AUDIO COMPRESSION USING DCT & CS 1 MR. SUSHILKUMAR BAPUSAHEB SHINDE, 2 PROF. MR. RAKESH MANDLIYA 1 M. Tech. (VLSI), BMCT Indore, Madhya Pradesh, India, [email protected] 2 Head of EC Department, BMCT Indore, Madhya Pradesh, India, [email protected] ABSTRACT A large amount of techniques have been proposed to identify whether a multimedia content has been illegally tampered or not. Nevertheless, very few efforts have been devoted to identifying which kind of attack has been carried out, especially due to the large data required for this task. We propose a novel hashing scheme which exploits the paradigms of compressive sensing and distributed source coding to generate a compact hash signature, and we apply it to the case of audio content protection. At the content user side, the hash is decoded using distributed source coding tools. If the tampering is sparsifiable or compressible in some orthonormal basis or redundant dictionary, it is possible to identify the time‐frequency position of the attack, with a hash size as small as 200 bits/second; the bit saving obtained by introducing distributed source coding ranges between 20% to 70%. The audio content provider produces a small hash signature by computing a limited number of random projections of a perceptual, time‐frequency representation of the original audio stream; the audio hash is given by the syndrome bits of an LDPC code applied to the projections. By using the DCT as signal preprocessor in order to obtain a sparse representation in the frequency domain, we show that the subsequent application of CS represent our signals with less information than the well‐known sampling theorem. This means that our results could be the basis for a new compression method for audio and speech signals. Index Terms: — Audio Signal, DCT, Compressive Sampling, Sparsity. 1. INTRODUCTION Compressive Sampling (CS) is a new framework for sampling and compressing of audio and speech signal. In compressive sampling Nyquist sampling model is replaced by sparse model by assuming that signal can be represented efficiently using just few significant coefficients. The tremendous work by Candes et.al.[3] an Donoho [4] proved that, along with implying the potential of dramatic reduction of sampling rates, power consumption and computation complexity in digital data acquisition, signal can be reconstructed with smaller than Nyquist rate. For low power and low resolution imaging devices and or when measurement is very costly, compressive sampling is traditionally used.(e.g. Terahertz application). But there still exits a huge gap between CS theory and its application to audio signals [13][14].How to construct a sparse audio signal, especially when CS depends on two principal: sparsity (which pertains to signal of interest), and incoherence (which pertains to sensing modality), is still unknown[6]‐[8]. We have used DCT for sparse representation of an audio signal. It concentrates on the transformation content in relatively few coefficients, and it achieves a good data compression which causes its popularity [9]. Thus we can obtain a compressed version of audio signal by first obtaining a sparse representation in frequency domain, and then after processing the result with CS algorithm. 1.1 COMPRESSIVE SAMPLING For increasing amount of data in our modern technology, most of data we can throw away without any perceptual loss e.g. lossy compression formats for sound, image etc. Hence question arises that why to acquire all data when most of data we will throw away? Can we directly measure only that data which is necessary? A theory of signal recovery from highly incomplete information is developed in recent series of paper [3]‐[8]. Overview of results state that sparse vector x 0 ϵR N (e.g. Digital Signal) can be recovered from small number of linear measurement b=Ax 0 ϵR N or b=Ax 0 +e, where A is n x m matrix with far fewer rows than column (n<<m) and e is measurement noise by solving a convex program. Consider real valued signal x of length N and suppose that the basis function ψ provides k as sparse representation of x. In terms of matrix notation, we have x= ψ.f. In which f is sparse vector with only K non‐zero elements, which can be well approximated using only k<<N non zero entries and ψ is called as sparse orthogonal basis matrix i.e. { ψ1 , ψ 2 .....ψ N} [4] The CS theory sates that by taking only M=O(klogN) linear, non adaptive measurements shown below we can reconstruct signal x.[1],[2]: (1) SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015 ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015 INTERNATIONAL JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY- www.ijset.in 308
