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Maximum likelihood estimation of time delays in multipath acoustic channel
ARTICLE IN PRESS Signal Processing 90 (2010) 1750–1754
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Maximum likelihood estimation of time delays in multipath acoustic channel Tarkeshwar Prasad Bhardwaj , Ravinder Nath Department of Electrical Engineering, National Institute of Technology, Hamirpur-177005, Himachal Pradesh, India
a r t i c l e i n f o
abstract
Article history: Received 20 March 2009 Received in revised form 31 October 2009 Accepted 12 November 2009 Available online 27 November 2009
This paper presents a solution to the problem of time delay estimation in multipath acoustic channel. Here the multipath acoustic channel output signal is modelled as a superposition of the delayed, attenuated, and filtered version of the stationary Gaussian stochastic input signal. A maximum likelihood (ML) estimator is developed for determining time delays in multipath acoustic channel in the presence of uncorrelated noise. Accuracy percentage (AP) performance measure has been introduced to characterize the performance of the estimators. The performance of the ML estimator is compared via computer simulation, using AP, with a generalized autocorrelation estimator (GAE) and AP CRLB which is obtained by expressing the CRLB in terms of probability. Simulation results show that the performance of the ML estimator is superior to the GAE and approaches to AP CRLB . The robustness of the algorithm has also been studied via computer simulation. & 2009 Elsevier B.V. All rights reserved.
Keywords: Maximum-likelihood estimation Multipath Time delay estimator Acoustic channel
1. Introduction During the last three decades, considerable attention has been given to the problem of time delay estimation in which the received signal contains multipath. Multipath is observed when the emitted source signal is received at the receiver through more than one path. Multipath phenomenon is common in radar, sonar and wireless communication systems and also finds application in seismic exploration, computerized tomography, and nondestructive testing, etc. [1–5]. In wireless communication systems these extra paths are useful but at the cost of additional complexity of the receiver [6]. However, in most of the applications these extra paths are not desirable e.g. in acoustic echo, which commonly appears in hands-free telephony and teleconferencing, etc. Acoustic echo results due to the coupling of the received voice
Corresponding author. Tel.: + 91 1972 254532; fax: + 91 1972 223834. E-mail addresses: [email protected] (T.P. Bhardwaj), [email protected] (R. Nath).
0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.11.023
and the mouthpiece of a mobile handset or the coupling of the speaker and microphone in hands-free applications. When this coupling of speaker and microphone takes place in an enclosure say room, such arrangement is called loudspeaker-enclosure-microphone (LEM) system. In LEM system model, loudspeaker emitted signal reaches the microphone not only directly but also via reflections from neighbouring objects [7–9]. Therefore, the signal received at the microphone is a superposition of the delayed, attenuated, and filtered versions of the emitted signal and the same can be modelled as multipath [10,11]. Thus the received signal contains a direct path plus extra paths resulting acoustic echo. Acoustic echo is typically more complex than the hybrid or network echo, and its impulse response is much longer. The acoustic echocancellation problem has been studied by many authors [12] for more than 30 years. Acoustic echo-cancellation can be achieved by using a single long length adaptive filter but due to its long length it has slow convergence and poor tracking behaviour. In [13,14] it has been suggested that convergence and tracking behaviour can be improved by using multiple-sub-filters in the place of a single long length filter provided that acoustic echo
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1751
coefficients gi and delays Di , the likelihood function for these parameters can be written as
channel be modelled as a multipath channel. To realize multiple sub-filter based echo canceller knowledge of time delay associated with each path is required. In this paper we derive a maximum likelihood (ML) estimator for estimation of time delay associated with each path in a multipath acoustic channel and compare its performance with a generalized autocorrelation estimator (GAE) [5] using accuracy percentage (AP) as a performance measure. Further, the performance of the algorithm is tested for different signal-noise scenarios and observation lengths. The robustness of this algorithm is evaluated by considering different time delay differences. This paper is divided into five sections. In Section 2 multipath model is given and in Section 3 estimator analysis and performance measure are presented. In Section 4 simulation results are discussed and conclusions have been presented in Section 5.
