Optimal Binaural LCMV Beamforming in Complex Acoustic Scenarios: Theoretical and Practical Insights

O P T I M A L B I N A U R A L L C M V B E A M F O R M I N G I N C O M P L E X A CO U S T I C S C E N A R I O S : T H E O R E T I C A L A N D P R A CT I C A L I N S I G H T S N i c o G ¨ o ß l i n g 1 , D a n i e l M a r q u a r d t 1 , 2 , I v o M e r k s 2 , T a o Z h a n g 2 , S i m o n D o c l o 1 1 U n i v e r s i t y o f O l d e n b u r g, D e p a r t m e n t o f M e d i c a l P h ys i c s a n d A c o u s t i c s a n d C l u s t e r o f E x c e l l e n c e H e a r i n g 4 A l l , O l d e n b ur g , G e r m a n y 2 S t a r k e y He ar i n g T ec hn o l o g i e s , E d e n P r a i r i e , M N , 5 5 3 4 4 , U S A AB STR A CT Bi nau ral be amf orm i n g alg ori thm s fo r he ad -mo unt ed ass ist iv e l i s te n i ng de vi ces ar e c ru cia l to i mpr ov e sp e ec h qu al ity an d s p ee c h i n te lli gib ili t y in no isy en v ir onm ent s, wh ile ma int a i ni n g th e sp ati al imp r e ss i o n of th e ac o u st i c sc ene . Wh ile t h e we l l -k n o wn BM VDR b e am for mer i s ab le to pr ese rv e th e bi n a ur a l cu e s of on e d es i r ed s ou rce , th e BL C M V be amf orm e r use s a dd i t io n a l c o n st r a in t s to al so p res erv e t h e b ina ura l cu es of in ter f e ri n g so ur c e s. In th i s pap er , w e pr o v id e t he ore tic a l an d pr act ica l in sig hts o n h o w t o op t i ma l l y se t t he i n te rfe ren ce sc a l in g p ar ame t e rs in th e BL CM V be am for mer fo r an arb i t ra r y num ber of in t er f e ri n g so urc es. In ad d it ion , s in c e in pr ac t i ce on l y a l i mi ted te m p or a l ob se rv ati o n in ter va l is a v a il a b le to e sti mat e al l re qui red be am for mer qu an tit ies , we pr ov i de an e x pe r i me n t al e v a l ua t i on in a co m pl e x a co ust ic sc e na r i o us ing mea s u re d i m p ul s e r es pon ses fro m he ar i n g a id s i n a c afe t e ri a f o r di ffe ren t o b s er v at i on in ter va ls. T he res ult s sh o w th at e ve n r a th e r s h or t ob ser va t i on in ter va ls a re su ffi cie n t t o a c hi e ve a de ce nt n oi s e r e du c t io n pe rfo rma n c e a n d tha t a p rop o s ed th r es h o ld on th e o pti mal in t er f e re n c e sc ali ng pa ram ete rs le ads to sm a l le r bi na ura l c ue er r o rs in pr a ct ice . In dex T er m s — He ari ng aid s , bi na u r al cu e s , no ise re duc tio n , be am - fo rmi ng, BL C M V , R TF 1. IN TR OD UC T I ON F o r h ead -mo unt ed a ssi sti v e li st e n in g de vi ce s (e .g ., hea r i ng ai d s, co c h le a r im pla nts ) , al go r i th m s tha t u se th e mi c ro p h on e si gn als fr om bot h th e lef t an d t h e ri ght h e ar i n g de vi c e ar e ef f e ct i ve t e ch niq ues t o im pro ve s p ee ch in - te lli gib i l it y , a s th e spa tia l in f or mat i o n cap tur ed by all mi cro pho nes ca n b e e x pl oit e d [1 , 2] . Be s id e s red uci ng un des ire d s ou r c es an d li mi tin g s pe e c h di sto rti o n , ano the r i mp ort a n t obj ect iv e o f bi n au r a l spe ech en han c e me n t al gor ith m s is th e pr e se r v at io n of th e l is ten er’ s p e rc ept ion o f the a co ust ica l sc ene , i n or de r t o e xp loi t th e bi n a ur a l he a ri n g ad v an ta g e [3 ] an d to re d uc e co nfu sio n s due to a m i sm atc h be twe en a cou sti cal an d vi su a l inf orm ati o n . T o a c hi e ve b in a u ra l no i se re d u ct i o n wit h b in aur al cue pr ese rv ati on, tw o ma in co n ce pts ha v e bee n d e ve l op e d . I n t he fi rs t c on c e pt , a co mm on re al- va l u ed sp e c tr o - te m p or a l ga in is ap p li e d to the re f e re n c e m icr oph o n e si gna ls i n the le ft an d t he r i g ht he ar ing de vi ce [4 – 10 ], e n su rin g pe r fe ct pr ese rv ati o n of t he i nst a n ta n e ou s bi na u r al cu e s b ut in e vit abl y in tro duc ing sp eec h di sto rti on. Th e se co n d con cep t, wh ich is co n si der ed in th i s pap er , is to ap p ly a c omp l e x - v a l ue d fi lt er t o a ll a v ai la b l e m i c ro p h on e si gn als on th e l ef t a nd th e r ig ht hea rin g de vi c e