SLIDE 1
The Prediction Error Signal
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Prediction Error Signal Behavior
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The Prediction Error Signal 1 Prediction Error Signal Behavior 2 - - PowerPoint PPT Presentation
The Prediction Error Signal 1 Prediction Error Signal Behavior 2 - - PowerPoint PPT Presentation
The Prediction Error Signal 1 Prediction Error Signal Behavior 2 LP Speech Analysis file:s5, ss:11000, frame size (L):320, lpc order (p):14, cov method Top panel: speech signal Second panel: error signal Third panel: log magnitude
SLIDE 2
SLIDE 3
LP Speech Analysis
Top panel: speech signal Second panel: error signal Third panel: log magnitude spectra of signal and LP model Fourth panel: log magnitude spectrum of error signal
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file:s5, ss:11000, frame size (L):320, lpc order (p):14, cov method
SLIDE 4
LP Speech Analysis
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Top panel: speech signal Second panel: error signal Third panel: log magnitude spectra of signal and LP model Fourth panel: log magnitude spectrum of error signal
file:s5, ss:11000, frame size (L):320, lpc order (p):14, ac method
SLIDE 5
LP Speech Analysis
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Top panel: speech signal Second panel: error signal Third panel: log magnitude spectra of signal and LP model Fourth panel: log magnitude spectrum of error signal
file:s3, ss:14000, frame size (L):160, lpc order (p):16, cov method
SLIDE 6
LP Speech Analysis
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Top panel: speech signal Second panel: error signal Third panel: log magnitude spectra of signal and LP model Fourth panel: log magnitude spectrum of error signal
file:s3, ss:14000, frame size (L):160, lpc order (p):16, ac method
SLIDE 7
Properties of the LPC Polynomial
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SLIDE 8
Minimum-Phase Property of A(z)
Proof: Assume that is a zero (root) of A(z) The minimum mean-squared error is Thus, A(z) could not be the optimum filter because we could replace z0 by and decrease the error
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SLIDE 9
PARCORs and Stability
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It is easily shown that –ki is the coefficient of z-i in A(i)(z), i.e., Therefore If |ki |1, then either all the roots must be on the unit circle or at least one of them must be outside the unit circle
- |ki |<1 is a necessary and sufficient condition for A(z) to be a
minimum phase system and 1/A(z) to be a stable system Proof:
SLIDE 10
Root Locations for Optimum LP Model
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SLIDE 11
Pole-Zero Plot for Model
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SLIDE 12
Pole Locations
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SLIDE 13
Pole Locations (FS=10,000 Hz)
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SLIDE 14
Estimating Formant Frequencies
- compute A(z) and factor it
- find roots that are close to the unit circle.
- compute equivalent analog frequencies from the angles of the
roots.
- plot formant frequencies as a function of time.
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SLIDE 15
Spectrogram with LPC Roots
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SLIDE 16
Spectrogram with LPC Roots
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SLIDE 17
Alternative Representations of the LP Parameters
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SLIDE 18
LP Parameter Sets
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SLIDE 19
PARCOR
- PARCORs to Prediction Coefficients
– assume that ki, i=1,2, …, p are given. Then we can skip the computation
- f ki in the Levinson recursion.
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SLIDE 20
PARCOR
- Prediction Coefficients to PARCORs
– assume that j, j=1,2, …, p are given. Then we can work backwards through the Levinson Recursion.
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SLIDE 21
Log Area Ratio
- log area ratio coefficients from PARCOR coefficients
with inverse relation
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SLIDE 22
Roots of Predictor Polynomial
- roots of the predictor polynomial
where each root can be expressed as a z-plane i.e.,
- important for formant estimation
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SLIDE 23
Impulse Response of H(z)
- IR of all pole system
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SLIDE 24
LP Cepstrum
- cepstrum of IR of overall LP system from predictor coefficients
- predictor coefficients from cepstrum of IR
where
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SLIDE 25
Autocorrelation of IR
- autocorrelation of IR
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SLIDE 26
Autocorrelation of Predictor Polynomial
- autocorrelation of the predictor polynomial
with IR of the inverse filter with autocorrelation
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SLIDE 27
Line Spectral Pairs
- Quantization of LP Parameters
- consider the magnitude-squared of the model frequency
response where g is a parameter that affects P.
- spectral sensitivity can be defined as
which measures sensitivity to errors in the gi parameters
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SLIDE 28
Line Spectral Pairs
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spectral sensitivity for ki parameters; low sensitivity around 0; high sensitivity around 1 spectral sensitivity for log area ratio parameters, gi – low sensitivity for virtually entire range is seen
SLIDE 29
Line Spectral Pairs
- Consider the following
- Form the symmetric polynomial P(z) as
- Form the anti-symmetric polynomial Q(z) as
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SLIDE 30
LSP Example
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Line Spectral Pairs
- properties of LSP parameters
- 1. all the roots of P(z) and Q(z) are on the unit circle
- 2. a necessary and sufficient condition for |ki |< 1, i = 1, 2, …,
p is that the roots of P(z) and Q(z) alternate on the unit circle
- 3. the LSP frequencies get close together when roots of A(z)
are close to the unit circle
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SLIDE 32
Applications
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SLIDE 33
Speech Synthesis
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SLIDE 34
Speech Coding
- 1. Extract αk parameters properly
- 2. Quantize αk parameters properly so that there is little
quantization error – Small number of bits go into coding the αk coefficients
- 3. Represent e(n) via:
– Pitch pulses and noise—LPC Coding – Multiple pulses per 10 msec interval—MPLPC Coding – Codebook vectors—CELP
- Almost all of the coding bits go into coding of e(n)
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SLIDE 35
LPC Vocoder
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