Image Processing Reference
In-Depth Information
It is evident that complex DSP is more complicated than real DSP. Complex DSP transforms
are highly theoretical and mathematical; to use them efficiently and professionally requires
a large amount of mathematics study and practical experience.
Complex math makes the mathematical expressions used in DSP more compact and solves
the problems which real math cannot deal with. Complex DSP techniques can complement
our understanding of how physical systems perform but to achieve this, we are faced with
the necessity of dealing with extensive sophisticated mathematics. For DSP professionals
there comes a point at which they have no real choice since the study of complex number
mathematics is the foundation of DSP.
1.2 Complex representation of signals and systems
All naturally-occurring signals are real; however in some signal processing applications it is
convenient to represent a signal as a complex-valued function of an independent variable.
For purely mathematical reasons, the concept of complex number representation is closely
connected with many of the basics of electrical engineering theory, such as voltage, current,
impedance, frequency response, transfer-function, Fourier and z-transforms, etc.
Complex DSP has many areas of application, one of the most important being modern
telecommunications, which very often uses narrowband analytical signals; these are
complex in nature (Martin, 2003). In this field, the complex representation of signals is very
useful as it provides a simple interpretation and realization of complicated processing tasks,
such as modulation, sampling or quantization.
It should be remembered that a complex number could be expressed in rectangular , polar and
exponential forms:
Ae
j
a bA

cos
j
sin
.
(1)
The third notation of the complex number in the equation (1) is referred to as complex
exponential and is obtained after Euler's relation is applied. The exponential form of complex
numbers is at the core of complex DSP and enables magnitude A and phase θ components to
be easily derived.
Complex numbers offer a compact representation of the most often-used waveforms in
signal processing - sine and cosine waves (Proakis & Manolakis, 2006). The complex number
representation of sinusoids is an elegant technique in signal and circuit analysis and
synthesis, applicable when the rules of complex math techniques coincide with those of sine
and cosine functions. Sinusoids are represented by complex numbers; these are then
processed mathematically and the resulting complex numbers correspond to sinusoids,
which match the way sine and cosine waves would perform if they were manipulated
individually. The complex representation technique is possible only for sine and cosine
waves of the same frequency, manipulated mathematically by linear systems.
The use of Euler's identity results in the class of complex exponential signals:

j

n
j
 
xn
A
Ae e
x n
jxn
0
0
.
(2)
R
I
e  
j
AAe
j
and
are complex numbers thus obtaining:
0
0
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