The ML estimate of parameters g and D is obtained by maximizing the log-likelihood function. N max½lnðpdjG;D ðdjG; DÞÞ ¼ max lnð2ps2x Þ G;D G;D 2 8 2 #2 9 # < = N 1 M X X 1 4 2 dðnÞ gi Hi ðzÞxðnDi Þ ; 2sx :n ¼ 0
2. Multipath model
This is equivalent to
The simplified multipath model for M extra paths apart from direct path is shown in Fig. 1. The observed signal dðnÞ is a linear function of filter coefficients and attenuation but non-linear function of delay. The observed signal dðnÞ (in mixed notation) can be expressed as [11]
Further simplifying we get
dðnÞ ¼
M X
gi Hi ðzÞxðnDi Þ þ xðnÞ;
n ¼ 0; 1; . . . ; N1
ð1Þ
pdjG;D ðdjG; DÞ ¼
8 2 #2 9 1 M < = X 1 NX 4dðnÞ exp g H ðzÞxðnD Þ i i i : 2s2x n ¼ 0 ; ð2ps2x ÞN=2 i¼0
Taking natural logarithm of both the sides we get lnðpdjG;D ðdjG; DÞÞ ¼
Hi ðzÞ ¼
LX i 1
hi ðnÞzn
n¼0
where gi and hi ðnÞ are the attenuation coefficient and impulse response respectively of the i th path. Further, by Hi ðzÞxðnDi Þ we mean hi ðnÞ xðnDi Þ where ‘ ’ denotes convolution. 3. Estimator analysis and performance measure We have assumed input xðnÞ to be Gaussian and channel parameters to be unknown. So to obtain the joint estimate of the channel parameters attenuation
8 2 #2 9 M = X X N 1 < N1 4dðnÞ gi Hi ðzÞxðnDi Þ lnð2ps2x Þ 2 ; 2 2sx :n ¼ 0 i¼0
i¼0
8 2 2 #2 9# 1 M = X N 1 < NX 2 4 4 lnð2psx Þ þ min dðnÞ gi Hi ðzÞxðnDi Þ ; G;D 2 2s2x :n ¼ 0 i¼0 8 2 !2 M X X< N 1 N1 2 min4 lnð2ps2x Þ þ dðnÞ þ g H ðzÞxðnD Þ i i i G;D 2 2s2x n ¼ 0 : i¼0
2 dðnÞ
M X
!)#
gi Hi ðzÞxðnDi Þ
i¼0
i¼0
where xðnÞ is the emitted source signal, xðnÞ is the ambient noise with zero mean and variance s2x , Hi ðzÞ denotes the low-pass filter of order Li corresponding to the i th path and
1
Now removing the terms independent of parameters g and D, the above can be further written as 8 2 !2 N 1 < X M X gi Hi ðzÞxðnDi Þ min4 : G;D n¼0 i¼0 !)# M X 2 dðnÞ gi Hi ðzÞxðnDi Þ ð2Þ i¼0
For single path case i.e. M ¼ 1, the minima of the above expression can be easily obtained. For M 41, a closed form solution is difficult to obtain. Here, we assume that paths are non-overlapping for the sake of obtaining a closed form solution. Mathematically this implies that 9 8 = N 1 < X M X M X Hi ðzÞxðnDi ÞHj ðzÞxðnDj Þ ¼ 0 for iaj ; : n¼0 i¼0j¼0
Estimator designed using non-overlapping condition, when used in real life applications, may have deterioration in its performance. By applying the non-overlapping assumption, (2) gets decoupled for delay and attenuation parameters of the multipath channel and can be written as " ( N 1 M X X min ðgi Hi ðzÞxðnDi ÞÞ2 2 dðnÞ G;D
n¼0 M X
i¼0
!)#
gi Hi ðzÞxðnDi Þ
i¼0
Fig. 1. Multipath model.
