IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 1998 979
Orthogonal Transmultiplexers
in Communication: A Review
Ali N. Akansu, Senior Member, IEEE, Pierre Duhamel, Fellow, IEEE, Xueming Lin, and Marc de Courville
Abstract— This paper presents conventional and emerging
applications of orthogonal synthesis/analysis transform config-
urations (transmultiplexer) in communications. It emphasizes
that orthogonality is the underlying concept in the design of
many communication systems. It is shown that orthogonal filter
banks (subband transforms) with proper time–frequency features
can play a more important role in the design of new systems.
The general concepts of filter bank theory are tied together
with the application-specific requirements of several different
communication systems. Therefore, this paper is an attempt to
increase the visibility of emerging communication applications of
orthogonal filter banks and to generate more research activity in
the signal processing community on these topics.
I. INTRODUCTION
S
IGNAL processing and communications have been com-
plementary fields of electrical engineering for a long time.
Although most of the basic processing tools utilized in the
design of communication systems clearly come from the signal
processing discipline, e.g., Fourier transform and modulation
schemes, others are specifically designed for communication
purposes, such as information theory and error-correcting
codes. In turn, signal processing experts have also been influ-
enced by this cross-fertilization and expanded their research
activities into various communication applications.
This mutual influence and interaction, however, has not
been as strong in the area of discrete-time multirate signal
processing. Highlighting the fundamentals of orthogonal sub-
band transforms from a time–frequency perspective, this paper
illustrates how both disciplines would benefit from a stronger
cooperation on this topic. Several popular communication
applications can be described in terms of synthesis/analysis
configuration (transmultiplexer) of subband transforms. Code
division multiple access (CDMA), frequency division multiple
access (FDMA), and time division multiple access (TDMA)
communication schemes can be viewed from this perspective.
In particular, FDMA [which is also called orthogonal fre-
quency division multiplexing (OFDM)] or discrete multitone
Manuscript received February 15, 1997; revised November 30, 1997. The
associate editor coordinating the review of this paper and approving it for
publication was Prof. Mark J. T. Smith.
A. N. Akansu and X. Lin are with the Department of Electrical and
Computer Engineering, New Jersey Center for Multimedia Research, New
Jersey Institute of Technology, Newark, NJ 07102 USA.
P. Duhamel is with the
´
Ecole National Superieure des
T
´
el
´
ecommunications/SIG, Paris, France.
M. de Courville was with the
´
Ecole National Superieure des
T
´
el
´
ecommunications/SIG, Paris, France. He is now with Motorola
CRM, Paris, France.
Publisher Item Identifier S 1053-587X(98)02554-9.
(DMT) modulation-based systems have been more widely used
than the others.
The orthogonality of multicarriers was recognized early on
as the proper way to pack more subchannels into the same
channel spectrum [20], [35], [36], [68]. This approach is
meritful particularly for the communication scenarios where
the channel’s power spectrum is unevenly distributed. The
subchannels (subcarriers) with better power levels are treated
more favorably than the others. Therefore, this approach
provides a vehicle for an optimal loading of subchannels
where channel dynamics are significant. The subcarrier or-
thogonality requirements were contained in a single domain
in conventional communication schemes. Namely, they are the
orthogonality in frequency (no interference between different
carriers or subchannels) and the orthogonality in time (no
interference between different subsymbols transmitted on the
same carrier at different time slots). If this property is ensured,
multichannel communication is achieved naturally.
Originally, the multicarrier modulation technique was pro-
posed by using a bank of analog Nyquist filters, which
provide a set of continuous-time orthogonal functions. How-
ever, the realization of strictly orthogonal analog filters is
impossible. Therefore, the initial formulation was reworked
into a discrete-time model. The steps of this discrete-time
model are summarized as follows. A digital computation
first evaluates samples of the continuous signal that is to
be transmitted over the channel. Then, these samples drive a
digital-to-analog converter (DAC), which generates the actual
transmitted signal. This discrete model makes explicit use of
a structure that is similar to the orthogonal synthesis/analysis
filter bank or transmultiplexer displayed in Fig. 1(b).
Transmultiplexers were studied in the early 1970’s by
Bellanger et al. [37] for telephony applications. Their sem-
inal work was one of the first dealing with multirate signal
processing, which has matured lately in the signal processing
field. Since complexity is an important issue in all of these
applications, the discrete Fourier transform (DFT) basis is
usually chosen as the set of orthogonal subcarriers [31], [38],
[66]. In addition, it has been shown that the DFT-based
transmultiplexers allow efficient channel equalization, which
make them attractive [see Section III-A1)].
