88

Computationa

Sabna

Department of Electronics,

sabnan@yaho

Abstract: Compressivfewer samples than thepromising reconstructprocessing, biomedicaCompressive sensing concepts or dynamic simple and computatunderwater signals usitraditional systems. Thhas been ascertained bby a 3 blade engine.

Index Terms— Comcompression vector.

1. Introduction

Nyquist sampling theoremof nearly all the signal acquisiand reconstruction techniquealmost all the systems. SamplNyquist rate and transmitting in situations where the bandresource may adversely afthroughput and efficiency. Tneed and requirement for devtechniques for transmitting alower than the Nyquist ratetransmitted at par with the Nmight result in too manynecessitates the need for compbefore it is stored or transmittcompressed can be reconstr

PROCEEDINGS

ally Efficient Sparse ReconstructUnderwater Signals

N., Supriya M.H and P.R. Saseendran Pillai

Cochin University of Science and Technology, Kochi-

oo.com, supriyadoe@gmail.com, prspillai@cusat.ac.in

ve sensing provides a means to reconstruct certain signe traditional methods use. Its popularity is increasing dtion capabilities in various applications, such as

al signal processing, underwater acoustic communicatproblems are usually handled with linear progrprogramming methods. This paper presents a sp

tionally efficient method for the sparse reconstruing fewer samples than are necessary for reconstructiohe suitability of this method for efficient sparse reconby using a wave file containing the underwater noise g

mpressive sensing, l1 minimization, compression

m forms the basis ition, transmission es widely used in ling a signal at the it at the same rate

dwidth is a scarce ffect the system This warrants the vising systems and a signal at a rate

e. If the signal is Nyquist criteria, it y samples. This pressing the signal ted. The signals so ructed at a later

stage following certain techniques.

Underwater acoustic channerecognized as one of the mochallenging communication med[1,2]. High data rate comunderwater acoustic channel challenging task due to many repropagation delays result in Moreover, current research in and technologies necessitates requirements for in-situ onlintime, monitoring of the oceanicgeneral perspective. There was demand to apply the senstechniques in ocean environmwaves, which are used as communication in underwater networks, suffers from innumer

OF SYMPOL 2013

tion of

682022, India

n

nals from due to its s speech tion, etc.. ramming ecialized

uction of on in the

nstruction generated

matrix,

optimization

els are generally st difficult and dia in use today

mmunication in has been a

easons. The high n data latency.

ocean systems the need and

ne, almost real c processes in a also a growing

sor networking ments. Acoustic

the mode of wireless sensor

rable constraints

Sabna N et al.: Computationally Effic

such as low bandwidth, hifailure rate, etc. The availableunderwater acoustic channeltransmission loss, which incrand frequency. This imtransmission loss limits bandwidth for underwcommunication. The bandwiunderwater acoustic communorder of 1kHz, which is mucavailable for RF communicainformation could be stored fewer numbers of samples rate, it would have been underwater acoustic communi

A new method called comis used to represent and reclasses of signals at a rate brate [3-5]. Compressive senfact that when representeappropriate basis functions, mrelatively few non-zero csignals are called sparse signahas a sparse representation, oinsignificant coefficients andreconstructed with the sparsehave vanishingly small distort

2. Background

2.1.Compressive SensingMany signals, when

appropriate basis functionrepresentations. A discrete with N elements, can be vievector with n=1,2,....N. Compapplicable to only sparsaccordingly it becomes necesssignal x into a sparse domainconsider a basis function ψ, sparse representations of constraint that K < N such tsparsity ratio. x(n) can be rmatrix form as x=ψf, whematrix of order N x N and fcoefficient vector of order N x

cient Sparse Reconstruction of Underwater Signals

igh latency, high e bandwidth of an l depends on the reases with range

mplies that the the available

water acoustic idth available for nication is of the ch lower than that ation links. If the or transmitted by than the Nyquist advantageous in

ication scenario.

mpressive sensing econstruct certain below the Nyquist nsing rests on the ed in terms of many signals have oefficients. Such als. When a signal ne can discard the d the signal thus e data is found to tion.

g expressed using

ns have concise time signal x(n)

ewed as an N x 1 pressive sensing is se signals and sary to convert the n. To achieve this, which provides K x(n), with the

that α=K/N is the represented in the re ψ is the basis f is the weighting x 1.

In Compressive sensing, generated such that y=�x, wmeasurement matrix of order MN [3],[5]. This implies thatnumber of measurements y is dimensions of the input signal in Figure 1.

The reconstruction of the from the compressed one is teconsuming, since � is rectangcolumns than the rows. Thuessential to solve an underdetermsimultaneous linear equationsreconstruct the original signal.

Fig. 1: Concept of compres

The reconstruction of the

optimization problem, which compressibility of x with respefunction ψ. For effective recomandatory that � be incohereIncoherence implies that the mor the maximum magnitude oproduct matrix �ψ is relativCoherence is measured accordinμ , √ , 1If � and ψ contain correlatedcoherence is large and it takes trange [1,√ ].