The energy of the emitted signal is given by P 1 2 Ex ¼ N n ¼ 0 ½Hi ðzÞxðnDi Þ , where N1 is the duration of the i th path. Therefore, we can simplify the above
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autocorrelation function we can obtain Sdd ðwÞ as ! M X 2 jw jw gi Hi ðe ÞHi ðe Þ Sdd ðwÞ ¼ Sxx ðwÞ þ Sxx ðwÞ
expression as " ( )# M N1 M X X X 2 gi 2 dðnÞ gi Hi ðzÞxðnDi Þ min Ex G;D
n¼0
i¼1
i¼0
or "
M X
N 1 X
2 g fdðnÞ Hi ðzÞxðnDi Þg Ex n ¼ 0 i ¼ 0 i " # M M X 2 X 2 ^ gi g R dx ðDi Þ ¼ min G;D Ex i ¼ 0 i i¼0
min G;D
gi2
þ 2Sxx ðwÞ
#
M X
gi gj Hi ðejw ÞHj ðejw Þ
cos½wðDj Di Þ
i¼0
^ iÞ ¼ R^ dx ðD
ð3Þ
N1 X
fdðnÞ Hi ðzÞxðnDi Þg
n¼0
From (3), it follows that ^ iÞ R^ dx ðD Ex
and ^ i ¼ maxR^ dx ðDi Þ D Di
for i ¼ 0; 1; 2; . . . ; M
ð4Þ
Number of correct estimation 100 Total number of trials
In general, an important measure of the performance of an estimator is its efficiency and CRLB is considered as a standard. We have expressed CRLB in terms of probability as AP CRLB for the comparisons of the performance of the estimator in AP. For any unbiased estimator, the CRLB can be obtained from the inverse of the Fisher-information matrix (J) which is known to satisfy the inequality ^ ZJ 1 [15]. The elements of the Fisher-information CovðDÞ matrix can be obtained by the formula [5] @Sdd ðwk Þ @Sdd ðwk Þ @Di @Dj Ji;j ¼ ; 2 ðw Þ S k dd k¼0 N1 X
i; j ¼ 0; 1; 2; . . . ; M
ð5Þ
where M X @Sdd ðwÞ ¼ gi gm Hi ðejw ÞHm ðejw Þ sin½wðDm Di Þ @Di m¼1
where Sdd ðwÞ is the PSD of the received signal and can be obtained by taking the Fourier transform of its autocorrelation function: ! M X 2 1 gi Hi ðzÞHi ðz Þ Rdd ðlÞ ¼ Rxx ðlÞ þ Rxx ðlÞ i¼0
þ
where Sxx ðwÞ and Sxx ðwÞ are the PSDs of the noise and the emitted signal respectively. CRLB can be expressed in terms of AP as the probability of occurrence of the estimated value within a certain tolerance band about its actual value. Let us assume that the distribution of the ^ i is Gaussian with NðDi ; s2 Þ. The estimated delay D i parameters Di and s2i denote the mean and variance of the estimate. For a given SNR the variance s2i can be calculated from the Fisher information matrix. We define the AP CRLB in terms of the probability that the estimated ^ i lies within 1 sample of the actual delay. delay D ^ i Di j r1Þ 100 AP CRLB ¼ PrðjD ! ^ i Di j jD 1 100 r ¼ Pr
si
In (4) the extrema of cross-correlation of the received signal with input is occurring at time delays associated with multipath and number of these extrema is M þ1. Accuracy percentage as performance measure: The accuracy percentage (AP) is defined as [16] AP ¼
i¼0 M X
i ¼ 1 j ¼ iþ1
where
g^ i ¼
M 1 X
M M X X
gi gj Hi ðzÞHj ðz1 Þ Rxx ðlðDi Dj ÞÞ
i ¼ 0iaj j ¼ 0
where Rxx ðlÞ and Rxx ðlÞ are autocorrelation functions of transmitted signal xðnÞ and ambient noise xðnÞ respectively. By taking the Fourier transform of the above
si
1 1 ¼ F F 100 si s i 1 1 100 ¼ 2F
si
ð6Þ
where FðxÞ is the standard normal distribution and s2i is the i th diagonal element of the matrix J 1 . 4. Simulation results Simulations have been carried out on the following data. A low pass FIR filter of order 12 with frequency response close to the impulse response spectrum has been used to construct each path of the multipath acoustic echo model. The received signal is obtained by applying white Gaussian process with zero mean and unity variance on these filters. We consider three multipaths with attenuation factors g0 ¼ 1:0, g1 ¼ 0:9, g2 ¼ 0:7 and delays D0 ¼ 0, D1 ¼ 115, D2 ¼ 200 respectively. These delays have been chosen for a typical room assuming that the path lengths travelled by the second and third path signals are 5.0 and 8.5 m respectively. Assuming the speed of sound to be 343 m/s and sampling rate to be 8 kHz, the number of samples will be given by 8000 5=343 C 115, hence D1 ¼ 115. The value of D2 can be explained similarly. The simulation results have been obtained after averaging over 400 independent runs and the observation length of each run was taken to be N ¼ 4096. In Fig. 2, AP of the ML estimator has been plotted along with AP CRLB and GAE. Here the average of all the delay estimates have been plotted. The plot shows that the estimator is reasonably good even for low SNRs as compared to that of the GAE. Also we can say is that in the relatively low SNR (say 0 dB) region the performance of ML estimator approaches to APCRLB . In Fig. 3, AP of ML estimator and GAE has been plotted for different observation lengths and at SNR ¼ 5 dB. It is clear from
ARTICLE IN PRESS T.P. Bhardwaj, R. Nath / Signal Processing 90 (2010) 1750–1754
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Fig. 2. Comparison between APCRLB , and AP of ML estimator and GAE.
