Estimation of non Distortion Audio Signal Compression

Transcription

1 Estimatio of o Distortio Audio Sigal Compressio M. BAK, S. PODOXI, V. TSIGOUZ Departmet of Commuicatio Egieerig Ceter for Techological Educatio Holo 5 Golomb street, POB 35 Holo 58, Tel: ISRAEL Abstract - There are two o-distortio audio sigal compressio methods (i some papers the term lossless is used rather tha the o-distortio oe): waveform codig ad spectrum codig. The ew combie method of audio sigal compressio is recommeded. Firstly, oe chooses the time itervals such that the sigal iside these itervals is statioary to the greatest degree. Secodly, for every time iterval the sigal is aalyzed i both time ad frequecy domais. Oe chooses the domai where sigal etropy is smaller. Thirdly, sigal codig is performed i the chose domai. Key Words: Audio Sigal, compressio, huma hearig system.. Itroductio. O the Audio Sigal Spectral Method Selectio The cocept of spectrum codig is based o the spectral aalysis of the sigal accordig to which the relevat sigal characteristics are separated ad ecoded. The, the aalysis is used for subsequet sythesis of a sigal similar to the iitial oe. So far, the most widespread discrete trasform methods used i digital soud sigal processig have bee those of cosie trasform (DCT), Fourier trasform (DFT), Hartley trasform (DHT) ad Walsh trasform (DWT). The lesser-kow sie-cosie trasform (DSCT) is expressed as follows []: g( ), = k ( + ) πk ( ) g( )cos, = S( = k ( ) g( ), = k ( + ) πk ( ) g( )si, = k = < k k = < k () where is the umber of samples (eve umber i a case of DCT ad DSCT), S( is the amplitude of the k-th spectral compoet. The orthogoality of the trasform follows from expressios likig it with DFT: i c( Re e S( = i c( Im e πk πk F(, k F(, k (), k =, / where c ( =, k, / ad F( is the DFT of the sequece g(). The abovemetioed methods approximate the Karhue-Loeve trasform (KLT), regarded as optimal for redudacy reductio, to various extets. For audio ad video sigals (excludig radom processes simulatig these sigals), the degree to which the first four of the abovemetioed trasform techiques approximate KLT has bee studied sigificatly, ad are listed here i the order of their efficiecy, DCT beig the most efficiet. As for high-quality audio sigals, these ratigs are somewhat ambiguous ad should be proved experimetally. I this respect, agai, the differeces i efficiecy betwee these trasform methods also deserve attetio. For the experimetal compariso of the trasform above, the method used icludes the calculatio of the correlatio fuctio. This is permissible i the case of audio sigals if a appropriate time iterval is used for the aalysis. Cosiderig the kow optimal

2 properties of KLT applicable to the give characteristic of the sigal, preferece should be give to this trasform method, to esure the highest degree of power cocetratio of the spectral compoets resultig from the trasform of the correlatio fuctio of the sigal. To measure power cocetratio, oe uses the followig expressio, which is based o the otio of etropy: where is the trasform dimesio (block legth), ad σ is the coefficiet of the ormalized spectral distributio of the audio sigal correlatio fuctio. Aalyzig expressio (3), it is easy to see that it attais the highest value for the uiform distributio of (3) σ, which correspods to zero power cocetratio. The values of E (Table ), obtaied by computeraided processig of a real audio sigal are averaged values derived from a set of typical soud sigal sequeces. The calculatio shows that DCT maitais its lead as a meas of efficiet redudacy reductio esuig from the correlatio liks existig betwee the samples of the audio sigal. Icidetally, similar results are preseted i where the audio sigal was simulated by a radom process with a predetermied statistic. Table : Compariso of power cocetratio values E of spectral compoets - the greater degree of cocetratio correspods to the lower value of E. Block Legth E = = σ log ( σ, DFT DCT DSCT DWT DHT To reduce redudacy due to huma auditory peculiarities revealed durig earlier experimetatio with simple harmoic sigals, it is worth cosiderig discrete trasforms i which the spectral ), σ represetatio of simple sigals has the least degree of redudacy. The time iterval selectio for etropy calculatio i the case of waveform codig ad spectrum codig ca be based o the mathematical model of audio iformatio processig by the huma auditory system [, 3]. The cetral processor cotiually compares the spectra obtaied with the previous samples ad geerates a feedback sigal so that the spectrum obtaied will be more sparse. That is why sparsig i uit 3 (see Figure ) has a lesser ifluece o the obtaied spectra. While listeig to musical sigals, the obtaied time itervals correspod to the rhythm, i case of speech, they correspod to syllable duratio. I both cases, the time itervals are approximately withi the rage of ms. The described process leads to etropy decrease, ad whe applied to soud sigal processig, causes less iformatio loss Resoators bak - Filter bak 3 - Sparsig chael 4 - Cetral processor 5 - Segmetator Figure : Mathematical model of audio iformatio processig by the huma auditory system The aalogy for the give method ca be based o the etropy calculatio of samples over the time iterval, icreasig util the etropy reaches its miimum value: mi H = M log (4) P ( E i ) where P ( Ei ) - is a probability of occurrece of sample value Ei, M - is a calculatio of samples expectatio.. Algorithm Developmet Developmet of the algorithm cotais followig 4

