Linear Prediction

1 / 39

# Linear Prediction - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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)

3.3.1 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:

3.3.4 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

3.3.7 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
• 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
• 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
• 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