Fig. 4. AP of ML estimator and GAE for different delay differences.
Fig. 3. AP of ML estimator and GAE for different observation lengths.
Fig. 5. AP of ML estimator for non-overlapping and overlapping multipath.
the figure that ML estimator performs comparatively better even at small observation length. Also it can be seen from the plots that longer the observation length, higher is the accuracy in estimating the delays. It can also be observed from these plots that, as expected, there is a trade-off between observation length and SNR for a given AP value. At low SNR, longer observation length is needed to achieve the same accuracy in delay estimation and vice versa. Fig. 4 shows the degradation, suffered by algorithm, if non-overlapping assumption is violated. Here difference between delays is plotted with AP of ML estimator and GAE. We have varied D2 while D1 is kept fixed. Since we have chosen the filter length 12, so non-overlapping assumption is violated till delay difference ðD2 D1 Þ r 12. The relative degradation in the performance of GAE, in overlapping region, is not only due to noise but also due to the bias associated with the estimator. However when this difference exceeds 12, multipath becomes nonoverlapping. From the figure, it is seen that when the delay difference is 4, the AP for GAE is 24, whereas the
corresponding value for ML estimator is 84. This implies that the ML estimator is reasonably good even when assumption is violated. In Fig. 5, the performance of the ML estimator is examined for non-overlapping and overlapping situations with observation lengths at SNR ¼ 0 dB. From the figure, it is seen that the ML estimator is robust in the important case of overlapping multipath.
5. Conclusions In this paper, ML estimator has been derived for an acoustic echo channel model. Apart from delay and attenuation, the model provides filtering in each path. AP performance measure gives a measure for comparisons between the ML estimator, AP CRLB , and GAE via simulations. The performance of the ML estimator is found approaching to AP CRLB and is better than GAE even in relatively low SNR ranges. Further, the robustness of this
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algorithm in important case of overlapping multipath is examined. References [1] C.H. Knapp, G.C. Carter, The generalized correlation method for estimation of time delay, IEEE Trans. Acoust. Speech Signal Process. ASSP-24 (4) (August 1976) 320–327. [2] G.C. Carter (Ed.), IEEE Trans. Acoust. Speech Signal Processing Special issue on time delay estimation ASSP-29 (3) (June 1981). [3] J.P. Ianniello, Large and small error performance limits for multipath time delay estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-34 (2) (April 1986) 245–251. [4] J.P. Ianniello, High resolution multipath time delay estimation for broad-band random signals, IEEE Trans. Acoust. Speech Signal Process. 36 (3) (March 1988) 320–327. [5] P.P. Moghaddam, H. Amindavar, R.L. Kirlin, A new time-delay estimation in multipath, IEEE Trans. Signal Process. 51 (5) (May 2003) 1129–1142. [6] J.O. Smith, B. Friedlander, Adaptive multipath delay estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-33 (4) (August 1985) 812–822. [7] L. Benesty, Y. Huang, Adaptive Signal Processing Applications to Real World Problems, Springer, Berlin, 2003. ¨ [8] E. Hansler, The hands-free telephone problem—An annotated bibliography, Signal Processing 27 (June 1992) 259–271.
