COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 3

1.1.1 Why Complex-Valued Signal Processing?
Complex domain is the natural home for the representation and processing of many
signals we encounter in practice. There are four main scenarios in which complex
processing is needed.
† The signal can be natively complex, where an in-phase and a quadrature com-
ponent is the natural representation and enables one to fully take the relationship
between the two components into account. Examples include radar and MRI
signal [2] as well as many communication signals such as those using binary
phase shift keying (BPSK), quadrature phase shift keying (QPSK), and quadra-
ture amplitude modulation (QAM) as shown in Figure 1.1. The MRI signal
is acquired as a quadrature signal using two orthogonal detectors as shown
in Figure 1.2 [17]. Hence, the complex k-space representation is the natural
one for the MRI signal, which is typically inverse Fourier-transformed into
the complex image space in reconstruction resulting in complex-valued spatial
domain signal.
† Harmonic analysis, in particular Fourier analysis, has been one of the most
widely used tools in signal processing. More recently, complex wavelet trans-
forms have emerged as attractive tools for signal processing as well, and in all
these instances where the processing has to be performed in a transform
domain, one needs to perform complex-valued signal processing.
† Analytic representation of a real-valued bandpass signal using its complex enve-
lope is commonly used in signal processing, in particular in communications.
The complex envelope representation facilitates the derivation of modula-
tion and demodulation techniques, and the analysis of certain properties of the
signal.
† There are also cases where complex domain is used to capture the relationship
between the magnitude and phase or two channels of real-valued signals.
Examples include wind data where a complex-valued signal is constructed
using the strength and direction of wind data [37] and the magnitude of structural
MRI data where the white and gray matter are combined to form a complex
number to make use of their interdependence in the processing of data [116].
Figure 1.1 Signal constellations for BPSK, QPSK, and QAM signals.