COMPRESSION OF TRANSMULTIPLEXED ACOUSTIC SIGNALS

COMPRESSION OF TRANSMULTIPLEXED ACOUSTIC SIGNALS Mariusz Ziółko, Przemysław Sypka ad Bartosz Ziółko Departmet of Electroics, AGH Uiversity of Sciece ad Techology, al. Mickiewicza 3, 3-59 Kraków, Polad, {ziolko, sypka}@agh.edu.pl ABSTRACT Trasmultiplexers provide may sigals over a sigle trasmissio lie. A example of the 8-chael trasmultiplexer is preseted. The specific frequecy properties of the trasmultiplexed sigal make it possible to apply the methods effectively. Wavelet packets were itroduced to split the sigal spectrum ito its frequecy compoets. The four-level wavelet packet decompositio was provided by meas of the discrete Meyer wavelets. The diversity i the frequecy of the spectrum values justify applyig the etropy codig i lossless. The arithmetic ad Huffma codig were tested as examples of method for trasmultiplexed sigal. The codig was coducted separately for differet subbads. Fially, the advatages ad disadvatages of lossless methods were compared to each other. 1. TRANSMULTIPLEXERS I trasmultiplexig systems [1,3] the iput sigals are combied for the trasmissio by a sigle chael ad ext recovered by a receiver. Figure 1 shows the classical structure of the M = 8 chael trasmultiplexer. The N iput sigals s i i R, where i { 1,,...,M } is a sigal umber, were upsampled, the filtered ad fially M ( N 1) + 1 summed to obtai a composite sigal s R which is trasmitted over a sigle trasmissio chael. At the receiver ed, the sigal is relayed to the M- chaels of the detrasmultiplexatio part, filtered ad ext dowsampled [] to recover the origial sigals. The upsamplig procedure icreases M-times ad the dowsamplig procedure decreases M-times the umber of samples. Although the umber of samples is varyig, the time of sigal duratio does ot chage because the samplig desity decreases M-times for upsamplig ad icreases M-times for the dowsamplig procedure. Trasmultiplexig does ot make distortios because digital filters (examples of their frequecy characteristics are preseted i Figure ) satisfy the requiremets of a perfect recostructio.. COMPRESSION I a algorithm the umber of bits eeded to represet the sigal or its spectrum is miimized. Compressio algorithms have made commuicatio ad the storage of data effective ad efficiet. Our fudametal cocept of is to split up the frequecy bad of a sigal ad the use less bits to represet the most frequetly occurrig values, ad more bits for the less occurrig oes. If there is a severe skew i the frequecy distributio some gai must be obtaied. Over the past several years, the wavelet methods have gaied widespread acceptace i sigal. Multirate processig is related to sigal trasformatio usig wavelets. Wavelet packets are a way to aa- Figure 1. Scheme of 8-chael trasmultiplexer

4 H t 1 4 H t 4 H t 3 4 H t 4 4 H t 5 4 H t 6 4 H t 7 4 H t 8..1..3.4.5 Frequecy [f/f ] s 4 H d 1 4 H d 4 H d 3 4 H d 4 4 H d 5 4 H d 6 4 H d 7 4 H d 8..1..3.4.5 Frequecy [f/f ] s Figure. A example of the amplitude characteristics of filters: trasmultiplexer (left side) ad detrasmultiplexer (right side) lyze a sigal usig base fuctios which are well localized both i time ad i frequecy. The frequecy methods eables us to exploit the kowledge coected with the frequecy properties of huma hearig. Moreover the local character of wavelets does ot itroduce the block effects ad makes it possible to coduct i real time systems. This type of processig is the mai advatage i. Multirate processig ivolves digital filters that may be used to process a sigal i real time. Some frequecies of the audio trasmultiplexed sigal appeared frequetly i the spectrum while the other frequecies are almost ot detectable. Without the origial sigals ad their