Lecture 12_winter_2012

Lecture 12_winter_2012 - Digital Speech Processing Lecture...

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1 Digital Speech Processing— Lecture 12 Homomorphic Speech Processing
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2 General Discrete-Time Model of Speech Production [] [] [] [] [] [] Voiced Speech Unvoiced Speech = ∗− =⋅ =∗ LV VV LU UN pn pn hn hn A gn vn rn pn un hn hn Avn rn
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3 Basic Speech Model • short segment of speech can be modeled as having been generated by exciting an LTI system either by a quasi-periodic impulse train, or a random noise signal • speech analysis => estimate parameters of the speech model, measure their variations (and perhaps even their statistical variabilites-for quantization) with time • speech = excitation * system response => want to deconvolve speech into excitation and system response => do this using homomorphic filtering methods
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4 Superposition Principle 12 [] [ ] { [ ]} { [ ]} { [ ]} =+ == + LL L xn ax n bx n yn a x n b x n + + x n } { L ]} [ { ] [ n x n y L = ] [ ] [ 2 1 n x n x + {} { } ] [ ] [ 2 1 n x n x L L +
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5 Generalized Superposition for Convolution 12 for LTI systems we have the result [] [] [] [][ ] "generalized" superposition => addition replaced by convolution [ ] { [ ]} { [ ]} { [ ]} homomorphic system f =−∞ =∗= =∗ == Hx k yn xn hn xkhn k xn x n x n n n n or convolution * * x n { } H { } ] [ ] [ n x n y H = ] [ ] [ 2 1 n x n x { } { } ] [ ] [ 2 1 n x n x H H
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6 Homomorphic Filter 12 1 1 homomorphic filter => homomorphic system that passes the desired signal unaltered, while removing the undesired signal ( ) [ ] [ ] - with [ ] the "undesired" signal {[] } { [] } ⎡⎤ ⎣⎦ =∗ H HH xn xn xn xn x n 2 11 22 {[ ] } { [ ]} ( ) - removal of [ ] ] } [ ] } [] for linear systems this is analogous to additive noise removal δ = H H H H n n x n
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7 Canonic Form for Homomorphic Deconvolution any homomorphic system can be represented as a cascade of three systems, e.g., for convolution 1. system takes inputs combined by convolution and transforms them into additive outputs 2. system is a conventional linear system 3. inverse of first system--takes additive inputs and transforms them into convolutional outputs * * [] x n + ˆ xn + + ˆ yn + 12 xn xn ˆˆ + yn yn + {} D { } 1 D { } L
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8 Canonic Form for Homomorphic Convolution 12 1 1 2 [ ] [ ] [ ] - convolutional relation ˆˆ ˆ [ ] { [ ]} [ ] [ ] - additive relation ˆ ˆ ˆ [ ] { [ ] [ ]} [ ] [ ] - conventional linear system [] { [] } - =∗ == + =+ = + = D L D xn x n x n x n yn y n y n yn yn 1 inverse of convolutional relation => design converted back to linear system, - fixed (called the characteristic system for homomorphic deconvolution) - fixed (characteristic system for ⎡⎤ ⎣⎦ L D D inverse homomorphic deconvolution) * * x n + ˆ + + ˆ + xn xn + y n y n + {} D { } 1 D { } L
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9 Properties of Characteristic Systems 12 11 ˆ [ ] { [ ]} { [ ] [ ]} { [ ]} { [ ]} ˆˆ [] ˆ { [ ]} { [ ] [ ]} { [ ]} { [ ]} ∗∗ −− == =+ =∗ = DD xn x n x n x n x n yn y n y n y ny n y n y n
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Discrete-Time Fourier Transform Representations 10
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Canonic Form for Deconvolution Using DTFTs 12 need to find a system that converts convolution to addition [] () since ˆˆ ˆ { [ ]} [ ] [ ] [ ] ˆ => use log function wh ωω ω =∗ =⋅ =+ = ⎡⎤ = ⎣⎦ D D jj j j j xn x n x n Xe X e x n 1 2 1 2 ich converts products to sums ˆ l o g ()l o g l o g l o g ˆ ˆ ˆ e x p ( == = + = + L j j j j j j j j Ye Y e )( ) ( )
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12 Characteristic System for Deconvolution Using DTFTs 1 2 () [ ] ˆ ( ) log ( ) log ( ) arg ( ) ˆ ˆ [] ( )
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Lecture 12_winter_2012 - Digital Speech Processing Lecture...

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