A dedicated codec for compression of Gravitational Waves Sound

A dedicated co dec for compression of Gra vitational W a v es Sound Laura Reb ollo-Nei ra Mathemat ics Department Aston Univ er sit y B4 7ET Birmingha m, UK Abstract A dedicated cod ec for compression o f gra vitational wa v es sound with high qualit y reco very is prop osed. The p erformance is tested on the a v ailable set of gravi tational sound signals th at has b een theoretic ally generated at the Massac h usetts Institute of T ec hnology (MIT). The appr oac h is based on a mo del for data reduction rend er in g high qualit y ap p ro ximation of the signals. The reduction o f dimensionalit y is ac hiev ed b y selecting ele m entary co m p onents f r om a redundant set ca lled a dictionary . Comparisons with t h e compression standard MP3 d emons trate the merit of the dedicated tec hniqu e f or compr essing this t yp e of sound. 1 In tro du ction After the celebrated first detection of a g ra vitational w av e (GW) on Sept 2015 [1] five more detections [2 – 6] ha v e confirmed Einstein’s theory of general relativit y . Predictions from the Laser Interferomete r Gra vitat io nal-W a v e Observ atory (LIGO) and Virgo Scien tific Collab oration assert that w eekly or ev en more often detections can b e expected in the near future. Scien tists en visage the prospect as the b eginning of a new era in astronomy , whereb y the ve ry early Univ erse will b e studied by the sound that w as made in its formation. Theoretically , G Ws are modelled and generated using t ec hnique s of num erical relativit y [7 – 9], from whic h audio represen tation can b e dev elop ed. In particular, the group of Prof. Hughes at MIT has pro duced and made av ailable the gravitational w av es sound (GWS) signals whic h hav e b een used in this study . The nu merically sim ulated signals are supp orted b y a n um b er of publications [11 – 15] and nice ly presen ted on a w ebsite [16 ]. This w ork uses those signals to demonstrate a dedicated sc heme for the compression of GWS with high quality reco very . The proposal falls within the usual transform co ding sch eme. It c o nsists of three main steps: 1) T ransformation of the sound signal. 2) Quan tization of the transformed data. 3) Bit- stream en tropy co ding. Ho w eve r, it differs fro m the tra dit io nal compression t echniq ues from the b eginning. Instead of considering an orthogonal tra nsformation, the first step is realized by appro ximating the signal using a redundan t dictionary from where the elemen tary comp onen ts for represen ting the signal are selected t hrough a greedy pursuit strategy . This strategy giv es r ise to what is kno wn as sparse represen tat io n of a signal. In the area o f audio pro cessing a num b er o f differen t tasks hav e b een sho wn to b enefit b y the sparsit y of a represen tatio n. The relev an t literatur e is extensiv e. As a sample w e refer to [17 – 27]. Here w e focus on comp r ession of G WS signals, whic h do require a dedicated framew ork for their sparse represen tation. In part icular, the dictionary w e use for the prop osed co dec w as introduced in a recen t publicatio n as p oten tially go o d to ac hieve sparse represen tation of G WS b y partitioning the signal [28]. No w it stands as a crucial comp onen t of a co dec for hig h quality p oin t-wise reco very of the compressed signal. 1 Since t he G WS under consideration is in the range of h uman hearing, it is inte resting to realize comparisons with the p opular compression standard MP3. The results demonstrate a significant impro v ement in compression p erformance for the equiv alen t quality of the reco v ered signal. More precisely , the signal is required to yield a Signal to Noise R atio (SNR) competitive with MP3 outcomes with respect to b oth, the whole signal and the elemen ts of the signal partition. The b enefits of the prop osed dedicated fo rmat for enco ding GWS go b ey ond the compression p erformance. Certainly , the underlying represen tation of the data g enerates a reduced set whic h con tains information ab out the elemen tary comp onen ts o f the signal. Hence, in addition to r ecov ering the signal from a compressed file with high quality , one can also r ecov er