Piya Kovintavewat's Homepage

Master Thesis (Sept'97 - Nov'98)

Modeling the impulse response of an office room

In hands-free operation of the cellular, an undesired acoustical feedback of the loud- speaker occurs. To avoid transmitting the loudspeaker signal via the microphone, an acoustic echo cancellation algorithm must be employed. However, since the loudspeaker's signal propagates in an enclosed environment, e.g. a room, the room acoustics then acts as a filter. In order to eliminate the acoustic echo, an estimate of the room impulse response is needed.

The impulse response of an office room are often modeled as a Finite Impulse Response (FIR). A drawback with a FIR model is that the number of parameters to be estimated is considerably large, probably in the order of 4000. This will absolutely affect the complexity of any acoustic echo cancellation algorithms. To reduce the number of estimated parameters, alternative models are needed to investigate.

Therefore, the aim of the thesis project is to try alternative representations for room impulse response which require fewer parameters than a FIR model.

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Abstract

Repetition of signals due to reflection from ceiling, wall, floor and other objects in an enclosed environment can be perceived as the acoustic echo. It is resulted from the hands-free operation on the cellular, teleconferencing application and hearing aid system. Since the echo propagates in an enclosed environment, e.g. a room, the room acoustics acts as a filter. To remove the acoustic echo successfully, an estimate of the room impulse response is needed.

Given experimental data, a system identification technique is employed as a tool to build the estimated model of the room impulse response. Herein, its model parameters can be estimated by means of an off-line or batch method in the least squares sense. Several traditional linear model structures have been presented. However, they often lead to an approximation of very high order. In order to reduce the number of estimated parameters, alternative methods for modeling the room impulse response are needed to investigate.

Approximation of the room impulse response by means of the so-called Laguerre and Kautz functions, which are the z-transform of a class of orthonormal exponentials, is then studied and examined. One of the most important parameters of these two functions is their dominating pole location. It can be shown that the number of estimated parameters is substantially reduced and the numerical accuracy is improved when the dominating pole is chosen properly. For a fixed model order, there exists the optimum choice of a dominating pole which gives the best approximation. Many methods to find a dominating pole are also given. Furthermore, since the typical echo response can be decomposed into two parts, i.e. the first part has rapid time variation and the second part is slowly decaying towards zero, the use of a two-stage echo canceller is then noteworthy to introduce and analyze.

As a preliminary step to investigate the possibility of reducing the number of estimated parameters by the proposed models, we shall therefore not consider the on-line (recursive or adaptive) method to find the values of the model parameters. Instead, the data used in this thesis project has been collected from the system (echo path) that is made as near stationary as possible in order to be able to use the off-line method to estimate the model parameters.

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Conclusions

As known in the literature, the effect of acoustic echo is quite complicated as discussed in section 1. FIR model is widely used to model the room impulse response due to simplicity and stability. However, it leads to the approximation of very high order, probably in the order of 4000. By exploiting a priori information about the dominating pole of the system, an approximation of the room impulse response by means of the Laguerre and Kautz functions is proposed. In general, such a priori information about the dominating pole of the system may sometimes not be available. To deal with this problem, we have presented three methods to find such an optimal dominating pole in section 7.

The first method given in section 7.1 is restricted only in the case of a real-valued dominating pole, and it requires a given set of the system impulse response to calculate an optimal Laguerre pole by minimizing the performance index J as given in Eq.(7.11). The advantage of using such a performance index is that it forces rapid convergence of the Laguerre spectrum and gives an analytical solution. The resulting optimal dominating pole will primarily depend on the characteristics of the system impulse response, such as its rate of decay, its smoothness and the time delay.

The second method is much more efficient than the first one with an expense of the higher computational complexity. This is because, since this method is based on minimizing the loss function which is a function of the model order and its dominating pole, there exists an optimal dominating time at each model order. The optimal dominating pole is needed to calculate at each model order by Eq.(7.23) and its value can be either real-valued or complex-valued depending on our intended use. The third method relies on the fact that the squared error or loss function is a function of a dominating pole. The optimal dominating pole can then be derived mathematically by setting the derivative of the squared error with respect to a dominating pole equal to zero. Given the nth model order, the optimal condition of Laguerre model is given in Eq.(7.34), while that of Kautz model is given in Eq.(7.42)-Eq.(7.43). Obviously, the major shortcoming of these two methods is the necessity of subdividing the whole interval of a dominating pole into smaller subintervals of more manageable size, in order to avoid the computational complexity of approximation at very high order.

According to the results in simulated examples given in section 8, when modeling a stable LTI system, all proposed models discussed in section 6 have performed quite better than a FIR model, especially for a Kautz model. Therefore, Kautz model seems to have large potential in real applications such as in signal processing and in control system, because the system in reality is likely to be resonant system. In addition, the performance of two-stage echo canceller is comparable to that of Laguerre and Kautz models but it has gained in terms of reducing the computational complexity. The concept of model reduction given in section 7.4 has also shown that the model order of the proposed models can be further reduce but still preserve enough energy of the system at the expense of a higher fit of error. Evidently, this is a compromise between the model order and the fit of error that one should take into consideration for each specific purpose.

Finally, we obtain small DB improvement on the real acoustic echo data by using the proposed models if compared with a FIR model This might be because the system (acoustic echo path) may have some non-linearity and time variation.

Supervisors

Jonas Sjoberg (Assistant Professor, Dept. of Signals and Systems, CTH.)
Ulf Lindgren (Ericsson Mobile Communication AB, Lund, Sweden.)
Lester S. H. Ngia (Phd. student, Dept. of Signals and Systems, CTH.)

Download

[MSc. Thesis Report] (508KBytes) [PDF Format] (Click here to get Adobe Acrobat Reader)
[Matlab script files] (15.8KBytes)
[The acoustic echo data (WGN input)] (1.16MBytes)

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