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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 9, SEPTEMBER 2004 1823
Complex Signal Processing is Not Complex
Kenneth W. Martin, Fellow, IEEE
Abstract—Wireless systems often make use of the quadrature re-
lationship between pairs of signals to effectively cancel out-of-band
and interfering in-band signal components. The understanding of
these systems is often simplified by considering both the signals
and system transfer functions as “complex” quantities. The com-
plex approach is especially useful in highly integrated multistan-
dard receivers where the use of narrow-band fixed-coefficient fil-
ters at the RF and high IF must be minimized. This paper first
presents a tutorial review of complex signal processing for wire-
less applications. The review emphasizes a graphical and picto-
rial description rather than an equation-based approach. Next, a
number of classical modulation architectures are described using
this formulation. Finally, more recent developments such as com-
plex filters, image-reject mixers, low-IF receivers, and oversam-
pling analog–digital converters are discussed.
Index Terms—Communication systems, complex signal pro-
cessing, filters, frequency modulation, mixers, wireless receivers.
I. INTRODUCTION
T
HIS paper is intended to be used as a tutorial introduction
to complex signal processing, and also as an illustration of
its wide use in a number of different applications, both analog
and digital, in recent communication systems.
A
complex signal consists of a pair of real signals at an in-
stant in time. If one denotes the complex signal
,as
where ; then, a Hilbert space can be de-
fined using appropriate definitions for addition, multiplication,
and an inner product and norm. In an actual physical system,
the signals are both real (but are called the real and imaginary
parts) and are found in two distinct signal paths. The multiplier
“
“ is used to help define operations between different complex
signals, and is to some degree imaginary; i.e., it does not actu-
ally exist in actual systems. Often, the dependence of signals on
time is not shown explicitly.
The use of complex signal processing to describe wireless sys-
tems is increasingly important and ubiquitous for the following
reasons: it often allows for image-reject architectures to be de-
scribed more compactly and simply; it leads to a graphical or
signal-flow graph (SFG) description of signal-processing sys-
tems providing insight, and it often leads to the development of
new systems where the use of high-frequency highly selective
image-reject filters is minimized. The result is more highly in-
tegrated systems using less power and requiring less physical
space.
Manuscript received November 25, 2003; revised March 2, 2004. This
work was supported in part by the Natural Sciences and Engineering Research
Council (NSERC) and in part by MICRONET, Canada. This paper was
recommended by Associate Editor U.-K. Moon.
The author is with the Department of Electrical and Computer Engi-
neering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:
martin@eecg.toronto.edu).
Digital Object Identifier 10.1109/TCSI.2004.834522
Fig. 1. SFG of an quadrature mixer using (a) a real SFG and (b) a complex
SFG.
As an example, consider the popular “quadrature” mixer with
a real input as shown in Fig. 1. In the complex SFG, two real
multiplications have been replaced by a single complex multi-
plication. Furthermore, since in the time domain, we have
(1)
then, taking the Fourier transform, we have
(2)
and we see that an ideal quadrature mixer results in a simple
frequency shift of the input signal with no images occurring, a
conclusion that is obvious from Fig. 1(b), but is certainly not
obvious from Fig. 1(a). If this mixer is followed by a low-pass
filter in each path, then, it can be used to directly demodulate
carrier-frequency signals without images as is discussed in Sec-
tion III-A. For this reason, it often called an image-reject mixer
although if one is being accurate, the term should denote the
combination of the quadrature mixer and low-pass filters.
The use of complex signal processing for wireless appli-
cations has blossomed recently. This is especially true for
high-bit-rate standards, such as local-area networks (LANs),
and for multistandard transceivers. Evidence of this prolifera-
tion is that in 2001, 2002, and 2003, the majority of the papers
in IEEE International Solid-State Circuits Conference sessions
on wireless transceivers for LAN applications employed com-
plex signal processing.
Much, but not all of this paper deals with the already men-
tioned tutorial review of complex signal processing, especially
as it is applicable to wireless systems. A great deal of the tutorial
material on complex analog signal processing is directly based
on the research of Snelgrove, as described in [1], [6], [7], and in
some theses by his graduate students [9], [11]. Unfortunately, a
journal submission describing this early work was rejected, but
a reprint of the originally submitted paper can be obtained from
[8]. Some of the early published work on complex digital filters
is described in [2] and [5].
1057-7122/04$20.00 © 2004 IEEE
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