¨ [9] C. Breining, P. Dreiseital, E. Hansler, A. Mader, B. Nitsch, H. Puder, T. Schertler, G. Schmidt, J. Tilp, J.S. Lee, Acoustic echo control—an application of very high order adaptive filters, IEEE Signal Process. Mag. 16 (4) (July 1999) 42–69. [10] R.N. Sharma, A.K. Chaturvedi, G. Sharma, Acoustic echo cancellation using multiple sub-filters, in: IEEE TENCON2003 Conference on Proceedings, India, October 2003, pp. 393–396. [11] R.N. Sharma, A.K. Chaturvedi, G. Sharma, Multipath delay estimation for acoustic echo channel, in: IEEE TENCON2004 Conference on Proceedings, Thailand, November 2004, pp. 21–24. [12] W. Kellermann, Current topics in adaptive filtering for hands-free acoustic communication and beyond, Signal Processing 80 (9) (September 2000) 1695–1696. [13] R.N. Sharma, A.K. Chaturvedi, G. Sharma, Tracking behaviour of acoustic echo canceller using multiple sub-filters, in: Proceedings of the European Signal Processing Conference EUSIPCO-2006, Italy, September 2006. [14] R. Nath, Adaptive Echo Cancellation Based on a multipath model of acoustic channel, Ph.D. Thesis, Department of Electrical Engineering Indian Institute of Technology Kanpur, Kanpur, UP, India, October 2005. [15] B. Friedlander, On the Cramer–Rao bound for time delay and doppler estimation, IEEE Trans. Inform. Theory IT-30 (3) (May 1984) 575–580. [16] P.P. Moghaddam, H. Amindawar, Estimation of multipath parameters for nonresolvable time delays, in: IEEE NORDIC Signal Processing Symposium, Kolmoarden, Sweden, NORSIG 2000, June 2000.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Maximum likelihood estimation of time delays in multipath acoustic channel Tarkeshwar Prasad Bhardwaj , Ravinder Nath Department of Electrical Engineering, National Institute of Technology, Hamirpur-177005, Himachal Pradesh, India
a r t i c l e i n f o
abstract
Article history: Received 20 March 2009 Received in revised form 31 October 2009 Accepted 12 November 2009 Available online 27 November 2009
This paper presents a solution to the problem of time delay estimation in multipath acoustic channel. Here the multipath acoustic channel output signal is modelled as a superposition of the delayed, attenuated, and filtered version of the stationary Gaussian stochastic input signal. A maximum likelihood (ML) estimator is developed for determining time delays in multipath acoustic channel in the presence of uncorrelated noise. Accuracy percentage (AP) performance measure has been introduced to characterize the performance of the estimators. The performance of the ML estimator is compared via computer simulation, using AP, with a generalized autocorrelation estimator (GAE) and AP CRLB which is obtained by expressing the CRLB in terms of probability. Simulation results show that the performance of the ML estimator is superior to the GAE and approaches to AP CRLB . The robustness of the algorithm has also been studied via computer simulation. & 2009 Elsevier B.V. All rights reserved.
Keywords: Maximum-likelihood estimation Multipath Time delay estimator Acoustic channel
1. Introduction During the last three decades, considerable attention has been given to the problem of time delay estimation in which the received signal contains multipath. Multipath is observed when the emitted source signal is received at the receiver through more than one path. Multipath phenomenon is common in radar, sonar and wireless communication systems and also finds application in seismic exploration, computerized tomography, and nondestructive testing, etc. [1–5]. In wireless communication systems these extra paths are useful but at the cost of additional complexity of the receiver [6]. However, in most of the applications these extra paths are not desirable e.g. in acoustic echo, which commonly appears in hands-free telephony and teleconferencing, etc. Acoustic echo results due to the coupling of the received voice
Corresponding author. Tel.: + 91 1972 254532; fax: + 91 1972 223834. E-mail addresses: [email protected] (T.P. Bhardwaj), [email protected] (R. Nath).
