Ch6-HomomorphicSignalProcessing

Ch6-HomomorphicSignalProcessing - Speech Processing...

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Speech Processing Homomorphic Signal  Processing
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2/13/12 Veton Këpuska 2 Outline u Principles of Homomorphic Signal  Processing u Details of Homomorphic Processing u Variants of Homomorphic Processing u Investigation of Homomorphic systems  to speech analysis and synthesis
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2/13/12 Veton Këpuska 3 Principles of Homomorphic Processing u Superposition  Property of Linear Systems: L x1[n] x2[n] x[n] L(x[n]) L x1[n] x2[n] a1L(x1[n]) L(x[n]) L a2L(x2[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 a1 a2 a2 a1
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2/13/12 Veton Këpuska 4 Principles of Homomorphic Processing u Example 6.1: n If signals fall in non-overlapping frequency bands then they  are separable. n x[n]=x1[n]+x2[n] n X1(w )= {x1[n]} & X1(w ) [0,  p /2], n X2(w )= {x2[n]} & X2(w ) [ p /2,  p ], y[n] = h[n] (x1[n]+x2[n]) = h[n] x1[n] + h[n] x2[n] y[n] = h[n] x2[n] = x2[n] 0 for [0,**/2] 1 for [*/2, *]
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2/13/12 Veton Këpuska 5 u Generalized Superposition n Concept that would support separation of nonlinearly  combined signals. n Leads to the notion of  Generalized Linear Filtering . n Properties: u H(x1[n] x2[n])=H(x1[n]) H(x2[n]) u H(c:x [n])=c H( x [n]) 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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2/13/12 Veton Këpuska 6 Principles of Homomorphic Processing u 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. u 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
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2/13/12 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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2/13/12 Veton Këpuska 8 Homomorphic Systems u 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
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2/13/12 Veton Këpuska 9 Multiplicative Homomorphic Systems u Consider Homomorphic Multiplicative System depicted below: u Use D to convert MULT into ADD.
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Ch6-HomomorphicSignalProcessing - Speech Processing...

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