1. Speech Processing Final Project
Estimation of pole and zero model in voiced speech
by Rafael A Alvarez

2. Introduction
This presentation shows the results of speech estimation using an pole-zero model. The pole-zero model was derived using the Linear prediction coding and homomorphic filtering methods. When both methods are combined the combined method is called the homomorphic prediction method. Several signals will be analyzed and the problems encounter in each case will be presented.

3. Objectives This project attempts to address the following areas:
Modeling of speech using a pole-zero model
Modeling of speech using Linear prediction and homomorphic filtering methods.
Results of estimating poles using Hommorphic Prediction.
Results of Zeros estimation using inverse-filtering and Homomorphic Prediction.
Problems estimating the model.
Other possible applications .

4. Speech modeling The complete discrete-time speech production model.
Speech source
Gain
Mixer
Vocal track and lips radiation

5. Speech modeling Periodic or voiced speech can be modeled with:
Speech signal
Gain
Glottal flow
Vocal track (poles and zeros)
Radiation impedance

6. Speech modeling
Complete transfer function of speech signal for voiced sound
Radiation impedance
Vocal tract zeros (min and max phase)
Glottal flow
Vocal tract poles

7. Linear Prediction Analysis
Linear prediction coding approximates the system using an an all pole model.
The zero produced at the radiation of the lips is approximated by long set of poles (Not efficient)
The resulting Transfer function:

8. Linear prediction analysis
Linear combination of past values
Time domain representation
When train of unit samples ug[n] = 0 then the above equation results in:

9. Linear prediction analysis
Two implementations of this analysis are:
Covariance method
Considers the value outside the window
Autocorrelation method
Considers the values outside the window to be zero.
Uses a window like the hamming window

10. Linear Prediction analysis
Results of LPC using autocorrelation methods

11. Homomorphic filtering
Base on the concept of superposition
Can easy separate linearly combine systems.
Generalized superposition
Can separate non-linearly combine systems.
The following properties must apply
Canonical formulation of homomorphic system + + + + L
y[n]
x[n] : . . . .

12. Homomorphic filtering
Homomorphic system for convolution
Applying the results from before on systems resulting from convolution gives: . . + +
log + + + + . .
exp

13. Homomorphic filtering
Algorithm to combined homomorphic filtering and LPC
w[n] x
s[n]
cepstrum
liftering
inv-cepstrum
LPC
With the liftering operation (filtering) convolutionaly combined signals can be separated in the quefrency domain.

14. Homomorphic filtering
Homomorphic filtering combined with LPC, Homomorphic prediction

15. Homomorphic prediction
Combining the previous we can derived a pole-zero model estimation method.
Remembering from before the zero-pole model of speech was given by
Remember that multiplication in the frequency domain transform into convolution in the time domain.

16. Homomorphic prediction
From the previous equation we have.
S(z) = original signal
P(z) = glottal flow train
B(z)= vocal tract zeros
A(z) = vocal track poles
* The system zeros and glottal flow poles can be obtained by filtering the signal with the inverse vocal track poles. (Inverse filtering)

17. Homomorphic prediction
An algorithm to estimate poles and zeros can be derived
First we obtain an approximation of our vocal tract poles as presented before,
w[n] x
s[n]
LPC
w[n] must window an area free of zeros and glottal flow poles
The resulting impulse response should represent all the poles in the system. Then this result can be used to inverse filter the original signal.

18. Homomorphic prediction
Zero and glottal flow deconvolution.
Separate the glottal flow from the zeros.
Separate min and max phase zeros.
B(z)P(z)
S(z)
Inverse- filtering
cepstrum
P(z)
inverse-cepstrum
High liftering
Bmax(z)
B(z)
Inverse-cepstrum
High liftering
Low liftering
Bmin(z)
Inverse-cepstrum
Low liftering

19. Homomorphic prediction
Example of “quefrency” domain signal of a voiced signal.

20. Results
Signal #1
First we examine a simple case of a synthesized signal
Pitch period

21. Homomorphic prediction
Zero free area of original signal, pitch-synchronized method
L = glottal width (19)
M= number of vocal tract zeros (2)
I = zero free area of vocal tract
(2 poles) L M I

22. Results
Frequency response of the estimated poles of the vocal tract

23. Results Inverse-Filtered signal
After filtering only the glottal flow train and zeros remain

24. Results Cepstrum of inverse-filtered signal
By liftering(filtering) the high and low part of the cepstrum the glottal flow can be separated from the zeros.

25. Results
Approximated glottal flow and zeros

26. Results
Signal #2
Second a more realistic signal.

27. Results Signal #2. Problems: How many zeros?
What is the length of the glottal flow?
How many poles?

28. Results
Signal #2
Why are the results so different?

29. Results Signal #2 Signal #1
The autocorrelation function on Signal #2 shows aliasing. The method will not work if the signal wasn’t sample at a high enough frequency.

30. Results Signal #3. Voice recorded at 20Khz
Area extracted for processing

31. Results Problems: How many zeros?
Extract area free of zeros
Problems:
How many zeros?
6 zeros
What is the length of the glottal flow?
38 since its 20khz
How many poles?
10 poles

32. Results
Spectrum of zero-free area, all-pole approximation and original signal

33. Results Resulting inverse-filtered signal
Area should be flat if all the poles where approximated accurately.
Compare with other areas it seem flat.

34. Possible enhancements
Implement a iterative algorithm that optimizes the results by combining different values for the different variables.
Length of glottal pulse
number of zeros
number of poles
Try to different approaches using the HF and LPC tools to get a better approximation.
Use homomorphic filtering to remove the zeros and/or glottal flow first.
Use pitch estimation algorithm to better establish the pitch period
Established a better relationship between the zeros and poles in the quefrency domain.

35. Problems Problems in the method:
Requires a good approximation of the area free of zeros
Requires a good approximation of the the number of zeros, poles and length of glottal flow
Requires a good approximation of the all pole approximation
Requires a high sampling rate of the original signal
May not work for high pitch voice

36. Applicatoins Possible applications include:
Speech synthesis : recreate human voice
Speech processing: machine human interaction
Speaker recognition: extraction of key features of the speaker