its reduced represen tatio n in terms of elemen tary comp onen ts. Since the reduced represe ntation giv es rise to an accurate appro ximation of the signal, it can b e of assistance to signal pro cessing tech niques relying on a reduction of dimensionalit y a s a first step of further pro cessing. The main goals o f the n umerical tests of Section 3 are: • T o pro duce strong evidence that high quality approximation of tw o differen tly generated types of GWS can b e achie ved as a sup erp osition o f elemen tary compo nents of differen t nature. A comp onen t of elemen ts generated b y a discrete v ersion of trigonometric functions, and a comp onen t consisting of pulses of small supp ort. • T o demonstrate that the ab o v e describ ed decomp osition can b e stored in a file whic h is signif- ican tly smaller than that obtained with the compression standard MP3. The go als are ac hieve d b y testing the metho d on t he av ailable audio r epre sentation of GWs which are group ed, according to their theoretical mo deling, as follo ws [16]: • Extreme mass ratio inspiral (EMRI). Gra vitatio na l w av es pro duced when a relativ ely lig ht compact ob ject orbits around a m uc h hea vier black hole and gra dually deca ys. • Binaries. Emitted during the merger of tw o b o dies of roughly the same mass. The first direct detection of a G W b elongs to t his category . 2 Metho d The principal constituen t of the prop osed co dec is the mathematical mo del of the sound signal. As already men tioned, the mo del is realized by selecting elemen tar y comp onen ts of a dedicated dictionary containing atoms of differen t nature. Th us, the metho d for selecting t he suitable atoms for the appro ximation of a g iv en signal is also relev an t to the mo delling. F or this task w e use the Optimized Orthogonal Matchin g Pursuit (OOMP) metho d [30], whic h is step wise optimal in the sense o f minimizing the norm of the appro ximation error a t eac h iteration step. Throughout the description of the metho ds R and N stand for the sets of real and natural n um b ers, resp ectiv ely . Boldface letters a re used t o indicate Euclidean vec to rs a nd their corresp onding comp onen ts are represen ted using standar d mathematical fonts, e.g., f ∈ R N , N ∈ N is a ve ctor of comp onen ts f ( i ) , i = 1 , . . . , N . The inner pro duct op eration is indicated as h · , ·i . A partition of a signal f ∈ R N is realized by a set of disjoin t pieces f q ∈ R N b , q = 1 , . . . , Q , whic h for simplicity a re assumed to b e all of the same size and suc h that QN b = N , i.e., it holds t ha t f = ˆ J Q q =1 f q , where the concatenation op eration ˆ J is defined as fo llows: f is a v ector in R QN b ha ving comp onen ts f ( i ) = f q ( i − ( q − 1 ) N b ) , i = ( q − 1) N b + 1 , . . . , q N b , q = 1 , . . . , Q . Hereinafter eac h elemen t of the signal partition f q will b e refereed to as a ‘blo c k’. 2 2.1 OOMP app ro ximat ion Giv en a signal f partitioned in to Q blo c ks f q ∈ R N b , q = 1 , . . . , Q , , t he k q -term approx imat io n of eac h blo c k is mo delled b y the sup erposition f k q q = k q X n =1 c q ( n ) d ℓ q n , q = 1 , . . . , Q. (1) The elemen ts d ℓ q n , n = 1 , . . . , k q in (1), called ’atoms’ are selected here from a dedicated dictionary D = { d n ∈ R N b , k d n k = 1 } M n =1 through the OO MP approach [30] whic h, for eac h blo c k q op erates a s follo ws. The algo rithm is initia lize d b y setting: r 0 q = f q , f 0 q = 0, Γ q = ∅ and k q = 0. The first atom for the atomic decomp osition of the q -th blo c k is selected as the one corresp onding to the index ℓ q 1 suc h that ℓ q 1 = arg max n =1 ,...,M   h d n , r k q q i   2 . (2) This first atom is used to assign w q 1 = b 1 ,q 1 = d ℓ q 1 , calculate r 1 q = f q − d ℓ q 1 h d ℓ q 1 , f q i and itera t e as prescribed b elo w. 1) Upgrade the set Γ q ← Γ q ∪ ℓ k q +1 , increase k q ← k q + 1, and select the index of a new ato m for the appro ximation as ℓ q k q +1 = arg max n =1 ,...,M n / ∈ Γ q |h d n , r k q q i| 2 1 − P k q i =1 |h d n , ˜ w q i i| 2 , with ˜ w q i = w q i k w q i k . (3) 2) Compute the correspo nding new v ector w q k q +1 as w q k q +1 = d q ℓ k q +1 − k q X n =1 w q n k w q n k 2 h w q n , d q ℓ k q +1 i . (4) including, for n umerical accuracy , t he re-orthog o nalizing step: w q k q +1 ← w q k q +1 − k q X n =1 w q n k w q n k 2 h w q n , w q k q +1 i . (5) 3) Upgrade v ectors b k q ,q n as b k q +1 ,q n = b k q ,q n − b k q +1 ,q k q +1 h d q ℓ k q +1 , b k q +1 ,q n i , n = 1 , . . . , k q , b k q +1 ,q k q +1 = w q k q +1 k w q k q +1 k 2 . (6) 4) Calculate r k q +1 q = r k q q − h w q k q +1 , f q i w q k q +1 k w q k q +1 k 2 . (7) 5) If for a give n ρ the condition k r k q +1 q k < ρ has b een met compute the co efficien ts c k q ( n ) = h b k q n , f q i , n = 1 , . . . , k q . Otherwise rep eat steps 1) - 5). 