Reverberation model which followed α-stable distribution

The reverberation probability density function has the same characteristics of a single peak, bell shape and thick tail as the SαS distribution, which is the most widely used and representative special distribution of the α-stable distribution. Therefore, the SαS distribution can be used to describe the statistical characteristics of underwater acoustic signal noise under a reverberation background [17].

In 1925, Levy proposed α-stable distribution which is also called non-Gaussian stable distribution based on generalized central limit theorem. Except for a few special cases, there is no unified and closed analytical expression for the probability density function of the distribution; characteristic function is generally used to describe its distribution characteristics. The random variable obeys α-distribution law, if and only if the characteristic function is

$$varphi left( t right) = exp left{ {{text{i}}delta t – left| {gamma t} right|^{alpha } left[ {1 + {text{i}}beta {text{sgn}} left( t right)omega left( {t,alpha } right)} right]} right}$$


$$omega left( {t,alpha } right) = left{ {begin{array}{*{20}c} {tan left( {frac{pi alpha }{2}} right),} & {alpha ne 1} \ {frac{2}{pi }log left| t right|,} & {alpha = 1} \ end{array} } right.$$


$${text{sgn}} left( t right) = left{ {begin{array}{*{20}c} {1,} \ {0,} \ { – 1,} \ end{array} } right.begin{array}{*{20}c} {t > 0} \ {t = 0} \ {t < 0} \ end{array}$$


where (0 < alpha le 2, – 1 le beta le 1,gamma > 0, – infty < delta < + infty), ({text{i}}) is the imaginary unit, and ({text{sgn}} left( t right)) is a symbolic function.

It is seen that the α-stable distribution is uniquely determined from four characteristic exponents. α is the characteristic index that determines the trailing thickness of the probability density function. Distinct from the Gaussian distribution, the α-stable distribution decays algebraically, and the attenuation velocity is related to α. The β parameter determines the degree of distribution symmetry. When β = 0, the distribution is also called an SαS. ϒ is the scale parameter that measures the dispersion of the distribution. The distribution is regarded as Gaussian if (S_{2} left( {Upsilon ,1,delta } right) = Nleft( {Upsilon ,1,delta } right)), Cauchy if (S_{1} left( {Upsilon ,0,delta } right)), and Levy if (S_{0,5} left( {Upsilon ,1,delta } right)).

The time-domain expression of active sonar emission waveform is (sleft( t right)). Assuming the number of discrete scatterer elements satisfying the i.i.d. condition, reverberation model was built by using the element scattering model based on the generating process of the seafloor reverberation. It divided the scattering element by the scattering coefficient. Therefore, the signal received at the receiver can be expressed as

$$rleft( t right) = a_{0} sleft( {t – tau_{0} } right) + sumlimits_{i = 1}^{N} {a_{i} g_{i} ssleft( {t – tau_{i} } right)} + n_{w} left( t right)$$


In Eq. (4), the first term on the right of the equation represents the received target signal, where (a_{0}) is the attenuation coefficient of the target signal which is related to the propagation loss of the acoustic signal of the excitation target in the back-and-forth sound path and the absorption loss of the medium to the sound energy. (tau_{0}) is the time delay of target echo signal. In this paper, the Doppler effect caused by target movement is not discussed. The second term represents the reverberation. (ssleft( {t – tau_{i} } right)) is the waveform expression of sound wave scattered by the ith scatterer element. (a_{i}) is the attenuation coefficient of the scattering signal of the ith scatterer element which is related to the propagation loss and absorption loss before and after the scattering of the sound wave; (g_{i}) represents the scattering intensity coefficient of the ith scatterer element which is related to the variation coefficient of scattering intensity caused by different scattering coefficient and incident grazing angle when acoustic wave enters the ith scatterer element. The third term represents marine environmental noise.

Fractional lower-order moments for signal enhancement in reverberation environment

In this part, the basic theory of FLOM is introduced firstly. As it can only be used to process the signal which satisfies SαS distribution [18], an energy redistribution method which is applied to redistribute the received signal energy in fractional domain is proposed in detail. Then, the target echo broadening and compression in fractional domain is analyzed.

Basic knowledge of fractional lower-order moments

Assume random variable (x sim Salpha S), (0 < alpha le 2), the fractional-order moment of the SαS random variable (x sim S_{alpha } left( {beta ,Upsilon ,0} right)) with zero location parameter (left( {delta = 0} right)) is given by

$${rm E}left[ {left| X right|^{p} } right] = frac{{2^{p + 1} Gamma left( {frac{p + 1}{2}} right)Gamma left( { – frac{p}{alpha }} right)}}{{alpha sqrt pi Gamma left( { – frac{p}{2}} right)}}Upsilon^{alpha /p} ,quad p < alpha$$


where (Gamma left( bullet right)) is the gamma function,

$$Gamma left( x right) = int_{0}^{infty } {t^{x – 1} e^{ – t} {text{d}}t}$$


If the random variables X and Y obey SαS distribution, the range of α is from 1 to 2. The fractional lower-order correlation (FLOC) can be expressed as

$$R_{XY}^{p} = Eleft[ {XY^{ < p – 1 > } } right],;;;1 le p < alpha$$


Based on the Wigner–Ville distribution (WVD) and FLOM theories, if the random variables obey SαS distribution, the fractional lower order of WVD (FLOM-WVD) expression can be written as

$$WVD_{FLOS} left( {t,f} right) = intlimits_{ – infty }^{infty } {x^{leftlangle P rightrangle } left( {t + tau /2} right)} x^{ – leftlangle P rightrangle } left( {t – tau /2} right)e^{ – j2pi ftau } dtau$$


where (x^{leftlangle P rightrangle } = left| x right|^{p} {text{sign}}left( x right),p < alpha /2).