spectra use 16 bits accuracy. Both, the Huffma ad arithmetic algorithms are based o statistical codig. The probability of a spectrum value has a direct bearig o the legth of its represetatio. The more probable the occurrece of a value is, the shorter will be its bit-size represetatio. Sequeces of spectrum values are represeted by idividual codes, accordig to their probability of occurrece. De takes the compressed bit stream as iput ad produces decompressed output which exactly matches the iput. The periodicity, preseted i Figure 4, of the spectrum of the trasmultiplexed sigal, result from the periodicity ivolved by upsamplig the sigals which have Fourier spectra preseted i Figure 3. The samplig desity i the frequecy domai for both cases, preseted i Figure 3 ad Figure 4, are the same but the umber of samples are differet. The spectrum of upsampled sigal cosists of the origial spectrum ad moreover its shifted frequecy compoets are added. 3. MULTILEVEL DECOMPOSITION The wavelet packet trasform gives a tool that ca be used to aalyze time varyig sigals. The wavelet packet algorithm geerates [] a set of subbad spectra that are derived from a sigle trasmutiplexed sigal. By usig a filter bak the subbad spectra are produced by cascadig the filterig ad dowsamplig operatios. As show i Figure 5, the wavelet packet trasform ca be viewed as a tree. The coefficiets of wavelet series ( t) = +, m+ 1, s c m 1 ϕ ( t) (1) for the origial sigal s (t) are the root of the tree, where { ϕ m +1, } are the orthogoal scale fuctios of the (m+1) resolutio level. The ext level of the tree is the result of a oe step of the wavelet discrete trasform c m, = hk cm+ 1, k d m, = g k cm+ 1, k where h ad m g m are the costat coefficiets which deped o the assumed scale fuctio ϕ (t). Subsequet levels i the tree are costructed by recursively applyig the wavelet trasform step () to the low (A - approximatio) ad (3) to the high (D - detail) pass filter results of the previous wavelet trasform step. Multirate processig (see the left part of Figure 5) ivolves the applicatio of filterig ad dowsamplig. The mai subject lies i the desig of lowpass ad highpass filters which produce useful trasformatios ad allow the recovery of almost the etire origial sigal. To recover the sigal from the wavelet spectrum, the iverse discrete wavelet filter bak (the right part of Figure 5) is used. 1. s i 1.5. 5 1 15 1. s i.5. 5 1 15 () (3) 1. s i 3 1. s i 4 Amplitude.15.1.5. 44.1 88. 13.3 176.4.5. 5 1 15 1. s i 5.5.5. 5 1 15 1. s i 6.5 Phase [rad/π] - -4-6. 44.1 88. 13.3 176.4 Frequecy [khz] Figure 3. The amplitude ad phase spectra of the trasmitted sigal.. 5 1 15 5 1 15 1. s i 1. 7 s i 8.5.5.. 5 1 15 5 1 15 Frequecy [khz] Frequecy [khz] Figure 4. Spectra of the iput sigals

Figure 5. Decompositio of trasmultiplexed sigal, ad sigal sythesis 4. EXAMPLE To verify the frequecy properties of trasmitted sigals ad the efficiecy of method, some examples (speech, acoustic ad artificial sigals) were aalyzed. Oe of them is preseted below. The followig was assumed: trasmultiplexer cosists of 8-chaels, all filters are of 4-th order, discrete Meyer wavelet was applied, all acoustic sigals cosist of N = 131 7 samples (3 secods approximately). The costructio of wavelet packet tree is show i Figure 5. Wavelet packets are well localized both i the time ad frequecy domai. Figure 5 is a example of the four-level packets geerated by usig the discrete Meyer wavelet packet filters. The absolute values of the coefficiets of the wavelet packet decompositio are preseted i Figure 6. The frequecy is plotted o the vertical axis ad time o the horizotal axis. The stages of a decompositio of the trasmulitiplexed sigal with equal bads are preseted. Each