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 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. 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 : 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 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). Thus the predicator coefficient given us immediate access to the poles and zeros of H(z).

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

6 Linear Prediction (Derivation of LP equations) : Thus we require that Thus we require that Or, Or, Interchanging the operation of averaging and summing, and representing by summing over n, we have 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 required predicators 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 The orthogonality principle also states that resulting minimum error is given by Or, Or, We can minimize the error over all time : We can minimize the error over all time : where where

8 Linear Prediction (Applications) : Autocorrelation matching : Autocorrelation matching : We have a signal y(n) with known autocorrelation. We model this with the AR system shown below : 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 : 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. For a normal vocal tract, there is an average of about one formant per kilohertz of BW. One formant require two complex conjugate poles. One formant require two complex conjugate poles. Hence for every formant we require two predicator coefficients, or two coefficients per kilohertz of bandwidth. Hence for every formant we require two predicator coefficients, or two coefficients per kilohertz of bandwidth.

10 Linear Prediction (AR Modeling of Speech Signal) : True Model: 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 : 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

LINEAR PREDICTIVE CODING MODEL FOR SREECH RECOGNITION

The LPC Model Convert this to equality by including an excitation term:

LPC Analysis Equations The prediction error: Error transfer function:

LPC Analysis Equations We seek to minimize the mean squared error signal:

16 Terms of short-term covariance: (*) With this notation, we can write (*) as: A set of P equations, P unknowns

LPC Analysis Equations The minimum mean-squared error can be expressed as:

The Autocorrelation Method w(m): a window zero outside 0≤m≤N-1 The mean squared error is: And:

The Autocorrelation Method

The Autocorrelation Method

The Autocorrelation Method

The Autocorrelation Method

The Autocorrelation Method

The Covariance Method

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

Examples of LPC Analysis

Examples of LPC Analysis

LPC Processor for Speech Recognition

LPC Processor for Speech Recognition Preemphasis: typically a first-order FIR, To spectrally flatten the signal Most widely the following filter is used:

LPC Processor for Speech Recognition Frame Blocking:

LPC Processor for Speech Recognition Hamming Window: Hamming Window: Windowing Windowing Autocorrelation analysis Autocorrelation analysis

LPC Processor for Speech Recognition LPC Analysis, to find LPC coefficients, reflection coefficients (PARCOR), the log area ratio coefficients, the cepstral coefficients, … LPC Analysis, to find LPC coefficients, reflection coefficients (PARCOR), the log area ratio coefficients, the cepstral coefficients, … Durbin’s method Durbin’s method

LPC Processor for Speech Recognition

LPC Processor for Speech Recognition LPC parameter conversion to cepstral coefficients LPC parameter conversion to cepstral coefficients

LPC Processor for Speech Recognition Parameter weighting Parameter weighting Low-order cepstral coefficients are sensitive to overall spectral slope Low-order cepstral coefficients are sensitive to overall spectral slope High-order cepstral coefficients are sensitive to noise High-order cepstral coefficients are sensitive to noise The weighting is done to minimize these sensitivities The weighting is done to minimize these sensitivities

LPC Processor for Speech Recognition

LPC Processor for Speech Recognition Temporal cepstral derivative Temporal cepstral derivative

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

39 333K Q 10108p 100 (10 msec) 80 (10 msec) 100 (15 msec) M 300 (30 msec) 240 (30 msec) 300 (45 msec) N Typical Values of LPC Analysis Parameters for Speech- Recognition System