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Unformatted text preview: Speech Processing Homomorphic Signal Processing 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 a 1 a 2 a 2 a 1 February 13, 2012 Veton Këpuska 4 Principles of Homomorphic Processing Example 6.1: If signals fall in nonoverlapping frequency bands then they are separable. x[n]=x 1 [n]+x 2 [n] X 1 ( ϖ )= ℱ {x 1 [n]} & X 1 ( ϖ ) [0, π /2], X 2 ( ϖ )= ℱ {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 ◈ 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 subsystems 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] □ : + ....
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This note was uploaded on 02/11/2012 for the course ECE 5525 taught by Professor Staff during the Fall '10 term at FIT.
 Fall '10
 Staff
 Signal Processing

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