What is Linear Prediction?
Basis for most modern coding algorithms (across channels e.g. zoom, phones, military)
Consists of:
- Analysis (determine parameters)
- Synthesis (re-synthesising speech based on estimated parameters)
Aim of Analysis
- Assumes each sample of a speech signal can be predicted from a weighted sum of the P preceding samples
- ŝ[k] = sum ( a(i) * s[k-i] )
where ai are prediction coefficients - Exploits stationary assumption, allows a speech frame of several hundred samples to be represented using 10-15 prediction coefficients (incredible compression!)
- Outputs error signals and weights
Prediction Error Function
Investigates relationship between samples, parameters and the error
Analysis Transfer Function
e[k] = s[k] - ŝ[k] = s[k] - prediction coefficients
where:
- e[k] = error
- s[k] = sample
- ŝ[k] = prediction
- prediction coefficients = sum ( a(i) s[k-i] )
Z-Transform of both sides:
E[z] = [1 - sum ( a(i) * z^-i ) ] S[z]
where:
- E[z] is error in z-space
- S[z] is sequence in z-space
Rearrange as ratio of output to input:
H[z] = E[z] / S[z]
Describes what exactly error filter is doing. In this case, it is an all-zero filter
Calculating Prediction Error
- Optimal coefficients found by minimising prediction error
- Standard cost function is squared error so:
E = sum ( s[k] - sum (a(i) * s[k-i]) )^2
Optimising Prediction Coefficients
- Want to minimise prediction error by solving δE/δa(i) = 0 for all i
- Want error and weights to be uncorrelated
- Uncorrelated noise over time is white noise
Aim of Synthesis
Put into inverse of analysis filter (synthesis filter) to rebuild signal (not quite the same as original signal)
Analysis Filter
- All-zero filter
- Stable, doesn’t use previous outputs as inputs
- Finite Impulse Response (FIR) filter
Synthesis Filter
- All-pole filter (inverse of analysis)
- Unstable, uses previous outputs as inputs
- Infinite Impulse Response (IIR) filter
Analysis Stage Outputs
- Error over time (error signal)
- Can transmit only error signal and weights which is smaller than entire signal
- Error signal = source
- Weights = vocal tract filter