0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.11.023
and the mouthpiece of a mobile handset or the coupling of the speaker and microphone in hands-free applications. When this coupling of speaker and microphone takes place in an enclosure say room, such arrangement is called loudspeaker-enclosure-microphone (LEM) system. In LEM system model, loudspeaker emitted signal reaches the microphone not only directly but also via reflections from neighbouring objects [7–9]. Therefore, the signal received at the microphone is a superposition of the delayed, attenuated, and filtered versions of the emitted signal and the same can be modelled as multipath [10,11]. Thus the received signal contains a direct path plus extra paths resulting acoustic echo. Acoustic echo is typically more complex than the hybrid or network echo, and its impulse response is much longer. The acoustic echocancellation problem has been studied by many authors [12] for more than 30 years. Acoustic echo-cancellation can be achieved by using a single long length adaptive filter but due to its long length it has slow convergence and poor tracking behaviour. In [13,14] it has been suggested that convergence and tracking behaviour can be improved by using multiple-sub-filters in the place of a single long length filter provided that acoustic echo
ARTICLE IN PRESS T.P. Bhardwaj, R. Nath / Signal Processing 90 (2010) 1750–1754
1751
coefficients gi and delays Di , the likelihood function for these parameters can be written as
channel be modelled as a multipath channel. To realize multiple sub-filter based echo canceller knowledge of time delay associated with each path is required. In this paper we derive a maximum likelihood (ML) estimator for estimation of time delay associated with each path in a multipath acoustic channel and compare its performance with a generalized autocorrelation estimator (GAE) [5] using accuracy percentage (AP) as a performance measure. Further, the performance of the algorithm is tested for different signal-noise scenarios and observation lengths. The robustness of this algorithm is evaluated by considering different time delay differences. This paper is divided into five sections. In Section 2 multipath model is given and in Section 3 estimator analysis and performance measure are presented. In Section 4 simulation results are discussed and conclusions have been presented in Section 5.
The ML estimate of parameters g and D is obtained by maximizing the log-likelihood function. N max½lnðpdjG;D ðdjG; DÞÞ ¼ max lnð2ps2x Þ G;D G;D 2 8 2 #2 9 # < = N 1 M X X 1 4 2 dðnÞ gi Hi ðzÞxðnDi Þ ; 2sx :n ¼ 0
2. Multipath model
This is equivalent to
The simplified multipath model for M extra paths apart from direct path is shown in Fig. 1. The observed signal dðnÞ is a linear function of filter coefficients and attenuation but non-linear function of delay. The observed signal dðnÞ (in mixed notation) can be expressed as [11]
Further simplifying we get
dðnÞ ¼
M X
gi Hi ðzÞxðnDi Þ þ xðnÞ;
n ¼ 0; 1; . . . ; N1
ð1Þ
pdjG;D ðdjG; DÞ ¼
8 2 #2 9 1 M < = X 1 NX 4dðnÞ exp g H ðzÞxðnD Þ i i i : 2s2x n ¼ 0 ; ð2ps2x ÞN=2 i¼0
Taking natural logarithm of both the sides we get lnðpdjG;D ðdjG; DÞÞ ¼
Hi ðzÞ ¼
LX i 1
hi ðnÞzn
n¼0
where gi and hi ðnÞ are the attenuation coefficient and impulse response respectively of the i th path. Further, by Hi ðzÞxðnDi Þ we mean hi ðnÞ xðnDi Þ where ‘ ’ denotes convolution. 3. Estimator analysis and performance measure We have assumed input xðnÞ to be Gaussian and channel parameters to be unknown. So to obtain the joint estimate of the channel parameters attenuation
8 2 #2 9 M = X X N 1 < N1 4dðnÞ gi Hi ðzÞxðnDi Þ lnð2ps2x Þ 2 ; 2 2sx :n ¼ 0 i¼0
i¼0
8 2 2 #2 9# 1 M = X N 1 < NX 2 4 4 lnð2psx Þ þ min dðnÞ gi Hi ðzÞxðnDi Þ ; G;D 2 2s2x :n ¼ 0 i¼0 8 2 !2 M X X< N 1 N1 2 min4 lnð2ps2x Þ þ dðnÞ þ g H ðzÞxðnD Þ i i i G;D 2 2s2x n ¼ 0 : i¼0
2 dðnÞ
M X
!)#
gi Hi ðzÞxðnDi Þ
i¼0
i¼0
where xðnÞ is the emitted source signal, xðnÞ is the ambient noise with zero mean and variance s2x , Hi ðzÞ denotes the low-pass filter of order Li corresponding to the i th path and
1
Now removing the terms independent of parameters g and D, the above can be further written as 8 2 !2 N 1 < X M X gi Hi ðzÞxðnDi Þ min4 : G;D n¼0 i¼0 !)# M X 2 dðnÞ gi Hi ðzÞxðnDi Þ ð2Þ i¼0
For single path case i.e. M ¼ 1, the minima of the above expression can be easily obtained. For M 41, a closed form solution is difficult to obtain. Here, we assume that paths are non-overlapping for the sake of obtaining a closed form solution. Mathematically this implies that 9 8 = N 1 < X M X M X Hi ðzÞxðnDi ÞHj ðzÞxðnDj Þ ¼ 0 for iaj ; : n¼0 i¼0j¼0
Estimator designed using non-overlapping condition, when used in real life applications, may have deterioration in its performance. By applying the non-overlapping assumption, (2) gets decoupled for delay and attenuation parameters of the multipath channel and can be written as " ( N 1 M X X min ðgi Hi ðzÞxðnDi ÞÞ2 2 dðnÞ G;D
n¼0 M X
i¼0
!)#
gi Hi ðzÞxðnDi Þ
i¼0
Fig. 1. Multipath model.
