Adaptive Microphone Arrays for Noise Suppression in the Frequency Domain

Transcription

1 Seon Cost 9 Workshop on Aaptve Algorthms n Communatons, Boreaux, Aaptve Mrophone Arrays for ose Suppreo the Frequeny Doman K. U. Smmer, A. Wasleff Department of Physs an Eletral Engneerng Unversty of Bremen, P.O. Box D-8 Bremen 33, Germany Abstrat Several methos for aaptve multhael nose suppreon of speeh sgnals are ompare. A mofe weghtng funton s propose to reue lnear stortons of the esre sgnal whh may be ntroue by the postproeng system omponent. Expermental results usng an array of four mrophones plae at the orners of a PC montor show a goo agreement between theory an experment. Introuton In ths paper we su aaptve algorthms for mult-mrophone nose reuton an eho anellato reverberant envronments. The results may be apple to hans-free telephony systems an speeh reognton applatons uner nosy ontons. Conventonal (.e. non-aaptve) mrophone arrays [] requre many sensors to obtan a hgh spatal seletvty n the reton of the esre speaker. Aaptve sensor arrays, as propose by Frost [], Grffths-Jm [3] or Duvall [4] are able to steer the zeros of the spae-transfer funtoto the reton of the nose sgnal. Ths approah an be very effetve f the number of nose soures s smaller than the number of sensors. However, n reverberant rooms the number of unwante nose soures s very hgh ue to refletng walls. Experments uner realst ontons have shown that the performane of Frost s algorthm (ashe lne n fg. 3a) s not better than a non-aaptve beamformer that omputes the average of the nput sgnals (sol lne n fg. 3a). Allan [5] has evelope a reverberaton suppreon system usng the fat that the orrelaton of reverberaton reeve at two-ponts s small ompare wth that of the ret soure sgnal f the soure s lose to the array. Kanea an Tohyama [6] have propose a two sensor nose reuton system that s base on the same aumptons. Zelnsk [7] has extene ths metho to a four sensor onfguraton an ntroue an atonal suppreon fator to mprove the nose reuton. Ther ommon pattern s the bas struture onsstng of three steps. The elay fferenes between sgnal soure avual sensors are ompensate n the frst step (preproeng). Thus the array beam s steere towars the esre sgnal soure. In Allen s system the ompensaton s performe n the frequeny oman, whereas the preproeng s mplemente by Kanea an Zelnsk n the tme oman wth the a of orrelators. The estmaton error of ths orrelator an be reue by ntroung a oherene etetor [4]. In the seon step (non-aaptve beamformer) the sgnals at the sensors ( use by Kanea, 4 by Zelnsk) are summe an weghte by /. Ths step may also be mplemente ether n the tme oman (Zelnsk) or n the frequeny oman (Allan, Kanea). Ths alreay gves a nose reuton about log() B. The remanng nose of the non-aaptve beamformer s further reue n the thr step (postproeng) by a one mensonal nose suppreon flter. The output of the non-aaptve beamformer s multple wth the transfer funton of the postproeng flter to obtan the spetrum of the fltere sgnal. The fnal output sgnal s reonstrute by usng nverse FFT an the overlap a (OLA) metho [8]. 85

2 Speaker Mrophones ose Soure Fg. Harware onfguraton of the aaptve array Mrophones Delay Fourer analyss on-aaptve Compensaton (Hang wnow) beamformer A/D τ (fxe) FFT Wener flter Overlapp-a synthess A/D A/D τ τ FFT FFT Σ Flter OLA A/D τ 3 FFT W(f) Tme Delay Estmaton Estmaton of Wener Flter Fg. Blok agram of the propose system ose Reuton [B] (a) on-aaptve Frost Duvall Grffths-Jm Input SR [B] ose Reuton [B] (b) Zelnsk ew Input SR [B] Fg. 3 ose reuton of fferent algorthms as funton of nput SR n an offe room. a) ose anellaton algorthms: 4 sensor array 6x6m, 33 taps, normalze stepsze µ=. b) osesuppreon algorthms: 4 sensor array 6x6m, wnow length 56, α=.9 ([5] eq.5) 86