3 Remark 1: F or eac h v alue of q the criterion (3) in the metho d ab o v e giv es the index minimizing the lo cal residual norm k f q − f k q q k with r esp ect to the newly selected atom. Moreo ver, t he approx- imation o f eac h blo c k q is completed at once tot a lly indep enden t of the other blo c ks. In previous publications [28, 31] we hav e show n the b enefit in ranking the blo c ks for their sequen tial step wise appro ximation to minimize the total residual error k f − f K k , with K = P Q q =1 k q . Suc h a strategy is called Optimized Hierarc hical Blo c k Wise OOMP (OHBW- OOMP). How ev er, for the particular application to compression, the quan tization pro cedure plays also a role in the a ppro ximation b y mapping small co efficien ts to zero and quantiz ing the others. Consequen tly , for the set of signals considered in this work the compression results rendered b y b oth metho ds are pra ctically equiv a len t. 2.2 The Dictionary The dictionary used for the approximation consists o f t w o sub-dictionaries of differen t nature. One of them is a trigonometric dictionary D T , whic h is t he union of the dictionaries D C and D S giv en b elo w. D x C = { w c ( n ) cos π (2 i − 1)( n − 1) 2 M , i = 1 , . . . , N b } M n =1 D x S = { w s ( n ) sin π (2 i − 1)( n ) 2 M , i = 1 , . . . , N b } M n =1 , where w c ( n ) and w s ( n ) , n = 1 , . . . , M are normalization f actors. In the n umerical sim ulatio ns we ha v e considered M = 2 N b and N b = 2048 . The o ther sub-dictionary is constructed by translation of the prototype atoms, p 1 , p 2 and p 3 in Fig. 1. 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 Figure 1: Protot yp e atoms p 1 , p 2 and p 3 , wh ic h generate the dictionaries D P 1 , D P 2 and D P 3 b y sequential translations of one p oin t. Denoting b y D P 1 , D P 2 and D P 3 the dictionaries ar ising b y translations of the a t oms p 1 , p 2 , a nd p 3 , resp ectiv ely , the dictionary D P is built as D P = D P 1 ∪ D P 2 ∪ D P 3 . The whole dictiona r y is then built as D = D T ∪ D P , with D T = D C ∪ D S . This dictionary w as previously in tro duced a s p oten tially suitable for a ppro ximation of GWS [28]. In this study it stands out as b eing essen tial for a c hieving high quality approx imatio n of most o f the signals whic h hav e b een compressed with t he prop osed co dec. 4 2.3 Co ding Strategy Previously to en tropy enco ding the co efficien ts resulting fro m approx imating a signal b y partitioning, the real num b ers need to b e con verted in to in t egers. This op eration is kno wn as quan tizatio n. F or all the n umerical cases w e ha ve ado pt ed a simple uniform quan t izat io n tec hnique. The a bsolute v alue co efficien ts | c q ( n ) | , n = 1 . . . , k q , q = 1 , . . . , Q, are con ve rted to in tegers a s follows : c ∆ q ( n ) = ⌊ | c q ( n ) | ∆ + 1 2 ⌋ , (8) where ⌊ x ⌋ indicates the largest in t eger n um b er smaller or equal to x and ∆ is the quantization parameter. The signs of the co efficien ts, represen ted a s s q , q = 1 , . . . , Q , are enco ded separately using a binary alphab et. F or creating the stream of n um b ers to enco de the signal mo del we pro ceed as in a recen t w ork [2 7]. The indices of the atoms in the atomic decomp ositions of eac h blo c k f q are first sorted in ascending order ℓ q i → ˜ ℓ q i , i = 1 , . . . , k q , whic h guaran tees that, for eac h q v alue, ˜ ℓ q i < ˜ ℓ q i +1 , i = 1 , . . . , k q − 1. This order of the indices induces an order in t he unsigned co efficien ts, c ∆ q → ˜ c ∆ q and in the corr esp onding signs s q → ˜ s q . The ordered indices are stored as