By Jacob Benesty, Constantin Paleologu, Tomas Gänsler, Silviu Ciochină

Single-channel hands-free teleconferencing structures have gotten well known. with a view to increase the communique caliber of those platforms, progressively more stereophonic sound units with loudspeakers and microphones are deployed. as a result of the coupling among loudspeakers and microphones, there's robust echoes, which make real-time communique very tricky. the way in which we all know to cancel those echoes is through a stereo acoustic echo canceller (SAEC), which might be modelled as a two-input/two-output approach with genuine random variables. during this paintings, the authors recast this challenge right into a single-input/single-output process with advanced random variables due to the commonly linear version. From this new handy formula, they re-derive an important facets of a SAEC, together with id of the echo paths with adaptive filters, double-talk detection, and suppression.

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Converge to the optimal ﬁlter) in one iteration but, rather, in a ﬁnite number of iterations. 51) instead of the minimum 2 -norm solution, hoping that with BP less iterations will be required to converge to the same solution. We can now derive a proportionate-type NLMS algorithm following the steps of our interpretation of the NLMS [5]. First, let us solve the optimization problem ← − min h (n) ← − h (n) subject to ← −H d(n) = h (n)x(n). 86) where ← − ← − ← − ← − G(n) = diag h 0 (n) , h 1 (n) , .

4, pp. 421–434, Apr. 2010. Chapter 4 A Class of Stochastic Adaptive Filters In this chapter, we derive, study, and analyze a class of stochastic adaptive ﬁlters for SAEC with the WL model. All developed algorithms try to converge to the optimal Wiener ﬁlter. We start with the classical stochastic gradient algorithm, which is a good approximation of the deterministic algorithm studied in the previous chapter. 1 Least-Mean-Square (LMS) Algorithm The least-mean-square (LMS) or stochastic gradient algorithm, invented by Widrow and Hoﬀ in the late 50’s [1], is certainly the most popular algorithm that can be found in the literature of adaptive ﬁlters.

Using the misalignment vector, m(n) = ht − h(n), the LMS update equation becomes m(n) = I2L − μx(n)xH (n) m(n − 1) − μx(n)v ∗ (n). 7), we get E [m(n)] = E I2L − μx(n)xH (n) m(n − 1) . 8) can be rewritten as E [m(n)] = E I2L − μx(n)xH (n) E [m(n − 1)] = I2L − μRx E [m(n − 1)] . 10) where t(n) = QH m(n) (see Chapter 3). We say that the LMS converges in the mean to the true impulse response if lim E [t(n)] = 0. 11) lim E h(n) = ht . 10) that a necessary condition for the LMS to converge in the mean to the desired solution is that 0<μ< 2 .