Direct Under-Sampling Compressive Sensing Method for Underwater Echo Signals and Physical Implementation

applied sciences

Article Direct Under-Sampling Compressive Sensing Method for Underwater Echo Signals and Physical Implementation

Tongjing Sun 1,*, Ji Li 1 and Philippe Blondel 2 1 Department of Automation, Hangzhou Dianzi University, Xiasha Higher Education Zone, Hangzhou 310018, China; [email protected] 2 Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK; [email protected] * Correspondence: [email protected]; Tel.: +86-571-8771-3597

Received: 29 September 2019; Accepted: 24 October 2019; Published: 29 October 2019

Featured Application: The direct under-sampling compressive sensing method presented in the article can be applied to many types of acquisition systems for underwater echo signals.

Abstract: Compressive sensing can guarantee the recovery accuracy of suitably constrained signals by using sampling rates much lower than the Nyquist limit. This is a leap from signal sampling to information sampling. The measurement matrix is key to implementation but limited in the acquisition systems. This article presents the critical elements of the direct under-sampling—compressive sensing (DUS–CS) method, constructing the under-sampling measurement matrix, combined with a priori information sparse representation and reconstruction, and we show how it can be physically implemented using dedicated hardware. To go beyond the Nyquist constraints, we show how to design and adjust the sampling time of the A/D circuit and how to achieve low-speed random non-uniform direct under-sampling. We applied our method to data measured with different compression ratios (volume ratios of collected data to original data). It is shown that DUS-CS works well when the SNR is 3 dB, 0 dB, 3 dB, and 5 dB and the compression ratio is 50%, 20%, and 10%, − − and this is validated with both simulation and actual measurements. The method we propose provides an effective way for compressed sensing theory to move toward practical field applications that use underwater echo signals.

Keywords: compressive sensing; under-sampling; measurement matrix; echo signals

1. Introduction The core of signal acquisition technology is the transformation of analog signals into digital signals. The Nyquist sampling theorem [1,2] forms the critical basis of modern approaches to signal sampling and processing; it requires that the sampling frequency is greater or equal to twice the highest frequency of the effective signal, thus avoiding aliasing. In this way, the sampled signal can contain the information of the original signal and can be restored without distortion. In practical applications, the sampling frequency is usually 5–10 times the highest frequency of the signal. As signal frequency and bandwidth increase in modern technology (up to MHz rates for some sonars), the Nyquist sampling rate also increases linearly. This makes the amount of data increase dramatically. Not only does this bring difficulties to the acquisition system, but for signals with underlying repeatability, it also generates a large amount of redundant data, which increases the cost of signal transmission, storage, and processing. Compressive sensing (also called compressive sampling) [3–6] differs from Nyquist sampling in that it combines the compression and sampling stages. It utilizes the characteristics of sparse

Appl. Sci. 2019, 9, 4596; doi:10.3390/app9214596 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4596 2 of 14 representation of signals on a set of bases and realizes the perception of high-dimensional signals through non-correlated observations of low-dimensional space, low-resolution, and under-Nyquist sampled data to reconstruct the original signal by combining optimization methods. The compressive sensing method directly converts the samples from a continuous time signal while discarding redundant information in the Nyquist sampling. Then optimization is used to process the compressed samples. How to achieve the sampling of information using compressive sensing theory is a critical issue that needs a solution at present. Many studies have been carried out on this issue, such as: the observation matrix construction [7–9], analog broadband signal sampling [10–12], radar signal compression [13,14], image compression technology [15–18], and ultrasound transducer fields [19]. Some researchers [20–22] designed ultra-high-speed analog-to-digital converters, but their high level of complexity makes them difficult to implement in practice. Some researchers [11,23] used an AIC (analog-to-information converter) structure to realize random observations. The main problem is that the clock frequency of the random symbol function is greater than the Nyquist sampling rate, which increases the difficulty of frequency mixing. The sampling method of AIC structure is quite different from the traditional sampling method; there is no compatibility between the AIC structure circuit and a traditional sampling circuit. Therefore, there is an urgent need for a compressive sensing acquisition method that is convenient for physical implementation. In this article, we first present the direct under-sampling compressive sensing (DUS-CS) method, summarizing the key points of the theory and how they can be implemented in hardware (Section2). Next, we apply DUS-CS to simulated high-resolution data and compare different compression ratios (Section3). The hardware designed for under-sampling and compressive sensing is presented in Section4, and its application is presented in Section5. The advantages and limits of this approach are summarized in Section6.