3 stages: Etropy divisio of time domai to time itervals; Etropy calculatio of each time iterval; Cosie trasform of the itervals; Etropy calculatio of frequecy domai of each iterval; Selectig the domai correspodig to a miimal etropy value for each iterval; Miimal etropy average value calculatio. I our simulatio we used MATLAB tools... Etropy Divisio Method ad Etropy Calculatio of the Time Itervals After Etropy Divisio We suggest here The Etropy Divisio Method of the time domai. This method is based o the fact that, withi differet parts of a audio sigal, the etropy is also differet. The parts uder cosideratio are syllables ad the segmets betwee the syllables (i the case of music, we ca iterpret the syllables ad the segmets betwee the syllables as musical rhythm). The fact is that the etropy withi a syllable is less tha the etropy betwee syllables because the sigal withi the syllable is more statioary tha that betwee the syllables. If we advace from the begiig of the sigal to its ed we ecouter ot oly syllables ad the iter-syllable segmets (these are the sigal parts that are either clearly related to a syllable or to a iter-syllable part), but also the parts that are placed betwee the ed of each syllable ad the begiig of the itersyllable segmet. Here, withi these sigal parts, we would expect to fid etropy peaks. Usig these etropy maximas, we ca perform divisio of a audio sigal ito syllables, i.e. Etropy Divisio. To perform this task, we take a time widow ad ru it alog the sigal. After each step, etropy calculatio of the sigal withi the curret widow is executed. What should the widow size be? To aswer this questio, we'll eed to ru the widow a umber of times usig a rage of sizes; for example from 5ms to ms. It is sufficiet to do use 5ms icremets to discover the best results for the etropy divisio. The greater the correspodece betwee etropy peaks ad begiig/ed of the syllables, the better the results... Compariso Betwee the Sigal Etropy i the Time ad Frequecy Domais, - Selected Domai Method Here we perform the spectral trasform of each time iterval. ext, we calculate the etropy withi the freof each iterval. I the last step, the compariso betwee time ad frequecy domai etropy values is performed. The purpose of this compariso is to select the miimal etropy value betwee two of the followig: time domai ad the correspodig frequecy domai. The secod method we suggest (the first beig the etropy divisio method) hadles the selectio of the sigal domai that correspods to the miimal etropy value. We call this method The Selected Domai Method. (After the domai is selected, it is fed to the coder for the further trasmissio) ext, the miimal etropy vector is costructed ad the Miimal Etropy Average Value is calculated..3. Etropy Divisio ad Radom Divisio Compariso The last simulatio compares the miimal etropy average value that is obtaied by the etropy divisio with the miimal etropy average value that is obtaied by the radom divisio. The calculatio of the average values is performed usig the selected domai method. I order to accurately compare betwee two these values, we had to equate the itervals of the divisio both i the etropy ad the radom. To do this, we simply shifted the etropy divisio borders to the ed directio of the sigal so that the ew order of the divisio became the followig: middle of the itersyllable part, middle of the syllable, middle of the iter-syllable part ad so o (the etropy divisio gave us the kow divisio: syllable, iter-syllable part, syllable, iter syllable part ad so o). We expect to get a lower miimal etropy average value i the case of etropy divisio tha i radom divisio. 3. The Sigals To achieve the best results, we used sigals from various sources, e.g.: Widows '95 operatig system built-i ".wav" files, iteret dowloaded ".wav" files, CDs with music records, huma speech etc. 3

