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On Robust Control of Adaptive Algorithms for Acoustic Echo CancellersVydáno dne 31. 12. 2008 (4964 přečtení)The main purpose of the paper was to investigate robust adaptive echo cancellation algorithms in handsfree scenarios to achieve an understanding of its performance and also its effects on Doubletalk (DT). 1. IntroductionDT is a major issue present in acoustic echo cancellation (AEC). It occurs when both the farend and the nearend speakers speak simultaneously. Many attempts have been made to alleviate the isruptive effects of DT; these include the use of DT detection schemes, robust daptive AEC, stepsize control, statespace representation ([1], [2]). The scheme f extending robust algorithms to an acoustic handsfree environment as proposed in [3] became the motivation for this paper.
Robust adaptive echo cancellation in particular, is an interesting approach
because if appropriately combined with a good doubletalk detector (DTD) it
could offer a noteworthy solution in overcoming the problems of DT [2]. This
approach was used for line echo cancellers (LEC) [3]. Thus, by extending the
method to acoustic handsfree scenarios, it may prove very valuable in the
long term goal of eliminating the problems caused by DT. 2. Adaptive Algorithms and NotationsIn derivations and descriptions the following notations are used (see also Fig. 1). Fig. 1 Block diagram of the echo canceller and DTD [3]. The excitation vector is denoted x_{n}=[x_{n},...,x_{nL+1}]^{T}, where x_{n} is the farend speech signal. v_{n}+ w_{n} is the background noise mixed with the nearend speech. The microphone signal is denoted y_{n}. The residual echo is e_{n}=y_{n}h_{n}^{T}x_{n}, where h_{n}=[h_{0,n},...,h_{L1,n}]^{T} is the estimated echo path. Here L is the length of the adaptive filter. 2.1 The Proportionate Algorithms
The stepsizes in the PNLMS algorithm are calculated from the last estimate of the filter coefficients so that a larger coefficient receives a larger weight, thus increasing the convergence rate of that coefficient. This has the effect that active coefficients are adjusted faster than non active coefficients (i.e. small or zero coefficients). This algorithm is described by the following equations:
G_{n} is a diagonal matrix which adjusts the stepsizes, μ is the overall stepsize parameter, and δ is a regularization parameter which prevents division by zero and stabilizes the solution when speech is used as input signal. The diagonal elements of G_{n+1} are calculated as follows, [4],
Parameters δ_{p} and ρ are positive numbers with typical values δ_{p} = 0.01, ρ= 5/L. A variant of this algorithm is called the PNLMS++ [5]. There, for oddnumbered time steps the matrix G_{n} is derived as above, while for evennumbered steps it is chosen to be the identity matrix ( G_{n} = I), which results in the NLMS algorithm. The alternation between NLMS and PNLMS iterations has several advantages compared to using just the PNLMS technique (in the case of pure PNLMS, as the large weights adapt, the remaining small coefficients adapt at a rate slower than NLMS), e.g., the PNLMS++ algorithm is much less sensitive to the assumption of a sparse impulse response. 2.2 The Robust NLMS and PNLMS++ Algorithms
The NLMS and PNLMS++ algorithm can both be made robust to large disturbances by modification of the criteria on which they are based [3]. The robust NLMS algorithm is given by,
where Ψ(^{.}) is a limiter function,
and the adaptive scale factor, s_{n}, is defined as,
where λ controls the scale factor. β is a normalization constant, it dependents on k_{0},
where erfc(^{.}) is a complementary error function defined as,
As with the Geigel DTD (which is an essential component in the robust algorithms) [8], it is useful to introduce a hangover time to control scale updating. When the DTD detects DT, adaptation of s_{n }should be inhibited for a specific time, preferably al long as the DTD hangover time, Thold, [10]:
The PNLMS algorithm can be made robust in an exactly analogous manner, yielding the equation
Alternating the iterations with G_{n} as given in (3) and the identity matrix then yields the robust PNLMS++ algorithm. 2.3 The Robust Proportionate APA (PAPA)
Let y_{n}=[y_{n},...,y_{np+1}]^{T}, be a vector of samples y_{n} and X_{n}=[x_{n},...,x_{nL+1}]^{T} the excitation matrix, where p is the projection order. A residual echo vector e_{n}=y_{n}X_{n}^{T}h_{n}, and PAPA is then given by,
where G_{n} is as defined in equation (2) and R
is a weighted estimate of the inverse correlation matrix of the input
signal. This matrix “whitens” the input data, X_{n}, and thus the
convergence rate of the adaptive filter is increased. With G_{n} = I
and δ=0, the equation (14)
reduces to the standard APA [6].