3 . Plaement of Sensors The stue nose suppreon systems are base on the aumpton of a low orrelaton of the stortng sgnals at fferent sensors of the array. Ths supposton has been verfe above a ertan lmtng frequeny by measurements n reverberant rooms lke usual offe rooms [3]. Informaton about the spatal oherene (orrelaton) of the nose soures as funton of frequeny s requre to etermne the sensor stane of the mrophone array. The refletng walls of the room may be onsere as aoust soun soures whh are almost statstally nepenent ue to the large elays. Therefore a ffuse an sotrop soun fel s fou reverberant rooms wth strong refletng walls, where the orrelaton s only a funton of stane an oes not epen on the loal poston. Suh a soun fel may be moele by an nfnte number of nose soures plae on the surfae of a sphere wth the mrophones n the enter ([] p.6). Cron an Sherman [] have erve on ths onton the followng relaton for the ro-spetrum ( between the sensor sgnals n n sn(π ) λ (, λ) = = sn ( ) () λ π λ s the stane between the sensors, λ the wave length an ( the power ensty spetrum of the nose sgnals. Ths moel has been orgnally use to esrbe the volume nose n eep water soun propagaton. Morrow has prove the valty of the same relaton () for the ffuse soun fel n a retangular reverberaton hamber [5]. Usng the relaton λ = between wavelength λ, frequeny f an soun veloty we may wrte: f (, = sn (. () We get the omplex frequeny oherene funton Γ ( ( Γ ( =. (3) ( ( Sne we have the same power ensty spetrum at all sensors = = n a ffuse, sotrop soun fel the oherene of ths soun fel s gven by Γ ( = sn (, (4) an the "magntue-square oherene funton" (MSC) s ompute as ( ( ) = = sn Γ f (. ( ( (5) Ths equaton gves the spae frequeny oherene of an sotrop soun fel as funton of the sensor stane. The oherene funton has ts frst zero at f =, the 3B pont Γ =. 5 s gven at f =. ose Canellaton algorthms yel only a nose reuton f there s a hgh oherene between the stortng sgnals. They fal f the oherene s low. The nose suppreon systems sue n ths paper behave n an opposte way. The nose suppreon works better wth lower oherene of the stortng sgnals. Equaton (4) an the fgure 5 show a low oherene for an sotrop soun fel only for frequenes 87

4 Cro-orrelaton Cro-orrelaton f >. Ths has to be onsere for the postonng of the mrophones. The stane of the sensors has to be etermne by > (6) f mn wth f mn the lowest storton frequeny that has to be suppree by the system. The mrophone stane however aot be arbtrarly nrease. Wth nreasng mrophone stane we eventually get areasng stane between array an esre sgnal (speaker). Thus we get a erease of ts spatal oherene an hene a erease of qualty of the output sgnal [6]. An upper lmt for the mrophone stane s foun from the neety of "elay-ompensaton". The beam of the mrophone array has to be steere nto the reton of the speaker an has to follow hs movements. The tme elays between soure an the partular mrophones have to be estmate for ths purpose. Due to the quasperoty of the voe portons of the speeh sgnal (fg. 4b) a unambguous estmaton of the elay τ s only poble wthn the half pth pero t = / f pth pth τ < (7) f If one oes not want to restrt the range of the speakers movements, one has to restrt the stane of the mrophones. The maxmum elay τ max (wth speaker 9 to plane of the mrophones) s gven as pth τ = ± (8) max soun The maxmum stane of mrophones that allows an unambguous loaton of the speaker s gven therefore as < f soun pth λ = To fulfll the gven ontons (6) an (9) of sensor stane, the lower ut-off frequeny of the nose suppreon system shoul not exee the hghest pth frequeny. The eal sensor stane s therefore pth soun soun = = () f f mn.8 (a).8 (b) tpth Delay (Samples) Fg. 4a Cro-orrelaton of unvoe onsonant [s] True elay samples. pth max (9) Delay (Samples) Fg. 4b Cro-orrelaton of voe vowel [a] True elay samples. 88

5 Coherene (a) 5 m Mrophone Dstane Measure Theory Coherene (b) 3 m Mrophone Dstane Measure Theory Frequeny [khz] Frequeny [khz] Fg. 5. Real part of the omplex oherene funton (eq. 3 an 4) measure n an offe room.. Performane of the on-aaptve Beamformer To etermne the effeny an to evelop the optmal flter funton of the postproeng flter (Wener flter) we nee to know the nose reuton of the non-aaptve beamformer as funton of frequeny. For that purpose we frst etermne the autoorrelaton funton R of the nose n at the output of the beamformer. R ( l ) = E n[ k] n[ k + () = = l] R = ( R nn + R nn R + R + R R nn + R + R R n n n n ) The power ensty spetrum = = = R at the output of the beamformer s gven as = = () ( = n n. (3) If we aume ental autospetra of the partular nose sgnals, we an wrte, usng the omplex oherene funton (equaton 3) ( = Γ nn. (4) = = The oherene between pars of ental sensors s gven as = ) ( f Γ 89

