Linear Prediction Simple first- and second-order systems Inputs/outputs Estimating filter coefficients Z-transforms and characteristic polynomials Inverse filtering LP Analysis of Speech The LP model

First Order Model An exponential signal/system en sn z -1 a1 sn-1

Estimation Assume zero-input and zero-output prior to time 0 Apply an impulse at the input at time 0 This “excites” the system Measure output at any time after time 0

Z-transform Convert signals to functions of z Delays are represented as power of z

Characteristic Polynomial The numerator of the transfer function H(z) is known as the characteristic polynomial Set it equal to zero, find the roots and plot them Using an “x”, plot the root(s) on the complex plane which has also has a unit circle marked A first order system will have a real root, but can be +ve or -ve.

The z-plane

Second Order Model en sn z -1 a1 sn-1 z -2 a2

Second Order Model How can we estimate the ai coefficients? Assume zero-input and zero-output prior to time 0 Apply an impulse at the input at time 0 This “excites” the system Measure outputs at any times after time 0

Second Order Example

z-plane plot

In General

Complex Conjugate Poles

Exponentially Decaying Sinusoids The frequency of oscillation f is proportional to the angle the poles make with the real axis The magnitude of the roots is inversely related to the rate of decay β:

Noisy Signals s = exp(-50*t).*(cos(2*pi*100*t) +0.1*randn(1,length(t)));

Parameter Estimation of Noisy Signals We overdetermine our system of linear equations Use Least Squares estimation (i.e. we clculate the pseudoinverse)

Parameter Estimation in Noise

Higher Order Signals If we model a signal as being the sum of two exponentially decaying sinusoids, then we allow each to have a pair of complex conjugate poles This requires a 4th-order charactyeristic polynomial We model any signal value as being a weighted sum of the previous 4 signal values

Real Speech Given a signal modelled as the sum of m exponentially decaying sinusoids, we will need 2m LP coefficients We model speech as having approximately 1 formant per 1kHz So given a sampling frequency of 10kHz, we would expect to see 5 formants - i.e. we model the speech as being the sum of 5 exponentially decaying sinusoids This implies an analysis order of 10 In fact we usually add 2 or 3 to this figure To see why let’s look at the actions of the larynx