Image Processing Reference
In-Depth Information
1
Complex Digital Signal Processing
in Telecommunications
Zlatka Nikolova, Georgi Iliev,
Miglen Ovtcharov and Vladimir Poulkov
Technical University of Sofia
Bulgaria
1. Introduction
1.1 Complex DSP versus real DSP
Digital Signal Processing (DSP) is a vital tool for scientists and engineers, as it is of
fundamental importance in many areas of engineering practice and scientific research.
The “alphabet” of DSP is mathematics and although most practical DSP problems can be
solved by using real number mathematics, there are many others which can only be
satisfactorily resolved or adequately described by means of complex numbers.
If real number mathematics is the language of real DSP, then complex number
mathematics is the language of complex DSP. In the same way that real numbers are a part
of complex numbers in mathematics, real DSP can be regarded as a part of complex DSP
(Smith, 1999).
Complex mathematics manipulates complex numbers - the representation of two variables
as a single number - and it may appear that complex DSP has no obvious connection with our
everyday experience, especially since many DSP problems are explained mainly by means
of real number mathematics. Nonetheless, some DSP techniques are based on complex
mathematics, such as Fast Fourier Transform (FFT), z-transform, representation of periodical
signals and linear systems, etc. However, the imaginary part of complex transformations is
usually ignored or regarded as zero due to the inability to provide a readily comprehensible
physical explanation.
One well-known practical approach to the representation of an engineering problem by
means of complex numbers can be referred to as the assembling approach : the real and
imaginary parts of a complex number are real variables and individually can represent two
real physical parameters. Complex math techniques are used to process this complex entity
once it is assembled. The real and imaginary parts of the resulting complex variable
preserve the same real physical parameters. This approach is not universally-applicable and
can only be used with problems and applications which conform to the requirements of
complex math techniques. Making a complex number entirely mathematically equivalent to
a substantial physical problem is the real essence of complex DSP. Like complex Fourier
transforms, complex DSP transforms show the fundamental nature of complex DSP and such
complex techniques often increase the power of basic DSP methods. The development and
application of complex DSP are only just beginning to increase and for this reason some
researchers have named it theoretical DSP.
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