Adaptive algorithms in the frequency domain are, in general, extremely effi-cient since they use the fast Fourier transform (FFT) as an intermediary step. As a result, they are now implemented in many prototypes and products for acous-tic echo cancellation. In this section, we briefly explain how these algorithms can be derived rigorously from a block error signal.
From now on and for simplification, we drop the parameter q in all equations.
With this simplification, the error signal at time is now:
where is the estimated impulse response of the channel,
We now define the block error signal (of length For that, we assume that L is an integer multiple of N, i.e., L = KN. We have:
where is the block time index, and
It can easily be checked that is a Toeplitz matrix of size (L × N).
We can show that for K = L/N, we can write
where is an (N × N) Toeplitz matrix and
its size, to a circulant matrix Also, a circulant matrix is easily decomposed
as follows: where is the Fourier matrix [of
size (2N × 2N)] and is a diagonal matrix whose elements are the discrete Fourier transform of the first column of If we multiply (5.86) by [Fourier matrix of size (N × N)], we get the error signal in the frequency domain (denoted by underbars):
where
The size of the matrix is (2N × PL) and the length of is PL.
By minimizing the criterion
where denotes conjugate transpose and is an exponential forgetting factor, we obtain the normal equations for the multichannel case:
where
is a (PL × PL) matrix,
and
is a (PL × 1) vector. Assuming that the P input signals are not perfectly pairwise coherent, the normal equations have a unique solution which is the optimal Wiener solution.
Define the following variables:
Depending of the approximations we make on matrix we obtain different algorithms. There is a compromise between performance (convergence rate) and complexity. Several algorithms can be deduced from the previous form, some of them are well-known while others are new. See [29] for a general derivation of adaptive algorithms in the frequency domain. See also [30] and [31] for new algorithms derived directly from the above equations.
9. CONCLUSIONS
In this chapter, we have given an overview on multichannel adaptive algo-rithms in the context of multichannel acoustic echo cancellation. We have first derived the normal equations of a MIMO system and discussed the identifi-cation problem. We have shown that, in the multichannel case and when the input signals are linearly related, there is a nonuniqueness problem that does not exist in the single-input single-output (SISO) case. In order to have a unique solution, we have to decorrelate somehow the input signals without affecting the spatialization effect and the quality of the signals. We have then derived many useful adaptive algorithms in a context where the strong correlation be-tween the input signals affects seriously the performance of several well-known algorithms.
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V-385–V-388.
Tomas Gänsler
Agere Systems [email protected]
Jacob Benesty
Université du Québec, INRS-EMT [email protected]
Abstract Double-talk detectors (DTDs) are vital to the operation and performance of acous-tic echo cancelers. In this chapter, we highlight important aspects needed to be considered when choosing and designing a DTD. The generic double-talk de-tector scheme along with fundamental means of performance evaluation are dis-cussed. A number of double-talk detectors suitable for acoustic echo cancelers are presented and objectively compared.
Keywords: Double-Talk Detector, Acoustic Echo Canceler, Adaptive Algorithms, LMS, NLMS, Coherence, Cross-Correlation, Robust Statistics
1. INTRODUCTION
The design of a good double-talk detector is much more of an art than the design of the adaptive filter itself. [ 1 ]1
Ideally, acoustic echo cancelers (AECs) remove undesired echoes that result from coupling between the loudspeaker and the microphone used in full-duplex hands-free telecommunication systems. Figure 6.1 shows a basic AEC block diagram. The far-end speech signal goes through the echo path represented by a filter then it is picked up by the microphone together with the near-end talker signal and ambient noise The microphone signal is denoted Most often the echo path is modeled by an adaptive FIR filter,
that generates a replica of the echo. This echo estimate is then subtracted
from the return channel and thereby cancellation is achieved. This may look like a simple straightforward system identification task for the adaptive filter.
However, in most conversation there are so called double-talk situations that make the identification much more problematic than what it might appear at a first glance. Double-talk occurs when the speech of the two talkers arrive simultaneously at the echo canceler, i.e. and (the situation with near-end talk only, and can be regarded as an “easy-to-detect” double-talk case). In the double-talk situation, the near-end speech acts as a large level uncorrelated noise to the adaptive algorithm. The disturbing near-end speech may cause the adaptive filter to diverge. Hence, annoying audible echo will pass through to the far-end. The usual way to alleviate this problem is to slow down or completely halt the filter adaptation when presence of near-end speech is detected. This is the very important role of the so called double-talk detector (DTD). The basic double-talk detection scheme is based on computing a detection statistic, and comparing it with a preset threshold, T. The important issues that have to be addressed when designing a DTD are:
(i) What basic knowledge is needed in order to devise a sufficient DTD solution?
(ii) What characterize “good” double-talk detectors?
Primarily, we must know under what circumstances double-talk disturbs the adaptive filter. The performance of AECs are most often evaluated through their mean-square error (MSE) performance or preferably, through their mis-alignment in different situations. The misalignment formula
the adaptive algorithm. That is, the level of convergence (misalignment perfor-mance) is completely governed by echo-to-background (near-end speech and noise) power ratio, EBR. If we have equally strong talkers during double-talk, we find that when the echo path has a high attenuation, the adaptive filter will be very sensitive to double-talk. On the contrary, a low attenuation or amplification of the echo path means lower sensitivity. This is important to remember when chosing the parameters of the DTD. Furthermore, it naturally shows why we should lower the step-size parameter in high noise or double-talk conditions in order to maintain a certain performance. Equation (6.1) is easily found by assuming the far- and near-end signals are uncorrelated stochastic processes and the adaptive algorithm is NLMS [3], Furthermore, if we suppose that the sig-nals are white noises, the misalignment is exactly the inverse of the echo return loss enhancement (ERLE) which reflects the echo attenuation provided by the AEC. Even though these assumptions describe an oversimplified model of the AEC situation, it gives the valuable insight needed to what governs divergence of the adaptive algorithm.
Issue (ii) can be partially addressed by characterizing DTDs through the use of general detection theory [4, 5, 6]. By means of detection probability and false alarm rates we can objectively evaluate and compare the performance of different DTDs. Moreover, the theory justifies a method to select the threshold T which has been missing in the field of DTD design. One must, however, also accompany these performance measures with a joint evaluation of the DTD and the echo canceler.
A large number of DTD schemes have been proposed since the introduction of echo cancelers [7]. The Geigel algorithm [8] has proven successful in network echo cancelers; however, it does not always provide reliable performance when used in the acoustic situation. This is because it assumes a minimum echo path attenuation which may not be valid in the acoustic case. Other methods based on cross-correlation and coherence [9, 10,4, 5] have been studied which appear to be more appropriate for the acoustic application. Spectral comparing methods [11] and two-microphone solutions have also been proposed [12]. A DTD based on multi statistic testing in combination with modeling of the echo path by two filters is proposed in [13]. The objective of this chapter is to summarize some of the DTD proposals and present evaluation methods. Many results in this chapter are derived from the papers [5, 6].
The chapter is organized as follows: Section 2 introduces the AEC notations and describes the general DTD scheme. A number of double-talk detection algorithms that have been proposed and used in acoustic echo cancelers are
presented in Section 3. Section 4 compares a selected number of DTDs by means of their receiver operation characteristics. Section 5 gives a discussion of the abilities of different DTD schemes and summarizes the important aspects that need to be considered for a successful double-talk detector implementation.