Another property that � shthe Restricted Isometry P[3],[6],[7], i.e., 1 δ 1

89

a signal is where � is the M x N with M < t the available

fewer than the x, as elucidated

original signal edious and time gular with more us, it becomes mined system of s in order to

ssive sensing

e signal is an relies on the

ect to the basis nstruction, it is ent with ψ [3].

mutual coherence f entries of the vely small [4]. ng to , (1)

d elements, the the values in the

hould satisfy is Property (RIP)

δ (2)

90 PROCEEDINGS OF SYMPOL 2013

where, δs is the isometry constant of the matrix � and s is the sparsity of the signal x. A matrix � is said to obey RIP of order s if δs is not too close to one. While evaluating RIP for a particular matrix at hand is an NP-hard problem, certain random matrices such as the Gaussian and Bernoulli matrices are known to obey RIP with high probability [6],[7]. Random measurement matrices are largely incoherent with any fixed basis function ψ. These matrices are universal, which implies that the sparsity basis function need not even be known while designing the measurement system [5].

The signal can be reconstructed from what appears to be an incomplete set of measurements by taking M ≥ K log (N/K) measurements [5].

2.2. Data Recovery The sparse reconstruction of an

underdetermined system is not quite straightforward. To formulate the true sparsest representation, the condition that should be satisfied is

|| || , (3)

where ||x||0 is the number of non-zero elements of x [5].

One of the methods to alleviate the computational complexities associated with solving Eqn. (3) is to make use of the basis pursuit relaxation method which stipulates the computation of || || , (4)

where ||x||1 is the sum of the absolute values of the elements in x, which is also being referred as the l1 norm.

For solving underdetermined systems, though certain greedy algorithms [8] such as matching pursuits (MP) algorithm, orthogonal matching pursuit (OMP) algorithm, etc. were being used, the most versatile and widely accepted method involves the use of l1 norm

minimization which utilizes the concept of linear programming and solutions incorporating the traditional simplex algorithm or modern interior point methods. l1 norm minimization suffers from severe computational complexities and slow response times. In an attempt to improve the deficiencies of the l1 norm minimization, a novel technique which incorporates a method of stuffing or padding the matrix � at the transmitter followed by Moore Penrose inverse and LMS based processing at the receiver for signal reconstruction is being proposed. This approach does not have the iterative complexities compared to the l1 reconstruction and hence, is fast in convergence. Moreover, the proposed method shows an improvement in performance over the l1 minimization method.

2.3.Discrete Cosine Transform The performance of the proposed system

and its reconstruction capabilities depends on the front end transform used to represent the signal in the sparse domain. Towards achieving this, one-dimensional DCT is used in pre-processing the wave file under consideration.

The advantages of the DCT over DFT lies on the fact that it is real-valued and has better energy compaction and as such a sizeable fraction of the signal energy can be represented by a few initial coefficients.

The DCT of a 1-D sequence f(x) of length N is ∑ (5) for u = 0,1,2,…,(N −1). where

, 0, (6)

Sabna N et al.: Computationally Efficient Sparse Reconstruction of Underwater Signals 91

Here, the first coefficient, being the average value of the sample sequence, is referred to as the dc coefficient, while all other transform coefficients are called the ac coefficients [9].

Similarly, the inverse DCT is defined as ∑ (7) for x = 0,1,2,…,(N −1). 3. Methodology

The wave file under consideration is not sparse in the time domain. However, by transforming it to the frequency domain using FFT, the signal becomes sparse with real and imaginary parts, which makes reconstruction of the signal difficult. Conversion of the signal to DCT results in a signal which is sparse and also its coefficients are real valued, thus making the reconstruction easier [10].

The proposed technique converts the signal into sparse domain by applying DCT, followed by compressing it using a modified measurement matrix. This modification of the measurement matrix has been effected by padding it with a suitable sub matrix for resolving the singularity problems, while solving the underdetermined system of linear equations. Making use of this matrix padding technique, a computationally efficient sparse signal reconstruction has been achieved.

The performance of the proposed technique has been compared with the l1-magic [11] by computing the signal-to-noise ratio and correlation parameters.

The signal-to-noise ratio parameter is computed as 10 ∑∑ ′

..(8)

where x is the original signal and x' is the

Fig. 2: Compression operation

reconstructed signal.