successive stage decomposes its iput vectors twice (see Figure 5). Each output vector has half of the umber of samples of the iput vector. Thus, the frequecy domai of the sigal spectrum is partitioed N twice ad the trasform with N stages has spectra. Trasmultiplexed sigals do ot use the whole rage of frequecies. I 99,9 % the iformatio is trasmitted i subbads which costitute a half of the whole bad. It is easy to distiguish importat ad uimportat bads. I both, but especially i uimportat oes there occur specific values much more probable tha others (see Figure 8). That eables us to use the etropy codes to DDDD4 ADDD4 DADD4 AADD4 DDAD4 ADAD4 DAAD4 AAAD4 DDDA4 ADDA4 DADA4 AADA4 DDAA4 ADAA4 DAAA4 AAAA4 Figure 6. The absolute values of the timefrequecy spectrum. 1.5 1..5. 14 1 1 8 6 4 -. -1.5-1. -.5.5 1. 1.5. Figure 7. Histogram of the multiplexed sigal

Figure 8. Histograms of wavelet spectra i ed-odes. Upper ad lower plots preset the importat bads with 16-bit quatizatio ad the cetral plots presets the uimportat bads with 8-bit quatizatio compress the sigals. Importat subbads eed 16-bit quatizatio while uimportat subbads eed 8-bit quatizatio, oly. The quatizatio levels are described by the atural umbers. The spectrum of trasmultiplexed sigal cosists of 148 616 words ad 16-bit quatizatio gives 16 777 856 bits. The arithmetic codig decreases this bit stream to 1 551 bits. Its distributio betwee the differet bads is preseted i Table 1. Hece, the degree of the lossless is equal to 1.644. Table 1. The legth of codes obtaied after arithmetic quatizer 8 bit quatizer 16 bit Node Code legth Node Code legth ADDD4 458 964 DDDD4 93 651 DADD4 38 918 AADD4 99 333 ADAD4 354 18 DDAD4 91 879 DAAD4 354 941 AAAD4 94 147 ADDA4 349 84 DDDA4 918 59 DADA4 31 67 AADA4 885 788 ADAA4 315 675 DDAA4 98 733 ADAA4 344 66 AAAA4 96 39 Huffma codig ad commercial ZIP software (versio from Widows Commader 4.11 with parameter maximal ) has bee used to compress sigals i three differet cases. Firstly, we compressed the multiplexed sigal (quatized ad recorded by usig 16-bit words; its histogram is preseted i Figure 7) which previously cosisted of 97 31 bytes. The ZIP has decreased it to 1 4 179 bytes. Hece, the degree of is equal to 1.679. The Huffma codig has bee used for the same sigal. Because of the 16-bit quatizatio 3896 iteger umbers have bee used as symbols. We have obtaied etrophy 14.859 ad degree of 1.1331. Aother solutio is compressig all bads separately (see Figure 8). The multiplexed sigal has also bee quatized ad recorded by usig 16-bit words. The coefficiets of 4 level wavelet decompositio have bee calculated with usig Mayer wavelet ad quatized (16- bit). Such sigals have bee compressed by ZIP ad Huffma codig. Results of Huffma codig are preseted i Table. The average degree of is 1.1566.

Table. Huffma codig of decomposed sigal Node Number of symbols Etropy Degree of AAAA4 3458 14.31 1.17 AAAD4 897 14.1638 1.174 AADA4 35 14.1854 1.157 AADD4 31 14.1681 1.17 ADAA4 149 13.37 1.6 ADAD4 1449 13.784 1.4 ADDA4 13744 13.18 1.16 ADDD4 1771 13.651 1.1696 DAAA4 1757 13.6437 1.173 DAAD4 15434 13.393 1.194 DADA4 1569 13.463 1.189 DADD4 15874 13.548 1.185 DDAA4 87 14.179 1.161 DDAD4 38 14.1 1.143 DDDA4 67 14.4366 1.16 DDDD4 467 14.61 1.1197 Table 3. ZIP of decomposed sigal AAAA4 1.886 DAAA4.86 AAAD4 1.843 DAAD4.165 AADA4 1.855 DADA4.1551 AADD4 1.8656 DADD4.336 ADAA4.471 DDAA4 1.8187 ADAD4.156 DDAD4 1.891 ADDA4.1654 DDDA4 1.851 ADDD4.1716 DDDD4 1.866 The effect of ZIP of the same sigals is preseted i Table 3. The average degree of is.131. Fially we use 16-bit quatizatio for odes DDDD4, AADD4, DDAD4, AAAD4, DDDA4, AADA4, DDAA4 ad AAAA4 which are importat bads ad 8-bit quatizatio for others whose coefficiets mostly equal. For such sigals we applied ZIP ad Huffma codig. The result of Huffma codig is preseted i Table 4. The average degree of is 1.65. The results for case with usig 16-bit ad 8-bit quatizatio ad usig ZIP are show i Table 5. The average degree of is 1.846 for odes with 16-bit quatizatio, ad 6.798 for 8-bit oes. That results i total average 4.3195. 