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 (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 (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):

  • 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):

  • 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 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 equations):

  • The orthogonality principle also states that resulting minimum error is given by

    • Or,

  • We can minimize the error over all time :

    • where


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):

  • 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):

  • 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 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


3.3.1 The LPC Model

Convert this to equality by including an excitation term:


3.3.2 LPC Analysis Equations

The prediction error:

Error transfer function:


3.3.2 LPC Analysis Equations

We seek to minimize the mean squared error signal:


(*)

Terms of short-term covariance:

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

A set of P equations, P unknowns


3.3.2 LPC Analysis Equations

The minimum mean-squared error can be expressed as:


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 Method


3.3.3 The Autocorrelation Method


3.3.3 The Autocorrelation Method


3.3.3 The Autocorrelation Method


3.3.3 The Autocorrelation Method


3.3.4 The Covariance Method


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


3.3.7 LPC Processor for Speech Recognition


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 Recognition

Frame Blocking:


3.3.7 LPC Processor for Speech Recognition

  • Windowing

  • Hamming Window:

  • Autocorrelation analysis


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 Recognition


3.3.7 LPC Processor for Speech Recognition

  • LPC parameter conversion to cepstral coefficients


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 Recognition


3.3.7 LPC Processor for Speech Recognition

  • Temporal cepstral derivative


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


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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