Appendix A Gabor Expansion and GaborTransform
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 cm,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 Ls represents the length of the signal s[k] and L0 represents the period
of the sequence L0 is the smallest integer that is greater than or equal
to Ls. L0 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
cmn,s
˜k[]γ
∗km∆M–[]ej2πnk–N⁄
n0=
N1–
∑
=
s
˜k[]
s
˜kiL
0
+[]
sk[] 0 kL
s
<≤
0 LskL
0
<≤
=
s
˜k[].