2. Direct Under-Sampling Compressive Sensing Method

2.1. Basic Theory of Compressive Sensing Compressive sensing includes three core parts: signal sparse decomposition, uncorrelated signal observation, and signal reconstruction. The sparseness of the signal is a necessary condition for compressive sensing, uncorrelated observation is the key to compressive sensing, and signal reconstruction is the technique of compressively sensing reconstructed signals. If a discrete signal x cN is sparse or compressible, it can be represented as a linear combination ∈ of a set of standard orthogonal bases

XN x = Ψiαi = Ψα (1) i=1

In the formula, Ψ = [ψ , ψ , ... , ψN] is an N N matrix and is considered as the sparse basis of the 1 2 × signal x. If there are k(k N) non-zero coeﬃcients in α, then the signal x is called a k-sparse signal under the sparse basis Ψ. Compressive sensing projects a high-dimensional signal onto low-dimensional T spaces through an observation matrix Φ = [φ , φ , ... , φM] using M(M N) measurements in the 1 2 following way y = Φx = ΦΨα = Θα (2) where y is the observation and Θ = ΦΨ is called the perceptual matrix. According to the M number of y observations necessary to calculate α, this can be expressed as an optimization problem in the sense of an l0-norm min α s.t. y = Φx = ΦΨα = Θα (3) k k0 Appl. Sci. 2019, 9, 4596 3 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 14

Equation (3) is an NP-hardNP-hard problem,problem, soso thethe signalsignal reconstructionreconstruction solutionsolution is transformedtransformed into a ll11-norm-norm optimizationoptimization problem problem with with relaxation relaxation constraints. constraints. It It then then becomes becomes

minminααα sst.t .. . y = y=ΘΘα (4)(4) k k11 This problemproblem becomesbecomes a a convex convex optimization optimization problem. problem. The The typical typical reconstruction reconstruction algorithm algorithm can becan the be greedthe greed tracking tracking method method [24], [24], convex convex relaxation relaxation method method [25], [25], or combination or combination method method [26]. [26].

2.2. Direct Under-Sampling Compressive Sensing Method (DUS-CS)(DUS-CS)

2.2.1. Direct Under-Sampling (DUS) Implementation The direct under-sampling method [[27]27] is modifiedmodified based on the traditional sampling circuit. TheThe samplingsampling trigger time is controlled by pseudo-random timing control, which samples alongalong the rising edgeedge (or(or fallingfalling edge)edge) ofof eacheach triggertrigger signal,signal, andand thethe correspondingcorresponding signal is collectedcollected in thethe data bubuffer.ffer. Compared to traditional sampling, only only the the design design of of the the sampling sampling circuit circuit is changed. ApplyingApplying DUSDUS to to the the reception reception of echoof echo signals signals can reducecan reduce both both the complexity the complexity of the receivingof the receiving system andsystem its cost.and its Before cost. theBefore A/D the conversion, A/D conversion, the control the control of the signal of the samplingsignal sampling time has time low has speed low andspeed is non-uniform.and is non-uniform. At each At instance, each instance, the sampling the sampling trigger istrigger initiated is initiated at a different at a different location andlocation at di ffanderent at times,different which times, can which ensure can the ensure resulting the signalresulting is a signal discrete is a sample. discrete sample.

2.2.2. Direct Under-Sampling Compressive SensingSensing (DUS-CS)(DUS-CS) MethodMethod The basic idea of DUS-CSDUS-CS is that, in combination with the inherent characteristics of the signal, one cancan useuse thethe prior prior information information sparse sparse matrix matrix [28 [28,29],29] to representto represent the the original original signal. signal. The The signal signal can becan represented be represented by a complete by a complete prior atomic prior library atomic and library K-sparse and coe K-sparseﬃcient vector.coefficient Then, vector. according Then, to theaccording implementation to the implementation method of direct method under-sampling of direct under-sampling and the RIP (restricted and the RIP isometry (restricted property) isometry [30] conditionproperty) of[30] the condition measurement of the matrix measurement to constructed matr measurementix to constructed matrix, measurement signal observation matrix, sampling signal canobservation be performed sampling to obtain can be under-sampling performed to observationobtain under-sampling signals. Finally, observation based on signals. the orthogonal Finally, matchingbased on the pursuit orthogonal method, matching combined pursuit with the method, block sparsecombined property with ofthe the block signal, sparse it becomes property possible of the tosignal, reconstruct it becomes the originalpossible signal to reconstruct (see ﬂow the diagram original in signal Figure (see1). flow diagram in Figure 1).

