![Page 1: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/1.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
HMM - Basics
![Page 2: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/2.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Content
• Hidden Markov Model (HMM)• The Three Basic Problems for HMMs
– Problem 1 Solution: Forward/ Backward Algorithm– Problem 2 Solution: Viterbi Algorithm– Problem 3 Solution: Baum- Welch Algorithm
• An Overview: HMM in Speech Synthesis System
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 3: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/3.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
HMM
URN 2
URN 1 URN 3
P(red) = 0.8
P(green) = 0.1
P(blue) = 0.1
P(red) = 0.2
P(green) = 0.2
P(blue) = 0.6
P(red) = 0.5
P(green) = 0.4
P(blue) = 0.1
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 4: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/4.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Elements of an HMM
• N, number of states S = {S1,S2,S3, … , SN}• M, number of observation symbols
V = {v1,v2,v3, … , vM}• State transition probability distribution: A = {aij}• Observation symbol probability distribution in state j:
B = bj(k)• Initial state distribution: π = {πi}• T, number of observations in the sequence
O = O1 O2 O3… OT
HMM completely characterized by:
λ = (A, B, π)
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 5: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/5.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Why HMM?
• No one-to-one mapping: speech – word symbol
• Different symbols – same sound
• Large variation in speech– Speaker variability– Mood– Environment
• No explicit symbol boundary detection
Speech waveform is NOT a concatenation of static patterns
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 6: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/6.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
The Three Basic Problems: Problem 1
Content
HMM
Three Basic
Problems
Speech System
Overview
Solution: Forward - Algorithm
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Forward - Algorithm
Forward variable:
1) Initialization:
2) Induction:
3) Termination:
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 8: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/8.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
The Three Basic Problems: Problem 2
Content
HMM
Three Basic
Problems
Speech System
Overview
Solution: Viterbi - Algorithm
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Viterbi- Algorithm (1)
• Highest probability along a single path:
1) Initialization
2) Recursion
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 10: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/10.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Viterbi- Algorithm (2)
3) Termination
4) Path Backtracking
Content
HMM
Three Basic
Problems
Speech System
Overview
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
The Three Basic Problems: Problem 3
Content
HMM
Three Basic
Problems
Speech System
Overview
Solution: Baum – Welch Algorithm
(finds local maximum only)
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Baum – Welch - Algorithm(1)
• Define:
• Forward/backward variable: Content
HMM
Three Basic
Problems
Speech System
Overview
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Baum- Welch- Algorithm(2)
• Define:
• Relation:
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 14: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/14.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Baum- Welch- Algorithm(3)
• Reestimation formulas (use iteratively to local maximum!)
• Baum‘s auxiliary function:
Derive reestimation formulas directly
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 15: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/15.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
HMM - Based Speech Synthesis System
Content
HMM
Three Basic
Problems
Speech System
Overview
![Page 16: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/16.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
References
[1] „A tutorial on Hidden Markov Models and Selected Applications in Speech Recognition“. Lawrence R. Rabiner (1989)
[2] „An HMM-Based Speech Synthesis System Applied to English“.
Keiichi Tokuda et al.
[3] Talk About HMM-Based Speech Synthesis. Keiichi Tokuda (2006)
[4] HTK Book. Cambridge University Engineering Department (2006)
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Markov- Chain(1)
• Transition probability:
• Markov- property:
• Initial state probability:
Content
Markov-Chain
HMM
Three Basic
Problems
Speech System
Overview
![Page 19: Advanced Signal Processing 2, SE 1 Patrick Gampp Graz, 04/29/08 HMM - Basics](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649e9f5503460f94ba17a3/html5/thumbnails/19.jpg)
Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
Markov- Chain: An Example
Content
Markov-Chain
HMM
Three Basic
Problems
Speech System
Overview
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Advanced Signal Processing 2, SE
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Patrick Gampp Graz, 04/29/08 HMM - Basics
The Backward Variable
Backward variable:
1) Initialization:
2) Induction:
Content
HMM
Three Basic
Problems
Speech System
Overview