Advanced Digital Signal Processing

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Advanced Digital Signal Processing
• Lecture 11
• LPC. Cepstral Analysis.
• Alexander Kolesnikov

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Speech Production
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Speech production mechanism
Vocal tract pharengeal and oral cavities Nasal
tract nasal cavities Articulators vocal folds
(cords), soft palate (or velum), tongue, teeth
and lips.
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Speech production as filtering
Speech production can be thought as an acoustic
filtering operation 1) Excitation (input) is
provided by the organs below the vocal
tract (e.g., larynx, lungs) 2) Filter vocal
and nasal tracts. Articulators change the shape
of the filter.
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Simplified model of speech production
x(n)
h(n)
u(n)
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Linear Predictive Coding
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LPC model
Given speech sample at time n, x(n), can be
approximated as a linear combination of the past
p speech samples, such that
This approach is based on model of speech
production
u(n) is a normalized excitation, G is the gain
of the excitation.
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Transfer function
Recursive equation
Z-transform
Transfer function
H(z) is all-pole IIR filter.
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Prediction error
The linear combination of p past speech samples
is the estimate or prediction
Prediction error
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Model learning
The basic problem of linear prediction is to
determine the set of predictor coefficients
a1,,ap, directly from the speech signal
Find such a set of predictor coefficients
a1,,ap that minimize the mean-squared
prediction error
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Minimum of the prediction error
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Correlation function
where K(i,j) is correlation function
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Yule-Walker equations
Mean squared prediction error
Solution is given by the system of linear
equations
Yule-Walker equations
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Theory
Yule-Walker equations
Prediction error
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Practice
In the absence of of knowledge about probability
distribution... Estimate correlation function
over a short segment of speech waveform.
1) Autocorrelation method 2) Covariance method
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1) Autocorrelation method
Signal is windowed in order to minimize
discontinuities at beginning and end of the
interval
w(n) is a finite length window of length N
samples.
n0
nN-1
n0
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1) Autocorrelation method
This is Toeplitz matrix. The system has fast
algorithm for solution (see Levinson-Durbin
recursive algorithm, O(N2) operations)
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2) Covariance method
Fix the interval over which the mean-squared
error is computed
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2) Covariance method
The matrix is still symmetric, but is no longer
Toeplitz. The matrix inversion O(N3) operations.
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Three methods
1) The autocorrelation method 2) The covariance
method 3) The lattice method.
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Example
Covariation and autocorrelation methods
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Normalized prediction error
where
Prediction error as a function of predictor order
p.
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Spectral analysis via LPC
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Spectrogram
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Spectrogram with LPC roots
rk are poles.
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Cepstral Analysis
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A homorphic transformation
A homorphic transformation y(n)D(x(n)) is a
transformation that converts a convolution
into a sum
In simple words separate (high frequency) source
from the (low frequency) filter
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Fourier Transform
Given a discrete-time signal x(n).
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Cepstral transform
Speech
Cepstrum
DFT
Window
log
IDFT
Quefrency
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Example 1 Signal
t 00.011.27 x0 sin(2pi45t) x x0
0.5zeros(1,20) x0(1108) f145 Hz f0100
Hz ?t0.2 sec
Time, sec
0.2 sec
Signal contains an echo, with 0.5 amplitude, 0.2
seconds after the beginning of the signal.
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Example 1 DFT
Time, sec
S fft(x,128) f f0(063)/128 plot(f,abs(S(1
64)))
Frequency, Hz
45 Hz
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Example 1 Cepstrum
Frequency, sec
C cceps(x) plot(t,C)
Quefrency, sec
Cepstrum extracts periodicity in spectrum
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Example 1 How?
DFT
Time, sec
Frequency, Hz
IDFT
semilogy(f,abs(S(164)))
echo?oscilations in S(f)?peaks in C(q)
Quefrency, sec
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Example 2 Signal and DFT
Time, sec
yhamming(512) . x
Sabs(fft(y))
Frequency, Hz
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Example 2 DFT and Cepstrum
Quefrency, sec
Frequency, Hz
Cifft(log(S))
Sfft(y)
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Example 2 Cepstrum
The first 20 cepstral coefficients
The lower numbered coefficients provide the
envelope information. The remainder of the
detail is mostly contained in the peaks which
are separated by the pitch period ( 70 sample)
and provide the fine detail pitch information.
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Example Cepstrum of phones
Quefrency, sec
Quefrency, sec
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LPC ? Cepstrum
Transfer function of LPC filter
Take the logarithm
where coefficients h(n) can be obtained by the
recursion