The orthogonality conditions and implementation of
discrete-time (digital) function sets are much easier to use
than the ones in the continuous-time domain (analog case)
[6], [7]. This is the first point where DSP tools can be useful.
In addition, it has been shown in [66] that the only Nyquist
filter, which allows the time and frequency orthogonalities
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1998 IEEE
980 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 1998
(a)
(b)
Fig. 1. (a) Maximally decimated
M
-band FIR PR-QMF filter bank structure (analysis/synthesis configuration). (b)
M
-band transmultiplexer structure
(synthesis/analysis filter bank configuration ion).
when modulated by a DFT, is a rectangular window (time)
function. All other modulations using more selective window
functions can only approximate the orthogonality conditions.
There are some ways to circumvent such restrictions, which
are discussed below in Section III-A. On the other hand, more
general orthogonality conditions need to be satisfied for other
transform bases. This is currently an active research topic in
the field.
Two specific communication applications have recently gen-
erated significant research activity on multicarrier modula-
tion (OFDM or DMT) techniques. First, the coded orthog-
onal frequency division multiplexing (COFDM) has been
forwarded for terrestrial digital audio broadcasting (T-DAB)
in Europe. As a result, a European T-DAB standard has
been defined, and actual digital broadcasting systems are
being built [see Section III-A2)]. Second, a DFT-based DMT
modulation scheme has become the standard for asymmetric
digital subscriber line (ADSL) communications, which provide
an efficient solution to the last mile problem (e.g., providing
high-speed connectivity to subscribers over the unshielded
twisted pair (UTP) copper cables) [21], [22], [24], [25]. In
addition to these two successful applications of multicarrier
modulation, we also highlight several emerging application
areas, including spread spectrum orthogonal transmultiplexers
for CDMA [15] and low probability of intercept (LPI) com-
munications [48], [49] (which might further benefit from the
discussions presented). The time–frequency and orthogonality
properties of function sets, or filter banks, are the unifying
theme of the topics presented in the paper. It is shown from
a signal processing perspective that these entirely different
communications systems are merely variations of the same
theoretical concept. The subband transform theory and its ex-
tensions provide the theoretical framework that serves all these
variations. This unified treatment of orthogonal multiplexers
are expected to improve existing solutions.
The paper is organized as follows. The perfect recon-
struction (PR) (orthogonality) properties and synthesis/analysis
configuration of filter banks as a transmultiplexer structure
are presented in Section II. This section also discusses the
time–frequency interpretation and optimal design methodolo-
gies of transform bases for different types of transmultiplexer
platforms, e.g., TDMA, FDMA, and CDMA. Section III ex-
amines in detail several multicarrier communication scenarios,
namely, DMT for ADSL and T-DAB, the spread spectrum PR
quadrature mirror filter (PR-QMF) bank for CDMA, and an
energy-based LPI detector, where each one uses an orthogonal
transmultiplexer of the proper type as their common com-
ponent. This section also emphasizes the linkage of popular
discrete-time transmultiplexers with their analog progenitor.
Finally, in an effort to generate interest, Section III reviews
some of the problem areas that require future research. This
paper is concluded in Section IV.
AKANSU et al.: ORTHOGONAL TRANSMULTIPLEXERS IN COMMUNICATION: A REVIEW 981
II. SYNTHESIS/ANALYSIS FILTER BANK AND ORTHOGONALITY
An -band, maximally decimated, finite impulse response
(FIR), PR-QMF bank in an analysis/synthesis configuration is
displayed in Fig. 1(a). The PR filter bank output is a delayed
version of the input as
(1)
where
is a delay constant related to the filter duration. In
a paraunitary filter bank solution, the synthesis and analysis
filters are related as (similar to a match filter pair)
(2)
where
is a time delay. Therefore, it is easily shown that
the PR-QMF bank conditions can be written on the analysis
filters in the time domain as
(3a)
(3b)
PR analysis/synthesis filter banks were first introduced by
Smith and Barnwell for the case of two bands [1], [60]. The
authors in [2]–[4] provide a detailed treatment of PR filter
banks and their extensions. Analysis/synthesis filter bank con-
figurations are most often used for applications in image/video
processing, speech/audio processing, interference excision in
spread spectrum communications, and many others [5].