Sparse matrix Under sampling Blocks sparse Reconstructed Original signal xt() with priori measurement reconstruction signal information matrix algorithm xtˆ()

Figure 1. Direct under-sampling compressive sensing method. Figure 1. Direct under-sampling compressive sensing method. As shown in Figure1, the core of the direct under-sampling compression sensing method lies in As shown in Figure 1, the core of the direct under-sampling compression sensing method lies in the construction of the measurement matrix and the sparse representation and reconstruction method, the construction of the measurement matrix and the sparse representation and reconstruction leveraging the prior information. Therefore, the following two parts are described in detail. method, leveraging the prior information. Therefore, the following two parts are described in detail. (1) Construction of measurement matrix (1) Construction of measurement matrix The direct under-sampling method is used on an input signal x(t), with signal length N, sampling The direct under-sampling method is used on an input signal x()t , with signal length N , rate fs, to yield M observation signals sampling rate fs , to yield M observation signals T y = [y1, y2, ... , yM] (5) = [ ]T yyyy12, ,..., M (5) where where ym = x(tm) (6) = The sampling moment satisﬁes the followingyxtmm conditions() (6) The sampling moment satisfies the following conditions min( tm tm 1 ) = kTs (7) − − | −=| min(ttmm−1 ) kT s (7) Appl. Sci. 2019, 9, 4596 4 of 14

where Ts = 1/ fs is the sampling period of the signal and k 2 is an integer to ensure that the ≥ under-sampled measurements are low speed and random. We use the implementation of direct under-sampling and combined with the theory of compressive sensing to construct the observation matrix. The observable structure of the original signal, x, is expressed as y = Φx (8) where y RM 1 is the observed signal and Φ RM N is observation matrix (M N). ∈ × ∈ × According to the Whittaker interpolation theorem [31], continuous signals can be reconstructed by interpolation of their uniform discrete samples by

X∞ t x(t) = x sin c n (9) n T n= s − −∞ So, the observation matrix can be expressed as

t Φ(m, n) = sin c m n , 1 m M, 1 n N (10) Ts − ≤ ≤ ≤ ≤

According to the Nyquist sampling theorem, signal reconstruction is obtained by ﬁltering the periodic spectrum of the signal. Assuming a periodic signal such as Equation (10), each element in the observation matrix is calculated considering a certain signal period. According to the slow decay and non-compact support characteristics of the sin c function, the role of other cycles, while not dominant, cannot be simply ignored directly. Therefore, the R cycles are accumulated to obtain the observation matrix

R/2 X∞ t X t Φ(m, n) = sin c m n + rN = sin c m n + rN (11) Ts − Ts − r= r= R/2+1 −∞ − In this formula, the larger the period R, the higher the calculation accuracy. However, as the number of cycles increases to obtain high precision, it takes longer to reach convergence of the Equation (11). In order to guarantee a certain precision, the Poisson superposition summation formula is used to speed up the calculation summation ! ! X∞ 1 X∞ k k f (t + rN) = F exp 2πj t (12) N N N r= k= −∞ −∞ In the formula, the function, F, is the Fourier transform of the signal f . Owing to the nature of the sinc function, the continuous Fourier transform is a ﬁnitely supported rectangular pulse. The measurement matrix now becomes

N/2 ! 1 X k t Φ(m, n) = exp 2πj m n (13) N N Ts − k= N/2+1 − This approach improves the accuracy of observation of the signals and increases the calculation speed. (2) Sparse representation and reconstruction method incorporating prior information

O1 Sparse representation method incorporating prior information The key to signal sparse representation is the construction of a sparse dictionary and selection of the appropriate atom from the dictionary to represent the signal. The more similar the characteristics of the atom are to the original signal, the better the effect of signal observation and processing. Based on the Appl. Sci. 2019, 9, 4596 5 of 14 prior information of the incident signal combined with the principle of signal sparse decomposition, it is possible to construct an a priori complete dictionary. The specific construction method is as follows: the number of sample points, n generated by one pulse width forms a vector quantity, sp, of size N 1 and h × i normalization identity sp = 1. This defines the prototype atom sp = sp(Ts), sp(2Ts), ... , sp(NTs) . k k2 Each atom sp is treated as a block and recorded as Ψ1. Then, by blocking and shifting, the sparse complete h iT n n = T T T dictionary Ψ × ψ1 , ψ2 , ... , ψn is achieved. The block sparse decomposition is expressed as     ψ11 ψ12 ψ1N  α1     ···    ψ1   ψ ψ ψ N  α     21 22 ··· 2  2   .  T xN =  . .  .  =  . [α1, , αn] (14)  . .  .   .  ···  . .  .     ···   ψm ψ ψN ψNN αN N1 2 ··· where α is the block sparse coefficient vector in sparse decomposition theory.

O2 Reconstruction method of block sparse orthogonal matching pursuit This is ﬁrst performed by integrating the block sparseness of prior information into an orthogonal matching pursuit (OMP) algorithm, to form a block-sparse orthogonal matching pursuit (BOMP) reconstruction. Then, the most relevant sub-block of the signal is chosen from the sparse complete dictionary and its projection on the sub-block from the signal is subtracted to obtain a residual signal. This process is iterated until the energy of the residual signal is less than a given threshold, or the algorithm reaches its termination condition. The BOMP algorithm ﬂow is as follows:

h iT 1) Initialize the residual r = x. The block structure of the dictionary is Ψ = ψ ψ ψn . 0 1 2 ··· 2) Selection of atomic blocks. On the i-th iteration, select the index to be the largest inner product value of the residual, such that

T index = argmax mean( Θ [i]ri 1 ) i [1, K] (15) i − ∈

3) Update the set of support blocks (choose sub-blocks that best match the signal) and the residuals, such that I = Ii 1 ii ri = ri 1 ψix (16) − ∪ { } − − 4) Convergence conditions. When i < k, i = i + 1, return to the step 2 or stop iterating, then output the reconstructed signal.

