Image Processing Reference
In-Depth Information
The spectrum of a real discrete-time signal lies between -ω s /2 and ω s /2 (ω s is the sampling
frequency in radians per sample), while the spectrum of a complex signal is twice as narrow
and is located within the positive frequency range only.
Narrowband signals are of great use in telecommunications. The determination of a signal's
attributes, such as frequency, envelope, amplitude and phase are of great importance for
signal processing e.g. modulation, multiplexing, signal detection, frequency transformation,
etc. These attributes are easier to quantify for narrowband signals than for wideband signals
(Fig. 2). This makes narrowband signals much simpler to represent as complex signals.
Wideband signal x2(n)
Narrowband signal x1(n)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
20
40
60
80
100
120
0
20
40
60
80
100
120
Sample number
Sample number
(a) (b)
Fig. 2. Narrowband signal (a)  
 
;
xn
sin
60
n
4 cos
2
n
1
wideband signal (b)  
 
xn
sin
60
n
4 cos
16
n
2
Over the years different techniques of describing narrowband complex signals have been
developed. These techniques differ from each other in the way the imaginary component is
derived; the real component of the complex representation is the real signal itself.
Some authors (Fink, 1984) suggest that the imaginary part of a complex narrowband signal
can be obtained from the first  
and second  

R xn
R xn
derivatives of the real signal:
 

xn


R
.
(4)
xn

x n
I
R
xn

R
One disadvantage of the representation in equation (4) is that insignificant changes in the
real signal x R ( n ) can alter the imaginary part x I ( n ) significantly; furthermore the second
derivative can change its sign, thus removing the sense of the square root.
Another approach to deriving the imaginary component of a complex signal representation,
applicable to harmonic signals, is as follows (Gallagher, 1968):
 
0
xn

R
,
(5)
xn
I
where  0 is the frequency of the real harmonic signal.
Analytical representation is another well-known approach used to obtain the imaginary part
of a complex signal, named the analytic signal. An analytic complex signal is represented by
its inphase (the real component) and quadrature (the imaginary component). The approach
includes a low-frequency envelope modulation using a complex carrier signal - a complex
exponent
e
j
n
named cissoid (Crystal & Ehrman, 1968) or complexoid (Martin, 2003):
0
 
 
jn
 
 
jn
 
 
xn e
 
xn xne
xn
cos
nj
sin
n xn xn
.
(6)
0
0
R
R
R
0
0
R
I
In the frequency domain an analytic complex signal is:
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