Linear Prediction

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

sn
en
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.

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

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Second Order Example
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(No Transcript)
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z-plane plot
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In General
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Complex Conjugate Poles
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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 ß

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Noisy Signals
s exp(-50t).(cos(2pi100t)
0.1randn(1,length(t)))
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Parameter Estimation of Noisy Signals

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

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Parameter Estimation in Noise
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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 lets look at the actions of the larynx