Information Technology Reference
In-Depth Information
A Gabor frame is a set of functions g
mn
.x/ that can be a basis of the L
2
space.
The Gabor frame depends on two positive and bounded numbers b
0
and f
0
and on a
function g, which has compact support is contained in an interval of length
2
f
0
[
191
].
If there exist two real constants C
1
and C
2
such that 0<C
1
P
jg.xnb
0
/j
2
C
2
<
1, then the family of Gabor functions g
mn
;n;m;2Z is a frame in L
2
.R/, with
bounds 2
C
1
f
0
and 2
C
2
f
0
.
The Balian-Low theorem describes the behavior of Gabor frames at the critical
frequency f
0
defined by b
0
f
0
D 2. The theorem is a key result in time-frequency
analysis. It expresses the fact that time-frequency concentration and non-redundancy
are incompatible properties for Gabor frames of Eq. (
12.20
). Specifically, if for some
˛>0and g2L
2
.R/ the set fe
2lx=˛
g.x k˛/g
k;l2Z
2
is an orthonormal basis for
L
2
.R/, then
Z
2
dx
Z
2
df
R
j
xg
.x/j
R
jf g.f/j
D1
(12.21)
This means that for a Gabor frame which is contracted from the function g.x/ one
has necessarily
R
x
2
2
dx
D1(which means not fine localizaton on the space
jg.x/j
axis x)or
R
f
2
2
df
jg.f/j
D1(which means not fine localization on the frequency
axis f ).
In Sect.
11.3
a practical view of the application of the uncertainty principles to
wavelets (which generalize Gabor functions) has been presented.
12.5.2
Uncertainty Principles for Hermite Series
For a function
k
.x/2L
2
.R/ the mean .
k
/ and the variance
2
.
k
/ are defined
as follows [
89
]:
.
k
/ D
R
xj
k
.x/j
2
dx
;
2
.
k
/ D
R
jx .
k
/j
2
2
dx
(12.22)
j
k
.x/j
The Hermite basis functions
k
.x/ were defined in Eq. (
12.6
), while the Her-
mite polynomials H
k
.x/ were given in Eq. (
12.8
). The Hermite functions are an
orthonormal basis for L
2
.R/. It was also shown that Hermite functions are the
eigenfunctions of the solution of Schrödinger's equation, and remain invariant
under Fourier transform, subject only to a change of scale as given in Eq. (
12.13
),
i.e.
k
.s/ D j
n
k
.s/. Furthermore, the
mean-dispersion principle
provides an
expression of the uncertainty principle for orthogonal series [
89
]. This states that
there does not exist an infinite orthonormal sequence f
k
g
kD0
L
2
, such that all
four of .
k
/, .
k
/, .
k
/ and .
k
/ are uniformly bounded.
Using Eq. (
12.22
) for the Hermite basis functions one obtains .
k
/ D .
k
/ D
0, and
2
.
k
/ D
2
.
k
/ D
2k
C
1
4
which quantifies the results of the mean-
dispersion principle and formulates an uncertainty principle for Gauss-Hermite
basis functions (Fig.
12.4
).
