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Time delay estimation using the LMS adaptive filter
"It,c
Copyright© If AC Programmable Devices and Embedded Systems Bmo, Czech Republic, 2006
,.
c>.
Publications
TIME DELAY ESTIMATION USING THE LMS ADAPTIVE FILTER Jacck Izydorczyk'
• Silesian University oJ Technology, Poland
Abstracl: The text (;ondsely shows an idea for use of two coupled LMS filters to estimate time delay between two detectors (microphones). In the algorithm stiff shape of the impulse response is assumed. As a result LMS algorithm adapt time delay and gain only. Time complexity of the algorithm is the same order as time computational complexity of conventional LMS algorithm. The article is illustrated by computations on real-life data. Keywords: TDOA, LMS filter, estimation
a;;
1. TNTRODUCTTON
tect.ors) N, then eigenvalue of mat.rix (1) is the smallest eigenvalue, degenerated M - N times. Each eigenvector of a~ corresponds t.o a polynomial which has no less than 111 zeros on the unit. drdp. at. the point.s exp(j . k j ). Tt. is sufficip.nt to precise estimation of wave numbers k j and estimation of DOA of each of wave.
In the literature we find a lot of techniques and algorithm, to estimate direction of wave arrival (DOA) to linear array of antennas or sensors (Orfanidis, ]!J88). Most. of thcm havc bccn invented to track onc or more sources of signal using antenna array or for elimination of onc or lIlore interferers. The algorithms are slill improved ntustly to enlarge capacity of Illobile cOllllllunications systems by realization an idea of space division mnlt.iplex of radio channel (Sarkar Et al., 2003). The algorithms are constructed under assumption that considered signals are nanowband and biased by a white, additive gaussian noise. The autocorrelation matrix of such signal takes the form:
Practically wc (;an use only an c~timatc of autocorrclat.ion matrix (1). Spurious 7,eros of polynomia];; corresponding to eigenvedors of O'~ are elimillated by variollli forms of averaging. III such a way MUSTC, Min.-Norm, F;SPTRTr anel similar algorithms appeared (Orfanidis, 1988). For large number of samples each of the method achieves lower CraIIH~r-Rao bound (Stoica and Nehorai, 1990). Just. mentioned est.imators are insensitive to SNH. (signal to noise ratio) under assumption that noise signal elet.ect.ecJ hy each ant.enna (elet.ector) is 11ncorrelated with noise detccted by uny other antenna and that each noise signal has the same power a;'. In quite a lot practical applications such assumptions are unrealistic, so it.erative algorithms have been deVeloped to estimate DOA of waves and power of noise detected by each antellIla (detector) by the means of maximulll likelihooel (ML) principle (7,iskinel anel \Vax, 19HH; Pe-
(1) where a;' is average power of channel noise, I is identity matrix, P = diag{P1 , P 2 , . .. , P",} is a diagonal mat.rix containing power of signal sourCteS and IS]n,j ~ expU . 11 • k j ) describes direction of arrival for <".Itch wave - I.:J is a wave nnmher of jth wave in t.he direction elet.ermined hy antenna array. Eigenvalue dccomposition of autocorrclation matrix (1) i, recognizeel as the most effective method of DOA estimat.ion. If number of signal sources Al is lower then number of antennas (de-
298
savento and Gershman, 2(01). It has been proved recently (Madurasinghe, 200,1) t.hat. predict.ion error can be formulated without explicit use of noise power in e~'1ch ant.enna. Such DOA estimat.or is faster convergent then the traditional onc and has lower time comput.at.ional complexit.y.
2L,
------
__
If detected signals arc wideband onc possible s~ lulion is a narrowhand filtration. Tt inevit.ably leads to a lot of parallel narrow band DOA estimation ~y~lems. In ~uch ca.se e~timation of DOA by adaptively adjusted FIR filters demands less computational power even if ordinary LMS algorithm is used for coefficients adaptation (Ko and Siddharlh, 1(99).
Sonar systems and navigation systems of robots used baseband signals to detect time difference of arrival (TDOA) which effectively is equivalent to estimation of waVe's DOA . In such case glohal maximum of correlat.ion function of det.ect.ed signals is se.arched. Classificat.ion of met hods are hased on pre-filtering method chosen. As a result Roth Filtering, SCOT (Smoothed C0herence Transform), PHAT (Pha.~ Transform), Eckart Filtering and HT Filtering arc recognized. In (Knapp and Carter, 1976) h~ been proven that the last algorithm achieves lower CramcrRao bound. Similar algorithm ha~ been u~ed in (Piersol, 19S1) in the case when noise detected by ~en~om i~ ~patially correlated. In (Strobel and Rabenstein, 19(9) efforts has been reported t~ ward estimation of probability distribution of autocorrelation function. As a result ML estimator of delay time has been obtained. Just mentioned method has heen ext.ended t.o new delay time visualizat.ion met.hod named Ddayogram. (Silverman and Sachar, 2005) la.~t. t.ime.
~_
d
h
J~ J::-;-'.
r:-;-] :
~.e(l).
-~ ... ~ ~ ~ xo(t)
L
.
rig . I . Microphones sC'paratm by d makes hC'aring sense of the robot; acoustic wave arrives at angle 'f'; signal Xl (t) is processed by adaptive filter. one and the same effectiveness of time difference estimation and interferer elimination. The algorithm has been developed for speaker tracking by simple react.ivE' 1.E(;O robot.
