DT is a major issue present in acoustic echo cancellation (AEC). It occurs when both the far-end and the near-end 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, step-size control, state-space representation ([1], [2]). The scheme f extending robust algorithms to an acoustic hands-free 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 double-talk 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 hands-free scenarios, it may prove very valuable in the
long term goal of eliminating the problems caused by DT.

As shown in ([4], [5]), faster convergence can be achieved by using
algorithms like Proportional Normalized Least Mean Squares (PNLMS) and
PNLMS++. For LEC, it is reasonable to assume that the echo path is sparse
(i.e. many coefficients are zero), and try to identify only the non-zero
active coefficients. This is the idea behind the PNLMS algorithm which is a
modification of the NLMS. Thus, higher convergence rate is achieved by using
the fact that the active part of network echo path is usually much smaller
(4-8 ms) compared to 64-128 ms of the whole echo path that has to be
covered. In PNLMS, an adaptive individual step-size is assigned to each
filter coefficient.

However, this algorithm does not utilize the fact that the speech signal is
a correlated signal. If this is taken into account, further increase in
convergence rate can be achieved; this thought leads to the Affine
Projection Algorithm (APA) ([6], [7]).

Besides the convergence rate, an important aspect of an echo canceller (EC)
is its performance during DT. To inhibit the divergence of the EC the
standard procedure is to use a level based DTD [8]. Whenever DT is detected
the step-size of the adaptive filtering algorithm is set to zero thus
inhibiting the adaptation. Unfortunately, during the time required by the
DTD to detect DT, the EC often diverges. This is because even a few
undetected amplitude samples perturb the echo path estimate considerably
[3]. Once this happens, the coefficients are frozen in a poor state for at
least the hangover time before adaptation can be resumed.

The key to the solution is a combination of a DTD with robust statistics.
Such approach is based on introducing a scaled nonlinearity into the
adaptive algorithm. The nonlinearity limits the impact of large disturbances
on the coefficient setting, but also reduces the convergence rate. The idea
was developed for a subband echo canceller in [9].

The main objective of this work investigates the type of parameters that can
be used in robust algorithms to offer robust performance and their effect on
DT. As the performance of the robust Proportionate Affine Projection
Algorithm (PAPA) is shown to be better than other algorithms in network echo
cancellation ([3], [10], [11]), investigation is performed on the parameters
of the robust PAPA algorithm.

In 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_{n-L+1}]^{T}*, where

**2.1 The Proportionate Algorithms**

The step-sizes 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:

(1) |

(2) |

* G_{n}* is a diagonal matrix which adjusts the step-sizes,
μ is the overall
step-size 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

(3) |

(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
odd-numbered time steps the matrix * G_{n}* is derived as
above, while for even-numbered steps it is chosen to be the identity matrix
(

**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,

(5) |

where *Ψ(*^{.}*)* is a limiter function,

(6) |

and the adaptive scale factor, *s _{n}*, is defined as,

(7) |

where λ controls the scale factor.
β is a normalization constant, it dependents on *k _{0}*,

(8) |

where *erfc( ^{.})* is a complementary error function defined
as,

(9) |

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]:

(10) |

The PNLMS algorithm can be made robust in an exactly analogous manner, yielding the equation

(11) |

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_{n-p+1}]^{T}*,
be a vector of samples y

(12) |

where * G_{n}* is as defined in equation (2) and

A robust version of PAPA (and hence of APA) is obtained straightforwardly, by applying the principles presented previously, [3]:

(13) |

where ☺ denotes
elementwise multiplications and |^{.}| in |e_{n}| operates on
the individual elements.

This section examines the relationship and function of the robust algorithm.
In the robust PAPA *Ψ( e_{n})* is an

(14) |

(15) |

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,

(16) |

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,

(17) |

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.

Similarly, when the error is large, the minimum function will favour the second term of equation (18) and the scale factor becomes,

(18) |

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

The 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 far-end speaker and the near-end 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=|| h-h_{ep}||/||h_{ep}||*, where

The misalignment plot for

Detailed investigations were carried out to find out the effects of β and λ using

Practical values of the robust parameters were best to be [0.99, 1) for λ in [3]. Clear the misalignment (Fig. 5) was considerably smaller when λ

*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.*

The 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 fine-tuning 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 double-talk
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. *

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