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关于多载波通信的最完整最详细的介绍.pdf
I Introduction
II Preliminary Concepts: Symbols, Lattices, and Filters
II-A Fundamentals
II-B Symbols
II-C Filters
II-C1 Matched Filtering
II-C2 Orthogonality of Scheme
II-C3 Localization
II-D Lattices
II-D1 Lattice Geometry
II-D2 Lattice Density/Volume
II-E A Combined Approach: Lattice Staggering
II-F Summary
III Multicarrier Schemes
III-A Orthogonal Schemes
III-B Bi-orthogonal Schemes
III-C Non-orthogonal Schemes
III-D Multicarrier Schemes with Spreading Approaches
III-E Milestones for Orthogonal Schemes
IV Filter Design
IV-A Design Criterion: Energy Concentration
IV-A1 Prolate Window
IV-A2 Kaiser Function
IV-A3 Optimal Finite Duration Pulses
IV-A4 Gaussian Function
IV-A5 Isotropic Orthogonal Transform Algorithm
IV-A6 Extended Gaussian Function
IV-A7 Hermite Filter
IV-B Design Criterion: Rapid-Decay
IV-B1 Raised-Cosine Function (Hanning Filter)
IV-B2 Tapered-Cosine Function (Tukey Filter)
IV-B3 Root-Raised-Cosine Function
IV-B4 Mirabbasi-Martin Filter
IV-B5 Modified Kaiser Function
IV-C Design Criterion: Spectrum-nulling
IV-C1 Hamming Filter
IV-C2 Blackman Filter
IV-D Design Criterion: Channel Characteristics and Hardware
IV-D1 Rectangular Function
IV-D2 Channel-based Pulses
V Evaluation Metrics and Tools for Multicarrier Schemes
V-A Heisenberg Uncertainty Parameter
V-B Direction Parameter
V-C Ambiguity Function
V-D Signal-to-Interference Ratio in Dispersive Channels
VI Practical Implementation Aspects
VI-A Lattice and Filter Adaptations
VI-B Equalization
VI-C Time-Frequency Synchronization
VI-D Spatial Domain Approaches
VI-E Channel Estimation
VI-F Hardware Impairments
VI-G Cognitive Radio and Resource Sharing
VI-H Poly-Phase Network
VI-I Complexity Analysis
VI-J Testbeds and Extensions to Standards
VII Concluding Remarks
References
A Survey on Multicarrier Communications: Prototype Filters,
Lattice Structures, and Implementation Aspects
Alphan S¸ ahin1, ˙Ismail G¨uvenc¸2, and H¨useyin Arslan1
1Department of Electrical Engineering, University of South Florida, Tampa, FL, 33620
2Department of Electrical and Computer Engineering, Florida International University, Miami, FL, 33174
Email: alphan@mail.usf.edu, iguvenc@fiu.edu, arslan@usf.edu
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Abstract—Due to their numerous advantages, communications
over multicarrier schemes constitute an appealing approach for
broadband wireless systems. Especially, the strong penetration
of orthogonal frequency division multiplexing (OFDM) into the
communications standards has triggered heavy investigation on
multicarrier systems,
leading to re-consideration of different
approaches as an alternative to OFDM. The goal of the present
survey is not only to provide a unified review of waveform design
options for multicarrier schemes, but also to pave the way for
the evolution of the multicarrier schemes from the current state
of the art to future technologies. In particular, a generalized
framework on multicarrier schemes is presented, based on what
to transmit,
i.e., filters, and
where/when to transmit, i.e., lattice. Capitalizing on this frame-
work, different variations of orthogonal, bi-orthogonal, and non-
orthogonal multicarrier schemes are discussed. In addition, filter
design for various multicarrier systems is reviewed considering
four different design perspectives: energy concentration, rapid
decay, spectrum nulling, and channel/hardware characteristics.
Subsequently, evaluation tools which may be used to compare
different filters in multicarrier schemes are studied. Finally,
multicarrier schemes are evaluated from the view of the practical
implementation issues, such as lattice adaptation, equalization,
synchronization, multiple antennas, and hardware impairments.
lattice, multicarrier
schemes, pulse shaping, OFDM, orthogonality, waveform design.
Index Terms—FBMC, Gabor systems,
i.e., symbols, how to transmit,
I. INTRODUCTION
The explosion of mobile applications and data usage in
the recent years necessitate the development of adaptive,
flexible, and efficient radio access technologies. To this end,
multicarrier techniques have been extensively used over the
last decade for broadband wireless communications. This wide
interest is primarily due to their appealing characteristics, such
as the support for multiuser diversity, simpler equalization, and
adaptive modulation and coding techniques.
