White Noise Reduction of Audio Signal using Wavelets Transform with Modified Universal Threshold

Transcription

1 Whte Nose Reducton of Audo Sgnal usng Wavelets Transform wth Modfed Unversal Threshold MATKO SARIC, LUKI BILICIC, HRVOJE DUJMIC Unversty of Splt R.Boskovca b.b, HR 1000 Splt CROATIA Abstract: - Ths paper dscusses wavelet-based algorthm for audo denosng. We focused on audo sgnals corrupted wth whte nose whch s especally hard to remove because t s located n all frequences. We use Dscrete Wavelet transform (DWT) to transform nosy audo sgnal n wavelet doman. It s assumed that hgh ampltude DWT coeffcents represent sgnal, and low ampltude coeffcents represent nose. Usng thresholdng of coeffcents and transformng them back to tme doman t s possble to get audo sgnal wth less nose. We are proposng modfed unversal thresholdng of coeffcents whch results wth better audo sgnal. The man crteron for evaluaton of expermental results was objectve degree grade (ODG). Keywords: audo denosng, wavelets, thresholdng, nose reducton 1 Introducton Sgnal denosng usng wavelets was ntroduced by Donoho [1]. He developed lnear denosng for nose consstng of hgh frequency components and non-lnear denosng (wavelet shrnkage) for nose exstng n the low frequency as well. Schremmer, Haenselmann and Bömers [] mplemented a software for real-tme waveletbased denosng of audo sgnals. Denosng s acheved usng soft or hard thresholdng of DWT coeffcents. The crteron for successful nose removal s dfference between orgnal sgnal and denosed sgnal. If we hear dfference as slence, that means perfect reconstructon of orgnal sgnal. The nose that has been added to the sgnal has been removed and sgnal has not been modfed. If dfference sounds lke musc, part of orgnal sgnal has also been removed wth nose. Error estmaton s acheved usng rato between square root energy of dfference between orgnal and denosed sgnal and square root energy of added nose. Donoho s denosng method wth new threshold s value searchng method was mproved n [3]. Frst step of ther method s based on teraton: output sgnal from wavelet denoser s used as new nput sgnal and t s denosed agan wth same threshold value. In second step they use sgnal processng method based on the enhancement of dversty of sgnal to be processed. In ths case dversty can be enhanced computng for the same sgnal some dfferent DWT transforms (transforms wth dfferent wavelet mother and number of teratons). For every DWT transform thresholdng and IDWT s performed. Denosed sgnal s obtaned by computng the mean of all outputs. Novel speech enhancement system based on a wavelet denosng framework was ntroduced n [4]. In ths system, the nosy speech s frst preprocessed usng a generalzed spectral subtracton method to ntally lower the nose level wth neglgble speech dstorton. A perceptual wavelet transform s then used to decompose the resultng speech sgnal nto crtcal bands. Overvew of complex wavelets, and ther applcaton to audo sgnal processng (nose reducton and sgnal compresson) s presented n [8]. Ths paper s organzed as follows. In chapter the basc concepts of dscrete wavelet transform are descrbed. In chapter 3 denosng usng DWT s explaned. Smulaton results obtaned usng modfed unversal threshold are presented n Secton 4. We conclude n Secton 5.

2 Dscrete wavelet transform Fourer transform gves nformaton about frequency content of sgnal, but t does not show at what tmes frequency components occur. It s the reason why we use Short term Fourer transform and wavelet transform for analyss of sgnals lke audo. Wavelet transform has advantage over Short term Fourer transform because t analyzes the sgnal at dfferent frequency wth dfferent resolutons. Hgh frequency components have good temporal localzaton, but frequency resoluton s poor. Low frequency components have good frequency resoluton, but they are not localzed n tme well. Ths approach s called multresoluton analyss and t makes sense when sgnal has hgh frequency components for short duratons and low frequency components for long duratons. Ths approach has certan smlartes wth Bark-scale of human audtory system: human ear has better frequency resoluton at low frequences and lower frequency resoluton at hgh frequences. The dscretzed contnuous wavelet transform enables the computaton of the contnuous wavelet transform by computers, but t s hghly redundant and requres sgnfcant computaton tme and resources. Dscrete wavelet transform (DWT) provdes analyss and synthess of orgnal sgnal wth sgnfcant reducton n the computaton tme. Decomposton of the sgnal s obtaned by passng tme doman sgnal through half band low pass and hgh pass flters. Flterng the sgnal s equvalent to convoluton of sgnal wth mpulse response of flter: = y = x h = x h (1) k k = y = x h = x h () k k Procedure s started by flterng nput sgnal wth hgh-pass and low-pass flter. Outputs from hgh-pass flter are called detal coeffcents and they are kept apart as level 1 DWT coeffcents. Outputs from low-pass flter, approxmaton coeffcents, are fltered agan. Ths procedure s llustrated n Fg. 1. DWT of orgnal sgnal s then obtaned by concatenatng all coeffcents startng from last decomposton level. Inverse transform s done by flterng approxmaton and detal coeffcents wth lowpass and hgh pass synthess flter. Outputs from these two flters form approxmaton coeffcents of next level. f = p/ ~ p g[] level 1 N/ DWT coeffcents f = p/ 4 ~ p/ x[] g[] f = 0 ~ p h[] f = 0 ~ p/ h[] level N/4 DWT coeffcents... f = 0 ~ p/ 4 Fg. 1. Dscrete wavelet transform (h and g denotes lowpass and hghpass flter respectvely) where x s nput sgnal, h s mpulse response of flter and y s flter output. Ths procedure doubles frequency resoluton because frequency band of the output of flter spans over half the prevous frequency band. After flterng, half of the samples can be elmnated accordng to the Nyqust s rule because sgnal hghest frequency s now halved. Therefore the sgnal s subsampled by smply dscardng every other sample. Ths halves the tme resoluton because only half the number of samples represent entre sgnal. Because of that subsamplng relaton (1) s modfed: 3 Denosng of audo sgnals usng DWT We assume sampled nosy audo sgnal y y =x +σ n n =1,,3 N (3) where x represents orgnal sgnal, σ n s standard devaton of nose and n s array of random numbers generated accordng to Gaussan probablty densty functon wth µ=0 and σ =1.

