Preprocessing - Non-linear Filtering

Question 1

What is a recursive filter?

Answer 1

Re-uses one or more of its outputs in one step as an input in the subsequent computation step
Equivalent to the discrete convolution if bμ = 0 for μ = 1,2,…,n

Notes:
• Derivation of recursive definition of filter usually non-trivial
• Significant speed-up, effort well-worth in real life applications
• Derivation performed only once
• Benefit of speed-up each time recursive filter is applied
Popular recursive filters:
• Kalman filtering
• Particle filtering

1. Define q based on σ:
2. Compute the filter parameters based on q:
3. Forward filtering:
4. Backward filtering:

Question 2

What is Homomorphic Filtering?

Answer 2

• Linear system suited for additive composition of signal and noise
• Not suited for non-additive composition
• Homomorphic system: generalization of a linear system
Procedure:
1. apply a suited transformation Tc to transform a non-additive into an additive composition Tc: characteristic system
2. process transformed signal with a linear transformation TL as before
3. invert transformation using T−1

Mathematical operations with different symbols (signal+signals 􏰙􏰙square/circle, constant*signal diamond/triangle)

Question 3

What does cepstrum mean?

Answer 3

• Describes rate of change in different frequency bands
• Originally used for characterizing seismic echos from earthquakes or explosions
• Principle of standard features in speech processing: Mel-Frequency Cepstral Coefficients (MFCC)
  
  • used to remove harmonic structure resulting from vocal cord vibrations from the speech signal
  • keeps the formant filtering effects of vocal tract