Received signal redistributed in fractional domain

Considering the size of sonar arrays, as the bandwidth of transmitted acoustic signals increases, the sonar resolution unit decreases, and the number of effective scatterers in the resolution unit cell decreases. Thus, the received signal does not well satisfy an SαS random variable. However, the fractional lower order can only be used on received signals that are SαS random variables. To apply FLOM to suppress reverberations that have any characteristic exponents, an energy redistribution method based on the FrFT is proposed to redistribute the received signal to meet the required characteristic exponent.

The FrFT is a type of linear integral transformation that is performed on LFM signals and used to transform signals from one domain to another. This is known as the FrFT domain and is represented by u. Then, the pth-order FrFT on the function is denoted as (F^{p} left( u right)) and defined as

$$F^{p} left( u right) = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)sleft( t right){text{dt}}}$$


where (mathop Klimits^{ sim }_{p} left( {u,t} right) = sqrt {1 – icot left( {frac{ppi }{2}} right)} exp left[ {ipi left( {u^{2} cot left( {frac{ppi }{2}} right) – 2ucsc left( {frac{ppi }{2}} right) + t^{2} cot left( {frac{ppi }{2}} right)} right)} right]), (p ne 2n), (n in Z). With some simplification, the above equation can be written as

$$F^{p} left( u right) = left{ {begin{array}{*{20}c} {B_{p} int_{ – infty }^{ + infty } {exp left[ {ileft( {frac{{t^{2} + u^{2} }}{2}cot left( {frac{ppi }{2}} right) – frac{tu}{{sin left( {ppi /2} right)}}} right)} right]sleft( t right){text{dt,}}} } & {p ne 4n} \ {sleft( t right),} & {p = 4n} \ {sleft( { – t} right),} & {p ne 2left( {2n + 1} right)} \ end{array} } right.$$


where (B_{p} = sqrt {frac{{1 – cot left( {ppi /2} right)}}{2pi }}), and (sleft( t right)) in Eq. (10) is expressed as

$$sleft( t right) = exp left[ {ipi left( {2f_{0} t + mu t^{2} } right)} right], – frac{T}{2} < t < frac{T}{2}$$


where (f_{0}), (mu) and (T) are the initial frequency, frequency modulation rate and duration. In particular, when (p = 1), the FrFT on (sleft( t right)) can be regarded as Fourier transform.

When we applied (10) to Eq. (4), the received signal in fractional domain with transform (p) will be

$$F^{p} left( u right) = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)sleft( t right){text{d}}t = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)left[ {a_{0} sleft( {t – tau_{0} } right) + sumlimits_{i = 1}^{N} {a_{i} g_{i} ssleft( {t – tau_{i} } right)} + n_{w} left( t right)} right]{text{d}}t = Sleft( {u_{p} } right) + SSleft( {u_{p} } right) + Nleft( {u_{p} } right)} }$$



$$Sleft( {u_{p} } right) = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)a_{0} sleft( {t – tau_{0} } right){text{d}}t}$$


$$SSleft( {u_{p} } right) = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)sumlimits_{i = 1}^{N} {a_{i} g_{i} ssleft( {t – tau_{i} } right)} {text{d}}t}$$


$$Nleft( {u_{p} } right) = int_{ – infty }^{ + infty } {mathop Klimits^{ sim }_{p} left( {u,t} right)sumlimits_{i = 1}^{N} {n_{w} left( t right)} {text{d}}t}$$


(Sleft( {u_{p} } right)), (SSleft( {u_{p} } right)) and (Nleft( {u_{p} } right)) are the received target signal, reverberation and marine environmental noise in fractional domain, respectively.

Here, we give a reverberation model example that obeys the α-distribution (1.2176, 0.6, 0.0420, 0.0004). The characteristic exponents of the redistributed signal are shown in Table 1 with different transform p values.

Table 1 Characteristic exponents of the redistributed signal with different transform p values

The probability density curves with different transform p values (0, 0.02 and 0.22) are shown in Fig. 1. With the FrFT of different p values, the redistributed signal has different characteristic exponents.

Fig. 1
figure 1

α and β values of redistributed signal with various transform p values

Consider the two cases of p = 0, p = 0.22 and p = 0.02; Fig. 2a–c, respectively, shows the probability density functions. For p = 0.22, there is a thicker and less smooth tail than for p = 0.02. Furthermore, α = 1.163 and β = 0.003 for the redistributed signal (p = 0.02) are SαS random variables. More specifically, β is closest to 0 for α between 0 and 1 and is the optimal p selection principle, which is completely different from the traditional FrFT. Here, we give the influence of the FrFT on the target echo.

Fig. 2
figure 2

Probability density distribution of three cases

The target echo broadening and compression in fractional domain

In this part, we discuss the influence of the redistributed signal method on the target echo. Figure 3 shows the transformation relationship between the time domain and the fractional domain of the target echo.

Fig. 3
figure 3

Corresponding relation between echo signal fractional domain

According to Fig. 3, there are some changes in the bandwidth of the received signal in the fractional domain due to the FrFT. We assume that the frequency bandwidth of the transmitted signal is (B), the duration of the transmitted signal is (T) and the observation duration is (T_{1}). The transform angle of FrFT is (p). With the geometric relation, the bandwidth in the fractional domain can be written as

$$B_{u} = left( {B – chi T} right)sin left( {ppi /2} right)$$


where (chi = – frac{1}{{tan left( {{{2p} mathord{left/ {vphantom {{2p} pi }} right. kern-nulldelimiterspace} pi }} right)}}) is related to the chirp rate of the transmitted signal in the fractional domain. The target echo broadening or compression in the fractional domain is related to the observation signal length (T) and the transform angle (p). In addition, according to Fig. 3, the information and energy of the echo signal are not changed in the fractional domain.

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