2 PROBLEM FORMOLATION Figure 1 shows two microphones separated by d which makes hearing sense of the robot. Let us assume that signal source (speaker) is far from microphones, so acoustic wave is practically planar. Angle of arrival if' is planar angle between wave vector and a line connecting microphones. If we a.~SHmE' t hat wave propagat.es in dispersion-less medium I , time delay between signals detected by microphones is following: TI.p
d· sin if' =- --c-.- '
(2)
where c is group velocity of acoustic wave. Delay ohsE'rved in systE'm depicted in figHrE' 1 can not exceed rm " = die . Signals generated al the input of analog to di?;ital converters have the form:
In the literature reports can be found about use of FIR filters "'ith coefficients adaptively adjusted by LMS algorithm for estimation of TDOA (Reed et al., 1981). In the article (Feintuch et al., 1981) tracking of moving source with adaptive Fllt lilter has heen considered under Ilssumption t.hat power spectrum density of noise is exponentially decaying. Experimentally have been proven that timedomain fluctuations of signallE'.ads in such casE' to enlarged variance of autocorrelation matrix estimate. It results with divergent LMS algorithm and limits tracking abilit.y of the method (Omologo and Svaizer, 1994).
IO(l)
xdt)
= er· J(I - r
=
(:.la)
(3b)
where f{t) is time domain signal generated by sou rce, Q is coefficient describing different gain in both channels of the sys tem and no (t) and n , (t) are noises and int erferences detected by each microphone. Let us assume, that noise signals are uncorrelaled, white and Gaussian. Power of each one equals ()'~.
In t.he article an idea of est.imation of time difference of arrival between two detect.ors (microphones) by the mean~ of two coupled FU{ liltcrs is presented. Coefficients of filters arc adaptively adjusted by LMS algorithm. The alg~ rithm evolved from idea proposed by (Ko and Siddharth, 1999) for pass band signals. Here it is modified for ba.~ehand signals. Tt possesses lower time computational complexity then the original
3. TIME DELAY ESTIMATION The problem is to estima te two independent parameters of Xo (t) signal, ie. gain Q and time
1
Of cours e such assumption is valid for acoust ic wave
propagatinp; in the air.
299
delay T
Po = [ [( xo(nT) - w T X l (nT)) ~l
'
Minimal power of the signal
(4)
wherexl(nT) = [xj(nT+ LT), ... ,xj(nT- LTW
= Rj/r,
r
= f[xo(nT) xd nT) j,
Pe"'; ,, - Po - rTR ~,I r ,
If delay can not be reconstructed by Wiener filter exactly, then at least we can give upper bound:
(.'ia)
L
(5 b )
r ~ F[xo(nT) xj(nT) j, Po - r Tn ~j' r.
Si2(1r(C - -j-))
(6a)
Pr + <7~
1
:c:e. )2 · (9) T
The last two ~ UIll S can be boundc>d by the following integrab:
J < 00
(6 b)
I
2:
t =L+ J
(c - :c:e.)2 T
L+l
de
(C -
'fF
L+
'f' (lOa)
no
,J;.]
(c -
* J )2 <
L+ J
(C - ; : - 1)2 = L -
'f.
T
(lOb ) As a result minim a l mean power of the signa l c(t) is bounded by:
20- ~2 2
Pe mi "
(
2
Pr
)
< 1r 2(P'r +(7n2)(~ - IT
(11) It is suflident. to es tima te minimal value of L "order" of the adap tive FIR filter - in a su(!h a way tit a t power of w mponent of e(t) caused by incident wave is lower than power of component of e(t ) mused by noise no(t) a nd nJ(t) for TI' = O: 2()2SNR~
> 1r 2·[1 + (1 + <> 2) SNR' . (12) . J Pr I (7~ = 20dD estimated value of
L min
<7~
J
no
::; -fr I=L 2:l l (C + 2£-)2 + -fr 2: (C _ T I=L .l l
For SNR = parameter Lmitl> by the means of (12 ), is Llllin= 22.
" IT
- oPt· T
2
Si (r.(C - +-)) ::;
I = L+l
r.·xo{IlT)xJ(nT - CT) ]
H-
2:
1
no
The solut.ion (Ga) is known in the lit.erature as Wicncr filter for a long time (O rfa nidis, 1(88). linfortwlat ely it is not d ear in what manner est imated para me ters, ie. time delay T
Si 2(1r(t--j-))+
t-'-.-co
+
00
Wf _
2:
=
'_ _L
and following mean power of e(t) signal: I'emin -
- L- j
2:
1-
where Rn is a utocorrelation ma trix of sampled signal :q(nT) and Po is mean power of xo(nT). At t he minimum derivat.ive of mean squared distance should equal zero. It. gives us an equation fulfilled by optimal values of fil ter coefficients w :
w opt ~ R~,lr,
equals:
where Si(x) = . iIJX) If estimated delay time is integer multiple of sampling period T and -L ::; T
is a vect.or consisting of21- - 1 sampl es of t he .T,(t) signal, whereas w = ~1L' L, ... , U.'L]''' is a vector of aciaptive filter coefficients. Power of t.h e signal Elc2 (nT)] is a quadratic fun ction of vector w, so it. pos.,es one glohal minimum : W opt
~(t)
e-j·w(fT-T·,)dw
=
- "rF
0 1'r = Pf + <7~·
4. LMS ALGORITHM
~ in 1r(C - T
1r (f - T
C=
L , . .. ,T,.
Minimi~at.i on of the power of signal e(t) (!3.n be done by the means of adjusting of adaptive filter
(7)
300
coefficients w with LMS algorithm. C nfortunately in 8uch c~ there appear diIliculties, mentiolled earlier, in mapping filt er coeliicient8 to gain 90 and time delay T
P' I
>i:
Fig. 2. l'vlinimized mean power Pe as a function of T
Gradient of power of the signal ell) with respect to paramet ers i 9u takes the form:
T"
ae
2
(nT) _
( "')
- , , - - -e n1 (JT~
2
ae a(nT) -_ 9u
( T' en)
awT
~X l (lTif'
T
aw a9u
Xl
(''')
,
(13a)
(T) n ,
(13b)
n1
Constraints imposed on adaptation process le.ad to L;~:L rp.gmcllp.ss of value of pmameters Top and 90 which d escribe filter coefficients. It rp~~ u1t s with :
wi "" 96
wbere:
S" ( ("
aWt _
7[90
aWl -, - =
Si ( 7[(C -
UTop - Y . , I 7[ Ugo
T T
< ))
'f. )),
( l~c )
(16) If we assume that filt er coefficients a re sam ples of a hase hand signalwl - y(lT), t.hen 2.
for f =
Pc (T" , 90) ~
] _ [Top] [Top 9u n + l 9u n
I" \!r.",goc 2(nT),
", aD2 + 902 -
Pr
In the above equatiom; we used derivative Si' (x) CO'T _ ~. In each step of adaptation process the p:ramete rs Top and 90 are modifIed as folloW8:
In our case y(t) - yuSi t o'
(14 )
Pe (T"dJo)
Pr
2
(
)
(17)
20 0 y TO •
(f(1 - T.,)) and it leads
2
T
• ((
~ ° u - 9u - 2a090S1
TO -
T
))
.
