More On Linear Predictive Analysis

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More On Linear Predictive Analysis

• ??????

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Contents

• Linear Prediction Error
• Computation of the Gain
• Frequency Domain Interpretation of LPC
• Representations of LPC Coefficients
• Direct Representation
• Roots of Predictor Polynomials
• PARCO Coefficients
• Log Area Ratio Coefficients
• Line Spectrum Pair

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• Linear Prediction Error

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LPC Error
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Examples
Could be used for Pitch Detection
Premphasized Speech Signals
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Normalized Mean-Squared Error
General Form
Autocorrelation Method
Covariance Method
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Normalized Mean-Squared Error
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Experimental Evaluation on LPC Parameters
Frame Width
N
Filter Order
p
Conditions
1. Covariance method and autocorrelation method
2. Synthetic vowel and Nature speech
3. Pitch synchronous and pitch asynchronous
analysis
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Pitch Synchronous Analysis
Covariance method is more suitable for pitch
synchronous analysis.
/i/
The frame was beginning at the beginning of a
pitch period.
Why the error increases?
zero the same order as the synthesizer.
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Pitch Asynchronous Analysis
Both covariance and autocorrelation methods
exhibit similar performance.
/i/
Monotonically decreasing
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Frame Width Variation
The errors resulted by covariance and
autocorrelation methods are compatible when N gt
2P.
/i/
Why the errors jump high when the frame size
nears the multiples of pitch period?
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Pitch Synchronous Analysis
Both for synthetic and nature speeches,
covariance method is more suitable for pitch
synchronous analysis.
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Pitch Asynchronous Analysis
Both for synthetic and nature speeches, two
methods are compatible.
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Frame Width Variation
Both for synthetic and nature speeches, the
errors resulted by covariance and autocorrelation
methods are compatible when N gt 2P.
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• Computation of the Gain

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Speech Production Model (Review)
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Speech Production Model (Review)
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Linear Prediction Model (Review)
Linear Prediction
Error compensation
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Speech Production vs. Linear Prediction
Speech production
Vocal Tract
Excitation
ak ?k
Linear Predictor
Error
Linear Prediction
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Speech Production vs. Linear Prediction
Speech production
Linear Prediction
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The Gain
Generally, it is not possible to solve for G in a
reliable way directly from the error signal
itself.
Instead, we assume
Energy of Error
Energy of Excitation
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Assumptions about u(n)
Voiced Speech
This requires that both glottal pulse shape and
lip radiation are lumped into the vocal tract
model.
1/A(z)
Unvoiced Speech
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Gain Estimation for Voiced Speech
Voiced Speech
This requires that both glottal pulse shape and
lip radiation are lumped into the vocal tract
model.
1/A(z)
This requires that p is sufficiently large.
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Gain Estimation for Voiced Speech
1/A(z)
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Correlation Matching
Define
Assumed causal.
Autocorrelation function of the impulse response.
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Correlation Matching
Autocorrelation function of the speech signal
If H(z) correctly model the speech production
system, we should have
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Correlation Matching
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Correlation Matching
Assumed causal.
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Correlation Matching
The same formulation as autocorrelation
method.
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The Gain for voice speech
En
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More on Autocorrelation
Assumed Stationary
Define
The stationary assumption implies
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Properties of LTI Systems
Define
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Properties of LTI Systems
Define
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Properties of LTI Systems
Independent on n
y(n) is also stationary.
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Properties of LTI Systems
l?lk
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Properties of LTI Systems
Define
Estimated from input
Estimated from output
Filter Design
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The Gain for Unvoiced Speech
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The Gain for Unvoiced Speech
?
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The Gain for Unvoiced Speech
Why?
?
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The Gain for Unvoiced Speech
Estimated using rm
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The Gain for Unvoiced Speech
Once again, we have the same formulation as
autocorrelation method.
Furthermore,
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• Frequency Domain Interpretation of LPC

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Spectral Representation of Vocal Tract
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Spectra
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Frequency Domain Interpretation of Mean-Squared
Prediction Error
Parsevals Theorem
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Frequency Domain Interpretation of Mean-Squared
Prediction Error
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Frequency Domain Interpretation of Mean-Squared
Prediction Error
Sn(ej?) gt H(ej?) contributes more to the
total error than Sn(ej?) lt H(ej?).
Hence, the LPC spectral error criterion favors a
good fit near the spectral peak.
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Spectra
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• Representations of LPC Coefficients ---
• Direct Representation

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Direct Representation
Coding ais directly.
G/A(z)
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Disadvantages

• The dynamic ranges of ais is relatively large.
• Quantatization possibly causes instability
  problems.