smaller p ositiv e num b ers by taking differences b et w een tw o consecutiv e v alues. By defining δ q i = ˜ ℓ q i − ˜ ℓ q i − 1 , i = 2 , . . . , k q the follo w string stores t he indices fo r blo c k q with unique recov ery ˜ ℓ q 1 , δ q 2 , . . . , δ q k q . The n umber ‘0’ is then used t o separate the string corresp onding to different blo c ks and entrop y co de a long string, st ind , whic h is built as st ind = [ ˜ ℓ 1 1 , . . . , δ 1 k 1 , 0 , ˜ ℓ 2 1 , . . . , δ 2 k 2 , 0 , · · · , ˜ ℓ k Q 1 , . . . , δ Q k Q ] . (9) The corresp onding quan tized mag nitude of the co efficien ts are concatenated in the strings st cf as follo ws: st cf = [ ˜ c ∆ 1 (1) , . . . , ˜ c ∆ 1 ( k 1 ) , · · · , ˜ c ∆ k Q (1) , . . . , ˜ c ∆ k Q ( k Q )] . (10) Using ‘0’ to store a p ositiv e sign and ‘1’ to store negative one, the signs are placed in the string, st sg as st sg = [ ˜ s 1 (1) , . . . , ˜ s 1 ( k 1 ) , · · · , ˜ s k Q (1) , . . . , ˜ s k Q ( k Q )] . (11) The next enco ding / deco ding sc heme summarizes the ab ov e describ ed pro cedure. Enco ding • Giv en a partition f q ∈ R N b , q = 1 , . . . , Q of a signal, appro ximate eac h blo ck f q b y the a t o mic decomp ositions ( 1 ). • Quan tize, as in (8), the absolute v alue co efficien ts to obtain c ∆ q ( n ) , n = 1 , . . . , k q , q = 1 , . . . , Q . • F or each q , sort the indices ℓ q 1 , . . . , ℓ k q in ascending order to hav e a new order ˜ ℓ q 1 , . . . , ˜ ℓ k q and the re-ordered sets ˜ s q (1) , . . . , ˜ s q ( k q ), and ˜ c ∆ q (1) , . . . , ˜ c ∆ q ( k q ), to create the strings: st ind , as in (9), and st cf , and st sg as in (10) and (11), resp ectiv ely . All these strings are enco ded, separately , using adaptiv e arithmetic co ding. Deco ding • Rev erse the arithmetic co ding to recov er strings st ind , st cf , st sg . • In v ert the quantization step as | ˜ c r q ( n ) | = ∆ ˜ c ∆ q ( n ) . (12) 5 • Reco v er the approximated partition thr o ugh the linear com binatio n f r ,k q q = k q X n =1 ˜ s q ( n ) | ˜ c r q ( n ) | d ˜ ℓ q n . (13) • Assem ble the reco v ered signal as f r = ˆ J Q q =1 f r ,k q q . (14) 3 Results Giv en the appro ximation f r of a signal f , the quality o f suc h a n appro ximation is assessed b y the SNR calculated as SNR = 10 log 10 k f k 2 k f − f r k 2 , (15) where k · k indicates the 2- norm. The lo cal SNR with resp ect to ev ery blo ck in the partition, whic h w e indicate as snr( q ) , q = 1 , . . . Q , is calculated as snr( q ) = 10 log 10 k f q k 2 k f q − f r ,k q q k 2 , (16) where f r ,k q q is the approxim a t ion of the blo c k f q . Both, the mean v alue ( snr) and standard deviation (std) of these lo cal quan tities a r e relev an t to comparison of p oin t-wise qualit y reco ve ry . Accordingly , w e define snr = 1 Q Q X q =1 snr( q ) , and std = v u u t 1 Q − 1 Q X q =1 (snr( q ) − snr) 2 . (17) In order t o use the SNR and snr as measures of quality fo r comparison, the MP3 signal has to b e optimized in relation to those quantities. This is carried out b y the following op erations [27]. • Shifting: Since MP3 in tro duces a shift with resp ect to the o r iginal signal, to compute the SNR that shift should b e rev ersed. Denoting b y f M the nume rical signal retriev ed from the MP3 file, the optimal time shift ˆ τ is determined to b e the time shift maximizing t he cross-correlatio n with the o riginal signal, i.e. ˆ τ = arg max τ = − N/ 2 ,...,N / 2 N X n =1 f ( n ) f M ( n + τ ) . (18) • Scaling: In order to neutralize the effect of any m ultiplicativ e and/or a dditiv e constan t whic h could affect the SNR v a lue, we allow for suc h tw o constan ts and adjust them to maximize the SNR as follows: Denoting by ˆ f M the MP3 signal after the shifting o peration, w e consider the linear form a ˆ f M + b and fix the v alues of a and b for whic h k f − ( a ˆ f M + b ) k 2 tak es t he minim um v alue, i.e. a = h f , f M i − ( 1 N P N i =1 ˆ f M ( i ))( 1 N P N j =1 f ( j )) k ˆ f M k 2 − 1 N ( P N i =1 f M ( i )) 2 b = 1 N N X i =1 f ( i ) − a 1 N N X i =1 ˆ f M ( i ) . 