Speech Enhancement. A Signal Subspace Perspective by Jacob Benesty

By Jacob Benesty

Speech enhancement is a classical challenge in sign processing, but nonetheless mostly unsolved. of the normal methods for fixing this challenge are linear filtering, just like the classical Wiener clear out, and subspace tools. those ways have frequently been handled as various sessions of tools and feature been brought in a bit various contexts. Linear filtering tools originate in stochastic approaches, whereas subspace equipment have principally been in line with advancements in numerical linear algebra and matrix approximation concept.

This ebook bridges the space among those periods of equipment by means of displaying how the tips at the back of subspace equipment should be integrated into conventional linear filtering. within the context of subspace equipment, the enhancement challenge can then be noticeable as a classical linear filter out layout challenge. which means numerous options can extra simply be in comparison and their functionality bounded and assessed by way of noise aid and speech distortion. The ebook exhibits how numerous clear out designs may be got during this framework, together with the utmost SNR, Wiener, LCMV, and MVDR filters, and the way those could be utilized in numerous contexts, like in single-channel and multichannel speech enhancement, and in either the time and frequency domains.

    • First brief ebook treating subspace techniques in a unified method for time and frequency domain names, single-channel, multichannel, in addition to binaural, speech enhancement.
    • Bridges the space among optimum filtering equipment and subspace approaches.
    • Includes unique presentation of subspace equipment from various views.

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    P are arbitrary complex numbers with at least one of them different from 0, corresponds to the maximum SNR filtering matrix since oSNR Hmax = λ1 . 49) 0 ≤ oSNR H ≤ oSNR Hmax , ∀H . 50) We always have and Choosing properly the values of ς p , p = 1, 2, . . , P, is important in practice if we want to avoid a large distortion of the transformed desired signal vector. The best way to find these values is by minimizing distortion. 51) where h pH is the pth line of H . 53) p = 1. 54) We deduce that the optimal maximum SNR filtering matrix with minimum transformed desired signal distortion is ⎡ H ⎤ b1 ⎢ 01×M ⎥ ⎥ ⎢ Hmax = ⎢ .

    Besides a (slight) different weighting factor, HW considers all directions where speech is present, while Hmax relies only on the direction where the maximum of speech energy is present. 62) we easily find the MVDR filtering matrix: HMVDR = T H T = T H. 63) General Concept with the Joint Diagonalization of the Speech and Noise Correlation Matrices 39 We deduce that the MVDR for the estimation of x is HMVDR = T HMVDR = T T H. 64) Finally, the MVDR for the estimation of x is P HMVDR = bpbH p. , HMVDR = I M .

    1 To 58 Speech Enhancement is a diagonal matrix, with λx ,1 ≥ λx ,2 ≥ · · · ≥ λx ,P > 0 and λx ,P+1 = λx ,P+2 = · · · = λx ,L = 0. 73) where Rxs = E xs (k)xsT (k) = diag λx ,1 , λx ,2 , . . , λx ,P = xs . 78) Single-Channel Speech Enhancement in the Time Domain 59 is the residual noise. 79) H = Tx H. 83) Rvrn = HRv H . 24). 65), since we consider xi (k) as an interference and x (k) as the desired signal vector. We obtain iSNR = tr Rx tr Rin . 85) The output SNR and the noise reduction factor are, respectively, oSNR H = = tr Rxfd tr Rxri + Rvrn and ξnr H = 2 Eventually, T T xs Tx H HRin H T tr HTx tr tr Rin tr HRin H T only the first P components of z (k) are relevant.

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