In contrast, Fig. 1(b) displays a synthesis/analysis filter bank
where there are
inputs and outputs of the system. It
is shown for the critically sampled case that if the synthesis
and analysis filters satisfy the PR-QMF
conditions of (2) and (3), the synthesis/analysis filter bank
yields an equal input and output for all the branches of the
structure as
(4)
where
is a time delay [6]–[8]. The orthogonal transmul-
tiplexers configured as the synthesis/analysis PR-QMF bank
have been widely utilized in many communication applications
for single- and multiple-user scenarios. This paper focuses on
these conventional and emerging communication applications
of orthogonal transmultiplexers. It also highlights and analyzes
the application-specific requirements of basis design problems
from the perspective of subband transform theory. The paper
shows the linkages of the theoretical fundamentals and the
specifics of each application under consideration.
A. Time–Frequency Interpretation and Optimal Basis Design
for TDMA, FDMA, and CDMA Communications
The orthogonality and time–frequency properties of subcar-
riers are very critical for system performance. This subsection
links well-known time–frequency concepts and measures with
the applications under consideration.
The time and frequency domain energy concentration or
selectivity of a function has been a classic problem in the
signal processing field. The “uncertainty principle” states that
Fig. 2. Time–frequency plane showing resolution cell of a typical dis-
crete-time function.
no function can simultaneously be concentrated in both the
time and frequency domains [9]. The time spread of a discrete-
time function
is defined by [10]
(5)
The energy
and time center of the function are
given as
(6)
(7)
where its Fourier transform is expressed as
Similarly, the frequency domain spread of a discrete-time
function is defined as
(8)
where its frequency center is written as
(9)
Fig. 2 displays a time–frequency tile of a typical discrete-
time function where
and were defined in (5) and
(8), respectively. The shape and the location of the tile can
be adjusted by properly designing the time and frequency
centers and the spreads of the function under construction.
This methodology can be further extended in the case of
basis design. In addition to shaping time-frequency tiles, the
completeness requirements of (2) and (3) are imposed on the
basis being designed.
The engineering challenge in this context is to design the
most suitable basis for the application at hand [2], [11], [13].
Hierarchical filter bank structures lend themselves to more
flexibility and can be helpful in achieving design targets more
easily. There have been studies on the selection of best basis
and the optimality measure for different applications. In this
context, note that the block transforms like DFT belong to
a subset of subband transforms with the minimum possible
duration of basis functions. Subband transforms, however,
982 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 1998
allow a higher degrees of freedom than block transforms to
be utilized for optimal basis design purposes at the expense of
additional computational cost. Readers interested in this topic
are referred to [2], [5], and [11]–[14].
The orthogonal synthesis/analysis filter bank configuration
provides a solid theoretical foundation for the single and
multiuser communication scenarios that are widely used in
literature. In their most popular version, orthogonal discrete-
time transmultiplexers are of the FDMA type. This implies
that the synthesis and analysis filters
and ,
respectively, are frequency selective and brick-wall shaped
in their ideal cases. Therefore, ideally, the communication
channel is divided into disjoint frequency subchannels. These
subchannels are allocated among the users of multiuser com-
munications and T-DAB. This type has been the most pop-
ular use of orthogonal transmultiplexers. More recently, the
subchannel (multicarrier modulation) concept has also been
applied to single-user communication scenarios like ADSL
applications [21], [22], [24], [25]. In another scenario, the
subchannel structures are intelligently utilized via frequency
hopping for multiuser spread spectrum communication. In
addition, OFDM modulation is currently being pursued by
several research groups for the third generation of personal
communication systems (PCS) applications [64].
In contrast to FDMA, a TDMA communication scheme
allocates a dedicated time slot for each user. A user is allowed
to use the full frequency channel only during the given time
slot. Time slot allocation is a simple delay, with transform
This can be interpreted as the allpass-
like (spectrally spread) user codes or synthesis filters, i.e.,
for for the
ideal case used in the orthogonal transmultiplexer shown in
Fig. 1(b). The ideal user codes in the time and frequency
domains for FDMA and TDMA communications scenarios and
their time–frequency resolution cells are displayed in Fig. 3.
The emerging CDMA communication techniques employ
user codes (filters) that are simultaneously spread in both
the time and frequency domains. More recently, the filter
bank design problem was extended for this purpose, and
spread spectrum PR-QMF’s were proposed as an alternative
to the existing codes, e.g., Gold codes. These orthogonal
transmultiplexers utilize the user codes
, which are
allpass-like in frequency as well as spread in the time domain
[15]. It is shown that the popular pseudo-random noise (PN)
and Gold codes are binary valued and nearly orthogonal
special subset of more general spread spectrum PR-QMF bank
framework [16], [17], [65].