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• Representations of LPC Coefficients ---
• Roots of Predictor Polynomials

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Roots of the Predictor Polynomial
Coding p/2 zks.
Dynamic range of rks?
Dynamic range of ?ks?
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The Application
Formant Analysis Application.
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Implementation
G/A(z)
Each Stage represents one formant frequency
and its corresponding bandwidth.
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• Representations of LPC Coefficients ---
• PARCO Coefficients

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PARCO Coefficients
Step-Up Procedure
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PARCO Coefficients
Dynamic range of kis?
Step-Down Procedure
where n goes from p to p?1, down to 1 and
initially we set
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• Representations of LPC Coefficients ---
• Log Area Ratio Coefficients

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Log Area Ratio Coefficients
kis Reflection Coefficients
gis Log Area Ratios
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More On Linear Predictive Analysis

• Representations of LPC Coefficients ---
• Line Spectrum Pair

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LPC Coefficients
where m is the order of the inverse filter.
If the system is stable, all zeros of the inverse
filter are inside the unit circle.
Line Spectrum Pair (LSP) is an alternative LPC
spectral representation.
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Line Spectrum Pair

• LSP contains two polynomials.
• The zeros of the the two polynomials have the
  following properties
• Lie on unit circle
• Interlaced
• Through quantization, the minimum phase property
  of the filter is kept.
• Useful for vocoder application.

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Recursive Relation of the inverse filter
where km1 is the reflection coefficient of the
m1th tube.
Special cases
Recall that
Ai1??
ki 1
Ai10
ki ?1
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LSP Polynomials
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Properties of LSP Polynomials
Show that
The zeros of P(z) and Q(z) are on the unit circle
and interlaced.
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Proof
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Proof
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Proof
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Proof
gt0
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Proof
This concludes that the zeros of P(z) and Q(z)
are on the unit circle.
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Proof (interlaced zeros)
Fact
H(z) is an all-pass filter.
One can verify that ?(0) 0
and ?(2?) ?2(m1) ?
Phase
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Proof (interlaced zeros)
zeros of Q(z)
zeros of P(z)
Therefore, z1 is a zero of Q(z).
One can verify that ?(0) 0
and ?(2?) ?2(m1) ?
Phase
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Proof (interlaced zeros)
?(0) 0
?(2?) ?2(m1) ?
Is this possible?
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Proof (interlaced zeros)
?(0) 0
?(2?) ?2(m1) ?
Is this possible?
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Proof (interlaced zeros)
?(0) 0
?(2?) ?2(m1) ?
Group Delay
gt 0
?(?) is monotonically decreasing.
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Proof (interlaced zeros)
?(0) 0
?(2?) ?2(m1) ?
Typical shape of ?(?)
. . .
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Proof (interlaced zeros)
Q(ej?)0
P(ej?)0
Q(ej?)0
P(ej?)0
Q(ej?)0
P(ej?)0
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Proof (interlaced zeros)
There are 2(m1) cross points from 0 ? ? ? ??,
these constitute the 2(m1) interlaced zeros of
P(z) and Q(z).
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Quantization of LSP Zeros
Is such a quantization detrimental?
For effective transmission, we quantize ?is
into several levels, e.g., using 5 bits.
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Minimum Phase Preserving Property
Show that in quantizing the LSP frequencies, the
reconstructed all-pole filter preserves its
minimum phase property as long as the zeros has
the properties shown in the left figure.
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Find the Roots of P(z) and Q(z)
Symmetric
Anti-symmetric
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Find the Roots of P(z) and Q(z)
. . .
. . .
. . .
. . .
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Find the Roots of P(z) and Q(z)
We only need compute the values on 1 ? i ? m/2.
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Find the Roots of P(z) and Q(z)
. . .
. . .
. . .
. . .
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Find the Roots of P(z) and Q(z)
We only need compute the values on 1 ? i ? m/2.
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Find the Roots of P(z) and Q(z)
Both P(z) and Q(z) are symmetric.
P(z)
zero on ?1
Q(z)
zero on 1
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Find the Roots of P(z) and Q(z)
To find its zeros.
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Find the Roots of P(z) and Q(z)
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Find the Roots of P(z) and Q(z)
Define
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Find the Roots of P(z) and Q(z)
. . .
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Find the Roots of P(z) and Q(z)
Consider m10.
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Find the Roots of P(z) and Q(z)
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Find the Roots of P(z) and Q(z)
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Find the Roots of P(z) and Q(z)
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Find the Roots of P(z) and Q(z)
We want to find ?is such that
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Find the Roots of P(z) and Q(z)
Algorithm
We only need to find zeros for this half.
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Find the Roots of P(z) and Q(z)