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Linear Prediction. 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. - PowerPoint PPT Presentation
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Linear PredictionLinear Prediction
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Linear Prediction Linear Prediction (Introduction)(Introduction)::
The object of linear prediction is to The object of linear prediction is to estimate the output sequence from a estimate the output sequence from a linear combination of input samples, linear combination of input samples, past output samples or both :past output samples or both :
The factors The factors a(i)a(i) and and b(j)b(j) are called are called predictor coefficients.predictor coefficients.
p
i
q
j
inyiajnxjbny10
)()()()()(ˆ
3
Linear Prediction Linear Prediction (Introduction)(Introduction)::
Many systems of interest to us are Many systems of interest to us are describable by a linear, constant-describable by a linear, constant-coefficient difference equation :coefficient difference equation :
If If Y(z)/X(z)=H(z),Y(z)/X(z)=H(z), where where H(z)H(z) is a ratio of is a ratio of polynomials polynomials N(z)/D(z),N(z)/D(z), then then
Thus the predicator coefficient given us immediate Thus the predicator coefficient given us immediate access to the poles and zeros of access to the poles and zeros of H(z). H(z).
q
j
p
i
jnxjbinyia00
)()()()(
p
i
iq
j
j ziazDzjbzN00
)()( and )()(
4
Linear Prediction Linear Prediction (Types of (Types of
System Model)System Model):: There are two important variants :There are two important variants :
All-pole model (in statistics, All-pole model (in statistics, autoregressive (AR)autoregressive (AR) model ) : model ) :
The numerator The numerator N(z)N(z) is a constant. is a constant. All-zero model (in statistics, All-zero model (in statistics, moving-moving-
averageaverage (MA) model ) : (MA) model ) : The denominator The denominator D(z)D(z) is equal to unity. is equal to unity.
The mixed pole-zero model is called the The mixed pole-zero model is called the autoregressive moving-averageautoregressive moving-average (ARMA) (ARMA) model.model.
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Linear Prediction Linear Prediction (Derivation of (Derivation of
LP equations)LP equations):: Given a zero-mean signal Given a zero-mean signal y(n), y(n), in the AR in the AR
model :model :
The error is :The error is :
To derive the predicator we use the To derive the predicator we use the orthogonality orthogonality principleprinciple, the principle states that the desired , the principle states that the desired coefficients are those which make the error coefficients are those which make the error orthogonal to the samples orthogonal to the samples y(n-1), y(n-2),…, y(n-p).y(n-1), y(n-2),…, y(n-p).
p
i
inyiany1
)()()(ˆ
p
i
inyia
nynyne
0
)()(
)(ˆ)()(
6
Linear Prediction Linear Prediction (Derivation of (Derivation of
LP equations)LP equations):: Thus we require that Thus we require that
Or,Or,
Interchanging the operation of averaging and Interchanging the operation of averaging and summing, and representing < > by summing summing, and representing < > by summing over n, we haveover n, we have
The required predicators are found by solving The required predicators are found by solving these equations.these equations.
p..., 2, 1,jfor 0)()( nejny
p1,...,j ,0)()()(0
n
p
i
jnyinyia
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Linear Prediction Linear Prediction (Derivation of (Derivation of
LP equations)LP equations):: The orthogonality principle also states that The orthogonality principle also states that
resulting minimum error is given byresulting minimum error is given by
Or,Or,
We can minimize the error over all time :We can minimize the error over all time :
wherewhere
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0
)(
)()()(2 nenyneE
Enyinyian
p
i
)()()(0
, ...,p,jria ji
p
i
21 ,0)(0
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Linear Prediction Linear Prediction (Applications)(Applications):: Autocorrelation matching :Autocorrelation matching :
We have a signal y(n) with known We have a signal y(n) with known autocorrelation . We model this autocorrelation . We model this with the AR system shown below :with the AR system shown below :
)(nryy
p
i
ii zazA
zH
1
1)()(
)(neσ
1-A(z)
)(nz
9
Linear Prediction Linear Prediction (Order of Linear (Order of Linear
Prediction)Prediction):: The choice of predictor order depends on The choice of predictor order depends on
the analysis bandwidth. The rule of thumb is the analysis bandwidth. The rule of thumb is ::
For a normal vocal tract, there is an average of For a normal vocal tract, there is an average of about one formant per kilohertz of BW.about one formant per kilohertz of BW.