4 From each of these sigals, we selected itervals of approximately ms, due to persoal computer CPU limitatios. This legth was sufficiet to obtai reliable results for our ivestigatio, because the goal of our research was ot to develop a real time audio compressio system, but to show that there is a method that allows us to compress a audio sigal, usig Huffma codig, better tha usig prevalet compressio methods. The sigals chose are of varyig degrees of soud quality: from the high quality CD's music to the very low quality of huma speech recorded from a low-grade microphoe oto the hard drive of persoal computer. The mai object of our work is speech. Speech has may differet characteristics. For example, if a poet reads his lyrics it could be perceived as a variety of music. A live sports broadcast, however, could be perceived as a variety of oise. Because of this, we used a umber of audio sigals with differet characteristics. The followig are examples of the sigals we used: first four otes of the classical compositio "The Four Seasos" by Atoio Vivaldi - take from music CD, high quality sigal. speech male sigal "What is this?" - was take from the Widows '95 operatig system as a ".wav" file. Other speech ad music sigals. 4. Aalysis of the Results 4.. Etropy Divisio Method As stated above, the purpose of this simulatio is to fid the optimal time widow size ad the step of this widow ecessary to divide the sigal ito syllables. The widow size rage we used is from 5ms to ms. The rage of steps used is from ms to the widow size itself. For example, a 5 ms time widow was ru with steps of ms to 5ms. The mai result of the simulatio was that the optimal time widow for etropy divisio of a audio sigal is about 5ms. The optimal step size is optioal ad may be chose as 5ms (the smaller size of the step, the higher resolutio of the divisio). Figure shows the "Vivaldi" sigal with the etropy "evelope" above it. Etropy peaks are clearly visible above begiig ad ed of each syllable o the figure: Figure : Music sigal "Vivaldi" with the etropy evelope. Widow size - 5ms, step - 5ms. 4.. Etropy Calculatio After Etropy Divisio Etropy calculatio of each sigal segmet (syllable or the iter-syllable segmet) was performed accordig to the followig equatio: E = P( s ) log ( ) + ( ) log ( ( ) ) P ( s ) P s + P s (5)... + P( s ) log ( ) i P ( s i ) where: E - etropy value, P s ) - probability of the I-th sample, i - total umber of samples i the processed sigal part. Figure 3 shows the male speech sigal: "What is this?". The top part of the figure is the whole sigal, the middle part is the "is" syllable i real size ad the ( i Figure 3: The male speech sigal "What is this?" a - the whole sigal b - the "is" syllable i real size c - the "is" syllable i the scaled size bottom part is the scaled "is" syllable: It ca be show that the etropy of a syllable i frequecy domai is less tha the etropy of the syllable i time domai, but the etropy of a itersyllable sigal segmet is larger i frequecy domai tha i time domai. This is the basis of our suggestio that a audio sigal is more statioary withi syllables tha withi iter-syllable segmets ad that the etropy withi syllables is less tha withi the iter-syllable segmets. This meas that a b c 4

5 the amout of iformatio is less withi the syllable ad, therefore, we ca compress more i frequecy domai tha i time domai i the case of a syllable (i.e. to use fewer bits). O the other had, we ca compress i time domai more tha i frequecy domai i the case of the iter-syllable segmet. This allows us to get the average compressio value more readily tha by compressig i a regular maer usig Huffma code by the time domai codig oly, istead of comparig where the etropy value is less: i time or i spectrum, as we suggest here Etropy Calculatio Usig the Selected Domai Method The simulatio calculates a selected domai average etropy value of the etropy divided audio sigal. The average value of the etropy was calculated i the followig maer: Withi each etropydivided sigal segmet, the miimal etropy value is selected from amog the sigal time ad frequecy domais. Afterwards, each selected etropy value is multiplied by the weightig factor. I this way, we obtai the weighted etropy value for the particular sigal part. To explai what the weightig factor is, let us cosider that x is a etropy value of a certai sigal segmet, y is the segmet legth ad z is the total sigal legth. The weightig factor will be y/z ad the weighted etropy value will be x (y/z). I the last step, all weighted etropy values are summed ad the miimal etropy average value is obtaied. Here is the poit to report about the compressio rate. The miimal average etropy value of the etire sigal is approximately 3.3 bits, i.e. at least twice less tha the iitial 8 bits sigal presetatio! It meas: we ca compress a audio sigal with a ratio of. The mai result of this simulatio is the fact that the etropy of a audio sigal withi its differet parts i time domai is differet from its etropy i the frequecy domai. It becomes apparet that the etropy withi the syllable i the frequecy domai is cosistetly less tha the etropy i the time domai. The opposite is the case regardig the itersyllable segmet Compariso Betwee the Miimal Etropy Average Value Obtaied by Both Etropy ad Radom Divisios The last step of the simulatio is to check whether there is ay poit i etropy divisio. Perhaps there is o differece betwee etropy divisio ad regular radom divisio. I the followig simulatio, we compare the miimal etropy average value of the etropy divided audio sigal with a radomly divided sigal. As a example, we took the "What is this?" speech sigal whose miimal etropy average value is show i Figure 4: It is clear that the miimal etropy average value Figure 4: Speech sigal "WHAT IS THIS?" a - time domai b - frequecy domai c - etropy: thi cotiuous lie - time domai, dashed lie - frequecy domai, thick cotiuous lie - miimal etropy average value of the etropy divided audio sigal is less tha the value of the radomly divided sigal: etropy divisio - approximately 3.3 bits, radom divisio bits. 5. The Compressio System ad Audio Frame Structure - Sigal Protocol i Etropy divisio Audio frame DCT out Etropy calculatio Etropy calculatio Sigal structure shaper Huffma codig Comparator Figure 5: Audio sigal compressio system based o the etropy divisio ad selected domai methods S. ew audio frame begiig. Time or frequecy domai a b c 5