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AUDIOCOMPRESSIONUSINGDCT&CS
1MR.SUSHILKUMARBAPUSAHEBSHINDE,2PROF.MR.RAKESHMANDLIYA
1M.Tech.(VLSI),BMCTIndore,MadhyaPradesh,India,[email protected]
2HeadofECDepartment,BMCTIndore,MadhyaPradesh,India,[email protected]
ABSTRACT
A largeamountoftechniqueshavebeenproposedto identifywhetheramultimediacontenthasbeen illegallytamperedornot.Nevertheless,veryfeweffortshavebeendevotedtoidentifyingwhichkindofattackhasbeencarriedout,especiallyduetothelargedatarequiredforthistask.Weproposeanovelhashingschemewhichexploitstheparadigmsofcompressivesensinganddistributedsourcecodingtogenerateacompacthashsignature,andweapplyittothecaseofaudiocontentprotection.At the content user side, the hash is decoded using distributed source coding tools. If the tampering is sparsifiable orcompressibleinsomeorthonormalbasisorredundantdictionary,itispossibletoidentifythetime‐frequencypositionoftheattack,withahashsizeassmallas200bits/second;thebitsavingobtainedbyintroducingdistributedsourcecodingrangesbetween20%to70%.Theaudiocontentproviderproducesasmallhashsignaturebycomputingalimitednumberofrandomprojections of a perceptual, time‐frequency representation of the original audio stream; the audio hash is given by thesyndromebitsofanLDPCcodeappliedtotheprojections.ByusingtheDCTassignalpreprocessorinordertoobtainasparserepresentation in the frequency domain,we show that the subsequent application of CS represent our signalswith lessinformation than thewell‐knownsampling theorem.Thismeans thatourresultscouldbe thebasis foranewcompressionmethodforaudioandspeechsignals.
IndexTerms:—AudioSignal,DCT,CompressiveSampling,Sparsity.
1.INTRODUCTION
Compressive Sampling (CS) is a new framework forsamplingandcompressingofaudioandspeechsignal. Incompressive sampling Nyquist sampling model isreplacedbysparsemodelbyassumingthatsignalcanberepresented efficiently using just few significantcoefficients.
The tremendousworkbyCandes et.al.[3] anDonoho [4]provedthat,alongwithimplyingthepotentialofdramaticreduction of sampling rates, power consumption andcomputationcomplexity indigitaldataacquisition,signalcanbereconstructedwithsmallerthanNyquistrate.
Forlowpowerandlowresolutionimagingdevicesandorwhenmeasurement isverycostly,compressivesamplingistraditionallyused.(e.g.Terahertzapplication).
ButtherestillexitsahugegapbetweenCStheoryanditsapplication to audio signals [13][14].How to construct asparse audio signal, especiallywhen CS depends on twoprincipal: sparsity (which pertains to signal of interest),and incoherence(whichpertains tosensingmodality), isstillunknown[6]‐[8].
WehaveusedDCTforsparserepresentationofanaudiosignal. It concentrates on the transformation content inrelatively few coefficients, and it achieves a good datacompressionwhichcausesitspopularity[9].Thuswecanobtain a compressed version of audio signal by firstobtaining a sparse representation in frequency domain,andthenafterprocessingtheresultwithCSalgorithm.
1.1COMPRESSIVESAMPLING
Forincreasingamountofdatainourmoderntechnology,mostofdatawecanthrowawaywithoutanyperceptuallosse.g. lossycompression formats for sound, imageetc.Hencequestionarises thatwhy toacquirealldatawhenmost of data we will throw away? Can we directlymeasure only that data which is necessary? A theory ofsignal recovery from highly incomplete information isdeveloped in recent series of paper [3]‐[8]. Overview ofresults state that sparsevectorx0ϵRN (e.g.Digital Signal)can be recovered from small number of linearmeasurement b=Ax0ϵRN or b=Ax0+e, where A is n x mmatrixwithfarfewerrowsthancolumn(n<<m)ande ismeasurementnoisebysolvingaconvexprogram.
