Lecture 12_winter_2012_6tp

Lecture 12_winter_2012_6tp - General Discrete-Time Model of...

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1 1 Digital Speech Processing— Lecture 12 Homomorphic Speech Processing 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 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 4 Superposition Principle 12 [ ] { [ ]} { [ ]} { [ ]} = + == + LL L xn ax n bx n y nx n 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 + 5 Generalized Superposition for Convolution for LTI systems we have the result [][ ] "generalized" superposition => addition replaced by convolution [ ] { [ ]} { [ ]} { [ ]} homomorphic system f =−∞ =∗= =∗ == ∗ Hx k yn xn hn xkhn k 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 6 Homomorphic Filter 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 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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2 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 x n ++ ˆ yn + 12 xnxn ˆˆ x nx n + y ny n + ynyn {} D 1 D L 8 Canonic Form for Homomorphic Convolution 1 -c o n v o l u t i o n a l r e l a t i o n ˆ [ ] { [ ]} [ ] [ ] - additive relation ˆ [ ] { [ ] [ ]} [ ] [ ] - conventional linear system { [] } - =∗ == + =+ = + = D L D xn x n x n x n 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 x n + + ˆ + x n + + { } D { } 1 D L 9 Properties of Characteristic Systems 11 ˆ {[] } } { [ ]} { [ ]} ˆ { [ ]} { [ ] [ ]} {[ ] } ] } ∗∗ −− ==∗ =∗= DD xn xn yn yn yn Discrete-Time Fourier Transform Representations 10 Canonic Form for Deconvolution Using DTFTs need to find a system that converts convolution to addition () () () since ˆˆ ˆ { [ ]} [ ] [ ] [ ] () => use log function wh ωωω ωω ω =⋅ =+ = =+= D D jj j j j Xe X e 1 2 12 12 ich converts products to sums ˆ l o g()l o g() () l o o g () () () ˆ ˆ ˆ () () () () ()e x p() ( ==⋅ + = + L j j j j j j j j Ye Y e )( ) ( ) 12 Characteristic System for Deconvolution Using DTFTs 1 2 [ ] ˆ l o o g() a r g() ˆ ˆ ( ) ωω ω π =−∞ = ⎤⎡ + ⎦⎣ = n n jj j j n xne j
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This note was uploaded on 12/29/2011 for the course ECE 259 taught by Professor Rabiner,l during the Fall '08 term at UCSB.

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Lecture 12_winter_2012_6tp - General Discrete-Time Model of...

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