6 While the additiv e constan t b is not relev ant, the scaling constan t a pro duces an imp ortant correction whic h significan tly increases the SNR. The compression p o w er is determined b y the compression ratio (CR) defined as follows CR = Size of the file with the signal Size of the compressed file . (19) In all the n umerical examples the files of the signals are giv en in W A V format. The compression is realized on a single audio channel. The mean v alue of all the signals is pra ctically zero. As discus sed b elo w, after the shifting and scaling o peration for lo w v alues of CR (e.g. CR=2 a nd CR=4) MP3 reco vers signals of high qualit y with resp ect to b oth measures, t he SNR and the snr. The prop osed co dec, henceforth to b e refereed to as ‘dictionary co dec’ (D C), is set to pro duce a n appro ximation of the orig ina l signal matc hing either the SNR or the snr v alue ac hiev ed by the MP3 signal (whatev er quantit y is the largest one). As seen in the tables b elo w, by matc hing the largest quan tity of either SNR, or snr, the other measure is guaran teed to b e larger than the v alue a ttained b y the MP3 signal. 3.1 Numerical Case I As a first numeric a l example w e conside r the audio represen tation of a detected G W, the c hirp gw151226 [29]. This is a short signal, it consists of N = 65536 p oin ts with sampling frequency of 41 kHz. F or CR=2 the p oint-wise qualit y of the signal recov ered from the MP3 file is excellen t: SNR=74.6 dB and snr = 70 . 2 dB. These v alues should b e appreciated b y taking into account that SNR=74.6 dB corresp onds to a mean square erro r b et w een the approximation a nd the signal of order 10 − 9 . Since in t his case the SNR is greater than the snr the DC is set to pro duce the same v alue of SNR. With this restriction the resulting snr is 7 1.7dB, i.e. larg er than the v a lue yielded b y MP3. The huge difference b et w een the tw o enco ding pro cedures is the CR. Denoting by CR M the CR with resp ec t to the MP3 file a nd CR D that of the DC file, for SNR=74.6 dB one has CR M = 2 and CR D = 2 3! While for larger v alues of CR M the qualit y of the r ecov ered signal decreases, up to CR M = 10 the quality of the MP3 signal is still v ery go o d. T able 1 display s t he v alues of SNR and snr, a s w ell as the corresp onding CRs pro duced b y b oth formats. These a re differen tiated b y the notation SNR M and snr M , used to indicate the v alues pro duced b y the MP3 format, and SNR D and snr D , used to indicate the v alues pro duced by the DC format. SNR M SNR D snr M std M snr D std D CR M CR D 74.5 dB 74 .5 dB 70.2 dB 6.3 71.7 dB 5.9 2.1 23.2 73.8 dB 73 .8 dB 69.8 dB 6.4 71.1 dB 5.7 4.2 24.9 71.3 dB 71 .3 dB 66.9 dB 6.7 68.4 dB 5.8 8.4 61.2 69.4 dB 69 .4 dB 65.2 dB 6.2 66.4 dB 5.9 10.4 69.3 T able 1: Comparison of CR M and CR D , for the sound represen t a tion of the detected chirp gw151226 . The first column giv es the v alues of SNR M pro duced b y the MP3 signal reco vered from files with CR M as listed in the 7th column. The second column are the identical v alues of SNR D pro duced b y the signal reco vere d from the DC files with CR D as listed in the last column. The 3rd and 5th columns a r e the v alues of snr M and snr D , resp ectiv ely . The 4th and 6th columns are the corresp onding standard deviations. 7 3.2 Numerical Case I I In this case the group of GWS has b een n umerically simulated at MIT [16]. The GWs b elong to the EMRI category with circular orbit. The signals are organized in tw o la rge subgroups, according to the spin of the larg er black hole: spin 99 . 8% of the maxim um v alue and spin 35 . 94% of the maxim um v alue. Eac h subgroup contains 16 signals, each of which is c har a cteriz ed by t w o angles: the orbital plane and the viewing angle. The signals corresp onding t o spin 99 . 8% are listed in the first column T able 2 and T able 3. The signals corresp onding to spin 35 . 