(Ill) Surface described by (18) for Ou = I is illus trated ill Fig. 2.
where gradient has been briefly deno ted using nabla symbol. Symbol J1. denotes the parameter that controls the speed of adaptation. In each step filter coefficients are computed using (7). Computations are so simple tha t if we use proper fixf'd point algorithm , ego CORmC (!Tll , 1992), it can he implementf'd llsing FPGA or similar dev ice. Jus t described algorithm has similar computational tim e complexity as algorit hm proposed originally hy (T\o and Siddharth, 1999). Co mputational time complexity of proposed algorithm has been 8ummarized in table 1.
\[ initial value of delay time between microphones is too fa r from a t.rue value TO, then a da ptation process "gds s t.uck" in side one of "c1rainpipp.s" s hown in figm e 2. Tn such cn.~e param eter.'lo (gain) converges toward zero quickly. In tilE' me.ant.ime est.imat f'd df'lay time is np"~ r1y const.ant. ancl IS equal t.o odd mll1t.iple of sampling rat" nearp" t init ial value of t.his parameter. Such phenomenon has been observed a lot of tim es during simulations of tbe svst.em
5. ERROR St:RFA CE 6. COCP LEO /\ 0/\ PTTVE FTLTERS
Unfortunat ely just proposed LMS algorithm has 8eriou8 drawback. Adaptatioll proceS8 CO il verges to the true value of delay time T" only if initial value of delay time TO, assumed at start of adaptation, differs from true value T
r: =n~ - 2no L I'
f
I.
I.
/ -- L
'Wf·Si(,,(l' - .y,))+
L
To overcome just described drawback let. U8 "form" t.he shape of erro r surface re(T",yo) choosing adapt.ive filter coeffi cif'nts from t.he s ignal o t.her t.hen Si(7[t IT ). Recallsf' in our prohlpm delay time can not. be grea ter then Tmax die then ·'rpa.'ionahle" choice seems 1.0 he: Wp
= 90' h(lT - T
(lOa)
2 'Wf·
[-- L
2
(15 )
We mu st rem€>mber to preserve constant. value of the
ro\Jowillg sum
301
L! ~ :L
u:i ~ y5·
Table 1. Computational time complexity of proposed LMS algorithm N UMDER
or
MULTIPLICATIONS
(DIVISIONS) PROPOSED RV
OPERATION
AND SIDDHAIlTH, 1999)
Computation uf e.(nT)
Computation of Jacobi matrix using (13c)-(13d), without. computational complexity of Si(x) Comput.ation of grarlient. using (13a)-(13b) Pseudu-inversiun of Jucuui JJHllrix New values of Top i gO New filter coefficients w by (7)1 without cumputatiunal complexity of Si(x)
(Ko
N UMOER
OF MULTIPLICATIONS (mVISIONS) RV PROPOSED
ALGUH.JTHM
2L
2LI 1
2L -I 1
.1[,+ 4
.1 [,
2L
t.
t
2L -1 1
1
[:~L -
!1-'
~.
Conplfld aclaptive FTR filt.ers nsfld for clelay time ffitimation. where
{~ -
IIITmaxl
if 11·1 < Tmax> if 11.1 > Tmux.
I jj
·w;
fh 2 (nT)
('twIT
a;:t XI (nT),
n:yo,
T
(h' =T·h(n:(C-f))
for
[~L+l -
!1
[;L,
(21)
7. EXPERIMENTAL DATA The presented algorithm has been tested in the laboratory. The hearing sense of the robot has consisted of two capacitance microphone elements placed d-23cm apart. It re.sults with maximal time delay in the syst.em Tmax "" O,fi7fims. Detccted signal has been amplified by a simple, two channel, two stage transistor amplifier build with RC109 t.ransistors. Ra.~e characteristics of the amplifier are: gain 20dl::l, source voltage 5V (from USB socket), nonlinear distortions of signal lower then 0.8% for output amplitude 1 Vpp, root mean square voltage of output. noises lower then 150!1-V. Amplified signal has been converted to digital form by conventional music card of PC computer (line.ar input) . Sampling rate 8kHz has been used and Hi-bit resolution . Collected data h3.~ hP.f!n processed with Mathcacl. Fig. 4 shows time-domain plot of estimated time delay 7', T'f' and gain !lu for "very good syst.em" ut.t.erance. Spf!aker has heen placed ahout lOcm apart of the microphone and he has spoken in the direct.ion of t.he line connecting microphonffi. In such a way maximal delay in the system has been reached. Unfortunately the assumption that
(20a)
where: ow1(
'V,.~,90f2(n1')+
where (3 is coupling coefficient. Goal of filter LMS1 is to put filter LMS2 into convergence region, whereas LMS2 filter precisely estimates delay time and gain. Components of input signal caused by incident acoustic wave is in high degree eliminated from signal e( l) at the ou tput of LMS2 filter. It is important in case when estimated TDOA concerns source of unwanted interferences.