The biggest diﬀerence between the BOMP algorithm and the traditional method is that the inner product of each atom block with the current residual is used to select the most correlated atomic block. Now there is no longer an atom, but an atomic sub-block. This reduces the number of calculations and improves the reconstruction accuracy.

3. Application of Compressive Sensing Method with Direct Under-Sampling in Underwater Echo Signals According to the highlight model [32,33], an echo signal can be represented as the linear superposition of K sub-echoes XK x(t) = A s(t τ ) (17) i ∗ − i i=1

jωct where s(t) = e− is the emitted pulse signal, with a pulse time of 0.01 s and a frequency of 30 kHz, x(t) is the echo signal, Ai is the amplitude of a sub-echo, τi is the time-delay of a sub-echo. Acoustic echoes are simulated and shown in Figure2. Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 14 − jω t where st()= e c is the emitted pulse signal, with a pulse time of 0.01 s and a frequency of 30 kHz, − jω t where st()= e c is the emitted pulse signal, with a pulse τtime of 0.01 s and a frequency of 30 kHz, x(t) is the echo signal, Ai is the amplitude of a sub-echo, i is the time-delay of a sub-echo. τ Appl.x(t) Sci.Acoustic is 2019the ,echo9, 4596echoes signal, are A simulatedi is the amplitude and shown of ain sub-echo, Figure 2. i is the time-delay of a sub-echo. 6 of 14 Acoustic echoes are simulated and shown in Figure 2.

Figure 2. Simulation results of underwater echo signals. Figure 2. Simulation results of underwater echo signals. Figure 2. Simulation results of underwater echo signals. Generally, the probability density function of marine environmental noise obeys Gaussian Generally, the probability density function of marine environmental noise obeys Gaussian distribution,Generally, so Gaussianthe probability noise isdensity added infunction the generated of marine signals environmental with different noise signal-to-noise obeys Gaussian ratios distribution, so Gaussian noise is added in the generated signals with diﬀerent signal-to-noise ratios distribution,(SNRs) to simulate so Gaussian marine noise environmental is added in the noise, generated and the signals under-sampling with different measurement signal-to-noise matrix ratios is (SNRs) to simulate marine environmental noise, and the under-sampling measurement matrix is used (SNRs)used to to compress simulate marinethe echo environmental signals. Figure noise, 3 sh andows the the under-sampling reconstruction measurementof a signal whenmatrix the is to compress the echo signals. Figure3 shows the reconstruction of a signal when the compression usedcompression to compress ratio isthe 100% echo (no signals. compression). Figure 3Figu showsres 4 theand reconstruction5 are reconstruction of a signalresults when when thethe ratio is 100% (no compression). Figures4 and5 are reconstruction results when the compression ratio compressioncompression ratioratio is is100% 50% (no and compression). 20%, respectively Figures. 4In and Figures 5 are reconstruction3–5, DCT-OMP results represents when thethe is 50% and 20%, respectively. In Figures3–5, DCT-OMP represents the combination of the discrete compressioncombination ofratio the isdiscrete 50% andcosine 20%, transformati respectivelyon .and In orthogonalFigures 3–5, matching DCT-OMP pursuit represents method andthe cosine transformation and orthogonal matching pursuit method and DUS-CS represents the direct combinationDUS-CS represents of the thediscrete direct cosine under-sampling transformati compressiveon and orthogonal sensing methodmatching presented pursuit inmethod this paper. and under-sampling compressive sensing method presented in this paper. DUS-CS represents the direct under-sampling compressive sensing method presented in this paper.

(a) (b) (a) (b)

(c) (d) (c) (d) FigureFigure 3.3. ReconstructionReconstruction signal at a compression ratio of of 100% 100% (no (no compression). compression). (a (a)) SNR SNR = 33 dB; (Figureb(b)) SNR SNR 3.= Reconstruction=0 0 dB; dB; (c )(c SNR) SNR= signal =3 − dB;3 atdB; (d a) compression( SNRd) SNR= 8 = dB. − 8ratio dB. of 100% (no compression). (a) SNR = 3 dB; − − (b) SNR = 0 dB; (c) SNR = −3 dB; (d) SNR = −8 dB. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 14 Appl. Sci. 2019, 9, 4596 7 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 14 amplitude/v amplitude/v amplitude/v amplitude/v amplitude/v amplitude/v

(a) (b) (a) (b) echo signal with noise, SNR = -3dB 20 echo signal with noise, SNR = -3dB 20 0