9 4 % are listed in the first column of T able 4 and T able 5 . The frequency o f these signals is 8kHz and w ere compressed with MP3 a t the lo we st p ossible r ate, CR M = 2, in the first instance. The reco ve r ed signals pro duce the v alues of snr M and SNR M giv en in the second and sixth columns of T able 2. Since in this case snr M > SNR M the DC w as set to pro duce snr D = snr M . This guaran tees that SNR D > SNR M for all signals. The actual v alues of snr M v ary with the signals, but for a ll of them the reco v ery is of go o d quality . It corresp onds to mean square errors of order of 10 − 8 . As can b e observ ed in the t a ble, for all the signals the compression p erformance of the DC format is clearly sup erior to MP3. T able 3 displa ys the equiv a len t results but corresp onding to a larger compression ratio of MP3 (CR M = 4). Signal snr M std M snr D std D SNR M SNR D CR M CR D E1 67.9 2.1 67.9 0 .2 65.8 67.8 2 6.8 E2 65.0 2.9 65.0 0 .4 58.9 64.9 2 5.3 E3 63.7 3.3 63.7 0 .5 54.4 63.6 2 5.2 E4 66.9 3.4 66.9 0 .6 55.2 66.7 2 5.6 E5 63.4 1.8 63.4 0 .2 62.4 63.4 2 6.0 E6 62.2 2.9 62.3 0 .2 58.9 62.1 2 5.1 E7 61.6 2.9 61.6 0 .2 57.9 61.6 2 4.9 E8 62.8 2.9 62.8 0 .3 59.4 62.8 2 5.2 E9 65.7 2.2 65.7 0 .3 64.5 65.7 2 6.4 E10 64.1 2.1 64.1 0 .2 62.4 64.1 2 5.6 E11 62.7 1.8 62.7 0 .2 62.2 62.7 2 5.5 E12 63.7 2.2 63.7 0 .2 63.0 63.7 2 5.8 E13 68.5 4.4 68.5 0 .4 64.3 68.5 2 6.3 E14 66.8 2.4 66.8 0 .4 65.8 66.8 2 5.6 E15 63.7 2.7 63.7 0 .6 62.7 63.6 2 5.7 E16 64.5 2.9 64.5 0 .4 62.6 63.7 2 5.9 T able 2: Comparison of CR M and CR D v alues for GWs within the EMRI category for circular orbit and spin 99 . 8% of the maxim um v alue. Eac h signal listed in the first column corresp onds to a particular orbital plane and viewing angle. The second column shows t he snr M v alues pro duced b y the MP3 signals reco v ered from a file corresp onding to CR M = 2. The sixth column sho ws the corresp onding v alues of SNR M . All the other figures in the ta ble a rise when setting the D C to ach ieve snr D = snr M . 8 Signal snr M std M snr D std D SNR M SNR D CR M CR D E1 60.3 2.4 60.3 0 .3 58.2 60.2 4 9.7 E2 55.5 4.2 55.5 0 .4 47.9 55.4 4 7.9 E3 54.1 4.9 54.1 0 .4 45.5 54.0 4 7.6 E4 57.5 4.0 57.5 0 .6 51.5 57.4 4 8.3 E5 54.9 2.9 54.9 0 .2 52.4 54.8 4 8.8 E6 52.4 4.2 52.4 0 .2 44.9 52.3 4 7.6 E7 52.2 5.0 52.2 0 .2 39.7 52.1 4 7.1 E8 53.6 3.9 53.6 0 .3 43.0 53.6 4 7.5 E9 56.5 2.7 56.5 0 .2 55.3 56.4 4 9.6 E10 54.7 1.8 54.7 0 .2 53.8 54.6 4 8.2 E11 53.2 2.4 53.2 0 .2 51.9 53.2 4 8.1 E12 54.4 2.8 54.4 0 .2 52.0 54.4 4 8.4 E13 59.4 3.4 59.4 0 .4 57.3 59.4 4 8.8 E14 57.4 2.2 57.3 0 .3 56.7 57.4 4 7.7 E15 55.5 2.2 55.5 0 .3 51.2 55.5 4 7.7 E16 55.0 2.6 55.0 0 .5 51.2 55.0 4 8.4 T able 3: Same description as in T able 2 but fo r CR M = 4. T able 4 and T able 5 hav e equiv alent description as T able 2 and 3, resp ectiv ely , but the signals b elong t o the EMRI group with spin 3 5 . 94% of the maxim um v alue. Signal snr M std M snr D std D SNR M SNR D CR M CR D S1 71.8 3.2 71.8 0 .6 68.8 71.7 2 10.2 S2 70.2 2.4 70.2 0 .7 67.9 70.1 2 6.6 S3 69.7 2.2 69.7 0 .5 67.3 69.6 2 6.4 S4 69.5 2.4 69.5 0 .5 67.3 69.4 2 6.4 S5 70.8 2.3 70.8 0 .5 68.9 70.8 2 7.7 S6 68.8 1.9 68.8 0 .4 67.9 68.7 2 5.5 S7 65.0 3.3 65.0 0 .4 61.1 65.0 2 5.4 S8 64.8 1.9 64.8 0 .4 63.9 64.7 2 5.8 S9 69.9 2.5 69.9 0 .3 68.2 69.9 2 5.2 S10 67.1 1.7 67.1 0 .4 66.6 67.2 2 4.6 S11 63.4 2.3 63.4 0 .5 62.3 63.5 2 4.5 S12 61.7 2.0 61.7 0 .2 60.9 61.7 2 4.5 S13 66.0 3.2 66.0 0 .6 62.8 66.1 2 4.9 S14 63.0 7.2 63.0 1 .6 61.6 64.4 2 4.9 S15 59.0 8.4 59.0 1 .3 59.5 60.2 2 4.9 S16 51.8 2.0 51.8 0 .3 51.0 51.9 2 4.7 T able 4: Same description as for T able 2 but the signals b elong to the EMRI group with spin 35 . 94% of the maximum v alue. 9 Signal snr M std M snr D std D SNR M SNR D CR M CR D S1 66.3 3.2 66.3 0 .5 61.9 66.4 4 13.1 S2 62.9 2.7 62.9 0 .5 58.7 62.8 4 9.2 S3 61.4 2.9 61.4 0 .5 57.5 61.2 4 9.3 S4 61.6 2.4 61.6 0 .5 58.6 61.5 4 9.0 S5 64.1 2.9 64.1 0 .5 61.3 64.0 4 10.3 S6 61.1 2.1 61.1 0 .4 59.2 61.0 4 7.9 S7 57.3 2.5 57.3 0 .4 54.8 57.3 4 7.8 S8 56.8 2.5 56.8 0 .4 54.5 56.8 4 8.3 S9 61.3 2.6 61.3 0 .3 59.9 61.2 4 8.0 S10 58.1 2.2 58.1 0 .4 56.7 58.1 4 7.2 S11 54.2 2.6 54.2 0 .5 52.9 54.3 4 7.1 S12 53.8 2.0 53.8 0 .2 53.0 53.8 4 6.8 S13 56.8 2.9 56.8 0 .6 54.4 56.8 4 7.7 S14 55.1 7.4 55.1 1 .2 54.0 56.1 4 7.0 S15 51.0 7.5 51.0 1 .2 51.7 52.0 4 7.0 S16 44.7 2.7 44.7 0 .7 43.1 44.6 4 6.6 T able 5: Same as the description o f T able 4 but for CR M = 4. 