National Instruments Order Analysis Toolset - Manual (Page 55)

Appendix A Gabor Expansion and Gabor Transform
© National Instruments Corporation A-5 LabVIEW Order Analysis Toolset User Manual
4. Compute the new Gabor coefficients after you obtain the time
waveform.
5. Repeat steps 1 through 4 until the time waveforms converge.
Without a loss of generality, rewrite the Gabor expansion from
Equation A-1 and the Gabor transform from Equation A-2 in matrix form,
as shown in the following equations.
where H denotes the analysis matrix and G denotes the synthesis matrix.
Notice that for over sampling, the following relationships are true.
Therefore, in the case of over sampling, the iterative process is described
by the following equations.
C Gs =
s H
T
Gs =
H
T
G I = GH
T
I ≠
Cˆ
1
ΦC =
s
1
H
T
Cˆ
1
=
C
2
Gs
1
GH
T ΦC
= =
Cˆ
2
ΦC
2
=
s
2
H
T
Cˆ
2
=
C
3
Gs
2
GH
T ΦGH T ΦC GH
T Φ ( )
2
C = = =
…
C
k
GH
T
Φ ( )
k 1 –
C =