Ch6-HomomorphicSignalProcessing

# Ch6-HomomorphicSignalProcessing - Speech Processing...

This preview shows pages 1–10. Sign up to view the full content.

Speech Processing Homomorphic Signal  Processing

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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, ＊]
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
2/13/12 Veton Këpuska 9 Multiplicative Homomorphic Systems u Consider Homomorphic Multiplicative System depicted below: u Use D to convert MULT into ADD.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 99

Ch6-HomomorphicSignalProcessing - Speech Processing...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online