3.3 Numerical Case I I I This group of signals has also b een sim ulated at MIT. All the signals b elong to the Binary cat ego ry , for circular syste ms, a nd are differen tiated by the mass of the larger b o dy . The first t wo signals listed in T ables 6 and 7 corresp ond to b o dies of roughly the same mass. The first signal (B1) is generated b y binary neutron star s each of 1.5 solar masses. The second signal (B2) is generated by binary blac k ho les , eac h of 2.5 solar masses. Signals B3 a nd B4 are pro duced b y binary blac k holes with mass ra tio 3:1. The signal B3 do es not include spin effects while B4 includes rapid spinning o f b oth b o dies . The minim um p ossible v alues of CR M are giv en in the eigh th column of T able 6. In this case SNR M > snr M for all the signals. Hence, t he D C was set to ac hiev e SNR M = SNR D (second and third columns of T able 6). These v alues pro duced snr D > snr M for all the signals. The remark able p erformance of the DC emerges from the CR D figures listed in the last column of T able 6. Signal SNR M SNR D snr M std M snr D std D CR M CR D B1 70.2 70.2 66.2 8.5 67.2 8.0 2.7 51 .7 B2 70.3 70.3 65.2 10.4 66.5 9.5 2.6 34.2 B3 59.0 59.0 58.5 2.0 58.8 0.6 2.7 29 .3 B4 55.6 55.6 54.9 2.1 55.2 0.7 2.7 28 .9 T able 6: Comparison of the CR M and CR D v alues for the audible part of the signals in the Binary group. The second column give s the v alues of SNR M pro duced b y the MP3 signal recov ered f rom files with CR M as listed the 8th column. The 3rd column are the iden tical v alues of SNR D pro duced b y the signal reco vere d from the DC files with CR D as listed in the last column. The 4th and 6th columns a r e the v alues of snr M and snr D , resp ectiv ely . The 5th and 7th columns are the corresp onding standard deviations. 10 Signal SNR M SNR D snr M std M snr D std D CR M CR D B1 68.8 68.8 65.9 8.6 65.9 7.8 5.4 56 .2 B2 68.6 68.6 65.0 10.5 65.0 9.4 5.2 36.9 B3 53.4 54.1 53.9 2.9 53.9 0.4 5.5 37 .5 B4 49.3 49.6 49.7 3.4 49.7 0.4 5.5 39 .3 T able 7: Same description a s T able 6 but for double v alues of CR M . 4 Discuss ion The results o f T ables 1-7 demonstrate the remark able compression p ow er of the DC, in comparison to MP3 at equiv alen t high quality o f the reco v ered signal. F or all cases the relation CR D = γ CR M (20) holds for v alues of γ v arying from a minim um v a lue γ min = 1 . 7 for signal S12 in T able 5 to a maxim um v alue γ max = 1 9 . 2 for signal B1 in T a ble 6. T able 8 shows the mean v alue ¯ γ in eac h of the ab o ve tables. It also sho ws the corresp onding std as w ell as the v alues of γ min and γ max in eac h of the tables. T able ¯ γ std γ min γ max 1 7 .7 2.3 5.9 11.0 2 2 .8 0.3 2.4 3.4 3 2 .1 0.2 1.8 2.4 4 2 .9 0.7 2.2 5.1 5 2 .2 0.6 1.7 3.2 6 1 3.5 3.9 10.8 19.2 7 7 .7 1.4 6.8 9.7 T able 8: St a tistic of the fa ctor γ (c.f. (20)) for T able 1 -7. The second column corr esp onds to the mean v alue with resp ect to the signals in each table. The third column give s the corresp onding standard deviation. It is w o rth commenting that the comp onen t of the dictionary containing the atoms of small supp ort plays an essen tial role of ac hieving la rge v alues of CR D for high quality recov ery . This is m uc h mor e imp ortant f or the group of EMRI sound. F or example, if the compression of signal E1 in T able 2 is carried out in the same wa y but excluding the sub-dictionary with those ato ms, the v alue of CR D drops from 6.8 to 3.5. Nev ertheless, ev en if for appro ximating E1 at the quality of T able 2 t he p ercen tage of atoms of small supp ort is significant (41%) the contribution to the signal appro ximation is minor. The norm of the signa l E1 is 207.6 and t he norm of comp onen t generated b y the ato ms of small supp ort is only 2 .1 . This comp onen t of small norm is needed to ac hiev e the high v alues of snr D and SNR D required in this study . Con trar ily , for compression of low er quality reco v ery these atoms play no role for most signals. A t low er qualit y , ho we ver, the p o we r of the DC increases substan tially . F or example, when compressing with MP3 and CR M = 8 the signal E1 the recov ered signal pro duces snr M = 4 1 . 3 dB. The DC ac hiev es the same v a lue of snr D (and SNR D > SNR M ) for CR D = 53! 