where ☺ denotes elementwise multiplications and ^{.} in e_{n} operates on the individual elements. 3. Investigation of Robust ParametersThis section examines the relationship and function of the robust algorithm. In the robust PAPA Ψ(e_{n}) is an Pby1. The scale factor s_{n} uses the first element in the limiter matrix,
From a glance at the equations, the parameters β and λ act as a weighting on the previous scale factor values and the current error value. This relationship between the parameters and the current error value can be demonstrated through the substitution of equation (14) into (15), yielding,
If the error is small, meaning no double talk is active; the minimum function will favour the first term and this results in following equation,
From equation (17), it can be deduced that when the error is small, the
algorithm uses the previous scale factor and the current error to compute
the scale factor which will be used in the next iteration.
In this case, the scale factor relies on the previous scale factor and k_{0}. This implies that when DT occurs, the scale factor will use a smaller k_{0} value instead of the larger error value. 4. SimulationsThe simulations were provided using MATLAB environment. The robust PAPA was simulated under user defined acoustic scenarios. The data used comprised of a real car channel (length of 2048 taps), female and male synthetic speech. 6 dB was used for DT as the ratio between the farend speaker and the nearend speaker and a SNR of 20 dB was used for the noise as illustrated in Figure 2. The two main factors that were considered in the investigation of adaptive algorithms were rate of convergence and robustness.The misalignment (it was the main focus of this paper) is given by,
ε=hh_{ep}/h_{ep}, where
h_{ep} is the true echo path. The scale estimate in (7),
s_{n} , is never allowed to become smaller than 2. This inhibits bad behaviour in low noise situations. Fig. 2 Echo signal, local speaker and background noise. Fig. 3 Misalignment for k_{0}. Fig. 4 Misalignment for β. Fig. 5 Misalignment for λ. 5. ConclusionThe parameter k_{0} was the main robust parameter, which affected the level of divergence in the algorithm DT occurred. This parameter was the key to the limiter function, which limits the magnitude in the filter weight adaptation. The other two robust parameters β and λ could be regarded as finetuning parameters, which allowed users to adjust the level of robustness integral to the robust adaptive algorithms. The recommended values suggested in ([3], [10]) were found to be relevant to the acoustic environment used in the simulations. An increase in the values of β and λ was found to increase the memory of the algorithm, meaning that the algorithm demonstrated slower convergence and reacted slower to changes in the input signal. Conversely, decreasing the values of β and λ also decreased the memory of the algorithm thus allowing it to track changes more readily, as it was less robust; the algorithm was found to diverge when doubletalk occurred. It was concluded that to configure the robust adaptive echo cancellation algorithms, the required degree of robustness should be known such that one can set the correct parameters. As allowing more robustness would decrease the convergence of the algorithms, it was very important to find a balance between the two key factors. This paper has originated thanks to the support from the Ministry of Education, Youth and Sports of Czech Republic within the project MSM6840770014. Literature
[1] C. Breining, "Control of a handsfree telephone set", Elsevier Signal
Processing, vol. 61, pp. 131143, 1997. Autor: K. Sakhnov, B. Šimák Pracoviště: České vysoké učení technické v Praze, FEL 
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