0 -20 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -20 time/s 0 0.005 0.01 0.015 0.02 0.025DCT-OMP 0.03 0.035 0.04 0.045 0.05 10 time/s DCT-OMP 10 0

0 -10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -10 time/s 0 0.005 0.01 0.015 0.02 0.025DUS-CS 0.03 0.035 0.04 0.045 0.05 10 time/s DUS-CS 10 0

0 -10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 -10 time/s 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 time/s (c) (d) (c) (d) Figure 4. Reconstruction signal at a compression ratio of 50%. (a) SNR = 3 dB; (b) SNR = 0 dB; (c) SNR FigureFigure= −3 4.4.dB; ReconstructionReconstruction (d) SNR = −8 dB. signal signal at at a compression a compression ratio ratio of 50%. of 50%. (a) SNR (a) SNR= 3 dB;= (3b dB;) SNR (b )= SNR0 dB; =(c)0 SNR dB; (=c) − SNR3 dB;= (d)3 SNR dB; (=d −)8 SNR dB. = 8 dB. − −

(a) (b) (a) (b)

(c) (d) (c) (d) FigureFigure 5. Reconstruction 5. Reconstruction signals signals at a at compression a compression ratio ratio of 20%. of 20%. (a) SNR (a) SNR= 3dB; = 3dB; (b) SNR(b) SNR= 0 dB;= 0dB; (c) SNR(c) SNR =Figure=3 − dB;3dB; 5. ( dReconstruction ()d SNR) SNR= = 5−5dB. dB. signals at a compression ratio of 20%. (a) SNR = 3dB; (b) SNR = 0dB; (c) SNR = −3dB; (d) SNR = −−5dB. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 14 From Figure 3a, we can see that when the compression ratio is 100% and SNR = 3 dB, both Appl.methods Sci.From2019 can ,Figure9 ,reconstruct 4596 3a, we the can original see that signal when with the acompression noise reduction ratio effect. is 100% When and the SNR signal-to-noise = 3 dB,8 ofboth 14 ratiomethods is further can reconstruct reduced to the0 dB, original the DCT-OMP signal with meth a noiseod does reduction not reconstruct effect. When the theoriginal signal-to-noise signal as wellratio as is before, further while reduced the DUS-CSto 0 dB, themethod DCT-OMP still reco methnstructsod does the signalnot reconstruct well. When the the original SNR continues signal as From Figure3a, we can see that when the compression ratio is 100% and SNR = 3 dB, both methods towell decrease, as before, the while DCT-OMP the DUS-CS method method can hardly still reco renstructsconstruct the the signal original well . signal,When thebut SNR the continuesDUS-CS can reconstruct the original signal with a noise reduction effect. When the signal-to-noise ratio is methodto decrease, still performs the DCT-OMP well. The method correctness can hardly and th ere engineeringconstruct the achievability original signal, of the butunder-sampled the DUS-CS further reduced to 0 dB, the DCT-OMP method does not reconstruct the original signal as well as before, measurementmethod still performs matrix construction well. The correctness have been and verified the engineering in Figure 3. achievability Figures 4 and of 5the are under-sampled the results of while the DUS-CS method still reconstructs the signal well. When the SNR continues to decrease, themeasurement two methods matrix when construction the compression have beenratios verified are 50% in and Figure 20%, 3. respectively. Figures 4 and From 5 are the the results, results the of the DCT-OMP method can hardly reconstruct the original signal, but the DUS-CS method still performs DCT-OMPthe two methods method when is not the ideal compression for signal ratiosreconstruction, are 50% and but 20%, the DUS-CSrespectively. method From reconstructs the results, the the well. The correctness and the engineering achievability of the under-sampled measurement matrix originalDCT-OMP signal method well. is not ideal for signal reconstruction, but the DUS-CS method reconstructs the construction have been verified in Figure3. Figures4 and5 are the results of the two methods when originalTo compare signal well. the two methods quantitatively, we use the concept of matching rate [34] the compressionTo compare ratios the two are methods 50% and quantitatively, 20%, respectively. we Fromuse the the concept results, of the matching DCT-OMP rate method [34] is not ideal for signal reconstruction,Matching_rate but the DUS-CS=− 1 method()XXˆˆ′′ − reconstructs / XX ′′ + the original signal well. (18) To compare the two methodsMatching_rate quantitatively,=− 1 we()XXˆˆ use′′ − the22 / concept XX ′′ + of matching rate [34] (18) 22 ∧ ∧ ∧ ˆ ˆ ′ ∧' In which X is the originalMatching_ signal,rate = 1X isX the0 Xreconstruction0 2/ X0 + X0 2 result, and X and X(18)', In which X is the original signal, X− kis the− reconstruction∧k k k result, and X ′ and X , respectively, represent the absolute values of X and X∧ . We give the variation of the matching In which X is the original signal, Xˆ is the reconstruction result, and X0 and Xˆ0 , respectively, respectively, represent the absolute values of X and X . We give the variation of the matching representrate with the the signal-to-noise absolute values ratio of whenX and theXˆ .compression We give the ratio variation is 100% of and the 50%, matching respectively rate with (Figure the rate with the signal-to-noise ratio when the compression ratio is 100% and 50%, respectively (Figure signal-to-noise6a,b). It can be ratio seen when that thethe compression DUS-CS method ratio is has 100% obvious and 50%, advantages. respectively Therefore, (Figure6 a,b).the Itdirect can 6a,b). It can be seen that the DUS-CS method has obvious advantages. Therefore, the direct beunder-sampling seen that the DUS-CSstructure methodcan be hasused obvious to observ advantages.e underwater Therefore, echo signals. the direct The under-sampling framework of under-sampling structure can be used to observe underwater echo signals. The framework of structurecompressive can sensing be used can to observe effectively underwater recover echothe orig signals.inal signals, The framework which confirms of compressive the correctness sensing and can compressive sensing can effectively recover the original signals, which confirms the correctness and eeffectivenessffectively recover of the thedirect original under-sampling signals, which observation confirms method. the correctness and effectiveness of the direct effectiveness of the direct under-sampling observation method. under-sampling observation method.

