Image Processing Reference
In-Depth Information
A real DFT spectrum can be represented in a complex form. Forward real DFT results in
cosine and sine wave terms, which then form respectively the real and imaginary parts of a
complex number sequence. This substitution has the advantage of using powerful complex
number math, but this is not true complex DFT. Despite the spectrum being in a complex
form, the DFT remains real and j is not an integral part of the complex representation of real
DFT.
Another mathematical inconvenience of real DFT is the absence of symmetry between
analysis and synthesis equations, which is due to the exclusion of negative frequencies. In
order to achieve a perfect reconstruction of the time domain signal, the first and last samples
of the real DFT frequency spectrum, relating to zero frequency and Nyquist's frequency
respectively, must have a scaling factor of 1/ N applied to them rather than the 2/ N used for
the rest of the samples.
In contrast, complex DFT doesn't require a scaling factor of 2 as each value in the time
domain corresponds to two spectral values located in a positive and a negative frequency;
each one contributing half the time domain waveform amplitude, as shown in Fig. 7. The
factor of 1/ N is applied equally to all samples in the frequency domain. Taking the negative
frequencies into account, complex DFT achieves a mathematically-favoured symmetry
between forward and inverse equations, i.e. between time and frequency domains.
Complex DFT overcomes the theoretical imperfections of real DFT in a manner helpful to
other basic DSP transforms, such as forward and inverse z-transforms. A bright future is
confidently predicted for complex DSP in general and the complex versions of Fourier
transforms in particular.
2. Complex DSP - some applications in telecommunications
DSP is making a significant contribution to progress in many diverse areas of human
endeavour - science, industry, communications, health care, security and safety, commercial
business, space technologies etc.
Based on powerful scientific mathematical principles, complex DSP has overlapping
boundaries with the theory of, and is needed for many applications in, telecommunications.
This chapter presents a short exploration of precisely this common area.
Modern telecommunications very often uses narrowband signals, such as NBI (Narrowband
Interference), RFI (Radio Frequency Interference), etc. These signals are complex by nature
and hence it is natural for complex DSP techniques to be used to process them (Ovtcharov et
al, 2009), (Nikolova et al, 2010).
Telecommunication systems very commonly require processing to occur in real time,
adaptive complex filtering being amongst the most frequently-used complex DSP techniques.
When multiple communication channels are to be manipulated simultaneously, parallel
processing systems are indicated (Nikolova et al, 2006), (Iliev et al, 2009).
An efficient Adaptive Complex Filter Bank (ACFB) scheme is presented here, together with
a short exploration of its application for the mitigation of narrowband interference signals in
MIMO (Multiple-Input Multiple-Output) communication systems.
2.1 Adaptive complex filtering
As pointed out previously, adaptive complex filtering is a basic and very commonly-
applied DSP technique. An adaptive complex system consists of two basic building blocks:
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