ConsiderrealvaluedsignalxoflengthNandsupposethatthebasis functionψprovideskas sparse representationofx.Intermsofmatrixnotation,wehavex=ψ.f.InwhichfissparsevectorwithonlyKnon‐zeroelements,whichcanbewell approximated using only k<<N non zero entriesandψiscalledassparseorthogonalbasismatrixi.e.{ψ1,ψ2.....ψN}[4]
The CS theory sates that by taking only M=O(klogN)linear,nonadaptivemeasurementsshownbelowwecanreconstructsignalx.[1],[2]:
(1)
SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015
INTERNATIONAL JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY- www.ijset.in 308
WhereYrepresentsM×1sampledvectorand is anM×N
measurement matrix that is incoherent with i.e., themaximummagnitudeoftheelementin .ψissmall[7].
Along with this information we decide to recover thesignalbyL1‐minimisationisprobablyexact[1].
1.2OneDimensionalDCT
The most common DCT definition of a 1‐D sequence oflengthNis
(2)
foru=0,1,2,…,N−1.Similarly,theinversetransformationisdefinedas
(3)
Forx=0,1,2,…,N−1.Inbothequations(ii)and(iii)α(u)isdefinedas
Itisclearfrom(1)thatfor
Thus,thefirsttransformcoefficientistheaveragevalueofthe sample sequence.Thisvalue is referred toas theDCCoefficientandallother transformcoefficientsarecalledtheACCoefficientsinLiterature.[9]
1.3 THETIME‐FREQUENCYFILTERBANK
TheMP3standard[4]recommendstheuseofahighpassfilter.Ahighpass filterallows frequenciesaboveagivencutofffrequencytopassanddoesnotallowloweronestopass. Inotherwords, itattenuatesthelowerfrequencies.Thecutofffrequencyshouldbeintherangeof2Hzto10Hz.
1.4 THEPOLYPHASEFILTER
ThepolyphasefilterusedinMP3[8]isadaptedfroman earlier audio coder named Masking Pattern AdaptedUniversal Subband Integrated Coding and Multiplexing(MUSICAM). It is a cosinemodulated lowpass prototypefilter with uniform bandwidth parallel M‐channelbandpass filter. This achieves nearly perfectreconstruction and has been called a psuedo QMF(QuadratureMirrorFilter).
2.PROPERTIESOFDCT
2.1 ENERGYCOMPACTION
For highly correlated signals DCT exhibits excellentenergycompaction.Theuncorrelatedsignalhasitsenergyspreadout,whereastheenergyofthecorrelatedsignalispackedintothelowfrequencyregion.Usingtheabilitytopack input data efficiency of transformation scheme canbe directly gauged into as few coefficients as possible.Because of this quantizer allows to discard coefficientwith relatively small amplitudes without introducingvisualdistortioninreconstructedsignal.
2.2 ORTHOGONALITY
IDCT basis functions are orthogonal. Thus, the inversetransformation matrix of A is equal to its transpose i.e.invA=A'.WhereAisanyrandomnxnmatrix.Thereforein addition to its decorrelation characteristics, thisproperty results reduction in pre‐computationcomplexity.
2.3 SYMMETRY
Thisisextremelyusefulpropertysinceitimpliesthatthetransformation matrix can be precomputed offline andapplied to the signal thereby providing orders ofmagnitudeimprovementincomputationefficiency.
2.4 DECORRELATION
The principle advantage of signal transformation is theremovalof redundancybetweenneighboringpixels.Thisleadstouncorrelatedthetransformcoefficientswhichcanbe encoded independently. The amplitude ofautocorrelation after the DCT operation is very smallhence it can be inferred that DCT exhibits excellentdecorrelationproperties.
2.5 SEPARABILITY
Perform DCT operation in any of the direction first andthenapplyonoppositedirection,thenalsocoefficientwillnotchange.
3.METHODOLOGY
This section includes proposed techniques applied to anaudiosignalanddescribedthetechniqueforrepresentingittheformofsparse.
Figure1:ProposedScheme
As we can recover sparse signal from just a fewmeasurements,compressivesamplingneedstodealwith
SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015
INTERNATIONAL JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY- www.ijset.in 309
speech signals which are only approximately sparse. Toobtain an accurate reconstruction of such signal fromhighly under sampled measurement is the main issue.IdeallywehavetomeasurealltheNcoefficientsoff,butCSframeworkwillallowobservingasubsetoftheseonlyandcollectingthedata.