0.9

0.8 DCT-OMP DUS-CS 0.8 DCT-OMP DUS-CS 0.7

0.7

0.6

0.5

0.4 -5-3-1.50123 0.4 SNR/dB -5-3-1.50123 (a) (bSNR/dB) (a) (b) Figure 6. Variation of the matching rate with the SNR. (a) the compression ratio is 100%; (b) the FigurecompressionFigure 6.6. VariationVariation ratio is 50%.of of the the matching matching rate rate with with the SNR. the SNR. (a) the (a )compression the compression ratio is ratio 100%; is ( 100%;b) the (bcompression) the compression ratio is ratio 50%. is 50%. 4. DUS-CS Hardware Design 4.4. DUS-CSDUS-CS HardwareHardware DesignDesign In the direct under-sampling system, the most vital component is data acquisition (DAQ) InIn the the direct direct under-sampling under-sampling system, system, the most the vitalmost component vital component is data acquisition is data acquisition (DAQ) hardware. (DAQ) Thishardware. is designed This is according designed toaccording the receiving, to the collecting, receiving, and collecting, transmitting and transmitting functions of functions the underwater of the underwaterhardware. Thisacoustic is designed signal. Theaccording hardware to the componen receiving,ts collecting,include a preamplifier,and transmitting band functions pass filtering, of the acoustic signal. The hardware components include a preamplifier, band pass filtering, programmable programmableunderwater acoustic amplifier, signal. ADC The (analog-to-digitalhardware componen conversion),ts include athe preamplifier, DUS-CS processor band pass andfiltering, the amplifier, ADC (analog-to-digital conversion), the DUS-CS processor and the network interface networkprogrammable interface amplifier, (Figure 7). ADC Figure (analog-to-digital 8 is an image of theconversion), signal collection the DUS-CS card. processor and the (Figurenetwork7). interface Figure8 is (Figure an image 7). Figure of the signal8 is an collection image of the card. signal collection card.

Figure 7. Data acquisition (DAQ) components. Figure 7. Data acquisition (DAQ) components. Appl. Sci. 2019, 9, 4596 9 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 14

Figure 8. Photograph of the collection card.