usi ng bin aur a l ex t e ns i o ns of sp ati a l fi lte rin g te c hn i q ue s [1 1 –1 9 ] . Wh ile t h e we l l -k n o wn bi n a ur a l mi nim um v a ri anc e d ist ort ion l e ss r e sp ons e Th i s w ork wa s su p po r te d i n pa rt by a jo i nt Lo w er Sax o n y- I sr a el i P ro j ec t fi n an c ia l ly s u pp o rt e d by t h e St a te o f L o w er S a xo n y , G e rm a n y , by t h e Cl u st e r of Ex c el l en c e 1 0 77 H e ar i ng 4 al l , f u nd e d b y t he G er m an R e se a rc h F ou nda t io n (D F G) , an d a r es e ar c h g i ft f r om S t ar k e y He ari n g T ech n ol o gi e s. w L ( ω ) w R ( ω ) . . . . . . Y L , 1 ( ω ) Y R ,M ( ω ) Y R , 1 ( ω ) Y L ,M ( ω ) Z L ( ω ) Z R ( ω ) Fi g. 1 : Bi na ura l h ea r i ng de vi c e con fig ura t i on . (B MVD R) be a mf o r me r [ 1] p re ser ve s th e b in a u ra l c ue s (i .e. , t h e in t er aur al le v e l d if f e re n c e (I LD) an d in te r a ur a l ti me di f f er e n ce (I T D )) of on e de s ir e d so urc e, the bin aur a l li ne arl y c on str ain ed min imu m v ar ia nce (BL CMV ) be amf orm e r [1 5] is al s o abl e t o pr ese rv e the bi nau ral cu e s of int erf eri ng so urc es. Th i s is ac hi e v a bl e by im p os ing in ter f e re n c e sca lin g co n st rai n t s fo r th es e sou rce s. It sh o ul d be not ed tha t the BM V DR an d BLC MV be amf orm e r s r equ ire an es t im ate of th e cor rel ati on m atr ix th at s hou ld be mi nim ize d and an es t im ate of th e r el a ti v e tr an sf er fu nc tio ns (R TFs ) o f th e de sir ed an d i nt erf eri n g so ur c e s. Th e pe rf orm anc e of th e se be a m fo r m er s ma y s i g ni f i ca n t ly de te rio r a te in ca s e of es ti m a ti o n er r or s. Suc h e s t im a t io n er ror s oc cu r if on l y sh or t te mp o r al ob ser va tio n in t er va ls for e st i m at i o n can be us ed, e. g . , d ue to dy nam ic spa t i al sc e n ar i o s su ch as mo vi n g so urc es or he a d mo v eme nt. In th is pa pe r , we fi rs t de ri ve opt ima l va lue s for th e in t er f e re n c e sc a li n g pa - ra met ers in th e BL CM V be a mf orm er ba se d on the BMV DR be am f o rm e r wi th R TF pr es e r v a t io n (B M VD R - R T F) [1 3 , 1 4 ] f or an arb itr ary nu m b er of i n te r f er i n g so u rc e s . Se co n d ly , si nce t h es e va l u es a r e op ti mal i n th e se nse of no ise re d u ct i o n bu t no t rob us t ag a in st R TF es ti mat i o n err o r s in pr act ice , we pr op o s e t o ap ply an up p er an d lo we r t h r es h o ld on th e m. W e e va lua t e th e pe rfo rma n c e of th e B MV D R be am f o rm e r an d t he BLC MV be amf orm e r us in g t he tw o d i ff ere nt int erf e r en c e sc al ing pa ram e t er s an d me asu red im p ul se re spo nse s fr om he a r in g ai ds in a c a fe ter ia [ 20] fo r se v- er al te m po r a l ob se rv ati on in t er v al s . Th e re su lts s h o w th a t e v e n ra t he r sh o r t te mpo ral te m po ral ob s e rv at i o n i nte r v al s l ead to su f fi c i en t no is e re duc tio n pe rfo rma n c e an d t ha t th e imp ose d t hr e s ho l d on the op tim al in ter fer enc e sc ali ng pa ram ete rs ca n s ig n i fi c a nt l y re du c e bin aur al cue er r o rs . 2. CO NFI GUR A TI ON AN D NO T A TI ON Co nsi der th e bi na u r al he a ri ng de vi c e con fig ura tio n in Fi g. 1, co n si s t in g of a mi c ro p h on e ar ray wi th M mi cro pho n e s on th e l e f t an d th e ri g ht he a ri n g de vi ce. F o r an ac ou s t ic sc ena rio wi th on e des ire d s ou rce , P in ter fer i n g so urc es a nd i nco h e re n t bac kgr oun d no is e, t he m -t h mi cro pho n e sig nal of th e l ef t he ar ing de vi c e Y L , m ( ω ) ca n b e w rit ten in th e fr eq uen cy - do m a in as Y L , m ( ω ) = X L , m ( ω )+ P X p = 1 U L , p ,m ( ω )+ N L , m ( ω ) , (1 ) wi th X L , m ( ω ) th e de s ir e d sp e ec h co m po n e nt , U L , p ,m ( ω ) th e p -t h in ter fer e n ce co m po nen t a nd N L , m ( ω ) th e b ac k g ro u n d n ois e co mp one nt (e .g. , d iff use no i s e) in th e m -t h mic rop hon e sig nal . Th e m -t h mic rop hon e si gna l of the rig h t he ar ing de v i ce Y R , m ( ω ) is d ef i