Figure2:AudioSignal
Figure3:FFTAmplitudeofAudioSignal
As seen in Figure 2, the audio signal (funky.wav) isconsideredherefortheoperationinTimedomainbutnotsparse, hence we have applied Fast Fourier Transform(FFT) which represents our signal in frequency domainandintheformofSparseSignalasshowninFig.3.
Due to Matrix transformation on compressive samplingprogram,asdescribein[10],phaseanglechangesbecauseof representation in real and complex parts. Hence justapplyingInverseFourierTransform,originalsignalwon’tberecovered.
4.RESULTS
Figure4:OriginalAudioSignal
Figure5:SpectrogramofOriginalAudioSignal
Figure6:DCTofOriginalAudioSignal
Figure7:ThresholdingofsignalafterDCT
Figure8:RandomMeasurementMatrix
SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015
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Figure9:ObservationVector
Figure10:ReconstructionofSignalusingl1‐minimisation
Figure11:ReconstructedAudioSignalafterIDCT
Figure12:SpectrogramofReconstructedAudioSignal
As per above discussion, here we have taken a sampleAudiosignalasshowninFigure4.Aspectrogramisvisualrepresentation of spectrum of frequency in sound.[15].Forbetter visibility andunderstandingof this signal,we
have constructed spectrogram as shown in Figure 5.Spectrogram is nothing but graph of Time versusNormalizedfrequency.
Asour requirement is first togenerate sparsesignal,wehave taken DCT of audio signal shown in Figure 6. Forbetterperformanceandgoodcompressionofgivenaudiosignal, we can omit the unnecessary noisy samples bythresholding.Wecandecidetherangeforthresholdingasper our need. Here we have taken the range ofthresholdingas ‐0.06 to0.04.This rangeof thresholdinghasbeendecidedbytrialanderrormethodandselectingthethresholdrangewhichgivesbetteroutputasshowninFigure7.
Now for generationof observation vector (Figure9),wehave taken random samples from original audio signaland then reconstructed a matrix called ‘Randommeasurement matrix’ as shown in Figure 8. Bymultiplying threshold signal with random measurementmatrixwegetobservationvectorwhichisusedforfurtherprocessofreconstruction.
L1 minimization is theory of signal reconstruction fromhighly incomplete information[16].SoauthorshaveusedL1 minimization for reconstruction of audio signal. Thereconstructed audio signal using L1 minimization isshown in Fig. 10. Now to reconstruct this signal intooriginalaudiosignalwehavetakenIDCTofitasshowninFigure11anditsspectrogramisshowninFig.12.
Inthisexpt.,wehavetoconsidernumberofsamplesandcompressionratioisgivenby
Similarity=e2/E2(iv)
Where e2 is Matrix error given by norm of ||x‐xrec||dividedbyhislength.AndE2ismatrixpowergivenbybynormof||x||dividedbyhislength.
In Experiment‐I, Sparsity is kept constant at value of1000andCompressionFactoriskeptconstantatvalueof0.05. By varying Block Size, Compression Ratio, SNR,PSNRismeasured.
From the table Iwe can see that, for the less block sizemeasureofcompressionratioismore,butSNRandPSNRis less. But as Block size goes on increasing,measure ofcompressionratiogoesondecreasingwhereasSNRandPSNR goes on increasing. For Block size 8, Compressionratiois0.6877,SNRis3.3517dBandPSNRis15.5773dBwhereasforblocksize512,compressionratiois0.59688which is low as compare to compression ratio of blocksize8,SNRis4.0212dBandPSNRis16.4728dBwhichishighascomparetocompressionratioofblocksize8.
Table‐1:MeasureofCompressionRatio,SNRandPSNRforvariousvaluesofBlocksize
SR.No.
BlockSize
Measure ofCompressionRatio
SNR PSNR
1 8 0.6877 3.3517 15.5773
2 16 0.64371 3.1055 15.4329
SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
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3 32 0.60878 3.6844 16.2396
4 64 0.58672 3.4062 15.7336
5 128 0.55898 3.5541 15.7331
6 256 0.58086 3.4986 16.0433
7 512 0.59688 4.0212 16.4728
In Experiment‐II, Block Size is kept constant at value of128andCompressionFactor iskeptconstantatvalueof0.05.ByvaryingSparsity,CompressionRatio,SNR,PSNRismeasured.
