Image Processing Reference
In-Depth Information
 
 
 
j
n
j
n
j
n
Xe
Xe
jXe
.
(7)
C
R
I
The real signal and its Hilbert transform are respectively the real and imaginary parts of the
analytic signal; these have the same amplitude and /2 phase-shift (Fig. 3).
X R ( e j n )
(a)
- S
S 2
S
- S 2
jX I ( e j n )
(b)
- S
S
X C ( e j n )
(c)
- S
S
- S 2
S 2
X C * ( e -j n )
(d)
- S
S 2
S
- S 2
Fig. 3. Complex signal derivation using the Hilbert transformation
According to the Hilbert transformation, the components of the  
Xe
j
n
spectrum are
shifted by /2 for positive frequencies and by -/2 for negative frequencies, thus the
pattern areas in Fig. 3b are obtained. The real signal  
Xe
j
n
and the imaginary one
 
I Xe
j
n
multiplied by j (square root of -1), are identical for positive frequencies and -/2
phase shifted for negative frequencies - the solid blue line (Fig. 3b). The complex signal
 
C Xe
j
n
occupies half of the real signal frequency band; its amplitude is the sum of the
 
and  
Xe
j
n
e
j
n
jX
amplitudes (Fig. 3c). The spectrum of the complex conjugate
I
analytic signal  
Xe
j
n
is depicted in Fig. 3d.
C
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