1 Linear Prediction

2 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.

3 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 predictor coefficients give us immediate access to the poles and zeros of H(z).

4 Linear Prediction (Types of System Model) : There are two important variants : (AR)  All-pole model (in statistics, autoregressive (AR) model ) : The numerator N(z) is a constant. (MA)  All-zero model (in statistics, moving-average (MA) model ) : The denominator D(z) is equal to unity. (ARMA)  The mixed pole-zero model is called the autoregressive moving-average (ARMA) model.

5 Linear Prediction (Derivation of LP equations) : Given a zero-mean signal y(n), in the AR model : The error is : To derive the predictor 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).

6 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 predictors are found by solving these equations.

7 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

8 Linear Prediction (Applications) : Autocorrelation matching :  We have a signal y(n) with known autocorrelation. We model this with the AR system shown below : σ 1-A(z)

9 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 kilo Hertz of BW.  One formant requires two complex conjugate poles.  Hence for every formant we require two predictor coefficients, or two coefficients per kilo Hertz of bandwidth.

10 Linear Prediction (AR Modeling of Speech Signal) : True Model: DT Impulse generator G(z) Glottal Filter Uncorrelated Noise generator H(z) Vocal tract Filter R(z) LP Filter Voiced Unvoiced Pitch Gain V U U(n) Voiced Volume velocity s(n) Speech Signal

11 Linear Prediction (AR Modeling of Speech Signal) : Using LP analysis : DT Impulse generator White Noise generator All-Pole Filter (AR) Voiced Unvoiced Pitch Gain estimate V U H(z) s(n) Speech Signal