5. CONCLUSION Trasmultiplexatio chages the parallel trasmissio ito a serial trasmissio. I other words it coverts M N M N vectors s R i to a sigle vector s R which i i has M-times more elemets. The mai advatage lies i the great variety of realizatios which are available by the proper desigig of digital filters H t ad H d. Through a compariso of the iput ad output sigals, o distortios except the delay, were oticeable as a result of trasmultiplexatio. Table 4. Huffma codig of decomposed sigal with 16-bit ad 8-bit quatizatio for differet bads. 16 - bit 8 - bit Node Number of symbols Etrophy Degree of AAAA4 3458 14.31 1.17 AAAD4 897 14.1638 1.174 AADA4 35 14.1854 1.157 AADD4 31 14.1681 1.17 DDAA4 87 14.179 1.161 DDAD4 38 14.1 1.143 DDDA4 67 14.4366 1.16 DDDD4 467 14.61 1.1197 ADAA4 148 5.59 1.5136 ADAD4 187 5.566 1.488 ADDA4 194 5.4155 1.4689 ADDD4 16 6.8 1.387 DAAA4 5.9845 1.3319 DAAD4 1 5.75 1.3936 DADA4 198 5.7361 1.3863 DADD4 163 5.8119 1.3717 Table 5. ZIP of decomposed sigal with 16-bit ad 8-bit quatizatio for differet bads. 8-bit quatizatio 16-bit quatizatio ADAA4 8.398 AAAA4 1.88 ADAD4 8.713 AAAD4 1.8419 ADDA4 6.578 AADA4 1.8548 ADDD4 6.813 AADD4 1.8651 DAAA4 4.8961 DDAA4 1.8183 DAAD4 6.146 DDAD4 1.897 DADA4 5.944 DDDA4 1.847 DADD4 8.661 DDDD4 1.8656 The power of the wavelet packet aalysis is that it allows sigal frequecy variatio through time to be examied. The wavelet packets method offers a riche rage of possibilities for sigal. The spectrum of trasmultiplexed sigal has special properties. The lack of high frequecies i the iput sigals leads to small values of the trasmitted sigal spectrum for some frequecies. The locatio of these frequecies depeds o the umber of chaels ad ca be easily determie. This makes it possible to apply some effective methods to reduce the

sio methods to reduce the umber of bits of the trasmitted sigal. Aother relevat property is that arithmetic ad Huffma codig are lossless schemes. Table 6 depicts the differece i efficiecy of betwee commercial solutios, Huffma ad arithmetic codig. Stadard methods i geeral use (such as ZIP) code sigle digits of umeral data. We have used arithmetic codig for of iteger umbers. As show i Figure 7 ad 8, certai values of digital data are more commo tha others. These methods require a smaller amout of bits for the more commo umbers (peaks of histograms preseted i Figure 7 ad 8), which apparetly results i substatial. Geeral methods are desiged uiversally to such extet that they are also efficiet i of trasmultiplexed sigals. Table 6. Efficiecy of various methods of Compressio method Degree of arithmetic codig 1.644 ZIP for multiplexed sigal 1.679 Huffma codig of multiplexed sigal 1.1331 Huffma codig of decomposed sigal 1.1566 (16-bit) ZIP of decomposed sigal.131 (16-bit) Huffma codig of decomposed sigal 1.65 (16-bit ad 8-bit) ZIP of decomposed sigal 4.3195 (16-bit ad 8-bit) 6. ACKNOWLEDGEMENT This work has bee supported by KBN uder grat umber 4 T11D 5 3. REFERENCES [1] A. N. Akasu, P. Duhamel, X. Li, & M. Courville, Orthogoal Trasmultiplexers i Commuicatios: A Review, IEEE Tras. o Sigal Processig, vol. 46(4), pp. 979-995, 1. [] C. S. Burrus, R. A. Gopiath, & H. Guo, Itroductio to Wavelets ad Wavelet Trasform. Eglewood Cliffs, NJ: Pretice Hall, 1998 [3] M. Vetterli, J. Kovacevic, Wavelets ad Subbad Codi. Eglewood Cliffs, NJ: Pretice Hall, 1997.