COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 4

K-Space Image-Space

Figure 1.2 MRI signal is acquired as a quadrature signal using two orthogonal
detectors, hence is inherently complex.

In all these instances, and in many similar ones, complex domain allows one to fully
take advantage of the complete information in the real and imaginary channels of a
given signal and thus is the natural home for the development of signal processing
algorithms.
In this chapter, our focus is the description of an efficient framework such that
all (or most) of the processing can be performed in the complex domain without
performing transformations to and from the real domain. This point has long been a
topic of debate since equivalent transformations between the two domains can be
easily established, and since the real domain is the one with which we are more fam-
iliar, the question arises as to why not transform the problem into the real domain
and perform all of the evaluations and analyses there. There are a number of reasons
for keeping the computations and analysis in the complex domain rather than using
complex-to-real transformations.

(1) Most typically, when the signal in question is complex, the cost function is also
defined in the complex domain where the signal as well as the transformations
the signal goes through are easily represented. It is thus desirable to keep all of
the computations in the original domain rather than working with transform-
ations to and from the real-valued domain, that is, transformations of the
type: C

N

2 N .
(2) Even though real-to-complex transformations are always possible using
Jacobians, they are not always very straightforward to obtain, especially when
the function is not

7! R

invertible. In addition, when nonlinear functions are