applications of a nonlinear optimization solver and ...

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 9, Number 1, January 2013

doi:10.3934/jimo.2013.9.205 pp. 205–225

APPLICATIONS OF A NONLINEAR OPTIMIZATION SOLVER AND TWO-STAGE COMPREHENSIVE DENOISING TECHNIQUES FOR OPTIMUM UNDERWATER WIDEBAND SONAR ECHOLOCATION SYSTEM

Chien Hsun Tseng Department of Information Engineering Kun Shan University, Taiwan

(Communicated by Kok Lay Teo) Abstract. This paper focuses on empirical design and performs real data test of a novel algorithm that contributes to the purpose of solving a specific SIP problem arising from a classical wideband active sonar echo location system in noisy environment. The algorithm is achieved by firstly isolating potential contact signals of interest embedded in the scattered returns through the firststage denoising using an adaptive noise canceling (ANC) neuro-fuzzy scheme. The ANC output is then feed into an iterative target motion analysis (TMA) scheme composed of the second-stage denoising and optimal motion estimation. In the first-stage denoising, the adaptive neuro-fuzzy inference system (ANFIS) is the core processor of ANC for tracking both the linear and nonlinear relations among complex contact signals. The second-stage denoising is appealed for further noise compression and is accomplished via trimmed-mean (TM) levelization and discrete wavelet denoising (WDeN). The two-stage comprehensive denoising techniques yield fine tuned signals for the system deconvolution based on solving a semi-infinite programming (SIP) problem. These two schemes form an ANC-TMA(CWT) algorithm for rapid processing of target echoes and provide a higher degree of signal detection capability with an increased robustness against false signal detections. Advantages and simulation results are discussed in terms of detection performance and computational time consumption.

1. Introduction. The process of active sonar returns in multipath media is a core problem of underwater signal processing [1, 2]. Because the active sonar system is concerned with the estimation of targets’ motion parameters, it is well known that the implementation of such a system exploits the time-scale joint representation of target echoes [3, 4]. One technique used to measure time and scale of objects is the cross correlation or matched filtering technique [5, 6]. As in wideband applications [7, 8], this technique estimates the time-delay and scale-change by cross correlation of overlapping segments of the received complex signal with a set of basis functions matched to the transmitted signal. This technique is commonly referred to as the wideband replica correlation (WRC) method in the parameter 2010 Mathematics Subject Classification. 90C34. Key words and phrases. Wideband acoustical signal, active sonar echolocation, reverberation, adaptive noise canceler, adaptive neuro-fuzzy inference system, wideband cross correlation, optimal wavelet transforms.

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estimation problem where the statistic of the WRC tested against thresholds and the local maximum is an estimate of the round-trip time of arrival (TOA) of the target. From the linear (or linearized) relation between the estimated TOA and the reference one, the target’s motion including Doppler range, radial velocity, and radial acceleration can be measured. The standard WRC processing works well for most problems and has optimum performance with the noise-free signal or the maximum output signal-to-noise ratio (SNR) condition [3, 9]. In the presence of severe interference or in highly distorting media such as multipath channels [10, 11], the WRC processing, however, degrades the underlying back scattering returns. Together with the system characteristics such as beam patterns, the WRC processing makes the target detection or signal recovery even more difficult. This is because sharp peaks of the contact signal are more sensitive to a sidelobe correlation interference mainly arising from unwanted harmonics of the transmission, which correlate with the replica. Recently proposed schemes [12, 13] driven by an ANC concept [14] with the core processor of ANFIS [15] have offered a possible remedy to localize the target returns. The use of artificial neural networks and fuzzy logic for active control of wideband noise has been well explored in [16, 17] as an alternative to conventional filtering approaches [18]. The stage of noise reduction was to remove unwanted parts of the returns contained in the higher frequency ranges, and hence to effectively improve the target strength; the echo signals of interest are in the lower frequency ranges. Excellent Performances of those schemes exploits capabilities of ANFIS in tracking both linearity and nonlinearity in multidimensional input space and hence alleviating the sidelobe correlation interference. Combined with the CWT for the target mapping, the ANC(CWT) algorithm [12] was able to accurately estimate the Doppler motion parameters with a low level of target strength. This one-stage denoising algorithm, however, is not suitable for on-line real-time implementation as the ANFIS operation involves complex learning procedure with sets of network parameters adaptively estimated in order to identify most noise components at each periodic update. Furthermore, when the performance of the ANFIS degrades due to insufficient training patterns, more membership functions (MFs) are then required in order to model the noise channel generated via a certain unknown processes within and outside the receivers. As a result, more hardware memory and computational time are needed to accommodate and process excessive data sets arisen from the duplication of nodes during the fuzzy inference system (FIS) training [15]. All of this deteriorates the computational load of the ANFIS operation in noise pattern identification. In our previous study [12], for a pair of input training data containing 16 fuzzy rules there are 96 necessary parameters per epoch involved in the ANFIS operation. Thus, for a transit return signal with a set of 215 data points, ANFIS needs to process at least 3 × 220 parameters in each epoch. Since each process itself requires on average 103 epochs with each epoch containing 103 multiplications and additions, this then requires at least 3.2 × 1012 operations for processing a return signal. Thus, even for a common 3.0 GHz computer CPU with an approximate 0.33 gigasecond per operation, this would take 1056 seconds ≈ 17 minutes to estimate one set of motion parameters. In the case of a real-time signal detection system based on a digital signal processing-field programmable gate array (DSP-FPGA) [19] hardware implementation, the one-stage ANC(CWT) algorithm would not be a suitable choice for noise canceling as its computational load must be improved.

