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

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)

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)