1 linear prediction. outline windowing lpc introduction to vocoders excitation modeling pitch...
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Linear Prediction
Outline Windowing LPC Introduction to Vocoders Excitation modeling
Pitch Detection
Short-Time Processing Speech signal is inherently non-stationary For continuant phonemes there are stationary periods of
at least 20-25ms The short-time speech frames are assumed stationary The frame length should be chosen to include just one
phoneme or allophone Frame lengths are usually chosen to be between 10-
50ms We consider rectangular and Hamming windows here
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Rectangular Window
Hamming Window
Comparison of Windows
Comparison of Windows (cont’d)
Linear Prediction Coding (LPC) Based on all-pole model for speech production system:
In time domain, we get:
In other words, we can predict s[n] as a function of p previous signal samples (and the excitation).
The set of {ak} is one way of representing the time varying filter. There are many other ways to represent this filter (e.g., pole value, Lattice filter value, LSP, …).
p
k
kk za
AzH
1
.1)(
][][.][1
nAuknsans g
p
kk
LPC parameter estimation There are many methods to estimate the
LPC parameters:Autocorrelation method: results in the
optimization of a in a set of p linear equations. Covariance method
Procedures (such as Levinson-Durbin, Burg, Le Roux) obtain efficient estimation of these parameters.
LPC Parameters in Coding (vocoders)
DT impulse
generator
G(z)glottalfilter
whitenoise
generator
H(z)vocal tract
filter
R(z)lip radiation
filter
s(n)speechsignal
voiced
unvoiced
Θ0
gain
Θ0
gain
Pitch period, P
DT impulse
generator
whitenoise
generator
all-polefilter
s(n)speechsignal
voiced
unvoicedΘ0
gain
Pitch period, P
V
UV
V
UV
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Linear Prediction (Introduction): The object of linear prediction is to
estimate the output sequence from a linear combination of input samples, past output samples or both :
The factors a(i) and b(j) are called predictor coefficients.
p
i
q
j
inyiajnxjbny10
)()()()()(ˆ
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Linear Prediction (Introduction): Many systems of interest to us are describable
by a linear, constant-coefficient difference equation :
If Y(z)/X(z)=H(z), where H(z) is a ratio of polynomials N(z)/D(z), then
Thus the predictor coefficients give us immediate access to the poles and zeros of H(z).
q
j
p
i
jnxjbinyia00
)()()()(
p
i
iq
j
j ziazDzjbzN00
)()( and )()(
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Linear Prediction (Types of System Model): There are two important variants :
All-pole model (in statistics, autoregressive (AR) model ) :
The numerator N(z) is a constant.All-zero model (in statistics, moving-average
(MA) model ) : The denominator D(z) is equal to unity.
The mixed pole-zero model is called the autoregressive moving-average (ARMA) model.
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Linear Prediction (Derivation of LP equations): Given a zero-mean signal y(n), in the AR
model :
The error is :
To derive the predictor we use the orthogonality principle, the principle states that the desired coefficients are those which make the error orthogonal to the samples y(n-1), y(n-2),…, y(n-p).
p
i
inyiany1
)()()(ˆ
p
i
inyia
nynyne
0
)()(
)(ˆ)()(
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Linear Prediction (Derivation of LP equations):Thus we require that
Or,
Interchanging the operation of averaging and summing, and representing < > by summing over n, we have
The required predictors are found by solving these equations.
p..., 2, 1,jfor 0)()( nejny
0)()()(0
p
i
inyiajny
p1,...,j ,0)()()(0
n
p
i
jnyinyia
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Linear Prediction (Derivation of LP equations): The orthogonality principle also states that resulting
minimum error is given by
Or,
We can minimize the error over all time :
where
Eriap
ii
0
)(
)()()(2 nenyneE
Enyinyian
p
i
)()()(0
n
i inynyr )()(
, ...,p,jria ji
p
i
21 ,0)(0
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Linear Prediction (Applications): Autocorrelation matching :
We have a signal y(n) with known autocorrelation . We model this with the AR system shown below :
)(nryy
p
i
ii za
zAzH
1
1)(
)(
)(neσ
1-A(z)
)(ny
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Linear Prediction (Order of Linear Prediction): The choice of predictor order depends on
the analysis bandwidth. The rule of thumb is :
For a normal vocal tract, there is an average of about one formant per kilo Hertz of BW.
One formant requires two complex conjugate poles.
Hence for every formant we require two predictor coefficients, or two coefficients per kilo Hertz of bandwidth.
cBW
p 1000
2
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Linear Prediction (AR Modeling of Speech Signal): True Model:
DTImpulse
generator
G(z)GlottalFilter
UncorrelatedNoise
generator
H(z)Vocal tract
Filter
R(z)LP
Filter
Voiced
Unvoiced
Pitch Gain
Gain
V
U
U(n)
Voiced
Volume
velocity
s(n)
Speech
Signal
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Linear Prediction (AR Modeling of Speech Signal): Using LP analysis :
DTImpulse
generator
WhiteNoise
generator
All-PoleFilter(AR)
Voiced
Unvoiced
Pitch
Gain
estimate
V
U
H(z)
s(n)
Speech
Signal
Introduction to Vocoders
Beside the estimation of the vocal tract parameters, a vocoder needs excitation estimation.
In early vocoders, this has been achieved by the estimation of V/UV, pitch, and gain.
More modern vocoders involve more sophisticated estimation of the excitation, such as in CELP, where vector quantization is used.
vocoderanalysis
Channel(or storage)
vocodersynthesizer
ŝ(n)synthesized
speechsignal
V/UVpitch
filter parameterss(n)original speechsignal
Pitch Detection
Because speech signal in voiced frames is quasi-periodic (and not fully periodic), the pitch detection is not always easy.
Especially in some phonemes that manifest less periodic behavior, pitch detection is difficult.
Some pitch detection methods:AMDF (Average Magnitude Difference Function)Autocorrelation with center clipping