One formant require two complex conjugate One formant require two complex conjugate poles.poles.
Hence for every formant we require two Hence for every formant we require two predicator coefficients, or two coefficients per predicator coefficients, or two coefficients per kilohertz of bandwidth.kilohertz of bandwidth.
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Linear Prediction Linear Prediction (AR Modeling of (AR Modeling of
Speech Signal)Speech Signal):: True Model: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 Linear Prediction (AR Modeling of (AR Modeling of
Speech Signal)Speech Signal):: Using LP analysis :Using LP analysis :
DTImpulse
generator
WhiteNoise
generator
All-PoleFilter(AR)
Voiced
Unvoiced
Pitch
Gain
estimate
V
U
H(z)
s(n)
Speech
Signal
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3.3 LINEAR PREDICTIVE CODING 3.3 LINEAR PREDICTIVE CODING MODEL FOR SREECH MODEL FOR SREECH
RECOGNITIONRECOGNITION
)(nu
G
)(zA)(ns
13
3.3.1 The LPC Model3.3.1 The LPC Model
.)(
1
1
1)(
)()(
)()()(
,)()()(
),(...)2()1()(
1
1
1
21
zAzazGUzSzH
zGUzSzazS
nGuinsans
pnsansansans
p
i
ii
p
i
ii
p
ii
p
Convert this to equality by including an excitation term:
14
3.3.2 LPC Analysis Equations3.3.2 LPC Analysis Equations
.1)()()(
)()()()()(
).()(
).()()(
1
1
~
1
~
1
p
k
kk
p
kk
p
kk
p
kk
zazSzEzA
knsansnsnsne
knsans
nGuknsans
The prediction error:
Error transfer function:
15
3.3.2 LPC Analysis Equations3.3.2 LPC Analysis Equations
.)()(
)(
)()()()(
2
1
2
m
p
knknn
mnn
n
n
kmsamsE
meE
mnememnsmS
We seek to minimize the mean squared error signal:
16
pikiai
kmSimSki
kmSimSamsims
pkaE
p
knkn
mnnn
m mnn
p
kknn
k
n
,...,2,1),()0,(
)()(),(
)()()()(
,...,2,1,0
1
1
Terms of short-term covariance:
(*)
With this notation, we can write (*) as:
A set of P equations, P unknowns
17
3.3.2 LPC Analysis Equations3.3.2 LPC Analysis Equations
p
knkn
mnn
p
kk
mnn
ka
kmsmsamsE
1
1
2
).,0()0,0(
)()()(
The minimum mean-squared error can be expressed as:
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3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
.01
),()(),(
01
),()(),(
)(
.,010),().(
)(
)(1
0
1
0
1
0
2
pkpi
kimsmski
pkpi
kmsimski
meE
otherwiseNmmwnms
ms
kiN
mnnn
pN
mnnn
pN
mnn
n
w(m): a window zero outside 0≤m≤N-1The mean squared error is:
And:
19
3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
)(),(
:functionation autocorrel simple toreducesfunction covariance thek,-i offunction aonly is),( Since
.01
),()(),()(1
0
kirki
ki
pkpi
kimsmski
nn
n
kin
mnnn
20
3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
.
)()3()2()1(
)0(...)3()2()1()3(...)0()1()2()2(...)1()0()1()1(...)2()1()0(
:as formmatrix in expressed becan and
1),(|)(|
:)()( i.e. symmetric, isfunction ation autocorrel theSince
2
1
1
prrrr
a
a
a
rprprprprrrrprrrrprrrr
piirakir
sokrkr
n
n
n
n
pnnnn
nnnn
nnnn
nnnn
p
knkn
nn
21
3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
22
3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
23
3.3.3 The Autocorrelation Method3.3.3 The Autocorrelation Method
24
3.3.4 The Covariance Method3.3.4 The Covariance Method
1
1
0
1
0
2
.01
),()(),(
, variablesof changeby or,01
),()(),(
:as defined ),(with
)(
:directlyspeech unweighted theuse to and 10 error to computing of interval thechange
iN
imnnn
n
N
mnn
n
N
mnn
pkpi
kimsmski
pkpi
kmsimski
ki
meE
Nm
25
3.3.4 The Covariance Method3.3.4 The Covariance Method
.