n ed si mil a r ly . F o r co nci sen e s s w e w i l l o m it th e f re que nc y v ar i a bl e ω in th e r e ma i n de r of th e p ap e r . W e de fi ne th e 2 M -d i me n s io n a l st ack ed si gn al v e ct or y as y = [ Y L , 1 .. .Y L , M Y R , 1 .. .Y R , M ] T , (2 ) wh ere ( · ) T de not es th e t ra nsp o s e an d w hi c h ca n b e w ri tte n as y = x + v , v = P X p = 1 u p + n , (3 ) wh ere x , u p an d n ar e de f in ed si m il a r ly a s in ( 2) a nd v de not es th e o v e ra l l un des ire d c o mp one n t , i. e ., i n te rfe r e nc e p l us b ack gro u n d no ise c omp o - ne nts . Fo r t h e c oh ere nt d es i r ed so urc e S x an d th e c oh e r en t i n te rfe rin g so urc es S u , p , wi th p ∈ { 1 , ... ,P } , th e v e ct ors x an d u p ca n b e w rit t e n a s x = S x a , u p = S u , p b p , (4 ) wi th a an d b p th e ac ou s t ic tr an sf er fun cti ons (A TF s) bet wee n all mi - cr oph one s a nd t h e d es ire d an d t h e p -t h in te rfe rin g so u rc e , r es pec ti ve l y . W it h ou t lo ss o f g e n er a l it y , we c hoo s e t h e f i rs t mi c ro pho nes on t h e l e ft an d t he ri g h t h ear ing de vi c e as ref ere nce mi c r op h o ne s , i. e. , Y L = e T L y , Y R = e T R y , (5 ) wh ere e L an d e R ar e 2 M -d ime nsi o n al v ec tor s w it h on e el eme nt eq ual to 1 an d th e o th e r ele men ts e qua l t o 0 , i. e., e L [1 ] = 1 an d e R [ M + 1] = 1 . Th e co rre lat ion ma t r ic e s of t he ba ckg rou nd n ois e c om p o ne n t , t h e des ire d sp eec h c om p o ne n t , t he p -t h i nt e r fe r e nc e co mp one nt an d a ll in t e rf e r en c e co mpo nen t s ar e de fin ed as R n = E n n n H o , R x = E n x x H o = Φ x a a H , (6 ) R u , p = E n u p u H p o = Φ u , p b p b H p , R u = P X p = 1 R u , p , (7 ) wh ere E {· } de not es t he e x pe cta t i on op e ra tor , ( · ) H de not es t he co nju ga te tr ans pos e an d Φ x an d Φ u , p de not e t he pow e r sp ec tr a l den sit y (P SD ) of th e d e si r e d sou rce and the p -t h i n te r f er i n g sou rce , r es pec ti v el y . As su m i ng st ati sti c a l ind epe nde n c e bet wee n th e co mp one nts in (1 ), the cor r e la t i on ma tri x o f t he mic rop hon e si g na l s R y ca n b e w rit ten as R y = R x + R u + R n = R x + R v , (8 ) wi th R v th e c or r e la t i on ma t r ix of th e ov er al l u nd esi red co m p on e n t. Th e ou tp u t s ig nal at t he le ft h ea r i ng d ev ice Z L is obt a i ne d b y f i lt eri ng th e m ic r o ph o n e si gna ls wi th the 2 M - d im e n si o n al fi l te r w L , i.e . , Z L = w H L y = w H L x + P X p = 1 w H L u p + w H L n , (9 ) Th e ou tpu t si g n al at th e rig ht he ar i n g ai d Z R is si mi l a rl y de f in ed. Fu rth erm o r e, we de f i ne th e 4 M -d ime n s io n a l fi lte r v e c to r w as w =  w L w R  . (1 0) Th e R TF ve c to r s o f t he d es ire d an d t h e i n te r f er i n g s ou rce s ar e d e fi ned by rel a t in g th e A TF v e ct o r s to the A TF of th e re fe r e nc e mi c ro pho n e on th e l ef t an d the ri ght he a r in g de vi c e, i. e ., a L = a A L , a R = a A R , b L , p = b p B L , p , b R , p = b p B R , p . (1 1 ) Th e 2 M × P -d ime nsi o n al ma tri ces B L an d B R co nta ini n g th e R TF v e ct ors of al l in te rfe rin g s ou r c es ar e de fi ned as B L =  b L , 1 ,. .., b L , P  , B R =  b R , 1 ,. .., b R , P  . (1 2) Th e b in a u ra l in p ut and o ut p u t sig nal - t o- n o is e r at io (S NR ) is de f i ne d as th e r at io of th e av er ag e i np ut and out put PS Ds of the des ire d s pe ech co mpo nen t an d th e b ack gro und no i s e c omp one nt, i. e . , SN R i = e T L R x e L + e T R R x e R e T L R n e L + e T R R n e R , SN R o = w H L R x w L + w H R R x w R w H L R n w L + w H R R n w R . (1 3) Th e b ina ura l i npu t an d ou t pu t si gn a l -t o - in t e rf e r en c e r at io (S I R) is de fi n e d as th e r at io of th e av er ag e i np ut and out put PS Ds of the des ire d s pe ech co mpo nen t an d th e i nte rfe ren c e co mp o n en t s , i .e. , SI R i = e T L R x e L + e T R R x e R e T L R u e L + e T R R u e R , SI R o = w H L R x w L + w H R R x w R w H L R u w L + w H R R u w R . (1 4) Th e bi n au ral i np ut an d ou t pu t si g na l - to - i nt e r fe r en ce- plu s - no i s e ra tio (S INR ) is def ine d as the ra ti o of the a v e ra g e i n p ut an d o u tp u t P SDs of th e de sir ed s pee ch c omp one n t and th e ov er al l un des ire d co mpo nen t, i .e. , SI NR i = e T L R x e L + e T R R x e R e T L R v e L + e T R R v e R , SI N R o = w H L R x w L + w H R R x w R w H L R v w L + w H R R v w R . (1 5) 3. BI NA U R AL NO ISE RE D U CT I O N AL GOR ITH M S In S ec tio n 3. 