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Ch6-HomomorphicSignalProcessing

Ch6-HomomorphicSignalProcessing - Speech Processing...

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Speech Processing Homomorphic Signal Processing

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February 13, 2012 Veton Këpuska 2 Outline Principles of Homomorphic Signal  Processing Details of Homomorphic Processing Variants of Homomorphic Processing Investigation of Homomorphic systems  to speech analysis and synthesis
February 13, 2012 Veton Këpuska 3 Principles of Homomorphic Processing Superposition  Property of Linear Systems: L x 1 [n] x 2 [n] x[n] L(x[n]) L x 1 [n] x 2 [n] a 1 L(x 1 [n]) L(x[n]) L a 2 L(x 2 [n]) [ ] [ ] ( 29 [ ] ( 29 [ ] ( 29 [ ] ( 29 [ ] ( 29 [ ] [ ] ( 29 [ ] ( 29 [ ] ( 29 n x L a n x L a n x a n x a L n x L n x L n x L n x L n x n x L 2 2 1 1 2 2 1 1 2 1 2 1 + = + = + = + α α a 1 a 2 a 2 a 1

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February 13, 2012 Veton Këpuska 4 Principles of Homomorphic Processing Example 6.1: If signals fall in non-overlapping frequency bands then they  are separable. x[n]=x 1 [n]+x 2 [n] X 1 ( ϖ )= {x 1 [n]} & X 1 ( ϖ ) [0, π /2], X 2 ( ϖ )= {x 2 [n]} & X 2 ( ϖ ) [ π /2,  π ], y[n] = h[n] (x 1 [n]+x 2 [n]) = h[n] x 1 [n] + h[n] x 2 [n] y[n] = h[n] x 2 [n] = x 2 [n] 0 for  ϖ   [0, π /2] 1 for  ϖ   [ π /2,  π ]
February 13, 2012 Veton Këpuska 5 Generalized Superposition Concept that would support separation of nonlinearly  combined signals. Leads to the notion of  Generalized Linear Filtering . Properties: H(x 1 [n] x 2 [n])=H(x 1 [n]) H(x 2 [n]) H(c:x   [n])=c H( x   [n]) Systems that satisfy those two properties are referred to as homomorphic systems and are said to satisfy a generalized principle of superposition . Principles of Homomorphic Processing H() x[n] Input rule : y[n] Output rule

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February 13, 2012 Veton Këpuska 6 Principles of Homomorphic Processing Importance of homomorphic systems for speech processing lies  in their capability of transforming nonlinearly combined signals to  additively combined signals so that linear filtering can be  performed on them. Homomorphic systems can be expressed as a cascade of three  homomorphic sub-systems depicted in the figure below –  referred to as the  canonic representation : H D x[n] : + . y[n] L + . . + D + . -1 I II III [ ] n x ˆ [ ] n y ˆ
February 13, 2012 Veton Këpuska 7 Canonic Representation of a  Homomorphic System i. The  Characteristic  System : Transforms   into  add “+” ii. The linear system:  transforms “add” into “add” iii. The inverse system:  transforms add into    D x[n] : + . I [ ] n x ˆ L + . . + [ ] n x ˆ [ ] n y ˆ II y[n] D + . -1 III [ ] n y ˆ

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February 13, 2012 Veton Këpuska 8 Homomorphic Systems Let the goal be removal of undesired component of the  signal (e.g., noise):  Type of  combination rule System Operation Signal & Additive  noise Linear System Linear Filtering Signal & Multiplicative  noise Multiplicative  System Multiplicative  Filtering Signal & Convolutional  Noise Convolutional  System Convolutional  Filtering
February 13, 2012 Veton Këpuska 9 Multiplicative Homomorphic Systems

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