(J9b)
CoefIicient ho is sek>cted in such a way that L~:~ /, = g8· As a result. minimized mean power Pe decre.ases nearly linearly with absolut.e value of difference TO - T'f' for all delays TO reali7~'lhle in t.he syst.em, ie. -"d ~ TO -; ~ Unfortunately, despite of good estimation of delay time, at the output of t.he adaptive filter we ohserve grp,at contrihution from signal f(t). The filter poorly eliminates acoustic signal detected by microphone>. Variance of e>timated time delay is greater then in the original version of t.he filter too. This gave us rise to omit adaptation of gain go. It is assumed that for the filter go = 1. The idea can be extended to use of two adaptively adjusted FIR filters for time delay est.imation as it is shown in figure 3. Filter LM81 , described by saw tooth impulse response, adapts delay time only and delivers T' - coarse estimate of TDOA - to LMS2 filt.er. Adaptation process is described in such case by following equations: ~ = e(nT)
+4
If difference is less t.hen T, t.hen LMS algorithm of LM82 filter works according to (13) and (14). If ahsolute value of clifference hetween ffitimated delay times 7' - T'f' is greater then T, then equation (14) is modified - there appears, apart from gradient, a component which describes coupling het.ween LMS1 and LMS2 filter:
e(nT)
h(t) = ho .
1
4L • 5 4[, + 6
go
Fig.
~
2L 1 1
t =- L, ... ,L.
(20b) During each step of adaptation parameter T' is modified as follows: , oc 2 (nT) T~ . 1 = Tn -!1-1 . ~. (2Oc) Estimated by LMSl delay time T' is compared by LMS2 with it's own estimate of delay time.
302
Hu, Yu Hen (1992). CO RDIC-based VLSI architectmes for digital signal processing. IEEE Signal Processing Magazin e 9, 16· 35. Knapp , Ch.H. and C.C. Cart.er (1976). The generalized correlation method for estimation of time delay. IEEE Tran•. on Amltstir-.• , Spef.r-h and Signal Processing 24, 320-327. Ko, e.c. and C.S. Siddharth (1999). Rejection and tracking of an unknown broadband tiource ill a two-element array through leUtit square approximation of inter-element delay. lJ:)};J:) Signall'rocessing Letters 6 , 122-125. Madurasinghe, D . (2005). A new doa estimator in nonuniform noise. IEEE Signal P"octssing Letlen 12 , ::l::l7-3::l9. Omologo, M. and P. Svaizer (1994) . Acoustic event localization using a crosspowerspectrum phase based technique. In: Proc. IGASSP. Vol. II . pp. 273-.276. Adelaide, Australia. Orfanidis, 5 ..1. (19RR). Opti,mal Signal Proa .•.• ing. McCrnw-TTiIl Puhlishing Company. l\ew York. P esavento, M. and A.B. Cershman (200 1). Maximum-likelihood direction-of-arrival estimatioll in the presellce of ulIkllowll Ilollulliform noise. lEEE Tmns. on Signal Processing 49, 1310-1324. Piersol, A .G. (1981). Time delay estimation using pbase data . JJ:)J:,'E 'l'rans. on Acoustics, Speech and Signal Processing 29, 171- 177. Reed, F .A., P.L Feintuch and 1\.J. Bershad (1081). Time delay estimation using the lms adaptive filter - static behavior. IEEE Tm" •. on Aco'a stits, Spetch a"d Signal P1'Octssing 29 , ,~6 1 571. Sarkar, T.K., M.C. Wicks , M. Salazar-Palma and R .J . Bonneau (2003 ). Smart Antennas. J ohn Wiley & Sons, Tnc ./IEEE Press. Silverma n, I-I.F. and J.M. Sachar (2005). The time-delay graph and the dclayogram - new visuali.:atiolls for time delay. jJ:,'EE Signal P1'O('essi"g Lelie1'S 12, 122-125. Stoica, P . and A. Nehorai (1900). Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Tr·a" •. on Acoustics, :::j'pee.ch and Signal Processing 38 , 178::l- 1795. Strobel, N. and R. Rabenstein (1999). Classificat.ion of time-delay estimates for robust speaker localizat.ion. In: Proc. IGASSP. Vol. VI. pp. :JORl :JOR4. Ziskind , 1. and M. Wax (1988). Maximum likelihood localiwtioll of lIlultiple sourceti byalternating projection. IEEE Trans. on Acoustics, Speech and Signal Processing 36, 1553- 1560.
Fig. 4. Estimatro delay time T', Top and gain go for utterance 'very good system". detected acoustic wave is planar has been satisfied poorly, but it is not crucial for the time delay estimation . The adaptation process has been started wit.h the following estimat.es: T' = Top =Oms and 90 = 1.1. The assumed values of coefficients used for controlling the spero of adaptation have been Jil = 0.15, Ji- O.Ol. Coupling coefficient equals !~=O.5 . Order of adaptive filters equals 2 T, 4 1=R5. The algorithm has quickly found the true value of the delay time which equab approximately 5· T = 0.ii25ms. Meantime E'stimated gain variE'd in 1-1.5 range. Variations of gain are probably caused by tilllaU distallce of the tipeaker from micropholleti and lIolltitatiollary character of tipeecb tiigllal.
8 . CONCLUSIONS In the article the author presented an idea for use of t.wo coupled, adaptive, FIR filters to ffitimate the delay time in case of baseband signal. Thanks to atisumptioll about "stilf" tihape of illlpube response of filt.ers the prohlem has heen reduced t.o adaptation of gain and time delay of the filters. Time computational complexity of proposed algorithm is comparable with complexity of conventional LMS algorithm.
REFERENCES Benedetto, S. and E Biglieri (1999). Pr-inciples of Digital Transmission with Wireless Applications. Klmver Academic. New York. Feint.uch, P.L., N ..T. Bershnd and F.A. Reed (1981). Time delay estimation using the Ims adaptive filter - dynamic hehavior. fEIIE Trans. on Acoustics, Speech and Signal Processing 29, 571- 576.