11 4.1 Bey ond Compression The success of the DC in pro ducing a small file stems fro m t he ability of constructing a signal approx - imation of go o d qualit y , but inv o lving less elemen tary comp onen ts than the num b er of samples giving the signal. Let’s supp ose that to represen t at the desired qualit y the blo c k f q in the signal partition one needs k q dictionary atoms, and let’s normalize these v alues ˜ k q = k q / P Q q =1 k q for comparison purp oses. As sho wn in Fig. 2 the num b ers ˜ k q , q = 1 , . . . , Q render meaningful informat io n ab out the signal in ternal v ariations ov er time. In the low er graphs of Fig. 2 eac h of these v alues is lo cated in the horizontal axis at the cen t er of the corresp onding blo c k (of size N b = 2 0 48) and provides a condensed digital summary of the sound. The left upp er graph is the spectrogr a ms of the signal E1. The lines in the low er left graph join the v alues ˜ k q , q = 1 , . . . , 225 resulting when appro ximating this signal at t hree differen t qualities: snr D = 65, 60 and 55 dB. As observ ed in the graph, the three lines are v ery close to eac h other and accoun t for the no de in the sp ectrogram whic h o ccurs a t around 55 secs. The right upp er graph of Fig.2 is the sp ectrogram of the signal S1. This signal differs fro m the signal E1 only in the spin (35 . 9 4% of the ma ximum v alue). Also in this case, the three lines connecting the v alues ˜ k q , q = 1 , . . . , 66 for approximations of snr D = 65, 60, and 55 dB are close t o eac h other and accoun t for the main feature o f the sp ectrogram, whic h is c ha r a cteriz ed b y a rise of frequency to wards the end. It is a n intere sting feature of the digital summary , giv en b y the p oints in the low er g r a phs, the small cardinalit y in relation to the length of the signal: 2 25 p oin ts for the signal E1 giv en b y N = 4608 0 0 samples a nd 66 p oin ts for the signal S1 given by N = 1 3 5168 samples. 0 10 20 30 40 50 0 0.006 0.012 0.018 0.024 Time (sec) 0 2 4 6 8 10 12 14 16 0 0.04 0.08 0.12 0.16 Time (sec) Figure 2: The up per graph s are the sp ectrograms of the signals E1 (left) and S1 (right). The lines in the low er graphs join the v alues ˜ k q = 1 , . . . , Q resulting wh en appro ximating these signals at thr ee d ifferen t qualities: snr D = 65, 60, and 55 dB. In the left graph eac h of the thr ee lines joins Q = 225 p oin ts. In the righ t graph Q = 66 p oin ts. 12 5 Conclus ions A dedicated co dec for compression o f gravitational sound with high quality reco v ery has b een pro - p osed. The p ow er o f the format t o store this type of sound stems from the mathematical mo del for represen t ing t he signal. The mo del is based on recursiv e selection of elemen tary comp onen ts appro x- imating the signal with accuracy . The comp onen ts, called atoms, are c hosen from a redundant set, called a dictionary , whic h is the union of t wo sub-dictionaries consisting of atoms of differen t nature. One sub-dictionary con tains trigonometric a toms. The other sub-dictionary con tains pulses of small supp ort. While the con tribution o f the atoms of small supp ort to the signal approximation is minor, they are necessary to obtain approx imatio ns of high accuracy . A t lo wer quality recov ery (less than 50 dB) this sub-dictionary could b e av oided and the compression p o w er of the prop osed co dec would b e ev en more p oten t. Nonetheless, the study rep orted here fo cuses on compression with high qualit y p oin t-wise reco very . The prop osal was tested on t he sound represen tation of the detected short c hirp gw151226 , and on a set of longer signals numerically sim ulated at MIT. Comparisons with the compression standard MP3 resulted in a significant incremen t of compression p ow er for equiv a len t quality o f the reco v ered signal. As a b ypro duct, the co dec generates a condensed digital summary of the sound whic h could b e of assistance for iden tification ta sk s. Note: All the results in this pap er can b e repro duced using the MA TLAB softw ar e whic h has b een made av ailable on [32]. Ac kno wle dgmen t Thanks are due to Karl Skretting for making av aila ble the Arith06 f unction [33] for arithmetic co ding, whic h has b een used at the en tropy co ding step. The author is grateful to Prof Hughes and the p eople from his group who part icipated in the sim ulation of the signals used in this study . References [1] B. P . Abb ott et al. 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