3 Equaton (3) n wavelet doman s: W Ψ y =( W Ψ )(x +σ n n )= W Ψ x +σ n (W Ψ n ) (4) where W Ψ denotes wavelet transform. If the bass functons of wavelet transform are orthonormal, wavelet transform of the whte nose n s also whte nose w of same ampltude. Solvng for x gves: Ths means sharp dscontnutes on the edges whch can produce large coeffcents values n wavelet doman and make t harder to resolve sgnal from nose. Soluton whch was used n our work s to add some extra coeffcents to wndow of samples and so form extended wndow. orgnal sgnal x x ( W )( W y σ w ) = Ψ Ψ n (5) + whte nose n We don t know σ n w, so we estmate t by some value t whch gves: ( W )( W y t) x = Ψ Ψ (6) where x denotes estmated x. Relaton (6) ndcates that denosng s n fact removal of nose contrbuton t from wavelet coeffcents. Ths procedure s called soft thresholdng and t s defned wth followng expresson: sgn(z )( z - t), z > t n t ( z ) = (7) 0, else where n t ( z ) s threshold operator and z = W s wavelet coeffcent. x can be x = W n z. Ψ y ( ) calculated as ( ) ( ) Ψ There exst varous schemes for selecton of threshold t. Ther am s to fnd threshold value that wll effcently remove nose, but also preserve fdelty of orgnal sgnal. Too hgh threshold often cuts part of orgnal sgnal and causes audble artfacts n denosed sgnal. On the other hand, too low threshold doesn t remove nose very well. Denosng algorthm scheme s showed n Fg.. Frst step s wndowng of tme doman sgnal because t s usually too long to be processed entrely. Frst, wndow length must be chosen: too short wndow doesn't pck up mportant tme structures of audo sgnal. On the other sde, too long wndow wll loose mportant short transent detals n musc. Accordng to [5] we use wndow length of 4096 samples. Because of nature of DWT algorthm whch ncludes subsamplng by factor t s advsable that number of samples s equal to power of two. The smplest wndowng functon s square equal to 1 over wndowng nterval and zero elsewhere. t nosy sgnal y DW T Thresholdng ID W T denosed sgnal cx Fg.. Nose removal algorthm Nosy sgnal s obtaned by addng an array σ n n to array of orgnal sgnal x. Usng DWT degraded sgnal s transformed nto wavelet doman. One of the frst methods for selecton of threshold t was developed by Donoho and Jonstone [6] and t s called VsuShrnk (unversal threshold). Slghtly dfferent unversal threshold was proposed n [3]: t σ n = log (N) (8) where N denotes number of samples and σ n s standard devaton of nose. Durng our research t s notced that threshold obtaned by (8) s too hgh. It must be corrected f we want to get maxmum performance especally for low level nose. After thresholdng, nverse DWT s appled to get denosed tme doman sgnal.

