Maximum likelihood estimation of time delays in multipath acoustic channel

doi:10.1016/j.sigpro.2009.11.023

Signal Processing

Volume 90, Issue 5, May 2010, Pages 1750-1754

https://doi.org/10.1016/j.sigpro.2009.11.023 Get rights and content

Abstract

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.

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 non-destructive testing, etc. [1], [2], [3], [4], [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 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], [8], [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 echo-cancellation 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 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.

Section snippets

Multipath model

The simplified multipath model for M extra paths apart from direct path is shown in Fig. 1. The observed signal $d (n)$ is a linear function of filter coefficients and attenuation but non-linear function of delay. The observed signal $d (n)$ (in mixed notation) can be expressed as [11] $d (n) = \sum_{i = 0}^{M} g_{i} \times H_{i} (z) x (n - D_{i}) + ξ (n); n = 0, 1, \dots, N - 1$ where $x (n)$ is the emitted source signal, $ξ (n)$ is the ambient noise with zero mean and variance $σ_{ξ}^{2}$ , $H_{i} (z)$ denotes the low-pass filter of order $L_{i}$ corresponding to the i th path

Estimator analysis and performance measure

We have assumed input $x (n)$ to be Gaussian and channel parameters to be unknown. So to obtain the joint estimate of the channel parameters attenuation coefficients $g_{i}$ and delays $D_{i}$ , the likelihood function for these parameters can be written as $p_{d | G, D} (d | G, D) = \frac{1}{(2 π σ_{ξ}^{2})^{N / 2}} \exp \{- \frac{1}{2 σ_{ξ}^{2}} \sum_{n = 0}^{N - 1} {[d (n) - \sum_{i = 0}^{M} g_{i} \times H_{i} (z) x (n - D_{i})]}^{2}\}$ Taking natural logarithm of both the sides we get $\ln (p_{d | G, D} (d | G, D)) = - \frac{N}{2} \ln (2 π σ_{ξ}^{2}) - \frac{1}{2 σ_{ξ}^{2}} \{\sum_{n = 0}^{N - 1} {[d (n) - \sum_{i = 0}^{M} g_{i} \times H_{i} (z) x (n - D_{i})]}^{2}\}$ The ML estimate of parameters g and D is obtained by maximizing the

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 $g_{0} = 1.0$ , $g_{1} = 0.9$ , $g_{2} = 0.7$ and delays $D_{0} = 0$ , $D_{1} = 115$ , $D_{2} = 200$ respectively. These delays have been chosen

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 algorithm in important case of overlapping multipath is examined.

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Fast communicationMaximum likelihood estimation of time delays in multipath acoustic channel

Abstract

Introduction

Section snippets

Multipath model

Estimator analysis and performance measure

Simulation results

Conclusions

Signal Processing

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IEEE Trans. Acoust. Speech Signal Process.

High resolution multipath time delay estimation for broad-band random signals

IEEE Trans. Acoust. Speech Signal Process.

A new time-delay estimation in multipath

IEEE Trans. Signal Process.

Adaptive multipath delay estimation

IEEE Trans. Acoust. Speech Signal Process.

Fast communication
Maximum likelihood estimation of time delays in multipath acoustic channel