1. 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

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

3. 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

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

5. 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.

6. The z-plane

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

8. 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

9. Second Order Example

11. z-plane plot

12. In General

13. Complex Conjugate Poles

14. 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:

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

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

17. Parameter Estimation in Noise

18. 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

19. 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