Linear Predictive Analysis 主講人：虞台文

Contents Introduction Basic Principles of Linear Predictive Analysis The Autocorrelation Method The Covariance Method More on the above Methods Solution of the LPC Equations Lattice Formulations

Linear Predictive Analysis Introduction

Linear Predictive Analysis A powerful speech analysis technique. Powerful for estimating speech parameters. – Pitch – Formants – Spectra – Vocal tract area functions Especially useful for compression. Linear predictive analysis techniques is often referred as linear predictive coding or LPC.

Basic Idea A speech sample can be approximated as a linear combination of past speech samples. This prediction model corresponds to an all- zero model, whose inverse matches the vocal tract model we developed.

LPC vs. System Identification The linear prediction have been in use in the areas of control, and information theory under the name of system estimation and system identification. Using LPC methods, the result system will be modeled as an all-pole linear system.

LPC to Speech Processing Modeling the speech waveform. There are different formulations. The differences among them are often those of philosophy or way of viewing the problem. They almost lead to the same result.

Formulations The covariance method The autocorrelation formulation The lattice method The inverse filter formulation The spectral estimation formulation The maximum likelihood estimation The inner product formulation

Linear Predictive Analysis Basic Principles of Linear Predictive Analysis

Speech Production Model Impulse Train Generator Impulse Train Generator Random Noise Generator Random Noise Generator Time-Varying Digital Filter Time-Varying Digital Filter Vocal Tract Parameters G u(n)u(n) s(n)s(n)

H(z)H(z) Speech Production Model Impulse Train Generator Impulse Train Generator Random Noise Generator Random Noise Generator Time-Varying Digital Filter Time-Varying Digital Filter Vocal Tract Parameters G u(n)u(n) s(n)s(n)

Linear Prediction Model Linear Prediction: Error compensation:

Speech Production vs. Linear Prediction Speech production: Linear Prediction: Vocal Tract Excitation Linear Predictor Error a k =  k

Prediction Error Filter Linear Prediction:

Prediction Error Filter s(n)s(n)e(n)e(n)

s(n)s(n)e(n)e(n) Goal: Minimize

Prediction Error Filter Goal: Minimize Suppose that c ij ’s can be estimated from the speech sample. Our goal now is to find a k ’s to minimize the sum of squared errors.

Prediction Error Filter Goal: Minimize Fact: c ij = c ji Let and solve the equations. i = k j = k

Prediction Error Filter i = k j = k k = 1: k = 2: k = p: Fact: c ij = c ji

=1 Prediction Error Filter Fact: c ij = c ji k = 1: k = 2: k = p:

=1 Prediction Error Filter Fact: c ij = c ji k = 1: k = 2: k = p:

Prediction Error Filter Fact: c ij = c ji Remember this equation

Prediction Error Filter Fact: c ij = c ji Such a formulation in fact is unrealistic. Why?

Error Energy =0, i  0

Short-Time Analysis Original Goal: Minimize Vocal tract is a slowly time-varying system. Minimizing the error energy for whole speech signal is unreasonable.

Short-Time Analysis Original Goal: Minimize n New Goal: Minimize

Short-Time Analysis n New Goal: Minimize

Linear Predictive Analysis The Autocorrelation Method

n Usually, we use a Hamming window.

The Autocorrelation Method Error energy 0 N1N1 So, the original formulation can be directly applied to find the prediction coefficients.

The Autocorrelation Method What properties they have? For convenience, I’ll drop the sup/subscripts n in the following discussion.

Properties of c ij ’ s Property 1: Property 2: Its value depends on the difference |i  j|.

The Equations for the Autocorrelation Methods

A Toeplitz Matrix

The Error Energy

Linear Predictive Analysis The Covariance Method

n Goal: Minimize

The Covariance Method n Goal: Minimize The range for evaluating error energy is different from the autocorrelation method.

The Covariance Method Goal: Minimize c ij

The Covariance Method or Property:

The Covariance Method or 0 N1N1 ij ii Ni1Ni1

The Covariance Method or 0 N1N1 ij ii Ni1Ni1 c ij is, in fact, a cross-correlation function. The samples involved in computation of c ij ’s are values of s n (m) in the interval  p  m  N  1. The value of c ij depends on both i and j.

The Equations for the Covariance Methods Symmetric but not Toeplitz

The Error Energy

Linear Predictive Analysis More on the above Methods

The Equations to be Solved The Autocorrelation Method The Covariance Method

 n and  n for the Autocorrelation Method Define  n is positive definite. Why?

 n and  n for the Covariance Method Define  n is positive definite. Why?

Linear Predictive Analysis Solution of the LPC Equations

Covariance Method--- Cholesky Decomposition Method Also called the square root method.    Symmetric and positive definite.

Covariance Method--- Cholesky Decomposition Method A lower triangular matrix A diagonal matrix

Covariance Method--- Cholesky Decomposition Method Y Y can be recursively solved.

