to the ground truth.

4CONCLUSION

In this paper, a target detection method for a continuous-wave active sonar in the reverberant environment isdeveloped. The proposed target detection method is developed based on the BD-MC-NMF, and by utilizing apriori information of the received target echo. To evaluate the proposed algorithm, a simulation is performed
with a synthesized reverberation. The simulation results show that the proposed algorithm can estimate thebearing, time, and frequency basis matrices, and the target basis is selected well to detect the target echo.

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