Among many other multicarrier techniques, orthogonal fre-
quency division multiplexing (OFDM) dominates the current
broadband wireless communication systems. On the other
hand, OFDM also suffers from several shortcomings such as
high spectral leakage, stringent synchronization requirements,
and susceptibility to frequency dispersion. Transition from the
existing OFDM-based multicarrier systems to the next gener-
ation radio access technologies may follow two paths. In the
first approach, existing OFDM structure is preserved, and its
shortcomings are addressed through appropriate solutions [1].
Considering backward compatibility advantages with existing
technologies, this approach has its own merits. The second
approach follows a different rationale based on a generalized
framework for multicarrier schemes [2], [3], which may lead
to different techniques than OFDM. In this survey, we choose
to go after the second approach since it provides a wider
perspective for multicarrier schemes, with OFDM being a
special case. Based on this strategy, the goals of the paper
are listed as follow:
• To provide a unified framework for multicarrier schemes
along with Gabor systems by emphasizing their basic
elements: what to transmit, i.e., symbols, how to transmit,
i.e., filters, and where/when to transmit, i.e., lattices;
• To extend the understanding of existing multicarrier
schemes by identifying the relations to each other;
• To review the existing prototype filters in the literature
considering their utilizations in multicarrier schemes;
• To understand the trade-offs between different multicar-
rier schemes in practical scenarios;
• To pave the way for the further developments by provid-
ing a wider perspective on multicarrier schemes.
The survey is organized as follow: First, preliminary con-
cepts and the terminology are presented in Section II. Various
multicarrier schemes are provided in Section III, referring to
the concepts introduced in Section II. Then, known prototype
filters are identified and their trade-offs are discussed in
Section IV. Useful tools and metrics to evaluate the filter
performances are investigated in Section V, transceiver design
issues for multicarrier schemes are investigated in Section VI,
and, finally, the paper is concluded in Section VII.
II. PRELIMINARY CONCEPTS: SYMBOLS, LATTICES, AND
FILTERS
The purpose of this section is to provide preliminary
concepts related with multicarrier schemes along with the
notations used throughout the survey. Starting from the basics,
symbols, lattices, and filters are discussed in detail within the
framework of Gabor systems. For a comprehensive treatment
on the same subject, we refer the reader to the books by I.
Daubechies [4], H.G. Feichtinger and T. Strohmer [5], and
O. Christensen [6]. Also, the reader who wants to reach the
development of Gabor systems from the mathematical point
of view may refer to the surveys in [3], [7]–[10]. Besides, it
is worth noting [11]–[24] constitute the key research papers
which construct a bridge between the Gabor theory and its
applications on communications. These studies also reveal how
the Gabor theory changes the understanding of multicarrier
schemes, especially, within the two last decades. Additionally,
[25]–[28] are the recent complete reports and theses based on
Gabor systems.
Composite
Effect
2
RX Filter
Projection:
…
Channel
(a) A block diagram for communications via multicarrier schemes. Both the
transmitter and the receiver construct Gabor systems.
A. Fundamentals
In the classical paper by C. Shannon [29], a geometrical rep-
resentation of communication systems is presented. According
to this representation, messages and corresponding signals are
points in two function spaces: message space and signal space.
While a transmitter maps every point in the message space
into the signal space, a receiver does the reverse operation.
As long as the mapping is one-to-one from the message space
to the signal space, a message is always recoverable at the
receiver. Based on this framework, a waveform corresponds
to a specific structure in the signal space and identifies the
formation of the signals. Throughout this survey, the signal
space is considered as a time-frequency plane where time
and frequency constitute its coordinates, which is a well-
known notation for representing one dimensional signals in
two dimensions [11], [12]. When the structure in signal space
relies on multiple simultaneously-transmitted subcarriers, it
corresponds to a multicarrier scheme. It is represented by
x(t) =
Xmkgmk(t) ,
(1)
∞
Xm=−∞
N −1
Xk=0
where m is the time index, k is the subcarrier index, Xmk is
the symbol (message) being transmitted, N is the number of
subcarriers, and gmk(t) is the synthesis function which maps
Xmk into the signal space. The family of gmk(t) is referred
to as a Gabor system, when it is given by
gmk(t) = ptx (t − mτ0) ej2πkν0t ,
(2)
where ptx (t) is the prototype filter (also known as pulse
shape, Gabor atom), τ0 is the symbol spacing in time, and
ν0 is the subcarrier spacing. A Gabor system implies that a
single pulse shape is considered as a prototype and others
are derived from the prototype filter via some translations
in time and modulations in frequency, as given in (2). The
coordinates of the filters form a two dimensional structure in
the time-frequency plane, known as lattice. Assuming a linear
time-varying multipath channel h(τ, t), the received signal is
obtained as
y(t) =Zτ
h(τ, t)x(t − τ )dτ + w(t),
(3)
where w(t) is the additive white Gaussian noise (AWGN).
