Answer: Custom Designing Basis Functions

[Soma Dhavala (soma@ieee.org)]

Reply to WD 11.5 Topic 5160.

Hi,
There are many methods to decompose a signal into its individual atoms. Mallat's Matching pursuits (MP) is one of them. You can find its implementation in wavelab also. I think, here you choose the set of atoms that suit the signal's inner structure from a pre-determined dictionary (theoretically of infinite size). Each time you pick the best atom from the dictionary, then compute the residue; again project the residue onto the dictionary. You can work in this divide and conquer mode until, you find that the residue's energy is below a threshold or some such stopping criteria.

Another method called "four parameter atomic decomposition" uses matching pursuits to decompose a signal into chirplets. It is equivalent to the above method except that here the dictionary is formed by infinite chirplets. A Guassian chirplet can be represented by (roughly) Aexp(- (t-tc)^2)exp(-j (t-tc)^2)exp(-j( - c)t). Here tc is the time center and c is the frequency center, is the chirp rate, is (inversely) proportional to time-spread (window parameter) and A is a complex constant. One decomposes a signal into these chirplets. Essentially finding the basis functions becomes searching based on MP in the dictionary formed by these chirplets It forms a redundant representation. Another chirplets decomposition method was suggested by Jeff O'Neil and Flandrin: it adaptively estimates the best chirplets instead of searching for them from an infinite dictionary. I am inclined to use this for any sparse adaptive representation of wide varity of signals.

Caution: Chirplets are not basis functions.

References:
1. Bultan, A. (1999) A Four-Parameter Atomic Decomposition of Chirplets. IEEE Trans. on Signal Processing, 41, 731-745.
2. Mallat, S. G. and Zhang, Z. (1993) Matching Pursuits with Time-Frequency Dictionaries. IEEE Trans. on Signal Processing, 41, 3397-3415.
3. Mann, S. and Haykin, S. (1995) The Chirplet Transform: Physical Considerations. IEEE Trans. on Signal Processing, 44, 2745-2761.
4. O'Neill, J. C. and Flandrin, P. (1998) Chirp Hunting. Proc. of the IEEE-SP International Symposium on Time-Frequency and Time-ScaleAanalysis. 425-428.



Hope it helps..
with regards
Soma
soma@ieee.org