303
Copyright© If AC Programmable Devices and Embedded Systems Bmo, Czech Republic, 2006
,.
c>.
Publications
TIME DELAY ESTIMATION USING THE LMS ADAPTIVE FILTER Jacck Izydorczyk'
• Silesian University oJ Technology, Poland
Abstracl: The text (;ondsely shows an idea for use of two coupled LMS filters to estimate time delay between two detectors (microphones). In the algorithm stiff shape of the impulse response is assumed. As a result LMS algorithm adapt time delay and gain only. Time complexity of the algorithm is the same order as time computational complexity of conventional LMS algorithm. The article is illustrated by computations on real-life data. Keywords: TDOA, LMS filter, estimation
a;;
1. TNTRODUCTTON
tect.ors) N, then eigenvalue of mat.rix (1) is the smallest eigenvalue, degenerated M - N times. Each eigenvector of a~ corresponds t.o a polynomial which has no less than 111 zeros on the unit. drdp. at. the point.s exp(j . k j ). Tt. is sufficip.nt to precise estimation of wave numbers k j and estimation of DOA of each of wave.
In the literature we find a lot of techniques and algorithm, to estimate direction of wave arrival (DOA) to linear array of antennas or sensors (Orfanidis, ]!J88). Most. of thcm havc bccn invented to track onc or more sources of signal using antenna array or for elimination of onc or lIlore interferers. The algorithms are slill improved ntustly to enlarge capacity of Illobile cOllllllunications systems by realization an idea of space division mnlt.iplex of radio channel (Sarkar Et al., 2003). The algorithms are constructed under assumption that considered signals are nanowband and biased by a white, additive gaussian noise. The autocorrelation matrix of such signal takes the form:
Practically wc (;an use only an c~timatc of autocorrclat.ion matrix (1). Spurious 7,eros of polynomia];; corresponding to eigenvedors of O'~ are elimillated by variollli forms of averaging. III such a way MUSTC, Min.-Norm, F;SPTRTr anel similar algorithms appeared (Orfanidis, 1988). For large number of samples each of the method achieves lower CraIIH~r-Rao bound (Stoica and Nehorai, 1990). Just. mentioned est.imators are insensitive to SNH. (signal to noise ratio) under assumption that noise signal elet.ect.ecJ hy each ant.enna (elet.ector) is 11ncorrelated with noise detccted by uny other antenna and that each noise signal has the same power a;'. In quite a lot practical applications such assumptions are unrealistic, so it.erative algorithms have been deVeloped to estimate DOA of waves and power of noise detected by each antellIla (detector) by the means of maximulll likelihooel (ML) principle (7,iskinel anel \Vax, 19HH; Pe-
(1) where a;' is average power of channel noise, I is identity matrix, P = diag{P1 , P 2 , . .. , P",} is a diagonal mat.rix containing power of signal sourCteS and IS]n,j ~ expU . 11 • k j ) describes direction of arrival for <".Itch wave - I.:J is a wave nnmher of jth wave in t.he direction elet.ermined hy antenna array. Eigenvalue dccomposition of autocorrclation matrix (1) i, recognizeel as the most effective method of DOA estimat.ion. If number of signal sources Al is lower then number of antennas (de-
298
savento and Gershman, 2(01). It has been proved recently (Madurasinghe, 200,1) t.hat. predict.ion error can be formulated without explicit use of noise power in e~'1ch ant.enna. Such DOA estimat.or is faster convergent then the traditional onc and has lower time comput.at.ional complexit.y.
2L,
------
__
If detected signals arc wideband onc possible s~ lulion is a narrowhand filtration. Tt inevit.ably leads to a lot of parallel narrow band DOA estimation ~y~lems. In ~uch ca.se e~timation of DOA by adaptively adjusted FIR filters demands less computational power even if ordinary LMS algorithm is used for coefficients adaptation (Ko and Siddharlh, 1(99).
Sonar systems and navigation systems of robots used baseband signals to detect time difference of arrival (TDOA) which effectively is equivalent to estimation of waVe's DOA . In such case glohal maximum of correlat.ion function of det.ect.ed signals is se.arched. Classificat.ion of met hods are hased on pre-filtering method chosen. As a result Roth Filtering, SCOT (Smoothed C0herence Transform), PHAT (Pha.~ Transform), Eckart Filtering and HT Filtering arc recognized. In (Knapp and Carter, 1976) h~ been proven that the last algorithm achieves lower CramcrRao bound. Similar algorithm ha~ been u~ed in (Piersol, 19S1) in the case when noise detected by ~en~om i~ ~patially correlated. In (Strobel and Rabenstein, 19(9) efforts has been reported t~ ward estimation of probability distribution of autocorrelation function. As a result ML estimator of delay time has been obtained. Just mentioned method has heen ext.ended t.o new delay time visualizat.ion met.hod named Ddayogram. (Silverman and Sachar, 2005) la.~t. t.ime.
~_
d
h
J~ J::-;-'.
r:-;-] :
~.e(l).
-~ ... ~ ~ ~ xo(t)
L
.
rig . I . Microphones sC'paratm by d makes hC'aring sense of the robot; acoustic wave arrives at angle 'f'; signal Xl (t) is processed by adaptive filter. one and the same effectiveness of time difference estimation and interferer elimination. The algorithm has been developed for speaker tracking by simple react.ivE' 1.E(;O robot.