The different versions of discrete-time orthogonal trans-
multiplexers along with their communications scenarios are
discussed in the following section.
III. O
RTHOGONAL TRANSMULTIPLEXER:
S
INGLE AND MULTIUSER COMMUNICATIONS
The popular and emerging communication applications of
orthogonal transmultiplexers are discussed in this section.
Highlights include DMT (OFDM)-based ADSL/VDSL and
DAB, spread spectrum CDMA, and LPI communications tech-
niques and their performance issues. The application-specific
requirements of synthesis/analysis filters are interpreted from
a time–frequency perspective. State-of-the-art communication
systems are discussed, and several future research topics are
suggested which explore potential improvements.
A. DMT Modulation:
DMT or multicarrier modulation utilizes a set of frequency
selective orthogonal functions for digital communication.
Since functions in the orthogonal set are designed to be
frequency localized, DMT is of an FDM type [19], [20]. It is
also referred to as OFDM. The terms DMT and OFDM are
used interchangeably in this paper. Note that this technique
multiplexes the incoming bit stream of a single or multiple
users into the frequency-selective subcarriers or subchannels.
DMT modulation has been widely used in applications such as
ADSL, high bit-rate digital subscriber line (HDSL), and very
high bit-rate digital subscriber line (VDSL) communications
for the single-user case. The digital subscriber line is the local
UTP telephone line. In fact, ADSL communication techniques
provide the means for high-speed digital transmission, up to
7 Mbps, over plain old telephone service (POTS) for limited
distances. A size 512 DFT-based DMT modulation scheme
has been standarized for the ADSL communications [21],
[22].
The basic structure of a DMT modulation-based digital
ADSL transceiver is displayed in Fig. 4(a). The uneven
frequency response of a typical ADSL channel (CSA Loop
1) is displayed in Fig. 4(b). Instead of using a single
carrier, e.g., carrierless phase/amplitude (CAP) modulation
technique [23], the DMT-based system uses a set of
frequency-selective orthogonal functions The
subsymbols
are formed by
grouping blocks of an incoming bit stream via certain
constellation schemes like quadrature amplitude modulation
(QAM) or pulse amplitude modulation (PAM). The parsing
of the incoming bit stream to the subsymbols is determined
by the subchannel attenuation levels. Therefore, each of the
subcarriers (orthogonal functions) carries a different number
of bits per symbol commensurate with the corresponding
subchannel attenuation. Hence, orthogonal subcarriers that
suffer less attenuation through the communication channel
will carry more bits of information [24], [25].
In summary, the prominent advantages of multicarrier mod-
ulation over a classical single carrier system are the following.
• Adaptation of the data rates of subchannels based on the
possible variations of the channel (fadings) and noise
characteristics. In that sense, OFDM systems overcome
problems introduced by the inherent colored nature of
the channel noise in wide-band transmission systems.
This technique is also known as “adaptive loading,” and
it is ADSL specific (it cannot be used in broadcasting
systems).
• Combining different coding schemes including block
(e.g., Reed–Solomon) and trellis-based modulation
in order to increase the system’s robustness toward
transmission errors. The strength of OFDM-based system
AKANSU et al.: ORTHOGONAL TRANSMULTIPLEXERS IN COMMUNICATION: A REVIEW 983
(a) (b)
Fig. 3. Ideal user codes (filters) for the cases of (a) FDMA and (b) TDMA communication scenarios.
is that it allows a given symbol to be transmitted at
a precise location in the time–frequency plane. Thus,
it is easier for the system designer to scatter in the
time–frequency plane all elements of the channel coder
in such a way that they are seldom statistically impaired
by selective fadings at the same time.
Next, we present the mathematical foundation of an OFDM
system using the conventional continuous-time filter bank
structure. Since digital solutions are more desirable than
analog ones in practice, we attempt to show the theoretical
linkages of analog and digital orthogonal transmultiplexers in
order to highlight their commonalities and possible extensions.
Then, we focus on the DFT-based DMT transceiver and
show its inherent mathematical features, which makes it a
very attractive solution especially in robust communication
environments. In order to make the following discussions
easier to understand, the OFDM transceiver is separated into
its transmitter and receiver parts.
Transmitter (Synthesis Filter Bank): As previously men-
tioned, the basic idea behind such systems is to modify the
initial communication problem of transmitting a single (or
several) wideband signal into the transmission of a set of
narrowband orthogonal signals so that the channel effects
can be modeled more efficiently. These narrowband carrier
signals are transmitted with a maximum of spectral efficiency
(no spectral holes and even overlapping of the spectra between