The correlation parameter is computed using

′ ∑ ∑ ∑ ′ ..(9)

Framing of wave file

Conversion to DCT domain

Generation of modified Measurement matrix �'

Signal Compression

Generate yc (Compressed data) and yam (Auxiliary matrix)

Store / transmit yc and yav (Auxiliary vector) separately

Read the wave file

Start

Stop

Generation of Measurement matrix �

92 PROCEEDINGS OF SYMPOL 2013

3.1. Compression The data from a wave file is converted into

N' frames, each having N samples. Each frame is then converted to the frequency domain by using DCT which resulted in a sparse data representation. The compression matrix used is a random Gaussian measurement matrix � of size M x N with M < N. In order to make the computation of the matrix inverse feasible during the reconstruction phase at the receiver, it is padded with (N-M) x N ones which makes the matrix size to N x N. Operation of this modified matrix �' upon the framed sparse data results in a signal matrix y that has two sub matrices of which the first sub matrix yc gives the data pertaining to the matrix operation y=�x, while the other sub matrix yam provides certain redundant data consequent to the process of matrix padding. Removing the redundant data from yam results in a vector yav of size 1 x N'. The matrix yc of order M x N' and the vector yav of order 1 x N' are to be transmitted separately. Figure 2 depicts the flowchart for the compression operation at the transmitter.

3.2. Reconstruction The signal is reconstructed at the receiver by

generating the signal matrix y' by appending yc with yam, which is generated from the received yav by performing the inverse of the operation carried out at the transmitter. The Moore Penrose inverse of �' is taken and is multiplied with y' and the data so obtained is converted back to the time domain by the Inverse DCT operation. LMS based adaptation is also used to improve the performance of the method. Figure 3 illustrates the flowchart for reconstructing the original signal at the receiver.

4. Results and Discussions

The proposed approach for compressive sensing has been simulated under noiseless and noisy environments and the performance of this

approach has been vis-a-vis compared with that of the l1 minimization method.

Fig. 3: Reconstruction operation

Case 1: Noiseless Transmission Figure 4 shows the results of comparison of

the performances of the proposed method with the existing l1 minimization method for reconstruction at the receiver. The plot shows the SNR variation for reconstruction with different number of sparsely chosen samples per frame. The SNR parameter is found to quantify a measure of the distortion of the reconstructed signal.

Retrieve / receive the signals yc and yav

Generate yam from yav

Append yam to yc and generate y'

Generate Moore Penrose inverse of �' and multiply it

with y'

Perform Inverse DCT operation

Perform LMS based processing for reconstruction

Start

Stop

Sabna N et al.: Computationally Effic

Fig. 4 : Comparison of SNmethod with the l1 minimi

Figure 5 shows the correlation coefficient of the with the l1 minimization methcomparison further show thnumber of samples used for rbetter the SNR and the corre

Fig. 5 : Comparison coefficient of proposed mminimization method

are and thus the similarity betreconstructed signals increafrom these figures, the propsuperior to the l1 minimizacomputation time of the has also been compared wiminimization method. The effof samples used for reconstructhe computation time in secoFigure 6. In this context, it is that the computation time method is fairly stable, irr

cient Sparse Reconstruction of Underwater Signals

NR of proposed ization method

comparison of proposed method

hod. The results of hat, the more the reconstruction, the elation parameters

of correlation method with l1

tween original and ases. As evident osed method is ation method.The proposed method

ith that of the l1 fect of the number cting the signal on

onds is depicted in worth mentioning for the proposed respective of the

number of samples used for recsignal.

Fig. 6 : Comparison of computhe proposed method with lmethod

Case 2: Noisy Transmissio

Figure 7 shows the comperformances of the proposed mexisting l1 minimization met

Fig. 7 : Comparison of SNRproposed method and l1 method in noisy case

reconstruction at the receiveenvironment. The plot shows tfor different number of spsamples per frame, while figucomparison of correlation coeproposed method with the lmethod under noisy environmefrom the plots, the SNR a

93

constructing the

utation time for l1 minimization

on

mparison of the method with the thod for signal

R variation of minimization

er under noisy the SNR values parsely chosen ure 8 shows a efficient of the l1 minimization ent. As evident and correlation

94

performances of the proposignificantly superior comparl1 minimization method under

Fig. 8 : Comparison coefficient of proposed minimization method under

Figure 9 shows a comparisignal with the signals reconproposed method and the l1 first 500 samples with 50% c

Fig. 9 : Comparison of orithe signals reconstructed methods

noiseless environment. Thecomparison reveal that the proutperforms the l1 minimizatio

PROCEEDINGS

osed method are red to the existing r noisy scenario.

of correlation method with l1

r noisy case

son of the original nstructed using the

minimization for ompression under

ginal signal with using the two

e results of this roposed technique on methods.

5. Conclusions

Though there are many solvers such as l1-magic [1SPARLS [13], YALL1 [14], ehas been made in this paper performance of the proposed mmagic.

The proposed method is computationally efficient foreconstruction in both noiselsituations. Its performance is facompared to the l1 minimiespecially in the case in whicdistorted by additive Gaussian n

ACKNOWLEDGMENTSThe authors gratefully ac

Department of Electronics, Coof Science and Technology, fothe facilities for carrying out authors also acknowledge theCouncil for Science, TecEnvironment for the financial as References

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