6 an for - pars the followng relaton hols Γ ( + Γ ( = Γ ( + Γ ( = Re{ Γ ( }. Thereforeweanwrtethepowerenstyspetrumattheoutputofthebeamformeras ( f ) = ( = + Re{ Γ ( = + Re{ Γ } nn = + (5) The nose reuton s gven as - - ( R( = log( ) = log[ / (+ Re{ Γn ( = log - - }) - = + ( ) - log(+ Re{ }) (6) - Γ = + Lmtng ases: For very low frequenes f << all (-)/ elements of the ouble sum onverge aganst one ( lm sn ( = ), an there s no nose reuton. For large frequenes f > there f s a erease of oherene, therefore we get approxmately the spetrum of the average sgnal as ( =. (7) Ths orrespons to a nose reuton of log( ) whh s equvalent to 6 B f we use four sensors. Fg. 6 shows that the nose reuton s lose to ths value above 5 Hz. If the oherene s postve then the nose reuton s a bt smaller an for negatve oherene a bt larger than the average value. n })] ose Reuton [B] (a) 3 m Mrophone Dstane Measure Theory ose Reuton [B] (b) 6 m Mrophone Dstane Measure Theory Frequeny [khz] Frequeny [khz] Fg 6 ose reuton of the non-aaptve beamformer as a funton of frequeny (eq. ) 9

7 Γn n In an sotrop soun fel the oherene epens only on the stane of sensor pars. (eq. 4). The nose reuto that ase s gven as - - R( = log( ) - log(+ sn( f (/ ) ) (8) = + If we plae four sensors at the orners of a square, we get the lateral oherene between the four sensor pars as ( Γ + = sn ( = Γ(,. (9) The oherene between the two agonal pars s gven as Γ + (, = sn ( = Γ(,. () Ths yels the autospetrum of the average value of the sensor sgnals: ( = (4 + 8Γ(, + 4Γ(, ) The nose reuton R(f) of the non-aaptve beamformer of four sensors s gven as = + Γ(, + Γ(,. () ( R( = log ( ) = log(4) - log(+ Γ(, + Γ( ( 3. Postproeng, f )) () The remanng nose n of the average sgnal x may be further suppree usng a postproeng flter. To erve suh a Wener flter we etermne an mnmze the power ensty spetrum ee of the error sgnal = ( - ( - ( + ( (3) ee ( ˆ ˆ sˆ sˆ The output power ensty spetrum of the flter s gven by the Wener-Lee formula sˆ sˆ ( = x x ( W ( (4) The ro-spetra between esre sgnal s an output sgnal of the flter ŝ are gven as ( = ( W ( (5) ˆ an ˆ ( = xs ( W ( (6) The power ensty spetrum of the error sgnal follows as * ( = ( - ( W ( - ( W ( + ( W ( (7) ee sx f sgnal s an average nose n areunorrelate weget = ( = ( (8) sx xs an ( = ( + ( (9) xx sx xs xx 9

8 The power ensty spetrum of the output error of the Wener flter follows as ee * ( = ( (- W ( -W ( + W ( ) + ( W ( Usng equatons (5) we get the error power ensty of the ombne system of beamformer an Wener flter as - - ee ( = ( (- Re{ W ( }+ W ( ) + (+ Re{ Γn n }) W ( = + (3) The power ensty of the estmaton error s mnmze f the partal ervatves of real an magnary part of W(f) are set to zero: - - Re{ ee ( } = - ( + Re{ W ( }+ (+ Re{ Γ Re{ W ( } Im{ ee ( } Im{ W ( } = It follows that W opt ( gven as W opt ( Im{ W ( }+ (+ - = - = + = + Re{ Γ (3) })Re{ W ( }= (3) })Im{ W ( }= (33) s real. Ths optmal weghtng funto the sense of the least square error s ( f ) = Usng (7) we get for frequenes ( f ) + (+ ( f > / approxmately W opt - - = + Re{ Γ nn }) (34) ( ( (35) ( + ( 4. Estmaton of the Weghtng Funton from Data Zelnsk s estmaton of the postflter transfer funton may be generalze to sensors W ( = z ( -) - - = + - Re{ X X ( ( X * ( } were X (f) s the Fourer transform of the nput sgnal at sensor. Aumng no orrelaton between sgnal aose an the same power ensty of nose at all sensors we get W ( = z + ( -) = + Re{ Γ } (36) (37) 9

