Linear prediction
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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.

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Linear Prediction

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Linear prediction

Linear Prediction


Linear prediction introduction

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.


Linear prediction introduction1

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 predicator coefficient given us immediate access to the poles and zeros of H(z).


Linear prediction types of system model

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.


Linear prediction derivation of lp equations

Linear Prediction (Derivation of LP equations):

  • Given a zero-mean signal y(n), in the AR model :

    • The error is :

    • To derive the predicator 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).


Linear prediction derivation of lp equations1

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 predicators are found by solving these equations.


Linear prediction derivation of lp equations2

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


Linear prediction applications

σ

1-A(z)

Linear Prediction (Applications):

  • Autocorrelation matching :

    • We have a signal y(n) with known autocorrelation . We model this with the AR system shown below :


Linear prediction order of linear prediction

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 kilohertz of BW.

    • One formant require two complex conjugate poles.

    • Hence for every formant we require two predicator coefficients, or two coefficients per kilohertz of bandwidth.


Linear prediction ar modeling of speech signal

Linear Prediction (AR Modeling of Speech Signal):

  • True Model:

Pitch

Gain

s(n)

Speech

Signal

DT

Impulse

generator

G(z)

Glottal

Filter

Voiced

U(n)

Voiced

Volume

velocity

H(z)

Vocal tract

Filter

R(z)

LP

Filter

V

U

Uncorrelated

Noise

generator

Unvoiced

Gain


Linear prediction ar modeling of speech signal1

Linear Prediction (AR Modeling of Speech Signal):

  • Using LP analysis :

Pitch

Gain

estimate

DT

Impulse

generator

Voiced

s(n)

Speech

Signal

All-Pole

Filter

(AR)

V

U

White

Noise

generator

Unvoiced

H(z)


3 3 linear predictive coding model for sreech recognition

3.3 LINEAR PREDICTIVE CODING MODEL FOR SREECH RECOGNITION


3 3 1 the lpc model

3.3.1 The LPC Model

Convert this to equality by including an excitation term:


3 3 2 lpc analysis equations

3.3.2 LPC Analysis Equations

The prediction error:

Error transfer function:


3 3 2 lpc analysis equations1

3.3.2 LPC Analysis Equations

We seek to minimize the mean squared error signal:


Linear prediction

(*)

Terms of short-term covariance:

With this notation, we can write (*) as:

A set of P equations, P unknowns


3 3 2 lpc analysis equations2

3.3.2 LPC Analysis Equations

The minimum mean-squared error can be expressed as:


3 3 3 the autocorrelation method

3.3.3 The Autocorrelation Method

w(m): a window zero outside 0≤m≤N-1

The mean squared error is:

And:


3 3 3 the autocorrelation method1

3.3.3 The Autocorrelation Method


3 3 3 the autocorrelation method2

3.3.3 The Autocorrelation Method


3 3 3 the autocorrelation method3

3.3.3 The Autocorrelation Method


3 3 3 the autocorrelation method4

3.3.3 The Autocorrelation Method


3 3 3 the autocorrelation method5

3.3.3 The Autocorrelation Method


3 3 4 the covariance method

3.3.4 The Covariance Method


3 3 4 the covariance method1

3.3.4 The Covariance Method

The resulting covariance matrix is symmetric, but not Toeplitz,

and can be solved efficiently by a set of techniques called

Cholesky decomposition


3 3 6 examples of lpc analysis

3.3.6 Examples of LPC Analysis


3 3 6 examples of lpc analysis1

3.3.6 Examples of LPC Analysis


3 3 7 lpc processor for speech recognition

3.3.7 LPC Processor for Speech Recognition


3 3 7 lpc processor for speech recognition1

3.3.7 LPC Processor for Speech Recognition

Preemphasis: typically a first-order FIR,

To spectrally flatten the signal

Most widely the following filter is used:


3 3 7 lpc processor for speech recognition2

3.3.7 LPC Processor for Speech Recognition

Frame Blocking:


3 3 7 lpc processor for speech recognition3

3.3.7 LPC Processor for Speech Recognition

  • Windowing

  • Hamming Window:

  • Autocorrelation analysis


3 3 7 lpc processor for speech recognition4

3.3.7 LPC Processor for Speech Recognition

  • LPC Analysis, to find LPC coefficients, reflection coefficients (PARCOR), the log area ratio coefficients, the cepstral coefficients, …

  • Durbin’s method


3 3 7 lpc processor for speech recognition5

3.3.7 LPC Processor for Speech Recognition


3 3 7 lpc processor for speech recognition6

3.3.7 LPC Processor for Speech Recognition

  • LPC parameter conversion to cepstral coefficients


3 3 7 lpc processor for speech recognition7

3.3.7 LPC Processor for Speech Recognition

  • Parameter weighting

  • Low-order cepstral coefficients are sensitive to overall spectral slope

  • High-order cepstral coefficients are sensitive to noise

  • The weighting is done to minimize these sensitivities


3 3 7 lpc processor for speech recognition8

3.3.7 LPC Processor for Speech Recognition


3 3 7 lpc processor for speech recognition9

3.3.7 LPC Processor for Speech Recognition

  • Temporal cepstral derivative


3 3 9 typical lpc analysis parameters

3.3.9 Typical LPC Analysis Parameters

N number of samples in the analysis frame

M number of samples shift between frames

P LPC analysis order

Q dimension of LPC derived cepstral vector

K number of frames over which cepstral time derivatives are computed


Typical values of lpc analysis parameters for speech recognition system

N

300 (45 msec)

240 (30 msec)

300 (30 msec)

M

100 (15 msec)

80 (10 msec)

100 (10 msec)

p

8

10

10

Q

12

12

12

K

3

3

3

Typical Values of LPC Analysis Parameters for Speech-Recognition System


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