1 an d 3. 2 we br ie fly r e vie w th e BM VD R be a m fo r m er [1 , 2, 1 2] an d th e BLC MV bea mfo rme r [1 5 ]. Ba s ed on th e op ti mal ity of th e BM VD R-R TF b e am for mer [14 ] in o p ti miz ing the SIN R (o r S N R) wh ile pr ese rvi ng the bin aur a l cu es of all sou r c es , in Se c ti on 3. 3 we der iv e op tim al v al ues f o r th e in t er f e re n c e sc al ing p ar ame ter s in th e BL C MV be amf orm e r in the ca s e of an ar bi t r ar y num ber of in te rfe rin g so ur c e s. Fu rth erm o r e, in ord e r to ac h i e v e a ro bu s t bi n a ur a l cue pr e se rv ati o n pe rfo rma n c e in cas e of es t im a t io n er r or s of th e co rr e l at i o n mat r i ce s an d th e R T F v e ct ors (S e ct ion 3. 4 ), we pr o po s e to t hre sho ld t hes e in te rfe ren ce sc ali ng pa ram ete rs. 3. 1. BMV DR be amf orm er Th e BM VD R be am for m e r a im s at m i ni m i zi n g t h e o ut p u t P SD in bo th he ar i n g de v i ce s , wh il e pr e se rvi ng the d es ire d sp e ec h co m p on e n t in th e re fe ren ce m i c ro p h on e si gn als . Th e co r r es p o nd i n g c o ns tra ine d op tim iza t i on pr o b le m is gi ve n b y mi n w w H e R w su bje ct to w H C = g , (1 6 ) wi th e R =  R 0 2 M × 2 M 0 2 M × 2 M R  , (1 7) wi th R ei the r eq u a l t o the co r r el a t io n ma tr ix R y of th e mi cr o p ho n e si gna ls, t h e c or r e la t i on m atr ix R v of th e ov er al l und esi red co mpo nen t or th e co rr e l at i o n m atr ix R n of th e ba ck g r ou n d noi se c omp one nt. The co nst rai n t se t in (1 6 ) is gi ve n by C =  a L 0 2 M × 1 0 2 M × 1 a R  , g =  1 1  , (1 8) re qui rin g th e R T F ve c to rs of th e de si red sou rce . Th e s ol uti on to th e op tim iza t i on pro ble m i n (16 ) u s in g t h e co nst rai n t se t i n (18 ) i s equ al to [1 , 12 , 2 1] w MV D R , L = R − 1 a L a H L R − 1 a L , w MV D R , R = R − 1 a R a H R R − 1 a R . (1 9) Fr om a t he ore tic al poi nt of vie w , in th e ca se of per fec t l y est ima t e d qua nti - ti es (i . e. , co rre l a ti o n ma t r ic e s an d R T F v ec tor ), us in g R = R y or R = R v in (1 9) is op t im a l i n the SI N R s e ns e, w h er e a s u s in g R = R n in (1 9) is op tim al in th e S NR se n s e. Wh i le th e BM V DR be a mf orm er pre s e rv es th e bi nau ral cu e s of th e des ire d so u rc e , it s maj o r dr a w ba c k is th e dis t o rt i o n of th e bi na u r al c ues of th e in te r f er i n g s ou r c es ( and b ack gro und no ise ), su ch t h at al l s o ur ces ar e p e rc ei ve d a s com i n g f r om th e d i re cti o n o f th e de sir ed sou r c e. In pr a c ti c e , i t s ho uld al so be rea l i ze d th a t us ing R = R y ma y le ad to t ar ge t c a n ce l l at i o n i n t he cas e of R TF e st ima tio n er ro rs o f th e d es i r ed so u r ce [2 1 ] and th a t R v is no t st r ai g h tf o r w ar d t o e sti mat e. 3. 2. BLC MV be amf orm er In or der to als o ta k e bi na ura l c ue pr ese rv ati on of the in ter fer ing so urc es in to a c co unt as w e l l a s c o n tr o l t h e a m o un t o f in t er fer enc e su p pr e s si o n , it ha s be e n pro pos ed in [1 5] to ad d in t er fer e n ce sc a l in g co n st r a in t s to th e BM V D R bea mfo r m er , lea din g to th e BL C MV b ea m f or m e r . Th is co rre spo n d s to th e co n s tr a i ne d op ti m i za t i on pr ob l e m in (1 8) wi th th e co nst rai n t se t C 1 =  a L B L 0 2 M × 1 0 2 M × P 0 2 M × 1 0 2 M × P a R B R  , g 1 =  1 δ L 1 δ R  , (2 0) re qui rin g the R TF ve c to r s of the de si red so ur ce an d al l in t er fer ing so ur ces . Th e P -d ime nsi o n al v ec t o rs δ L =  δ L , 1 .. .