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The high cost of hardware implementation and computational time consumption, due to the complexity of the neuro-fuzzy learning procedures, is a common drawback encountered in the neural networks-based intelligent technology [20, 21]. To alleviate the computational burden of the ANC neuro-fuzzy scheme, while achieving the real-time processing, the load of balancing step is developed by introducing a trimmed mean and discrete wavelet transform (DWT)-based [22] denoising for efficiently and effectively localizing potentials. Together with the optimal CWT target mapping to identify the targets’ motion, the proposed TMA(CWT) scheme comprises three steps proceeded in an orderly way: TM-levelization, WDeN and optimal CWT target mapping. The step of the TM-levelization, similar to that of the TM normalization described in [23, 24], is a simple but dynamic level-based process controlled by two gauges within the second-stage denoising. The external gauge is employed to generate step-sizes and to achieve fast convergence towards an optimal target mapping in the CWT process. The internal gauge based on the current level of the external gauge is used to remove power of most of the sharp detail information in the sense of trimmed mean estimation. Processing with the TM-levelization in the first instance, the WDeN step, the second-stage denoising associated with the DWT proven to be efficient and can be viewed as a nonparametric estimation of the desired noise-free signal [22], is applied to further effectively suppress the remaining noise part of the training data and thus produce fine tuned test cells for the final step of target mapping with the optimal CWT. By integrating with the ANC neurofuzzy scheme as the first stage input signal denoising to significantly enhance the target strength following by the iterative TMA(CWT) scheme containing simple but effective second-stage denoising to achieve fast and optimum target mapping, the proposed two-stage ANC-TMA(CWT) algorithm can be of rapid processing and successfully achieve target mapping in the presence of severe interference, a combination of reverberation and ambient noise. To best implement the CWT in an optimum, effective and cost-efficient manner, the multi-channel continuous time-scale joint deconvolution process being formulated as a semi-infinite programming (SIP) problem is solved by a hybrid implementation: the abscissa time-domain is obtained by the high-performance of finite impulse response (FIR) matched filtering mechanism (i.e. training sets), while the fast-convergence of 1D nonlinear optimization technique is employed to achieve an optimum in the scale-domain. The resultant algorithm is novel and can benefit for more frequent ensonification of a water mass for targets at moderate ranges, giving a higher rate of return echoes for improved target tracking. Together with the real data set provided by the Defense Evaluation Research Agency (DERA UK), computer simulations using the two-stage ANC-TMA(CWT) algorithm for various target strength of SNR has shown a higher degree of signal detection capability with an increased robustness against false signal detections. 2. Modeling of wideband sonar signal. Let us consider a basic active sonar system where a ping (contact signal) is firstly projected into the water in a narrow beam towards the targets and the reflected signal corrupted by reverberation and ambient noise is sensed, processed, and displayed by a receiver. A mathematical model [2, 25] for each backscattered signal in a nondirectional sonar channel can be described as: PI g˜(t) = i=1 gi (t) + η(t). (1)

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This accounts for superposition of I contact signals known as target signatures gi (t), i ∈ I: √ (2) gi (t) = αi Si ψ(Si (t − Di )) 2 which are received Doppler distorted pulses √ in the L ([0, ∞)) Hilbert space of finite energy at time t ∈ Ω ⊂ |2 = ||˜ g (t)||22 |||ψs˜i ,τ (t)||22 cos(θ)2 ≤ ||˜ g (t)||22 |||ψs˜i ,τ (t)||22 .

(5)

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Here θ between the two vector is defined in the interval [0, +π/2], || · ||2 represents 4 R∞ the L2 norm with the definition of ||f (t)||2 = ( −∞ |f (t)|2 dt)1/2 , and the wideband 4 √ basis functions is defined by ψs˜i ,τ (t) = s˜i ψ(˜ si (t − τ )). Assume that the transmitted signal ψ(t) is completely specified and characterized by a modulation function m(t) having a unity power within an envelope (or window function) w(t), i.e., ψ(t) = w(t)m(t), subject to ||m(t)||22 = 1. (6) √ 4 Defining the wideband window function ws˜i ,τ (t) = s˜i w(˜ si (t−τ )) with the Doppler scale s˜i , the basis function yields ψs˜i ,τ (t) = ws˜i ,τ (t)m(˜ si (t − τ )).

