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In-Depth Information
a
b
Fig. 11.4
(
a
) Heisenberg boxes in the x-frequency plane for a number of superimposing wavelets
of the same length
x
along the x-axis and at three different spectral frequencies set to low,
medium, and high value. (
b
) Heisenberg boxes in the x-frequency plane for a number of Gaussian
basis functions which have the same variance and which are shifted on the x axis
G
(
X
kD1
)
1
X
.kc/e
ifkc
p
2e
kc
/
2
2
.
w
ij
1
2
1
2
2
f
2
e
a
k
D
kD1
where ˛
k
D kc; with c being the distance between the centers of two adjacent
fuzzy basis functions. If only the real part of G.f/ is kept, then one obtains
G
(
X
kD1
)
1
X
.
kc
/ cos.
fkc
/
p
2e
kc
/
2
2
.
w
ij
1
2
1
2
2
f
2
e
a
k
D
(11.39)
kD1
If the weight
w
ij
is decomposed in the x domain into shifted Gaussians of the
same variance
2
then, in the frequency domain,
w
ij
is analyzed in a superposition
of filters, which have a shape similar to the one given in Fig.
11.4
b.
Rayleigh's theorem states that the energy of a signal f.x/x 2 Œ1; C1 is
given by
R
C1
1
f
2
.x/
dx
. Equivalently, using the Fourier transform F.s/of f.x/,the
energy is given by
R
C1
1
F
2
.s/
ds
. The energy distribution of a particle is proportional
2
of finding the particle between
w
ij
and
w
ij
C
w
ij
.
Therefore, the energy of the particle will be given by the integral of the squared
Fourier transform
to the probability
j .
w
ij
/j
j
O
.f /j
2
. The energy spectrum of the weights of the quantum
associative memories is depicted in Fig.
11.4
b.