4 4 Smulaton and results We mplemented whte nose removal algorthm n Matlab whch has large collecton of functons for wavelet analyss (Wavelet toolbox). We choose 8-level DWT and appled soft thresholdng on all levels ncludng approxmaton coeffcents too. For DWT Daubeches flters wth 6 coeffcents are used. We notced that flter of hgher order ncrease computatonal complexty, but also mproves denosng results. Input of our smulaton s orgnal sgnal n Wave format. Nosy sgnal s created by addng random generated numbers σ n n to the orgnal sgnal samples (3). Accordng to wndow length, blocks of 4096 samples are processed ndvdually. Crteron used for evaluaton of results obtaned by denosng algorthm s objectve degree grade (ODG), varable obtaned accordng to [5]. We have also compared t wth mean square error (MSE) whch s wdespread used for estmatng sgnal qualty. Input MSE s defned as: 1 (x y ) (9) N where x s orgnal sgnal and y s nosy sgnal. Output MSE s defned as: 1 (x x ) (10) N where x s estmated x (nosy sgnal y passed through denosng algorthm (Fg. 3)). [ \ [Q Q wavelet denoser Fgure 3. Sgnals used for calculatng mean square error Lower MSE means the closer match between two sgnals. Accordng to ths crteron denosng s successful f output MSE s lower than nput MSE. In that case denosed sgnal x s closer to orgnal sgnal x than nosy sgnal y. \ x MSE s conventonal method for estmatng sgnal qualty, but for audo sgnal they aren t always n accordance wth lstenng tests. An example whch s often mentoned to llustrate lmtatons s so-called 13 db mracle. Supermposed nose wth spectral structure adapted to audo sgnal s almost naudble even f resultng SNR declnes to 13 db [7]. As a man crteron for qualty estmaton we have used Objectve degree grade (ODG) whch s calculated accordng to [7]. ODG corresponds to sbjectve lstenng tests. ODG values range from 0 (mperceptble mparment) to -4 (very annoyng mparment). Lower ODG means more annoyng mparment. We nvestgated maxmum gan n objectve degree grade for few nput sgnals wth dfferent levels of degradaton. Amount of whte nose added to orgnal sgnal s controlled wth varable σ n (3). Maxmum ODG gan s obtaned replacng threshold t (8) wth t σ n = k log (N) (11) where 0<k<1. Durng our work t s notced that unversal threshold t gven by equaton (8) s too hgh for audo sgnals and t cuts part of orgnal sgnal too. So t s modfed wth factor k n order to obtan hgher qualty output sgnal. The value of k was changed gradually wth steps of 0.1. In that way for every nose level we found k that gves the best result. Ths s shown n table 1 (example 1: 8. s long excerpt from song Sultans of swng, Dre strats.). Input output nput MSE output σ n k ODG ODG (10-6 ) MSE (10-6 ) Table 1. Results of nose removal algorthm for modfed soft thresholdng method (example 1) Table shows results for second example: 15.4 s long excerpt from Sprng (Antono Vvald).

5 Input Output nput MSE output σ n k ODG ODG (10-6 ) MSE (10-6 ) Table. Results of nose removal algorthm for modfed soft thresholdng method (example ) For dfferent audo examples sgnfcant enhancement s obtaned, but wth dfferent k values. It suggests that better denosng can be obtaned usng modfcaton of threshold but t also means that modfcaton factor k should be calculated for each audo sample. From results n Table 1 and Table (modfed soft thresholdng) t s obvous that gan n objectve degree grade doesn t correspond to MSE gan. MSE shows lttle enhancement or even loss whle ODG and also nformal lstenng tests prove sgnfcant enhancement of sgnal qualty. These results confrm that MSE s not always n accordance wth perceptual qualty of audo sgnal. Denosng algorthm works better for lower nose sgnals. For hgher amount of added nose hgher threshold must be set, but except nose t removes part of orgnal sgnal causng audble artfacts n denosed sgnal. 5 Concluson In ths paper we have used wavelet transform for denosng audo sgnal corrupted wth whte nose. Audo denosng s performed n wavelet doman by thresholdng wavelet coeffcents. It s shown that unversal threshold must be corrected n order to get maxmum performance especally for low level nose. Unversal threshold s modfed wth factor that should be calculated for each audo example. For dfferent audo examples sgnfcant enhancement s obtaned comparng to unversal threshold. However, the perfect denosng sn't possble: hgher threshold removes nose well, but the part of orgnal sgnal s also lost. Objectve degree grade (ODG) was used as man crteron and threshold was adjusted to fnd maxmum ODG gan. For every maxmum output ODG MSE gan s also calculated. The results confrm that MSE s not always n accordance wth perceptual qualty of audo sgnal. References: [1] Donoho, D. L., Denosng va soft thresholdng'', IEEE Trans. on Informaton Theory, 41: , 1995 [] Clauda Schremmer, Thomas Haenselmann, Floran Bomers: Wavelet based audo denoser, IEEE Internatonal Conference on Multmeda and Expo (ICME) 001, pp , Tokyo, Japan, August 001 [3] Alexandru Isar, Dorna Isar: Adaptve denosng of low SNR sgnals, Thrd Internatonal Conference on WAA 003, Chongqng, P. R. Chna, 9-31 May, 003, p.p [4] Fu Qang, Wan Erc A.: "Perceptual wavelet adaptve denosng of speech", EUROSPEECH- 003, Geneve, Swtherland, p.p [5] Jonathan Berger, Ronald R. Cofman, Maxm J. Goldberg: "Removng nose from musc usng local trgonometrc bases and wavelet packets", Journal of the Audo Engneerng Socety, 4(10): , October 1994 [6] D.L.Donoho: "De-nosng by Soft Thresholdng", Techncal Report no.409, Stanford Unversty, December 199 [7] Method for objectve measurements of perceved audo qualty, RECOMMENDATION ITU-R BS [8] P.J. Wolfe, S.J. Godsll: "Audo sgnal processng usng complex wavelets", presented at the 114 th Conventon of the Audo Engneerng Socety, March 003