2 PROBLEM FORMOLATION Figure 1 shows two microphones separated by d which makes hearing sense of the robot. Let us assume that signal source (speaker) is far from microphones, so acoustic wave is practically planar. Angle of arrival if' is planar angle between wave vector and a line connecting microphones. If we a.~SHmE' t hat wave propagat.es in dispersion-less medium I , time delay between signals detected by microphones is following: TI.p
d· sin if' =- --c-.- '
(2)
where c is group velocity of acoustic wave. Delay ohsE'rved in systE'm depicted in figHrE' 1 can not exceed rm " = die . Signals generated al the input of analog to di?;ital converters have the form:
In the literature reports can be found about use of FIR filters "'ith coefficients adaptively adjusted by LMS algorithm for estimation of TDOA (Reed et al., 1981). In the article (Feintuch et al., 1981) tracking of moving source with adaptive Fllt lilter has heen considered under Ilssumption t.hat power spectrum density of noise is exponentially decaying. Experimentally have been proven that timedomain fluctuations of signallE'.ads in such casE' to enlarged variance of autocorrelation matrix estimate. It results with divergent LMS algorithm and limits tracking abilit.y of the method (Omologo and Svaizer, 1994).
IO(l)
xdt)
= er· J(I - r
=
(:.la)
(3b)
where f{t) is time domain signal generated by sou rce, Q is coefficient describing different gain in both channels of the sys tem and no (t) and n , (t) are noises and int erferences detected by each microphone. Let us assume, that noise signals are uncorrelaled, white and Gaussian. Power of each one equals ()'~.
In t.he article an idea of est.imation of time difference of arrival between two detect.ors (microphones) by the mean~ of two coupled FU{ liltcrs is presented. Coefficients of filters arc adaptively adjusted by LMS algorithm. The alg~ rithm evolved from idea proposed by (Ko and Siddharth, 1999) for pass band signals. Here it is modified for ba.~ehand signals. Tt possesses lower time computational complexity then the original
3. TIME DELAY ESTIMATION The problem is to estima te two independent parameters of Xo (t) signal, ie. gain Q and time
1
Of cours e such assumption is valid for acoust ic wave
propagatinp; in the air.
299
delay T
Po = [ [( xo(nT) - w T X l (nT)) ~l
'
Minimal power of the signal
(4)
wherexl(nT) = [xj(nT+ LT), ... ,xj(nT- LTW
= Rj/r,
r
= f[xo(nT) xd nT) j,
Pe"'; ,, - Po - rTR ~,I r ,
If delay can not be reconstructed by Wiener filter exactly, then at least we can give upper bound:
(.'ia)
L
(5 b )
r ~ F[xo(nT) xj(nT) j, Po - r Tn ~j' r.
Si2(1r(C - -j-))
(6a)
Pr + <7~
1
:c:e. )2 · (9) T
The last two ~ UIll S can be boundc>d by the following integrab:
J < 00
(6 b)
I
2:
t =L+ J
(c - :c:e.)2 T
L+l
de
(C -
'fF
L+
'f' (lOa)
no
,J;.]
(c -
* J )2 <
L+ J
(C - ; : - 1)2 = L -
'f.
T
(lOb ) As a result minim a l mean power of the signa l c(t) is bounded by:
20- ~2 2
Pe mi "
(
2
Pr
)
< 1r 2(P'r +(7n2)(~ - IT
(11) It is suflident. to es tima te minimal value of L "order" of the adap tive FIR filter - in a su(!h a way tit a t power of w mponent of e(t) caused by incident wave is lower than power of component of e(t ) mused by noise no(t) a nd nJ(t) for TI' = O: 2()2SNR~
> 1r 2·[1 + (1 + <> 2) SNR' . (12) . J Pr I (7~ = 20dD estimated value of
L min
<7~
J
no
::; -fr I=L 2:l l (C + 2£-)2 + -fr 2: (C _ T I=L .l l
For SNR = parameter Lmitl> by the means of (12 ), is Llllin= 22.
" IT
- oPt· T
2
Si (r.(C - +-)) ::;
I = L+l
r.·xo{IlT)xJ(nT - CT) ]
H-
2:
1
no
The solut.ion (Ga) is known in the lit.erature as Wicncr filter for a long time (O rfa nidis, 1(88). linfortwlat ely it is not d ear in what manner est imated para me ters, ie. time delay T
Si 2(1r(t--j-))+
t-'-.-co
+
00
Wf _
2:
=
'_ _L
and following mean power of e(t) signal: I'emin -
- L- j
2:
1-
where Rn is a utocorrelation ma trix of sampled signal :q(nT) and Po is mean power of xo(nT). At t he minimum derivat.ive of mean squared distance should equal zero. It. gives us an equation fulfilled by optimal values of fil ter coefficients w :
w opt ~ R~,lr,
equals:
where Si(x) = . iIJX) If estimated delay time is integer multiple of sampling period T and -L ::; T
is a vect.or consisting of21- - 1 sampl es of t he .T,(t) signal, whereas w = ~1L' L, ... , U.'L]''' is a vector of aciaptive filter coefficients. Power of t.h e signal Elc2 (nT)] is a quadratic fun ction of vector w, so it. pos.,es one glohal minimum : W opt
~(t)
e-j·w(fT-T·,)dw
=
- "rF
0 1'r = Pf + <7~·
4. LMS ALGORITHM
~ in 1r(C - T
1r (f - T
C=
L , . .. ,T,.
Minimi~at.i on of the power of signal e(t) (!3.n be done by the means of adjusting of adaptive filter
(7)
300
coefficients w with LMS algorithm. C nfortunately in 8uch c~ there appear diIliculties, mentiolled earlier, in mapping filt er coeliicient8 to gain 90 and time delay T
P' I
>i:
Fig. 2. l'vlinimized mean power Pe as a function of T
Gradient of power of the signal ell) with respect to paramet ers i 9u takes the form:
T"
ae
2
(nT) _
( "')
- , , - - -e n1 (JT~
2
ae a(nT) -_ 9u
( T' en)
awT
~X l (lTif'
T
aw a9u
Xl
(''')
,
(13a)
(T) n ,
(13b)
n1
Constraints imposed on adaptation process le.ad to L;~:L rp.gmcllp.ss of value of pmameters Top and 90 which d escribe filter coefficients. It rp~~ u1t s with :
wi "" 96
wbere:
S" ( ("
aWt _
7[90
aWl -, - =
Si ( 7[(C -
UTop - Y . , I 7[ Ugo
T T
< ))
'f. )),
( l~c )
(16) If we assume that filt er coefficients a re sam ples of a hase hand signalwl - y(lT), t.hen 2.
for f =
Pc (T" , 90) ~
] _ [Top] [Top 9u n + l 9u n
I" \!r.",goc 2(nT),
", aD2 + 902 -
Pr
In the above equatiom; we used derivative Si' (x) CO'T _ ~. In each step of adaptation process the p:ramete rs Top and 90 are modifIed as folloW8:
In our case y(t) - yuSi t o'
(14 )
Pe (T"dJo)
Pr
2
(
)
(17)
20 0 y TO •
(f(1 - T.,)) and it leads
2
T
• ((
~ ° u - 9u - 2a090S1
TO -
T
))
.