The preamplifierpreamplifier isis thethe firstfirst stage of the collection card, which is controlled with a single resistor thatthat meetsmeets thethe requirements requirements of of the the system. system. The The bandpass bandpass filter filter uses uses a fourth-order a fourth-order high-pass high-pass filter filter and aand fourth-order a fourth-order low-pass low-pass filter filter in cascade in cascade implementation. implementation. The The programmable programmable amplifier amplifier is used is used for automaticfor automatic gain controlgain control of the underwaterof the underwater acoustic receiver.acoustic Accordingreceiver. toAccording the experimental to the experimental environment, itenvironment, is possible to it adjust is possible the amplification to adjust the factor amplification manually orfactor automatically manually toor improve automatically the signal-to-noise to improve ratiothe signal-to-noise of the receiver ratio in a of better the receiver state. The in a ADC better converts state. The the ADC signal converts from analog the signal to digital from analog and this to functiondigital and determines this function the qualitydetermines of the the acquired quality of signal. the acquired The processor signal. isThe the processor main component is the main of signalcomponent acquisition of signal and control,acquisition and itand is alsocontrol, the main and moduleit is also of thethe direct main under-sampling module of the method. direct Theunder-sampling network collection method. of theThe DAQ network is used collection for data of upload the DAQ and is network used for protocol data upload conversion. and network protocol conversion. 5. DUS-CS Data Acquisition and Analysis 5. DUS-CS Data Acquisition and Analysis 5.1. Data Acquisition 5.1. DataThe Acquisition direct under-sampling sensing system has multiple collection modes, such as Nyquist sampling, direct under-sampling, and pseudo-random sampling. Each specific implementation is The direct under-sampling sensing system has multiple collection modes, such as Nyquist defined as follows: sampling, direct under-sampling, and pseudo-random sampling. Each specific implementation is a)defined Nyquist as follows: sampling is directly taken from ADC, and put it into the data cache. b)a) NyquistThe direct sampling under-sampling is directly is taken based from on the ADC, coeffi andcient, put n, it of into under-sampling. the data cache. Starting the counter, b) Theeach direct time oneunder-sampling ADC data is received,is based theon counterthe coef isficient, incremented n, of under-sampling. by one. When the Starting number the of counter, each time one ADC data is received, the counter is incremented by one. When the Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 14 Appl. Sci. 2019, 9, 4596 10 of 14 Appl. numberSci. 2019, 9of, x countsFOR PEER equals REVIEW n, the current number is stored in the buffer and the counter is reset10 of to 14 zero. In this mode, the amount of data is 1/n of Nyquist sampling. c) Pseudo-randomcountsnumber equals of counts n, sampling the equals current n, is numberthe the current random is stored number acquisitio in theis stored bun offfer one in and theelement the buffer counter of and data isthe reset from counter to coefficient zero. is reset In this n.to Themode,zero. specific In the this amount implementationmode, ofthe data amount is method 1/n of of data Nyquist is isto 1/npre- sampling. ofgenerate Nyquist a sampling.pseudo-random sequence of numbers. c)c) IfPseudo-random the data volume sampling is 1/10 of is thethe Nyquistrandom saacquisitio acquisitionmpling, nand of one random element numbers of data between from coecoefficient 0 andfficient 9 are n. generated,The specificspecific then implementation the probability method of choosing is to pre-generatepre- a randomgenerate number a pseudo-random is the same. sequence Then, the of numbers.numbers areIf thethe saved datadata to volumevolume the ROM isis 11/10/ 10inside ofof thethe the NyquistNyquist FPGA. sampling,sa Thempling, counter andand randomisrandom started numbersnumbers and the betweenbetween count value 00 andand is 99 are10,are countinggenerated, from thenthen 0 thetothe 9. probability probability When the ofcount of choosing choosing value a is randoma thrandome same number numberas the is output theis the same. valuesame. Then, of Then, the the ROM, numbersthe numbers data are is putsavedare intosaved to the the to buffer. ROM the ROM insideWhen inside thethe FPGA. counter the FPGA. The reaches counter The 9, counter the is started output is andstarted of the ROM countand the is value movedcount is 10, valueto countingthe isnext 10, address.fromcounting 0 to This 9.from When is 0how to the 9. pseudo-random When count valuethe count is the sampling value same is as th of thee datasame output can as valuebethe achieved. output of the value ROM, of data the isROM, put into data the is buputff er.into When the buffer. the counter When reaches the counter 9, the outputreaches of 9, the the ROM output is movedof the ROM to the is next moved address. to the This next is 5.2. Datahowaddress. Processing pseudo-random This isand how Analysis pseudo-random sampling of data sampling can be achieved. of data can be achieved.

5.2.5.2. DataDataFirstly, ProcessingProcessing the Nyquist andand AnalysisAnalysis sampling data is compressed, observed, and reconstructed to verify the correctness of the method. Figure 9 shows the Nyquist sampling data (sampling rate of 500 kHz) and the reconstructionFirstly,Firstly, thethe NyquistNyquist results samplingofsampling the DCS-OMP datadata isis method compressed,compresse andd, the observed,observed, DUS-CS andandmethod. reconstructedreconstructed toto verifyverify thethe correctnesscorrectness ofof thethe method.method. FigureFigure9 9 shows shows the the Nyquist Nyquist sampling sampling data data (sampling (sampling rate rate of of 500 500 kHz) kHz) and and thethe reconstructionreconstruction resultsresults ofof the the DCS-OMP DCS-OMP methodmethod and and the the DUS-CS DUS-CS method. method.

Figure 9. Nyquist sampling and reconstruction results. Figure 9. Nyquist sampling and reconstruction results. To show the reconstructionFigure 9. Nyquisteffect more sampling clearly, and reconstructionwe cut off pa results.rt of the original signal and comparedTo show it to the the reconstruction under-sampling eﬀect observation more clearly, and we reconstruction cut oﬀ part of performed the original on signal the original and compared signal (Figureit to theTo 10). under-samplingshow the reconstruction observation effe andct reconstructionmore clearly, we performed cut off onpa thert of original the original signal (Figuresignal and10). compared it to the under-sampling observation and reconstruction performed on the original signal (Figure 10).

