Image Processing Reference
In-Depth Information
In the frequency domain the analytic complex signal, its complex conjugate signal, real and
imaginary components are related as follows:
1
2
1
2
 
   
jn
jn
jn
Xe
Xe
Xe
R
 
   
jn
jn
jn
jX
e
X e
X
e
(8)
I
 
 
jn
jn
2
Xe
2
jXe
,


0
 
2
 
R
I
S
jn
Xe
0,
2
 
0
S
Discrete-time complex signals are easily processed by digital complex circuits, whose
transfer functions contain complex coefficients (Márquez, 2011).
An output complex signal Y C ( z ) is the response of a complex system with transfer function
H C ( z ), when complex signal X C ( z ) is applied as an input. Being complex functions, X C ( z ),
Y C ( z ) and H C ( z ), can be represented by their real and imaginary parts:
 
 
Yz
H z
X z
C
C
C


(9)






Y
z
jY
z
  
H
z
jH
z
 
X
z
jX
z
 
R
I
R
I
R
I
After mathematical operations are applied, the complex output signal and its real and
imaginary parts become:
 
       
   
Yz

H z jHz X z jXz
H


 
C
R
I
R
I
 
 

zX
z
H zX z
 
j H zX
z
H
zX z

(10)
R
R
I
I
I
R
R
I



Yz
Yz
R
I
According to equation (10), the block-diagram of a complex system will be as shown in
Fig. 4.
X R ( z )
+
Y R ( z )
H R ( z )
H I ( z )
H I ( z )
X I ( z )
Y I ( z )
+
H R ( z )
Fig. 4. Block-diagram of a complex system
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