δ L , P  an d δ R =  δ R , 1 .. .δ R , P  co nta in the int erf er en ce sca lin g pa r a me t e r s , whi c h co nt rol the sup pre s - si on and th e bi n au r a l cu e p re s e rv at i o n o f t he P in ter fer i n g so urc es. Th e BL CMV be a m fo r m er is gi v en by w LC M V = e R − 1 C 1  C H 1 e R − 1 C 1  − 1 g H 1 . (2 1) Se tti ng δ L , p = δ R , p en sur es bi nau ral cu e pr es erv at i o n o f t he p -t h i nt e r fe r - in g s ou r c e, wh i le th e ab so lut e v alu es of δ L , p an d δ R , p di rec tly de t er min e th e S IR im pro v em e n t for th e p -t h i nt erf eri n g so ur ce. Fr o m a t h eo ret ica l po int of vi e w , i n t he ca s e of pe rf ect ly es t i ma t e d qu a nt i t ie s (i .e . , co r re lat ion ma tri ces an d R TF v ec to rs) , se t ti ng δ L , p = δ R , p = 0 in th e BLC M V be amf orm e r is opt ima l i n t he SIR se n s e, b u t n ot nec ess a r il y in th e SIN R or SN R s en se. Mo re o ve r , in co nt ras t to th e B MV DR b e am f o rm e r , th e ch oic e of th e co rre lat ion ma t ri x R ha s no im pa c t on th e SI NR, SN R an d S IR imp r o v eme nt an d the bin aur al cu e pre ser va tio n as th ese a re co mpl ete l y det erm ine d by the in t er fer enc e sc al i n g pa r a me t e rs . In pr ac t i ce , in t h e ca se of e s ti m a ti o n er ror s th e ch o ic e of th e co rr ela tio n m at rix R wi ll ob vi o us l y ha v e an in flu enc e on th e per for man ce of th e BL CMV be amf orm e r (c f. Se c ti on 4). 3. 3. Int erf e r enc e sca lin g par ame ter s As an e xt ens ion of th e me th od pre s e nt e d in [22 ] f or an ar b it rar y n um b e r of in ter fer i n g so urc es, in th is sec t i on we pr o p os e a m e t ho d to de te rmi ne th e i nt erf e r en c e sc al i n g pa ram ete rs tha t m ax i m iz e th e SI NR or th e SN R wh ile pre ser vin g t h e bi nau r a l cu e s of the in te r f er i n g so u rc es. T o thi s en d, we wi ll us e th e BM VD R b ea mfo rme r w it h R T F pr es erv at ion [1 4], de not ed as BM VD R - R T F bea mfo rme r , wh ic h is a spe cia l ca s e of th e BL CMV be a m fo r m er . In th e BMV DR- R TF be am f o rm e r th e c ons tra int s re lat ed to th e in t er fer i n g so ur c e s on ly c on t r ol t he b in aur a l cu e pr e se r v at io n wh ile th e am o u nt of de sir ed in t er fer enc e sup pre ssi on is no t sp e ci f i ed , i.e . , w H L b p w H R b p = B L , p B R , p ⇒ w H L b L , p w H R b R , p = 1 , (2 2) le adi ng to th e c on s t ra i n t s et C 2 =  a L B L 0 2 M × 1 0 2 M × 1 − B R a R  , g 2 =  1 0 1 × P 1  . (2 3) Th e B MV D R -R TF be am f o rm e r is gi ve n by [14 ] w R T F = e R − 1 C 2  C H 2 e R − 1 C 2  − 1 g H 2 , (2 4) an d ei the r ma x im i z es th e SI NR ( R = R y or R = R v ) or t h e SN R ( R = R n ), wh i l e pr ese rvi ng th e b in a u ra l cu e s of al l so u rc e s . He nce , th e op t i ma l i nt e r fe r e nc e s ca lin g p ar ame t e rs f o r th e BL CMV be amf orm e r (i n th e S IN R or SN R sen se) ca n be de te rmi ned as δ op t p = δ L , p = δ R , p = w H R T F , L b L , p = w H R T F , R b R , p (2 5) Ho we v e r , u s in g t he op t i ma l in te rfe ren ce sca lin g pa r am e t er s ma y lea d t o pr obl ems in pr act i c e due to es tim ati o n er ro r s of th e cor rel ati o n ma tr ice s an d R T F ve c t or s . Mo re in pa r t ic u l ar , in th e ca se of SI N R ma xi m i za t i on , th e c or r e sp o n di n g in te r f er e n ce sc a li ng par ame t e rs ma y be ra th er sma ll, le adi ng t o a d ecr eas ed b ina ura l cu e p res erv at ion pe r fo rma nce (c f . sim ula - ti ons in Se c ti on 4 ). On the ot h e r h and , in th e ca se of SN R ma xi m i za t i on , th e co rr esp o n di n g i n te r f er e n ce sca l i ng par a m et e r s m ay be rat her lar ge , de pen din g on th e po si tio n o f the int erf eri n g so ur ce, le adi ng to an uns at- is fyi ng SI NR im pro v em e n t. He n ce , we pr opo se t o e nf o r ce an up p er an d lo we r t hr e s ho l d on the op tim a l in te r f er e n ce sc a l in g pa ra met ers , i .e . , δ th r p =      | δ op t p | , i f δ mi n < | δ op t p | < δ ma x , δ mi n , if | δ op t p | ≤ δ mi n , δ ma x , if | δ op t p | ≥ δ ma x . (2 6) Th e th re s h ol d s ha v e be en ex per ime nta lly ob t ai n e d as δ mi n = 0 . 