National Instruments Order Analysis Toolset - Manual (Page 52)

Appendix A Gabor Expansion and Gabor Transform
LabVIEW Order Analysis Toolset User Manual A-2 ni.com
The sampled STFT is also known as the Gabor transform and is represented
by the following equation.
(A-2)
where
∆M represents the time sampling interval and N represents the total
number of frequency bins.
The ratio between N and
∆M determines the Gabor sampling rate. For
numerical stability, the Gabor sampling rate must be greater than or equal
to one. Critical sampling occurs when N =
∆M. In critical sampling, the
number of Gabor coefficients c
m,n
equals the number of original data
samples s[k]. Over sampling occurs when N/
∆M > 1. For over sampling,
the number of Gabor coefficients is more than the number of original data
samples. In over sampling, the Gabor transform in Equation A-2 contains
redundancy, from a mathematical point of view. However, the redundancy
in Equation A-2 provides freedom for the selection of better window
functions, h[k] and
γ[k].
Notice that the positions of the window functions h[k] and
γ[k] are
interchangeable. In other words, you can use either of the window functions
as the synthesis or analysis window function. Therefore, h[k] and
γ[k] are
usually referred to as dual functions.
The method of the discrete Gabor expansion developed in this appendix
requires
in Equation A-2 to be a periodic sequence, as shown by the
following equation.
(A-3)
where L
s
represents the length of the signal s[k] and L
0
represents the period
of the sequence L
0
is the smallest integer that is greater than or equal
to L
s
. L
0
must be evenly divided by the time sampling interval
∆M. For a
given window h[k] that always has unit energy, you can compute the
c
m n ,
s˜ k [ ]γ∗ k m∆M
–
[ ]e j2 πnk – N ⁄
n 0 =
N 1 –
∑
=
s˜ k [ ]
s˜ k iL
0
+
[ ]
s k [ ] 0 k L
s
< ≤
0 L
s
k L
0
< ≤ 


=
s˜ k [ ].