Figure 10. The original signal. Figure 10. The original signal. The amount of the data is 20% of the original signal after direct under-sampling. Noise is added to generateThe amount signals of with the data the di isﬀ 20%erent of SNRs.Figure the original Then,10. The thesi originalgnal original after signal. signaldirect isunder-sampling. reconstructed using Noise DCS-OMP is added toand generate DUS-CS signals (Figure 11with). the different SNRs. Then, the original signal is reconstructed using DCS-OMPThe amount and DUS-CS of the data(Figure is 20% 11). of the original signal after direct under-sampling. Noise is added to generate signals with the different SNRs. Then, the original signal is reconstructed using DCS-OMP and DUS-CS (Figure 11). Appl.Appl. Sci. Sci.2019 2019, 9,, 9 4596, x FOR PEER REVIEW 11 of11 14 of 14 Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 14

(a) (b) (a) (b)

(c) (d) (c) (d) Figure 11. Reconstruction results with compression ratio of 20%. (a) SNR = 3 dB; (b) SNR = 0 dB; (c) Figure 11. Reconstruction results with compression ratio of 20%. (a) SNR = 3 dB; (b) SNR = 0 dB; (c) FigureSNR = 11. −3 ReconstructiondB; (d) SNR = −8 resultsdB. with compression ratio of 20%. (a) SNR = 3 dB; (b) SNR = 0 dB; (c)SNR SNR == −3 dB;3 dB; (d ()d SNR) SNR = −=8 dB.8 dB. − − Figure 12 is a compressive sensing signal with a compression ratio of 10% with different SNRs. FigureFigure 12 12 is is a a compressive compressive sensing sensing signalsignal withwith a compression ratio ratio of of 10% 10% with with different diﬀerent SNRs. SNRs.

(a) (b) (a) (b)

(c) (d) (c) (d) Figure 12. Reconstruction results with compression ratio of 10%. (a) SNR = 3 dB; (b) SNR = 0 dB; (c) SNR = 3 dB; (d) SNR = 8 dB. − − Appl. Sci. 2019, 9, 4596 12 of 14

Figure 11 shows the compressive sensing signal with a compression ratio of 10% with different SNRs. As for the processing result of the compressive sensing signal, it can be seen from Figure 11 that when the compression ratio is 20% and SNR = 3 dB, the reconstruction effect of the DCS-OMP method is poor, but the prior information method works well. As the SNR continues to decrease, the DUS-CS still reconstructs the original signal very well. When the compression ratio is 10%, the DCS-OMP method has a poor processing effect, as seen from Figure 12. For the DUS-CS method, as the signal-to-noise ratio decreases, the amplitude of the reconstructed signal deviates from the original, but the overall reconstruction effect is still significant. This result shows that the compressive signal obtained by the direct under-sampling system can recover the original signal, which confirms the correctness and engineering feasibility of the direct under-sampling compressive sensing system.

6. Conclusions This paper studies the direct under-sampling compression sensing method from the perspective of theory and engineering implementation. The DUS-CS method is proposed based on the theory of compressive sensing and the combination of the characteristics of echo signals with the implementation of direct under-sampling. Underwater echo simulation data was used to verify the correctness and effectiveness of the proposed method. Engineering implementation was achieved by improving the Nyquist sampling circuit with the adoption of pseudo-random sampling timing control. The rising edge (or falling edge) of each trigger signal was sampled and the signal was stored in the data buffer to obtain observation signals with compression ratios of 100%, 50%, 20%, and 10%. Then the DCT-OMP and the DUS-CS methods were used to process the data. The results show that both methods can reconstruct the original signal well when the compression ratio is 100% (in Nyquist sampling mode). Based on this, the correctness of the two methods was verified. When the compression ratio was 20% and below, the reconstruction effect of the DCS-OMP method was weak, but the DUS-CS method proposed in this paper was found to reconstruct the original signal well, even with the SNR being exceptionally low. These results confirm the correctness and engineering practicability of the DUS-CS method and data acquisition system. The method proposed in this paper provides an effective way for compressive sensing to be employed in practical engineering applications.

Author Contributions: Conceptualization, methodology, validation, writing—original draft preparation, T.S. and J.L.; Writing—review and editing, P.B.; Formal analysis, J.L.; Funding acquisition, T.S. Funding: This research was funded by the pre-research field fund project from Chinese Military Equipment Development Department, grant number 6140243010116DZ04001; the Extension Fund from Underwater Test & Control Technology Key Laboratory, grant number YS24071802; and Chinese National Natural Science Foundation, grant numbers 11974084. Acknowledgments: This article used experimental data collected at Jiaxing Yisheng Electronic Technology Company (China). We gratefully acknowledge our colleagues for their experimental skills. T. Sun worked on this article as a Visiting Research Fellow to the University of Bath (UK), thanks to the Chinese Scholarship Council (State Scholarship Fund no. 201608330338). Conflicts of Interest: The authors declare no conflict of interest.

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