2 an d δ ma x = 0 . 4 , l imi tin g th e t he o r et i c al l y pos s i bl e SI R im pro v em e n t f or e ach in ter fer i n g so urc e b et w e en 8 d B an d 14 d B . 3. 4. Est ima t i on of co r r e l at i o n ma tri c e s an d R T Fs Al l c on sid ere d bi n a ur a l be a mf o r me r s re qu ire an es ti mat e o f th e R TF v e ct ors a L an d a R of th e de sir e d s ou rce (c f. (1 1) ). In add iti on, th e BL CMV an d BM VD R-R TF be a mf orm ers re qu i r e an es t im ate of th e R TF v e ct ors b L , p an d b R , p of eac h in t er fer ing so urc e. In th is pa per , we wi l l es tim ate th es e R TF s us i ng th e c o v ar i a nc e wh ite nin g ap p ro ach [2 3, 24 ], wh ich i s ba se d on t he g en e r al i z ed e ig e n v a lu e de com p o si t i on ( GE V D ) of t h e sp ee ch + no is e co rre lat ion m a tr i x R xn = R x + R n an d th e ba ckg rou n d no i se co rre lat ion mat rix R n or the GEV D o f th e in te rfe ren ce + no ise co r re lat ion ma t ri x R v , p = R u , p + R n an d R n . Wh il e R n ca n be es tim ate d e xp l o it i n g the as sum ed sta tio n a ri ty of the ba ckg rou nd noi se, es - ti mat ing R xn an d R v , p fr om t he a v a i la b l e m ixt u r e i s no t st rai ght for wa rd . Du e t o li m it ed so ur ce ac ti vi t y an d po s si b l e sp at i a l ch an g e s of th e ac o us t i c sc ena rio s , th e tem por al ob s er v a ti o n in te rva l th a t is av ai l a bl e in pr a ct i c e fo r e st i m at i n g t hes e c or r e la t i on ma t ri ces is ty p i ca l l y l imi ted . W e as s um e th at th e co rr e l at i o n ma tr i x R xn ca n be e st i m at e d fr o m an ob se rv ati on in ter va l co ns i s ti n g of T L fr ame s (c or r e sp o n di n g to L se con ds) wh e re on ly th e d es ire d s ou r c e a nd the ba c k gr o u nd no i s e a re act i v e, i. e ., ˆ R xn = 1 T L T L X t = 1  x ( t ) + n ( t )  x ( t ) + n ( t )  H , (2 7) wh ere t is th e fr am e in de x. Si m il a r ly , we ass ume th a t the co r r el a t io n ma tri x R v , p ca n b e e sti mat ed f rom an ob s e rv at i on in ter v al of T L fr ame s wh ere on ly the p -t h in te r f er i n g sou rce an d th e bac kgr oun d no is e ar e ac ti v e . 4. EX PER IME N T AL RE S UL TS In thi s se c ti on, we ex p er ime nta l l y in v es t i ga te the eff ect of the tem p o ra l ob ser va t i on in t er v al on th e per for man ce of th e BMV DR bea mfo rme r ( w MV D R ) and th e B LC MV bea mfo r m er us i n g eit h e r the op tim a l in te r - fe ren ce s cal ing pa r a me t e rs ( w LC M V ( δ op t ) ) or th e pr op ose d t hr e s ho l d ed in ter fer e n ce sc a l in g pa r am e t er s ( w LC M V ( δ th r ) ) (c f. Se ct i o n 3 .3) . W e c on sid er th ree di f f er e n t a cou sti c s ce n a ri o s com pri sin g o f o ne de sir ed so urc e, on e or t w o int erf eri ng s o ur ces an d dif fus e ba ck gro u n d n o is e (c f. T a bl e 1 fo r s o ur c e p o si t i on s ) . Th e d es i r ed s our ce w a s a m a le Ge r- ma n s pe ak er , th e fi r st in ter fer ing so urc e w a s a m a le Dut ch sp ea k e r an d th e se c o nd in t er f e ri n g s our ce w as a ma l e E ngl ish sp e ak er . Th e des i r ed sp eec h a nd int e r fe r e nc e co m po nen ts wer e g en era ted by con v ol vi n g th e de sir ed an d i nt erf eri n g so ur c e si gn a l s w ith me asu red im p u ls e re s po n s es of bi nau ral be hin d - th e - ea r he a ri n g ai ds mo unt e d on a d u mm y h ea d i n a ca fet eri a ( T 60 ≈ 12 50 ms ) [ 20] , wi th M = 2 mi cro pho n e s p er h ear ing ai d . F o r bac kgr oun d no i se we us ed rea l a mb ien t n oi se rec ord ed in the sa me ca fet eri a wi t h t he sam e se t up . Th e sam p l in g fr eq uen cy wa s 16 kH z . Al l si gna ls st art wi t h 2 s of no i s e- o n ly , f o ll ow ed by ab out 20 s of al l so u rc e s be ing ac ti ve . Th e br oad ban d inp ut S NR w as s e t to 5 d B an d th e SI Rs we re se t t o 0 dB . Th e n oi se cor rel ati on mat rix R n w as est i m at e d us in g t he 2 s no ise -on l y se gm e nt . T o e st i m at e th e c orr ela tio n ma tr ice s R y , R v , R xn an d R v , p , we con sid ere d di ff ere n t t e mp ora l o b s er v at i on in ter va ls ( st art i n g a t 2 s ), wh ose le n g th L ra nge d be twe en 0 . 