(Ill) Surface described by (18) for Ou = I is illus trated ill Fig. 2.
where gradient has been briefly deno ted using nabla symbol. Symbol J1. denotes the parameter that controls the speed of adaptation. In each step filter coefficients are computed using (7). Computations are so simple tha t if we use proper fixf'd point algorithm , ego CORmC (!Tll , 1992), it can he implementf'd llsing FPGA or similar dev ice. Jus t described algorithm has similar computational tim e complexity as algorit hm proposed originally hy (T\o and Siddharth, 1999). Co mputational time complexity of proposed algorithm has been 8ummarized in table 1.
\[ initial value of delay time between microphones is too fa r from a t.rue value TO, then a da ptation process "gds s t.uck" in side one of "c1rainpipp.s" s hown in figm e 2. Tn such cn.~e param eter.'lo (gain) converges toward zero quickly. In tilE' me.ant.ime est.imat f'd df'lay time is np"~ r1y const.ant. ancl IS equal t.o odd mll1t.iple of sampling rat" nearp" t init ial value of t.his parameter. Such phenomenon has been observed a lot of tim es during simulations of tbe svst.em
5. ERROR St:RFA CE 6. COCP LEO /\ 0/\ PTTVE FTLTERS
Unfortunat ely just proposed LMS algorithm has 8eriou8 drawback. Adaptatioll proceS8 CO il verges to the true value of delay time T" only if initial value of delay time TO, assumed at start of adaptation, differs from true value T
r: =n~ - 2no L I'
f
I.
I.
/ -- L
'Wf·Si(,,(l' - .y,))+
L
To overcome just described drawback let. U8 "form" t.he shape of erro r surface re(T",yo) choosing adapt.ive filter coeffi cif'nts from t.he s ignal o t.her t.hen Si(7[t IT ). Recallsf' in our prohlpm delay time can not. be grea ter then Tmax die then ·'rpa.'ionahle" choice seems 1.0 he: Wp
= 90' h(lT - T
(lOa)
2 'Wf·
[-- L
2
(15 )
We mu st rem€>mber to preserve constant. value of the
ro\Jowillg sum
301
L! ~ :L
u:i ~ y5·
Table 1. Computational time complexity of proposed LMS algorithm N UMDER
or
MULTIPLICATIONS
(DIVISIONS) PROPOSED RV
OPERATION
AND SIDDHAIlTH, 1999)
Computation uf e.(nT)
Computation of Jacobi matrix using (13c)-(13d), without. computational complexity of Si(x) Comput.ation of grarlient. using (13a)-(13b) Pseudu-inversiun of Jucuui JJHllrix New values of Top i gO New filter coefficients w by (7)1 without cumputatiunal complexity of Si(x)
(Ko
N UMOER
OF MULTIPLICATIONS (mVISIONS) RV PROPOSED
ALGUH.JTHM
2L
2LI 1
2L -I 1
.1[,+ 4
.1 [,
2L
t.
t
2L -1 1
1
[:~L -
!1-'
~.
Conplfld aclaptive FTR filt.ers nsfld for clelay time ffitimation. where
{~ -
IIITmaxl
if 11·1 < Tmax> if 11.1 > Tmux.
I jj
·w;
fh 2 (nT)
('twIT
a;:t XI (nT),
n:yo,
T
(h' =T·h(n:(C-f))
for
[~L+l -
!1
[;L,
(21)
7. EXPERIMENTAL DATA The presented algorithm has been tested in the laboratory. The hearing sense of the robot has consisted of two capacitance microphone elements placed d-23cm apart. It re.sults with maximal time delay in the syst.em Tmax "" O,fi7fims. Detccted signal has been amplified by a simple, two channel, two stage transistor amplifier build with RC109 t.ransistors. Ra.~e characteristics of the amplifier are: gain 20dl::l, source voltage 5V (from USB socket), nonlinear distortions of signal lower then 0.8% for output amplitude 1 Vpp, root mean square voltage of output. noises lower then 150!1-V. Amplified signal has been converted to digital form by conventional music card of PC computer (line.ar input) . Sampling rate 8kHz has been used and Hi-bit resolution . Collected data h3.~ hP.f!n processed with Mathcacl. Fig. 4 shows time-domain plot of estimated time delay 7', T'f' and gain !lu for "very good syst.em" ut.t.erance. Spf!aker has heen placed ahout lOcm apart of the microphone and he has spoken in the direct.ion of t.he line connecting microphonffi. In such a way maximal delay in the system has been reached. Unfortunately the assumption that
(20a)
where: ow1(
'V,.~,90f2(n1')+
where (3 is coupling coefficient. Goal of filter LMS1 is to put filter LMS2 into convergence region, whereas LMS2 filter precisely estimates delay time and gain. Components of input signal caused by incident acoustic wave is in high degree eliminated from signal e( l) at the ou tput of LMS2 filter. It is important in case when estimated TDOA concerns source of unwanted interferences.