)0,(
)0,3()0,2()0,1(
),()3,()2,()1,(
),3()3,3()2,3()1,3(),2()3,2()2,2()1,2(
),1()3,1()2,1()1,1(
3
2
1
pa
a
a
a
ppppp
pp
p
n
n
n
n
pnnnn
nnnn
nnnn
nnnn
The resulting covariance matrix is symmetric, but not Toeplitz,and can be solved efficiently by a set of techniques called Cholesky decomposition
26
3.3.6 Examples of LPC Analysis3.3.6 Examples of LPC Analysis
27
3.3.6 Examples of LPC Analysis3.3.6 Examples of LPC Analysis
28
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
29
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
).1()()(
.0.19.0,1)(~~
1~
nsansns
azazH
Preemphasis: typically a first-order FIR, To spectrally flatten the signalMost widely the following filter is used:
30
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech RecognitionFrame Blocking:
.1,...,1,01,...,1,0
),()(~
LNn
nMsnx
31
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
Hamming Window:Hamming Window:
,,...,1,0),()()(
.10,1
2cos46.054.0)(
.10),()()(
~1
0
~
~
pmmnxnxmr
NnN
nnw
Nnnwnxnx
mN
n
WindowingWindowing
Autocorrelation Autocorrelation analysisanalysis
32
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
1ifor ommitted is (*)in summation the:note,)1(
1(*),|)(|)(
)0(
)1(2)(
)1()1()(
)(
)1(1
1
)1(
)0(
ii
i
ijii
ij
ij
ii
i
iL
j
iji
EkE
k
k
piEjirirk
rE
LPC Analysis, to find LPC coefficients, LPC Analysis, to find LPC coefficients, reflection coefficients (PARCOR), the log reflection coefficients (PARCOR), the log area ratio coefficients, the cepstral area ratio coefficients, the cepstral coefficients, …coefficients, … Durbin’s methodDurbin’s method
33
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
.11
logtscoefficienratioarealog
tscoefficienPARCOR
1,tscoefficienLPC )(
m
mm
m
pmm
kk
g
k
pma
34
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
LPC parameter conversion to cepstral LPC parameter conversion to cepstral coefficientscoefficients
,
1
model LPCin gain term theis ln
1
1,
,
1
1
220
m
kkmkm
kmk
m
kmm
pmacmkc
pmacmkac
c
35
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
m
mjm
j
m
mjm
j
ecjmes
eceS
)()(log
|)(|log
Parameter weightingParameter weighting Low-order cepstral coefficients are sensitive to overall Low-order cepstral coefficients are sensitive to overall
spectral slopespectral slope High-order cepstral coefficients are sensitive to noiseHigh-order cepstral coefficients are sensitive to noise The weighting is done to minimize these sensitivitiesThe weighting is done to minimize these sensitivities
36
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
,1
).(
,|)(|log
, Qmcwc
jmcc
eces
mmm
mm
m
mjm
j
QmQmQwm
1,sin
21
37
3.3.7 LPC Processor for Speech Recognition3.3.7 LPC Processor for Speech Recognition
))(),...,(),(),(),...,(),((
Finally)()()( :optionallyor
)()()(
dowlength win-finite aover fit polynomial orthogonalan by eApproximat
)(|),(|log
:derivative time theoftion representa seriesFourier
2121 tctctctctctco
KtcKtctc
ktctct
tc
et
tctes
t
QQt
mmm
K
Kkmm
m
mj
m
mj
Temporal cepstral derivativeTemporal cepstral derivative
38
3.3.9 Typical LPC Analysis Parameters3.3.9 Typical LPC Analysis Parameters
N number of samples in the analysis frame N number of samples in the analysis frame M number of samples shift between framesM number of samples shift between framesP LPC analysis orderP LPC analysis orderQ dimension of LPC derived cepstral vectorQ dimension of LPC derived cepstral vectorK number of frames over which cepstral K number of frames over which cepstral
time derivatives are computedtime derivatives are computed
39
333333KK
121212121212QQ
1010101088pp
100 (10 100 (10 msec)msec)
80 (10 80 (10 msec)msec)
100 (15 100 (15 msec)msec)MM
300 (30 300 (30 msec)msec)
240 (30 240 (30 msec)msec)
300 (45 300 (45 msec)msec)NN
parameter kHzFs 67.6 kHzFs 8 kHzFs 10
Typical Values of LPC Analysis Parameters for Typical Values of LPC Analysis Parameters for Speech-Recognition SystemSpeech-Recognition System