The energy of the emitted signal is given by P 1 2 Ex ¼ N n ¼ 0 ½Hi ðzÞxðnDi Þ , where N1 is the duration of the i th path. Therefore, we can simplify the above
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autocorrelation function we can obtain Sdd ðwÞ as ! M X 2 jw jw gi Hi ðe ÞHi ðe Þ Sdd ðwÞ ¼ Sxx ðwÞ þ Sxx ðwÞ
expression as " ( )# M N1 M X X X 2 gi 2 dðnÞ gi Hi ðzÞxðnDi Þ min Ex G;D
n¼0
i¼1
i¼0
or "
M X
N 1 X
2 g fdðnÞ Hi ðzÞxðnDi Þg Ex n ¼ 0 i ¼ 0 i " # M M X 2 X 2 ^ gi g R dx ðDi Þ ¼ min G;D Ex i ¼ 0 i i¼0
min G;D
gi2
þ 2Sxx ðwÞ
#
M X
gi gj Hi ðejw ÞHj ðejw Þ
cos½wðDj Di Þ
i¼0
^ iÞ ¼ R^ dx ðD
ð3Þ
N1 X
fdðnÞ Hi ðzÞxðnDi Þg
n¼0
From (3), it follows that ^ iÞ R^ dx ðD Ex
and ^ i ¼ maxR^ dx ðDi Þ D Di
for i ¼ 0; 1; 2; . . . ; M
ð4Þ
Number of correct estimation 100 Total number of trials
In general, an important measure of the performance of an estimator is its efficiency and CRLB is considered as a standard. We have expressed CRLB in terms of probability as AP CRLB for the comparisons of the performance of the estimator in AP. For any unbiased estimator, the CRLB can be obtained from the inverse of the Fisher-information matrix (J) which is known to satisfy the inequality ^ ZJ 1 [15]. The elements of the Fisher-information CovðDÞ matrix can be obtained by the formula [5] @Sdd ðwk Þ @Sdd ðwk Þ @Di @Dj Ji;j ¼ ; 2 ðw Þ S k dd k¼0 N1 X
i; j ¼ 0; 1; 2; . . . ; M
ð5Þ
where M X @Sdd ðwÞ ¼ gi gm Hi ðejw ÞHm ðejw Þ sin½wðDm Di Þ @Di m¼1
where Sdd ðwÞ is the PSD of the received signal and can be obtained by taking the Fourier transform of its autocorrelation function: ! M X 2 1 gi Hi ðzÞHi ðz Þ Rdd ðlÞ ¼ Rxx ðlÞ þ Rxx ðlÞ i¼0
þ
where Sxx ðwÞ and Sxx ðwÞ are the PSDs of the noise and the emitted signal respectively. CRLB can be expressed in terms of AP as the probability of occurrence of the estimated value within a certain tolerance band about its actual value. Let us assume that the distribution of the ^ i is Gaussian with NðDi ; s2 Þ. The estimated delay D i parameters Di and s2i denote the mean and variance of the estimate. For a given SNR the variance s2i can be calculated from the Fisher information matrix. We define the AP CRLB in terms of the probability that the estimated ^ i lies within 1 sample of the actual delay. delay D ^ i Di j r1Þ 100 AP CRLB ¼ PrðjD ! ^ i Di j jD 1 100 r ¼ Pr
si
In (4) the extrema of cross-correlation of the received signal with input is occurring at time delays associated with multipath and number of these extrema is M þ1. Accuracy percentage as performance measure: The accuracy percentage (AP) is defined as [16] AP ¼
i¼0 M X
i ¼ 1 j ¼ iþ1
where
g^ i ¼
M 1 X
M M X X
gi gj Hi ðzÞHj ðz1 Þ Rxx ðlðDi Dj ÞÞ
i ¼ 0iaj j ¼ 0
where Rxx ðlÞ and Rxx ðlÞ are autocorrelation functions of transmitted signal xðnÞ and ambient noise xðnÞ respectively. By taking the Fourier transform of the above
si
1 1 ¼ F F 100 si s i 1 1 100 ¼ 2F
si
ð6Þ
where FðxÞ is the standard normal distribution and s2i is the i th diagonal element of the matrix J 1 . 