1 s an d 3 s . T o es ti m a te th e cor rel ati on ma tri ces R v , R xn an d R v , p th e a lg ori t h m ha d a cc ess to th e re s pe c t i v e mi xtu res . Th e R TF v ec tor s of th e de s ir e d so ur c e an d the in ter fer ing so urc e(s ) we r e the n c al c u la ted ba s e d on the se est i m at e d co rr ela t i on ma - tr ice s (c f . Se ct ion 3. 4). P le ase no t e th at sh o r te r te m po r a l ob se r v at io n in ter va l s co rr e s po n d to lar ge r es ti mat ion er r o rs . Th e mi c ro p h on e s ign als we re pro ces sed us in g a we igh ted ov er la p - ad d fr ame w or k wi th a bl ock le ngt h of 25 6 wi th 50% o v e r la p an d a sq u ar e - ro o t Ha nn wi nd o w . Th e BMV DR an d BLC MV bea mfo rme rs wer e c a lc u l at e d us ing th re e di ffe r e nt cor rel ati on ma tr ice s, i . e. , R = R y (m axi miz i n g SI NR wi t h po s s ib l e ta r ge t ca n ce lla tio n), R = R v (m axi miz i n g SI N R) an d R = R n (m axi miz i n g S NR ) . T he fil t e rs we re u se d a s f i x e d fi lte rs o v e r th e w hol e s ig n a l. As pe rfo r m an c e me as u r es we us e d the bi nau ral SI N R imp ro v eme n t an d th e b i n au r a l cu es er r or s , i. e ., IL D an d IT D er r o rs , th at we cal c u la t e d us i ng a mo del of bi n au ral au d i to r y pro ces sin g [2 5] . Al l per for man c e mea sur es we re a v era ged o v e r al l f req uen cie s an d al l a cou sti c s ce n a ri o s . Fi gur e 2 de pi c t s th e S IN R im p ro ve men t f or di f f er e n t le ngt hs of the te m - po ral o b se rv ati on in t er va l and f o r di ffe ren t c or rel ati on ma t ri ces , wh i l e Fi gur e 3 dep ict s t he bi n a ur a l cu e e rro rs of th e f ir s t in te r f er i n g s our ce fo r th e s am e te m p or a l ob s er va tio n in ter va ls an d R = R v . Fi rs t , it ca n be ob ser ve d tha t wh en u sin g R y or R v th e SI NR i mpr ov em en t is gen era lly la rg e r th an w he n us i n g R n . T hi s is e x pe cte d be cau s e us i ng th e no ise co rre lat i o n mat rix R n is ma xim izi ng the SN R a nd no t t he SI NR. Se con d, wh en us in g R y or R v , a n ap p a re n t di f fe ren c e ca n be se en for sm al l ob ser va tio n in te rv als be l o w 20 0 ms . The sm a l l o bse r v at io n in te rv als le a d to la r ge r es tim ati o n er ro r s for th e co rre lat ion ma t r ic e s and he nce al s o for th e R TF v ec t o rs , su c h th a t th e dr o p in SI N R im p ro ve men t o b se rv e d wh en us ing R y is pro bab ly at tr i b ute d to ta rg e t ca nc e l la t i on . Fo r l on ger obs erv a- ti on i nte rv als an d hen ce s mal ler es t i ma t i on er r or s , the di f fe ren ce b etw een us ing R y an d R v is sm a ll er . As ex p ec t e d, th e SIN R im pro v em e n t o f th e BL CMV be a m fo r m er us i n g th e t hr e s ho l d ed in t e rf e r en c e sca lin g p ar am- et ers δ th r is sm a l le r th an fo r th e B LCM V be am for mer us i ng th e op ti mal in ter fer e n ce sc ali ng par ame ter s δ op t . Alt hou gh, loo k i ng at the bi nau ral cu e e rr ors , u si ng δ th r in the BL CMV be amf orm er lea ds to mu ch be tte r bi nau ral c u e pr e se r v at io n , wh il e us i ng δ op t le ads t o si m i la r bi nau ral c u e er - ro rs a s fo r th e BM VDR be a mf orm er . Thi s di ffe ren ce i s es pec ial ly v i s ib le fo r t he IT D er r or at sm a ll ob ser va tio n in t er v al s and is al so con fir m e d by Sc ena rio 1 2 3 De sir ed − 3 5 ◦ 0 ◦ 0 ◦ In ter fer i n g 15 0 ◦ − 3 5 ◦ − 3 5 ◦ , 150 ◦ T ab l e 1 : S p at i a l s ce n a ri o s ( 0 ◦ : f r on t a l di rec tio n. − 9 0 ◦ : l e ft h and sid e. 90 ◦ : ri gh t h an d si de ). 2 4 6 8 2 4 6 8 0.1 0.2 0.3 0.4 0.5 1 2 3 2 4 6 8 Fi g. 2 : SIN R i mp ro v e me nt for dif fer ent te mpo r a l obs erv at i o n int e r v a l s fo r R = R y (t op) , R = R v (m id) an d R = R n (b ott om) . 