6 The audio frame is outputted from the compressio system show o the Figure 5. The compressio system works i the followig way: A audio sigal is trasmitted to the Etropy Divisio block, which performs the sigal divisio by syllables ad iter-syllable segmets accordito the divisio method described i the paragraph.. This block has two outputs. Each of them is coected to the Etropy Calculatio block, which just calculates etropy of the give sigal. The oly differece is that the first oe gets o its iput the time domai segmet, ad the secod oe the frequecy domai segmet. After that we have to compare the outputs of both etropy calculatio blocks i order to decide which of them will be take for further codig. For this purpose the Comparator ad the Switch are used. The comparator compares the etropy values obtaied by two etropy calculatio blocks, ad trasmits the cotrol pulse for the switch ad the selected domai parameters ad a ew audio frame begiig. Switch, i oe s tur, switches the sigals accordig to the comparator commads, - it trasmits the sigal with smallest etropy. This sigal is trasmitted to the Huffma Codig block, which, as it follows from its ame, performs the sigal codig. At the ed, the Sigal Structure Shaper builds a audio frame out of the sigal give by the Huffma codig block ad accordig to the comparator output parameters. The actual umber of bits cosumed by a real etropy codig will exceed the iformatio theoretic miimum (etropy). Overhead for side/cotrol parameters. Coclusios. Research has foud that the DCT trasform is the best spectral trasform method to achieve a miimal etropy value of a audio sigal. This coclusio was reached by comparig DCT, DFT, DHT ad DWT.. I curret audio compressio systems, Etropy Codig is implemeted usig either time or frequecy domai. Research has show that if the etropy is calculated withi each time ad correspodig frequecy domai (i.e. by usig the suggested Chose Domai Method), the a lower miimal etropy average value ca be obtaied. 3. The Etropy Divisio method for a audio sigal is recommeded. 4. The Miimal Etropy Average Value is lower whe divided by the Etropy Divisio Method rather tha the Radom Divisio Method. 5. By calculatig a wide rage of audio sigal data, umerical values of etropy i coclusios ad 4 ca be obtaied. 6. Etropy Divisio of a audio sigal ca improve compressio results based o psychophysiological redudacy reductio. This is because humas perceive audio sigals differetly withi ad betwee syllables. 7. I compariso to curret audio sigal compressio methods (based o time or frequecy domai Huffma Codig), the suggested domai method makes it possible to demad less data. 8. To obtai a complete codig algorithm we have to cotiue our work o some more extesive statistical base. Refereces: [] V.M.Kolesikov, M.U.Bak, V.A.Suchili, A.M.Siilikov, Spectral Methods of Redudacy Reductio i Broadcast-Quality Digital Audio Sigals, Radio ad Televisio, Vol.XXXIX, 989, Prague, Czechoslovakia. [] M.U.Bak, Bearbeitugder der Schalliformatio im meschliche gehorsystem ud i techische alage, Rudfuktechische Mitteiluge, o., 99. [3] M. Bak, O the algorithm of soud iformatio iformatio i a real coder ca be cosiderable, depedig o the shift size (update rate) for these processig by the huma auditory system ad compressio system, J.Audio Eg., 4375 L-4. 6