Then, the symbol ˜Xnl located on time index n and subcarrier
index l is obtained by the projection of the received signal
onto the analysis function γnl(t) as
˜Xnl = hy(t), γnl(t)i ,Zt
y(t)γ∗
nl(t)dt ,
where
γnl(t) = prx (t − nτ0) ej2πlν0t .
(4)
(5)
TX Filter
Time-frequency plane
Sampling the
time-frequency plane
(b) Sampling the time-frequency plane. Modulated pulses are placed into the
time-frequency plane, based on the locations of samples. In the illustration,
the product of 1/τ0ν0 corresponds to the lattice density.
Fig. 1. Utilization of the prototype filters at the transmitter and the receiver.
Similar to (1), γnl(t) given in (5) is obtained by a prototype
filter prx (t) translated in both time and frequency, constructing
another Gabor system at the receiver.
The equations (1)-(5) correspond to basic model for multi-
carrier schemes illustrated in Fig. 1(a), without stressing the
variables in the equations. In a crude form, a multicarrier
scheme can be represented by a specific set of equations
constructed in the time-frequency plane, i.e., these equations
are synthesized at the transmitter and analyzed at the receiver.
In the following subsections, symbols, filters, and lattices in a
multicarrier system are discussed in detail.
B. Symbols
Without
loss of generality,
the transmitted symbols are
denoted by Xmk ∈ C, where C is the set of all complex
numbers. As a special case, it is possible to limit the set of
Xmk to real numbers, i.e. Xmk ∈ R, where R is the set of all
real numbers. One may choose Xmk as a modulation symbol
or a part of the modulation symbol, e.g. its real or imaginary
part or a partition after a spreading operation. In addition, it
is reasonable to consider finite number of elements in the set,
based on the limited number of modulation symbols in digital
communications. Note that the set of the symbols may be
important for the perfect reconstruction of the symbols since
its properties may lead one-to-one mapping from message
space to the signal space [29], as in signaling over Weyl-
Heisenberg frames, faster-than-Nyquist signaling, or partial-
response signaling [23], [24], [30], [31].
and
ptx =Xm,k
R−(1−ρ)pmk ,
3
(8)
C. Filters
In digital communication, symbols are always associated
with pulse shapes (also known as filters). A pulse shape essen-
tially corresponds to an energy distribution which indicates the
density of the symbol energy (in time, frequency, or any other
domain). Hence, it is one of the determining factors for the
dispersion characteristics of the signal. At the receiver side, the
dispersed energy due to the transmit pulse shape is coherently
combined via receive filters. Thus, the transmit and receive
filters jointly determine the amount of the energy transfered
from the transmitter to the receiver. Also, they determine the
correlation between the points in the lattice, which identify
the structure of the multicarrier scheme, i.e., orthogonal, bi-
orthogonal, or non-orthogonal.
1) Matched Filtering: If the prototype filter employed at the
receiver is the same as the one that the transmitter utilizes, i.e.,
ptx (t) = prx (t), this approach corresponds to matched filter-
ing, which maximizes signal-to-noise ratio (SNR). As opposed
to matched filtering, one may also use different prototype
filters at the transmitter and receiver, i.e., ptx (t) 6= prx (t)
[16].
2) Orthogonality of Scheme: If the synthesis functions and
the analysis functions do not produce any correlation between
the different points in the lattice,
i.e., hgmk(t), γnl(t)i =
δmnδkl, where δ is the Kronecker delta function, the scheme
is either orthogonal or bi-orthogonal. Otherwise, the scheme
is said to be non-orthogonal, i.e., hgmk(t), γnl(t)i 6= δmnδkl
. While orthogonal schemes dictate to the use of the same
prototype filters at the transmitter and receiver, bi-orthogonal
schemes allow to use different prototype filters at the trans-
mitter and the receiver.
A nice interpretation on orthogonality and bi-orthogonality
is provided in [18]. Let R be a Gram matrix given by
R , QQH where QH is a block-circulant matrix in which the
columns consist of the modulated-translated vectors generated
by an initial filter p(t). Then, the relation between the filters
at the transmitter and the receiver for orthogonal and bi-
orthogonal schemes can be investigated by
Xmk = RR−1Xmk = R−ρRR−(1−ρ)Xmk
= R−ρQQHR−(1−ρ)Xmk
Transmitted signal
(R−(1−ρ)Q)HXmk
(6)
}|
Received symbol
z
= R−ρQ ×
|
{z
{
}
where [·]H is the Hermitian operator and ρ is the weighting
parameter to characterize orthogonality and bi-orthogonality.
