By Jacob Benesty

Additive noise is ubiquitous in acoustics environments and will impact the intelligibility and caliber of speech signs. for that reason, a so-called noise relief set of rules is needed to mitigate the influence of the noise that's picked up via the microphones. This paintings proposes a common framework within the time area for the one and a number of microphone instances, from which it's very handy to derive, learn, and study all type of optimum noise aid filters. not just that every one identified algorithms might be deduced from this process, laying off extra gentle on how they functionality, yet new ones might be came upon as well.

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Sørensen, Reduction of broad-band noise in speech by truncated QSVD. IEEE Trans. Speech Audio Process. 3, 439–448 (1995) 5. S. Doclo, M. Moonen, GSVD-based optimal filtering for single and multimicrophone speech enhancement. IEEE Trans. Signal Process. 50, 2230–2244 (2002) 6. B. Searle, Matrix Algebra Useful for Statistics. (Wiley, New York, 1982) 7. G. Strang, Linear Algebra and its Applications. 3rd edn (Harcourt Brace Jovanonich, Orlando, 1988) 8. Y. C. Loizou, A subspace approach for enhancing speech corrupted by colored noise.

Obviously, taking N = 1 (single-channel case), we find all the optimal filtering vectors derived in Chap. 2. 1 Maximum SNR Let us rewrite the output SNR: oSNR h = T h σx21 hT ρ xx1 ρ xx 1 hT Rin h . 59) 52 4 Multichannel Filtering Vector The maximum SNR filter, hmax , is obtained by maximizing the output SNR as given above. 59), we recognize the generalized Rayleigh quotient [6]. It is well known that this quotient is maximized with the maximum eigenvector of the −1 T . Let us denote by λ matrix σx21 Rin ρ xx1 ρ xx max the maximum eigenvalue corre1 sponding to this maximum eigenvector.

55) T R h and b = h T R h . We Proof Let us define the positive reals am = hm xd m m m in m have M m=1 am M m=1 bm M = m=1 am · bm bm M i=1 bi . 57) T bM ··· M i=1 bi . 59) which ends the proof. 1 The maximum SNR filtering matrix is given by ⎡ ⎤ β1 b1M T ⎢ β2 b M T ⎥ 1 ⎢ ⎥ Hmax = ⎢ ⎥, .. ⎣ ⎦ . 60) βm b1M T wher e βm , m = 1, 2, . . , M are real numbers with at least one of them different from 0. The corresponding output SNR is oSNR (Hmax ) = λ1M . 61) −1 We recall that λ1M is the maximum eigenvalue of the matrix Rin Rxd and its correM sponding eigenvector is b1 .