4. Simulation results Simulations have been carried out on the following data. A low pass FIR filter of order 12 with frequency response close to the impulse response spectrum has been used to construct each path of the multipath acoustic echo model. The received signal is obtained by applying white Gaussian process with zero mean and unity variance on these filters. We consider three multipaths with attenuation factors g0 ¼ 1:0, g1 ¼ 0:9, g2 ¼ 0:7 and delays D0 ¼ 0, D1 ¼ 115, D2 ¼ 200 respectively. These delays have been chosen for a typical room assuming that the path lengths travelled by the second and third path signals are 5.0 and 8.5 m respectively. Assuming the speed of sound to be 343 m/s and sampling rate to be 8 kHz, the number of samples will be given by 8000 5=343 C 115, hence D1 ¼ 115. The value of D2 can be explained similarly. The simulation results have been obtained after averaging over 400 independent runs and the observation length of each run was taken to be N ¼ 4096. In Fig. 2, AP of the ML estimator has been plotted along with AP CRLB and GAE. Here the average of all the delay estimates have been plotted. The plot shows that the estimator is reasonably good even for low SNRs as compared to that of the GAE. Also we can say is that in the relatively low SNR (say 0 dB) region the performance of ML estimator approaches to APCRLB . In Fig. 3, AP of ML estimator and GAE has been plotted for different observation lengths and at SNR ¼ 5 dB. It is clear from
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Fig. 2. Comparison between APCRLB , and AP of ML estimator and GAE.
Fig. 4. AP of ML estimator and GAE for different delay differences.
Fig. 3. AP of ML estimator and GAE for different observation lengths.
Fig. 5. AP of ML estimator for non-overlapping and overlapping multipath.
the figure that ML estimator performs comparatively better even at small observation length. Also it can be seen from the plots that longer the observation length, higher is the accuracy in estimating the delays. It can also be observed from these plots that, as expected, there is a trade-off between observation length and SNR for a given AP value. At low SNR, longer observation length is needed to achieve the same accuracy in delay estimation and vice versa. Fig. 4 shows the degradation, suffered by algorithm, if non-overlapping assumption is violated. Here difference between delays is plotted with AP of ML estimator and GAE. We have varied D2 while D1 is kept fixed. Since we have chosen the filter length 12, so non-overlapping assumption is violated till delay difference ðD2 D1 Þ r 12. The relative degradation in the performance of GAE, in overlapping region, is not only due to noise but also due to the bias associated with the estimator. However when this difference exceeds 12, multipath becomes nonoverlapping. From the figure, it is seen that when the delay difference is 4, the AP for GAE is 24, whereas the
corresponding value for ML estimator is 84. This implies that the ML estimator is reasonably good even when assumption is violated. In Fig. 5, the performance of the ML estimator is examined for non-overlapping and overlapping situations with observation lengths at SNR ¼ 0 dB. From the figure, it is seen that the ML estimator is robust in the important case of overlapping multipath.
5. Conclusions In this paper, ML estimator has been derived for an acoustic echo channel model. Apart from delay and attenuation, the model provides filtering in each path. AP performance measure gives a measure for comparisons between the ML estimator, AP CRLB , and GAE via simulations. The performance of the ML estimator is found approaching to AP CRLB and is better than GAE even in relatively low SNR ranges. Further, the robustness of this
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