(7)

From the Cauchy’s inequality, Eq. (5) for any s˜ > 0 and τ ∈ Ω, thus, becomes: |W Cψ˜g (˜ si , τ )|2 ≤ ||˜ g (t)||22 ||ws˜i ,τ (t)||22 .

(8)

Whereas, a lower bound of a normalized inner product is obtained with θ¯ ∈ [0, +π/2]: ¯ 2 = | h˜ ||˜ g (t)||22 ||ws˜i ,τ (t)||22 ≥ ||˜ g (t)||22 ||ws˜i ,τ (t)||22 cos(θ) g , ws˜i ,τ i |2 .

(9)

The cross correlation processing generally works well for most problems and has optimum performance with the noise free signal or the maximum output SNR condition [3, 7, 9]. In the presence of severe interference or in highly distorted media, e.g. spread or multipath channel [11], the correlation processing, however, degrades the underlying back scattering returns. This is known as the sidelobe correlation interference [29], which mainly arises from unwanted harmonics of the transmission correlating with the replica. The situation is even more deteriorated for a multi-target application where a single Doppler scale s˜i is replaced by a vector 4 form of m-target Doppler scale as ˜s = [˜ s1 , . . . , s˜m ]0 ∈ 0 and hence D training and mapping process will be discussed in Algorithm 1 to deal with the similarity magnitude ε in terms of measuring (euclidean) distance between the reference and target. Consequently, the designed single point target detection problem may be solved by seeking the local maximum of the following quadratic programming problem with the continuous time-scale joint representation: max

s>0,τ ∈[0,∞)

{|CW Tw g˜(s, τ )|2 }.

(13)

Replaced by an m-target Doppler scale ˜s in Eq. (12), the above single point target detection problem is extended to a quadratic SIP problem:  Pm max f (s, t) = i=1 |CW Tw g˜(si , τ (t))|2 (14) s.t. s(t) > 0m , ∀t ∈ [0, ∞) where s(t) ∈ C([0, ∞), Rm ) and C([0, ∞), Rm ) denotes the Banach space consisting of all continuous functions depending on variable t defined on [0, ∞) with value in Rm . From Eqs. (12)-(14), we note that the detection process using the CWT is independent of modulation function m(t), which is modulated by a carrier frequency. This advantage makes the CWT easier to implement digitally as only the window function (a lowpass waveform) needs to be sampled. In contrast, the carrier frequency in the process of cross correlation cannot be separated from the window function due to the wideband condition [7, 32]. 3.2. Algorithm for multi-target detection. To develop as applicable to a realtime signal detection system based on a DSP-FPGA hardware structure, the SIP problem with continuous time-scale joint variables of Eq. (14) (and thus Eq. (13)) must be approximated. Motivated by the high-performance of the FIR filtering technique and the fast convergence of nonlinear optimization solver, the SIP problem is solved iteratively by hybrid: FIR filtering provides the optimal delay, while nonlinear solver produces the optimal scale. To demonstrate this hybrid implementation, we assume g˜(t) and w(t) are in the sequence Hilbert space `2 to yield the CWT in the problem of Eq. (13) as R∞ CW Tw g˜(s, τ ) = √1s −∞ g˜(τ − t)w(−t/s)dt P R l+1 (15) = √1s l l g˜(τ − t)w(−t/s)dt. If g˜(t) = g˜(l) for t ∈ [l, l + 1], Eq. (15) is approximated as a discrete convolution for a specified CWT scale s, i.e. R l+1 R [ l+1 P √ P s ] √1 g ˜ (τ − l) w(−t/s)dt = s g ˜ (τ − l) w(−t)dt l l l s [ sl ] l  P   l+1  √ P = s l g˜(τ − l) F ( s ) − F ( s ) = l g˜(τ − l)h(s, l) = (˜ g ∗ h(s)) [τ ]. (16) The function h defined as an impulse response in Eq. (16) involves the cumulative density function F defined in terms of w: 4 √ 4 Rx h(s, l) = s[F ([(l + 1)/s]) − F ([l/s])], F (x) = −∞ w(−t)dt. (17) In practice, we have only an effective support t ∈ [−Tw , Tw ] for w(t) with the sampling rate fw . Following from Eq. (17), due to the symmetric structure of the

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mother wavelet, the impulse response h(s, l) can then be determined for a specified CWT scale s with indices as: l∈

[−sTw ,sTw ] (2Tw fsw sTw

4

− 1) = [0, L − 1], L < ∞.

(18)

Given time support t ∈ [0, T ] for g˜(t) and following from Eq. (16), a discretetime version of the CWT consists of breaking the time interval into N = m × (n + 4

L − 1) sub-intervals, and approximating the input signal as g˜ = [˜ g0 , . . . , g˜℘−1 ] with n samples in each signal segment g˜i , i = 0, . . . , ℘ − 1. The discrete-time CWT coefficient obtained for g˜ is then represented by the output response of a bank of finite impulse response (FIR) filter with coefficients h(s, `), i.e., 4

CW Tw g˜[s, k] = y˜[s, k] = [˜ y0 [s, k], . . . , y˜℘−1 [s, k]] ∈