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Linear Prediction

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Linear Prediction

- 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.

- 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).

- 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.

- All-pole model (in statistics, autoregressive (AR) model ) :

- 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).

- 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.

- 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)

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

- 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.

- 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

- 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)

Convert this to equality by including an excitation term:

The prediction error:

Error transfer function:

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

The minimum mean-squared error can be expressed as:

w(m): a window zero outside 0≤m≤N-1

The mean squared error is:

And:

The resulting covariance matrix is symmetric, but not Toeplitz,

and can be solved efficiently by a set of techniques called

Cholesky decomposition

Preemphasis: typically a first-order FIR,

To spectrally flatten the signal

Most widely the following filter is used:

Frame Blocking:

- Windowing

- Hamming Window:

- Autocorrelation analysis

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

- Durbin’s method

- LPC parameter conversion to cepstral coefficients

- 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

- Temporal cepstral derivative

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