Information Technology Reference
In-Depth Information
5
2
2
4
1
1
3
0
0
2
−1
−1
1
−2
−2
0
−50
0
50
−50
0
50
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
(a)
(b)
(a)
(b)
2
2
5
5
4
4
1
1
3
3
0
0
2
2
−1
−1
1
1
−2
−2
0
0
−50
0
50
−50
0
50
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
(c)
(d)
(c)
(d)
a
b
Fig. 11.2
(i) Real (continuous) and imaginary (
dashed
) part of Morlet wavelets for various
frequencies: (
a
) f
0
D 0:1 Hz (
b
) f
0
D 0:3 Hz (
c
) f
0
D 0:5 Hz, (
d
) f
0
D 0:7 Hz (ii) Energy
spectrum of the Morlet wavelet of Fig.
11.2
c for central frequency f
0
D 0:5 Hz and different
variances: (
a
)
2
,(
b
) 4
2
,(
c
) 8
2
,(
d
) 16
2
11.3
Spectral Analysis of the Stochastic Weights
Spectral analysis of associative memories with weights described by interacting
Brownian particles (quantum associative memories) will be carried out following
previous studies on wavelets power spectra [
2
,
42
,
149
,
168
]. Spectral analysis in
quantum associative memories shows that the weights
w
ij
satisfy the principle of
uncertainty.
11.3.1
Spectral Analysis of Wavelets
11.3.1.1
The Complex Wavelet
The study of the energy spectrum of wavelets will be used as the basis of the
spectral analysis of quantum associative memories. The Morlet wavelet is the most
commonly used complex wavelet and in a simple form is given by
.x/ D
4
e
i2f
0
x
e
.2f
0
/
2
e
x
2
(11.34)
2
This wavelet is simply a complex wave within a Gaussian envelope. The complex
sinusoidal waveform is contained in the term e
i2f
0
x
D cos.2f
0
x/Ci sin.2f
0
x/.
The real and the imaginary part of the Morlet wavelet for various central frequencies
are depicted in Fig.
11.2
a.
