Network Card User Manual

Appendix A Gabor Expansion and Gabor Transform

LabVIEW Order Analysis Toolset User Manual A-4 ni.com

Discrete Gabor-Expansion-Based Time-Varying Filter

Initially, discrete Gabor expansion seems to provide a feasible method

for converting an arbitrary signal from the time domain into the joint

time-frequency domain or vice versa. However, discrete Gabor expansion

is effective for converting an arbitrary signal from the time domain into the

joint time-frequency domain or vice versa only in the case of critical

sampling, ∆M = N. For over sampling, which is the case for most

applications, the Gabor coefficients are the subspace of two-dimensional

functions. In other words, for an arbitrary two-dimensional function, a

corresponding time waveform might not exist. For example, the following

equation represents a modified two-dimensional function.

where Φ

m, n

denotes a binary mask function whose elements are either

0 or 1. Applying the Gabor expansion to the modified two-dimensional

function results in the following equation.

The following inequality results from Gabor expansion.

The Gabor coefficients of the reconstructed time waveform are not

equal to the selected Gabor coefficients .

To overcome the problem of the reconstructed time waveform not equaling

the selected Gabor coefficients, use an iterative process. Complete the

following steps to perform the iterative process.

1. Determine a binary mask matrix for a set of two-dimensional Gabor

coefficients.

2. Apply the mask to the two-dimensional Gabor coefficients to preserve

desirable coefficients and remove unwanted coefficients.

3. Compute the Gabor expansion.

mn,

k[] c

mn,

n 0=

N 1–

∑

hk mT–[]e

j2πnk N⁄

∑

k[]γkmT–[]e

j2πnk– N⁄

mn,

≠

k[]

mn,