(J9b)
CoefIicient ho is sek>cted in such a way that L~:~ /, = g8· As a result. minimized mean power Pe decre.ases nearly linearly with absolut.e value of difference TO - T'f' for all delays TO reali7~'lhle in t.he syst.em, ie. -"d ~ TO -; ~ Unfortunately, despite of good estimation of delay time, at the output of t.he adaptive filter we ohserve grp,at contrihution from signal f(t). The filter poorly eliminates acoustic signal detected by microphone>. Variance of e>timated time delay is greater then in the original version of t.he filter too. This gave us rise to omit adaptation of gain go. It is assumed that for the filter go = 1. The idea can be extended to use of two adaptively adjusted FIR filters for time delay est.imation as it is shown in figure 3. Filter LM81 , described by saw tooth impulse response, adapts delay time only and delivers T' - coarse estimate of TDOA - to LMS2 filt.er. Adaptation process is described in such case by following equations: ~ = e(nT)
+4
If difference is less t.hen T, t.hen LMS algorithm of LM82 filter works according to (13) and (14). If ahsolute value of clifference hetween ffitimated delay times 7' - T'f' is greater then T, then equation (14) is modified - there appears, apart from gradient, a component which describes coupling het.ween LMS1 and LMS2 filter:
e(nT)
h(t) = ho .
1
4L • 5 4[, + 6
go
Fig.
~
2L 1 1
t =- L, ... ,L.
(20b) During each step of adaptation parameter T' is modified as follows: , oc 2 (nT) T~ . 1 = Tn -!1-1 . ~. (2Oc) Estimated by LMSl delay time T' is compared by LMS2 with it's own estimate of delay time.
302
Hu, Yu Hen (1992). CO RDIC-based VLSI architectmes for digital signal processing. IEEE Signal Processing Magazin e 9, 16· 35. Knapp , Ch.H. and C.C. Cart.er (1976). The generalized correlation method for estimation of time delay. IEEE Tran•. on Amltstir-.• , Spef.r-h and Signal Processing 24, 320-327. Ko, e.c. and C.S. Siddharth (1999). Rejection and tracking of an unknown broadband tiource ill a two-element array through leUtit square approximation of inter-element delay. lJ:)};J:) Signall'rocessing Letters 6 , 122-125. Madurasinghe, D . (2005). A new doa estimator in nonuniform noise. IEEE Signal P"octssing Letlen 12 , ::l::l7-3::l9. Omologo, M. and P. Svaizer (1994) . Acoustic event localization using a crosspowerspectrum phase based technique. In: Proc. IGASSP. Vol. II . pp. 273-.276. Adelaide, Australia. Orfanidis, 5 ..1. (19RR). Opti,mal Signal Proa .•.• ing. McCrnw-TTiIl Puhlishing Company. l\ew York. P esavento, M. and A.B. Cershman (200 1). Maximum-likelihood direction-of-arrival estimatioll in the presellce of ulIkllowll Ilollulliform noise. lEEE Tmns. on Signal Processing 49, 1310-1324. Piersol, A .G. (1981). Time delay estimation using pbase data . JJ:)J:,'E 'l'rans. on Acoustics, Speech and Signal Processing 29, 171- 177. Reed, F .A., P.L Feintuch and 1\.J. Bershad (1081). Time delay estimation using the lms adaptive filter - static behavior. IEEE Tm" •. on Aco'a stits, Spetch a"d Signal P1'Octssing 29 , ,~6 1 571. Sarkar, T.K., M.C. Wicks , M. Salazar-Palma and R .J . Bonneau (2003 ). Smart Antennas. J ohn Wiley & Sons, Tnc ./IEEE Press. Silverma n, I-I.F. and J.M. Sachar (2005). The time-delay graph and the dclayogram - new visuali.:atiolls for time delay. jJ:,'EE Signal P1'O('essi"g Lelie1'S 12, 122-125. Stoica, P . and A. Nehorai (1900). Performance study of conditional and unconditional direction-of-arrival estimation. IEEE Tr·a" •. on Acoustics, :::j'pee.ch and Signal Processing 38 , 178::l- 1795. Strobel, N. and R. Rabenstein (1999). Classificat.ion of time-delay estimates for robust speaker localizat.ion. In: Proc. IGASSP. Vol. VI. pp. :JORl :JOR4. Ziskind , 1. and M. Wax (1988). Maximum likelihood localiwtioll of lIlultiple sourceti byalternating projection. IEEE Trans. on Acoustics, Speech and Signal Processing 36, 1553- 1560.
Fig. 4. Estimatro delay time T', Top and gain go for utterance 'very good system". detected acoustic wave is planar has been satisfied poorly, but it is not crucial for the time delay estimation . The adaptation process has been started wit.h the following estimat.es: T' = Top =Oms and 90 = 1.1. The assumed values of coefficients used for controlling the spero of adaptation have been Jil = 0.15, Ji- O.Ol. Coupling coefficient equals !~=O.5 . Order of adaptive filters equals 2 T, 4 1=R5. The algorithm has quickly found the true value of the delay time which equab approximately 5· T = 0.ii25ms. Meantime E'stimated gain variE'd in 1-1.5 range. Variations of gain are probably caused by tilllaU distallce of the tipeaker from micropholleti and lIolltitatiollary character of tipeecb tiigllal.
8 . CONCLUSIONS In the article the author presented an idea for use of t.wo coupled, adaptive, FIR filters to ffitimate the delay time in case of baseband signal. Thanks to atisumptioll about "stilf" tihape of illlpube response of filt.ers the prohlem has heen reduced t.o adaptation of gain and time delay of the filters. Time computational complexity of proposed algorithm is comparable with complexity of conventional LMS algorithm.
REFERENCES Benedetto, S. and E Biglieri (1999). Pr-inciples of Digital Transmission with Wireless Applications. Klmver Academic. New York. Feint.uch, P.L., N ..T. Bershnd and F.A. Reed (1981). Time delay estimation using the Ims adaptive filter - dynamic hehavior. fEIIE Trans. on Acoustics, Speech and Signal Processing 29, 571- 576.
303