Covariance Method--- Cholesky Decomposition Method Y

How?

=1 Covariance Method--- Cholesky Decomposition Method Consider diagonal elements

Covariance Method--- Cholesky Decomposition Method =1

Covariance Method--- Cholesky Decomposition Method =1 The story is, then, continued.

Covariance Method--- Cholesky Decomposition Method Error Energy

Autocorrelation Method--- Durbin ’ s Recursive Solution The recursive solution proceeds in steps. In each step, we already have a solution for a lower order predictor, and we use that solution to compute the coefficients for the higher order predictor.

Autocorrelation Method--- Durbin ’ s Recursive Solution Notations: Coefficients for the n th order predictor: Error energy for the n th order predictor: The Toeplitz matrix for the n th order predictor:

Autocorrelation Method--- Durbin ’ s Recursive Solution The equation for the autocorrelation method: How the procedure proceeds recursively?

Permutation Matrix Row inversing Column inversing

Property of a Toeplitz Matrix A Toeplitz Matrix

Autocorrelation Method--- Durbin ’ s Recursive Solution

This is what we want.

Autocorrelation Method--- Durbin ’ s Recursive Solution

=0

Autocorrelation Method--- Durbin ’ s Recursive Solution

What can you say about k n ?

Autocorrelation Method--- Durbin ’ s Recursive Solution Summary: Construct a p th order linear predictor. Step1. Compute the values of r 0, r 1, , r p. Step2. Set E (0) = r 0. Step3. Recursively compute the following terms from n=1 to p.

Linear Predictive Analysis Lattice Formulations

The Steps for Finding LPC Coefficients Both the covariance and the autocorrelation methods consist of two steps: – Computation of a matrix of correlation values. – Solution of a set of linear equations. Lattice method: – Combine them into one.

The Clue from Autocorrelation Method Consider the system function of an n th order the linear predictor. The recursive relation from autocorrelation method:

The Clue from Autocorrelation Method Change index i  n  i A ( n  1) ( z )

The Clue from Autocorrelation Method A ( n  1) ( z  1 )

Interpretation e ( n  1) ( m ) b ( n  1) ( m  1 )

order n  1 Interpretation Forward Prediction Error Filter What is this?... s(m)s(m) s(m1)s(m1) s(m2)s(m2) s(m3)s(m3) s(m  n+3) s(m  n+2) s(m  n+1) s(mn)s(mn)

order n  1... s(m)s(m) s(m1)s(m1) s(m2)s(m2) s(m3)s(m3) s(m  n+3) s(m  n+2) s(m  n+1) s(mn)s(mn) Interpretation Backward Prediction Error Filter

Backward Prediction Defined Define

Backward Prediction Defined Define

Forward Prediction vs. Backward Prediction Define

The Prediction Errors The forward prediction error The backward prediction error

The Lattice Structure z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 kpkp z1z1

k i =? Throughout the discussion, we have assumed that k i ’s are the same as that developed for the autocorrelation method. So, k i ’s can be found using the autocorrelation method. z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 kpkp z1z1

Another Approach to Find k i ’s For the n th order predictor, our goal is to minimize So, we want to minimize z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 kpkp z1z1

Another Approach to Find k i ’s So, we want to minimize Set

Another Approach to Find k i ’s Set Fact:

PARCOR kpkp CORR k1k1 s(m)s(m) z1z1 k2k2 z1z1 z1z1 z1z1 Given k n ’s, can you find  i ’s?

All-Pole Lattice z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 kpkp z1z1

z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 knkn z1z1

z1z1 s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 knkn z1z1 knkn  k n z1z1

All-Pole Lattice knkn  k n z1z1 k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 z1z1 kpkp z1z1 kp1kp1 kp1kp1 11

All-Pole Lattice e(m)e(m) s(m)s(m) k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 z1z1 kpkp z1z1 kp1kp1 kp1kp1 11

Comparison k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 z1z1 kpkp z1z1 kp1kp1 kp1kp1 e(m)e(m)s(m)s(m) PARCOR k1k1 k1k1 z1z1 k2k2 k2k2 z1z1 z1z1 kpkp z1z1 kp1kp1 kp1kp1 11

Normalize Lattice knkn  k n z1z1 knkn knkn z1z1 Section n

Normalize Lattice Section 1 Section 1 Section 2 Section 2 Section p Section p knkn knkn z1z1 Section n 11

Normalize Lattice

Three multiplier form knkn  k n z1z1

Normalize Lattice Three multiplier form Let Four multiplier form knkn  k n z1z1 z1z1

Normalize Lattice Kelly-Lochbaum form knkn  k n z1z1 z1z1 knkn z1z1

Normalize Lattice Section 1 Section 1 Section 2 Section 2 Section p Section p 11 knkn  k n z1z1 z1z1 knkn z1z1