From the table II we can see that, for the less value ofSparsity, SNR and PSNR is less. But as Sparsity goes onincreasing,SNRandPSNRgoesonincreasing.
ForSparsity128,SNRis‐0.5677dBandPSNRis11.6113dB where as for Sparsity 2500, SNR is 20.1456 dB andPSNRis32.3249dBwhichistoohighascomparetoSNRandPSNRatSparsity128.
Table‐2:MeasureofCompressionRatio,SNRandPSNRforvariousvaluesofSparsity
SR.No.
Sparsity SNR PSNR
1 128 ‐0.56777 11.6113
2 300 ‐0.07818 12.1008
3 600 1.6737 13.8527
4 800 2.3328 14.5118
5 1000 3.2318 15.4109
6 1200 5.1285 17.3075
7 1500 6.8487 19.0277
8 1800 10.2104 22.3895
9 2000 14.0706 26.2496
10 2500 20.1459 32.3249
InExperiment‐III,BlockSize is kept constant at valueof128 and Sparsity is kept constant at value of 1000. ByvaryingCompressionfactor(CF),MeasureofCompressionRatio,SNR,PSNRismeasured.
From the table III we can see that, for the lessCompression factor, measure of compression ratio ismore but SNR and PSNR is Compression factor 0.01,Measureofcompressionratiois0.86133,SNRis3.395dBandPSNRis15.7545dBwhereasforCompressionfactor2.5,measureofcompressionratio is0.0019531which istoo low as compare tomeasure of compression ratio ofCompression factor 0.01. SNR is 17.927 dB and PSNR is32.8129dBwhichisalsotoohigh.
Table‐3:MeasureofCompressionRatio,SNRandPSNRforvariousvaluesofCompressionfactor(CF).
SR.No.
CFMeasure ofCompressionRatio
SNR PSNR
1 0.01 0.86133 3.395 15.7545
2 0.03 0.68516 3.9678 16.3307
3 0.05 0.55898 4.0602 16.2393
4 0.08 0.4375 3.7993 16.052
5 0.1 0.38125 3.7526 15.9674
6 0.5 0.080469 3.6443 16.7103
7 1 0.028516 3.9149 17.0527
8 1.5 0.0125 6.6961 19.4112
9 2 0.0054687 10.7931 24.5405
10 2.5 0.0019531 17.9247 32.8129
5.CONCLUSION
Inputdata ispacked into few coefficients inDCT speechsignal representation. This helps quantizer to removecoefficients with smaller amplitudes without generatingaudiodistortioninreconstructedsignal.
Compressivesampling ismainlyusedforcompressionofimagesbutwecanachievegoodresultsbypreprocessingtheaudiosignal.
This technique can achieve a significant reduction innumber of samples required to represent certain audiosignal and it reduces required number of bytes forencoding.
FurtherimprovementsarepossiblewithadvancedcodingtechniqueslikeWaveletorDWT[23].
ACKNOWLEDGEMENTS
I want to give my whole sincere to my supervisor andgrateful appreciation to Prof. Mr. Rakesh Mandliya, hetriedherbest tohelpme.Withoutherguidance Icannotbringthetheoriesintopractice.Ontheotherhand,Iwantto thank all my family members and friends for theiralwayssupportandspiritualmotivation.
Thanksalot!
REFERENCES
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SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015
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BIOGRAPHIES
Mr. Sushilkumar BapusahebShinde completed B.E. inElectronics & Telecommunicationin 2012 from SRESCOE,Kopargaon, PuneUniversity, PuneandcurrentlypursuingM.Tech.inVLSI from BMCT, Indore, RGPV,MP.
Prof Mr.RakeshMandliya isHeadofECDepartment,BMCTIndore,MadhyaPradesh,India.
SUSHILKUMAR BAPUSAHEB SHINDE et al. DATE OF PUBLICATION: FEB 20, 2015
ISSN: 2348-4098 VOL 3 ISSUE 1 JAN-FEB 2015
INTERNATIONAL JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY- www.ijset.in 313
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