9 There s no nose reuton at low frequenes whereas for hgh frequenes f > / one has W z ( (38) + A omparson of the optmal soluton (35) wth Zelnsk s metho shows an overestmaton of fator of nose power ensty gven by hs soluton. As both sgnal aose are weghte by W(f) we get a goo nose suppreon (sol lne n fg. 3b) but on the other han a lnear storton of the esre sgnal. Ths stortoreases wth number of sensors an wth nose power. To avo ths effet we propose a mofe verson (ashe lne n fg. 3b) of the postproeng flter: W sw ( f ) = ( -) - - = + - Re{ X X ( ( X * ( } Ths alternatve proeure estmates the power ensty spetrum of sgnal aose by alulatng the power spetrum of the sum of the output sgnals (ashe lne n fg.).uner the same aumptons as above we get W sw ( = + + ( -) ( = + - = + Re{ Γ Re{ Γ Although there s agao nose reuton for low frequenes, we get for W sw }) } (39) (4) f > the approxmaton: ( ( (4) ( + ( Power spetrum [B] Sgnal Sgnal+ose ew Zelnsk Frequeny [khz] Fg. 7. Power spetra of proee sgnals as a funton of frequeny 93

10 5. Results The fgure shows the results of the ompare methos. The sol lne shows the power spetrum of the esre (orgnal unstorte) sgnal. The ashe lne gves the power spetrum of the storte (orgnal an nose) sgnal. The ashe otte lne shows the power spetrum proee by the metho propose n paper [7]. The otte lne proves the goo agreement of the results of the proeure evelope n ths paper wth the esre spetrum. Aknowlegment The authors wsh to thank M. W. Rege for useful suons an helpful omments. Referenes [] J. L. Flanagan, J. D. Johnston, R. Zahn, G. W. Elko, "Computer-steere mrophone arrays for soun transuto large rooms", J. Aoust. So. Am., vol. 78, no. 5, 985, pp [] O. L. Frost, III, "An algorthm for lnearly- onstrane aaptve array proeng," Pro. IEEE, vol. 6, no. 8, pp , Aug. 97. [3] L. J. Grffths, C. W. Jm, "An alternatve approah to lnearly onstrane aaptve beamformng", IEEE Trans. Anteas Propagat. vol. 3, no., pp. 7-34, Jan. 98. [4] K. M. Duvall, "Sgnal anellato aaptve anteas: The phenomenon an a remey," Stanfor Unv., Stanfor, Calf., Aug. 983 (Ph.D. thess). [5] J. B. Allen, D. A. Berkley an J. Blauert, "Multmrophone sgnal-proeng tehnque to remove room reverberaton from speeh sgnals", J. Aoust. So. Am., vol. 6, no. 4, 977, pp [6] Y. Kanea, M. Tohyama, "ose suppreon sgnal proeng usng -pont reeve sgnals", Eletrons an Communatons n Japan, vol. 67-A, no., pp. 9-8, Apr [7] R. Zelnsk: "A mrophone array wth aaptve post-flterng for nose reuto reverberant rooms", Pro. Int. Conf. Aoust., Speeh an Sgnal Proeng, ICASSP-88, ew York, pp , 988. [8] J. B. Allen, L. R. Rabner, "A unfe approah to short-tme Fourer analyss an Synthess", Pro. IEEE, vol. 65, no, 977, pp [9] J. S Benat, A. G. Persol, "Engneerng applatons of orrelaton an spetral analyss", Wley Intersene, ew York, 98. [] Sven Fsher, "Enfluß er räumlhen Kohärenz akustsher Sgnale auf mehrkanalge Geräushunterrükungsverfahren". Dploma Thess, Unversty of Bremen, Sept. 99. [] B. F. Cron, C. H. Sherman, "Spatal-orrelaton funtons for varous nose moels", J. Aoust. So. Am., vol. 34, no., 96, pp [] C. T. Morrow, "Pont-to-pont orrelaton of soun preures n reverberant hambers", J. Soun Vb., vol. 6, no., 97, pp [3] K. U. Smmer, A. Wasleff: "Analyss an omparson of systems for aaptve array proeng of speeh sgnals n a nosy envronment", Trezème Colloque GRETSI, Juan-Les-Pns, pp , 99. [4] K. U. Smmer, P. Kuzynsk, A. Wasleff, "Tme elay ompensaton for aaptve multhael speeh enhanement systems", ISSSE-9, Pars -4 Sept