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

ISBN-10: 364222573X

ISBN-13: 9783642225734

ISBN-10: 3642225748

ISBN-13: 9783642225741

Single-channel hands-free teleconferencing platforms have gotten well known. so one can improve the communique caliber of those structures, increasingly more stereophonic sound units with loudspeakers and microphones are deployed. as a result coupling among loudspeakers and microphones, there's powerful echoes, which make real-time verbal exchange very tough. the way we all know to cancel those echoes is through a stereo acoustic echo canceller (SAEC), which are modelled as a two-input/two-output method with genuine random variables. during this paintings, the authors recast this challenge right into a single-input/single-output process with advanced random variables because of the commonly linear version. From this new handy formula, they re-derive crucial features of a SAEC, together with identity of the echo paths with adaptive filters, double-talk detection, and suppression.

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The system output (observation) signal, d(n), and the adaptive ﬁlter output, y(n). 49) can be rewritten as αp (n) = 1 − σd2 (n − p) − σ 2 (n − p) y σep (n) , p = 0, 1, . . , P − 1. 51), the previous relation is more suitable in practice. 50). Under our assumptions, we have E |d(n − p)|2 ≥ E |y(n − p)|2 and 2 2 2 E |d(n − p)| − E |y(n − p)| ≈ E |ep (n)| . 57). 5 Improved Proportionate APA (IPAPA) αp (n) = 1 − 59 σd2 (n − p) − σ 2 (n − p) y σep (n) , p = 0, 1, . . , P − 1. 58). 5 Improved Proportionate APA (IPAPA) As explained in Chapter 4, it makes more sense to use the minimum 1 -norm solution than the minimum 2 -norm solution in any type of adaptive ﬁlters when the impulse response is sparse.

M. Chabries, and F. A. Bishop, “A variable step (VS) adaptive ﬁlter algorithm,” IEEE Trans. , vol. ASSP-34, pp. 309–316, Apr. 1986. 8. R. H. Kwong and E. W. Johnston, “A variable step size LMS algorithm,” IEEE Trans. , vol. 40, pp. 1633–1642, July 1992. 48 4 A Class of Stochastic Adaptive Filters 9. V. J. Mathews and Z. Xie, “A stochastic gradient adaptive ﬁlter with gradient adaptive step size,” IEEE Trans. , vol. 41, pp. 2075–2087, June 1993. 10. J. B. Evans, P. Xue, and B. Liu, “Analysis and implementation of variable step size adaptive algorithms,” IEEE Trans.

64) 60 5 A Class of Aﬃne Projection Algorithms is a (non-orthogonal) projection matrix. 65) Making this algorithm more practical to avoid problems such as stalling of the coeﬃcients, will lead to the well-known proportionate APA (PAPA) and improved APA (IPAPA) [8], [9]. 68) is the normalized regularization parameter of the IPAPA. 58). 72) is a 2L × 2L diagonal matrix that assigns an individual step size to each ﬁlter coeﬃcient (in order to “proportionate” the algorithm behavior). 73) where g(n − 1) = g0 (n − 1) g1 (n − 1) · · · g2L−1 (n − 1) T is a vector containing the diagonal elements of G(n − 1) and the operator denotes the Hadamard product.

### A Perspective on Stereophonic Acoustic Echo Cancellation by Jacob Benesty, Constantin Paleologu, Tomas Gänsler, Silviu Ciochină

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