0.1 0.5 1 2 3 3.5 4 4.5 5 5.5 6 0.1 0.5 1 2 3 0.3 0.4 0.5 0.6 Fi g. 3 : Bi na ura l c ue er r o rs of th e fi rs t in t er f e ri n g sou rce ( R = R v ). in for mal li st e n in g te sts . Th i rd , wh e n us i ng R n , th e BL C MV bea mfo rme r ou tpe rfo r m s the BM VDR be amf orm e r fo r lon ger ob ser va t i on in ter va l s ab ov e 30 0ms be cau se of the add iti o n al co nst rai n t s. Ad dit i o na l l y , us i ng δ th r in th e B LC MV b e am f o rm e r a p pa ren tly le ad s t o ma r g i na l l y b et ter SI NR imp ro v e m en t in th is ca se. Be cau s e R v is in pr act ice v e ry ha r d to ac cur ate l y est ima te, it sh o ul d be rec omm end e d to u se R n wh en s hor t ob - se rv ati o n in te r v als ar e r eq u i re d (e .g ., in dy n a mi c ac o us t i c s cen ari o s ) an d to us e δ th r in th e BL C MV be a m fo r m er to pr e ve n t bi na u r al cu e er ro rs. 5. CO NCL USI O N S In th is pap er , we pr o po sed op tim al va lue s fo r th e int erf ere nce sc ali n g pa ram ete r s in th e BL C MV be a mf o r me r for an arb i t ra r y nu m b er of in ter fer i n g s our ces ba s ed on th e BM VD R - R T F b eam for m e r . W e sh o w e d ho w t o s et t h e se pa ra m e te r s i n pra cti ce s u ch th at a r ob us t p erf orm anc e in th e c as e o f es ti m a ti o n er ro r s ca n be ac hi e ve d. Ev alu a t io n re s ul ts in a co mpl ex ac o us t i c sc ena rio sh o we d th at e v en sho rt tem por a l ob se r v at io n in ter va ls f or e sti mat ing th e re qu i r ed co r re lat ion ma t ri ces an d R TF ve cto rs ar e suf fic i e nt to ac h ie v e a dec ent no i s e re d u ct i o n pe r f or m a nc e an d bi nau ral cu e pr e se r v ati on. 6. RE FER ENC E S [1 ] S. Do c lo , W . K ell erm ann , S. Ma ki n o , an d S . E . N o rd hol m, “M ult ich a n ne l Sig nal En ha nce men t Alg ori t h ms fo r As s i st e d Li ste nin g De vi c e s: Ex plo iti n g sp a ti al di ve r s it y us i n g mu lt ipl e mi cro pho n e s, ” IEE E Si gn a l Pr oc ess ing Ma ga zi ne , v o l . 32 , n o. 2, pp . 1 8– 3 0 , M ar . 20 15 . [2 ] S. D o cl o, S. Ga n no t, D. Ma r q ua r d t, an d E. H a da d , “ B i na u r al S p ee c h Pr oce ssi n g wit h A pp l i ca t i on to He a ri n g De vic es, ” i n A ud i o Sou r ce Se par at i on an d Sp ee ch En h an c e me n t , c hap ter 18 . W il e y , 2 0 18 . [3 ] A. W . B ro n k ho r s t a n d R. P l om p, “ T he ef f ec t of he a d- ind uce d in ter aur a l t i me an d l e ve l d i ff e r en c e s o n s pe e c h i nt ell i g ib i l it y i n no ise , ” Th e Jo urn a l of th e Ac ou s t ic a l So ci ety of Ame r i ca , v ol . 8 3 , no . 4 , p p. 150 8–1 516 , 1 98 8 . [4 ] T . Lo tte r an d P . V ar y , “D ual -ch a n ne l sp e e ch en han cem ent b y su per dir e c ti v e be a mf o r mi n g , ” EUR ASI P J our n a l on Ap p li ed Si gna l Pr oc e s si n g , v ol . 20 06 , p p. 1– 1 4 , 20 06. [5 ] G. G ri mm, V . H o h ma n n , a nd B. K o l lm e i er , “I n cr eas e a nd Su bje cti v e Ev alu ati on of Fe e db a c k S ta b i li t y in Hea rin g Aid s by a Bi nau r a l Co h e re n c e- b a se d N oi se Re duc tio n S c he m e , ” IE EE T r a ns act ion s on Au d i o, Sp e e c h, an d Lan gua ge Pr o c es s i ng , v ol . 1 7, no . 7 , p p. 140 8–1 419 , S ep . 20 09 . [6 ] A. H . K am k a r - P a rs i a nd M . B ou cha r d , “Im pro v ed N o is e P o wer Sp ect rum De n si ty Est i m at i o n f or Bi nau ral He a r in g Ai ds Op e r at i n g in a Di ff u s e Noi s e Fi el d En v i ro nme n t , ” IE E E T ra n sa cti o n s on A u d io , Sp e ec h, an d La n gu ag e Pr o c es s i ng , v o l. 17 , no. 4, pp . 52 1–5 33, Ma y 20 09 . [7 ] A. H. Ka m ka r- P ars i a n d M. Bo u c ha r d , “I n st a n ta n e ou s Bi n a ur a l T a r - ge t P SD Es t i ma t i on fo r He ar ing Ai d N oi s e Re du c t io n in Co mp le x Ac ous tic En v ir o n me n t s, ” I EE E T ra nsa c t io n s on In s t ru m e nt a t io n an d Mea sur e m en t , vo l . 60 , n o. 4, pp . 1 14 1 – 11 5 4 , Ap r . 2 01 1. [8 ] K. Re ind l , Y . Z he ng, A. Sc hw ar z, S. Me ier , R. Ma a s, A. Se h r , a n d W . 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