Using (6),
the transmitter and the
receiver can be obtained from the first rows of R−ρQ and
R−(1−ρ)Q, respectively, which yields
the prototype filters at
where pmk is the column vector generated by modulating and
translating p. As a simpler approach, it is also possible to
calculate (7) and (8) as
and
prx = S−ρp
ptx = S−(1−ρ)p
(9)
(10)
respectively, where S = QHQ. While the choice ρ = 1/2
leads to an orthogonal scheme, ρ → 0 or ρ → 1 result in bi-
orthogonal schemes [18]. When ρ = 1, minimum-norm dual
pulse shape is obtained.
Note that orthogonal schemes maximize SNR for AWGN
channel [18] since they assure matched filtering. On the
contrary, bi-orthogonal schemes may offer better performance
for dispersive channels, as stated in [16]. In addition, when
the scheme has receive filters which are not orthogonal to
each other, i.e., hγn′l′ (t), γnl(t)i 6= δn′nδl′l, the noise samples
becomes correlated, as in non-orthogonal and bi-orthogonal
schemes [16].
3) Localization: The localization of a prototype filter char-
acterizes the variances of the energy in time and frequency.
While the localization in time is measured by ktp(t)k2, the
localization in frequency is obtained as kf P(f )k2, where k·k
is the L2-norm and P(f ) is the Fourier transformation of p(t).
The functions where kf P(f )kktp(t)k → ∞ are referred as
non-localized filters; otherwise, they are referred as localized
filters.
D. Lattices
A lattice corresponds to an algebraic set which contains
the coordinates of the filters in the time-frequency plane [18],
[20], [21], [23], [25]. In other words, it is a set generated by
sampling the time-frequency plane as illustrated in Fig. 1(b).
It determines the bandwidth efficiency and the reconstruction
properties of a multicarrier scheme. Without loss of generality,
a lattice Λ can be described by a non-unique generator matrix
L as,
L =x y
z ,
0
(11)
where x, z 6= 0. The generator matrix contains the coordinates
of the first two identifying points of the lattice in its column
vectors, i.e., (0, x) and (y, z) [18]. The locations of other
points are calculated by applying L to [m k]T , where [·]T
is the transpose operation.
1) Lattice Geometry: Generator matrix L determines the
lattice geometry. For example, the choice
prx =Xm,k
R−ρpmk
(7)
0
L =T
0 F ,
(12)
yields a rectangular structure as in (2) and (5), with a symbol
duration of T and subcarrier width F . Similarly, a hexagonal
(or quincunx) pattern [18], [23], [25] is obtained when
L =T 0.5T
F .
0
(13)
Lattice geometry identifies the distances between the points
indexed by the integers m and k. For example, assuming that
F = 1/T , while the minimum distance between the points is 1
for the rectangular lattice in (12), it is √1.25 for the quincunx
lattice in (13) [25].
2) Lattice Density/Volume: Lattice density can be obtained
as
δ(Λ) =
1
vol(Λ)
=
,
(14)
1
|det (L)|
where |·| is the absolute value of its argument and vol(Λ) is the
volume of the lattice Λ calculated via determinant operation
det (·). It identifies not only the bandwidth efficiency of the
scheme as
ǫ = βδ(Λ),
(15)
where β is the bit per volume, but also the perfect recon-
struction of the symbols at the receiver. In order to clarify
the impact of the lattices on the perfect reconstruction of the
symbols, the concept of basis is needed to be investigated
along with Gabor systems.
A set of linearly independent vectors is called a basis if
these vectors are able to represent all other vectors for a
given space. While including an extra vector to the basis
spoils the linear independency, discarding one from the set de-
stroys the completeness. From communications point of view,
having linearly independent basis functions is a conservative
condition since it allows one-to-one mapping from symbols
to constructed signal without introducing any constraints on
the symbols. Representability of the space with the set of
{gmk(t)} is equivalent to the completeness property, which
is important in the sense of reaching Shannon’s capacity [15],
[16]. Gabor systems provide an elegant relation between the
linear independence and the completeness properties based on
the lattice density. This relation for Gabor systems is given as
follows [16], [22], [26], [32]:
• Undersampled case (δ(Λ) < 1): Gabor system cannot
be a complete basis since the time-frequency plane is not
sampled sufficiently. However, this case gives linearly in-
dependent basis functions. Well-localized prototype filters
can be utilized, but the bandwidth efficiency of the Gabor
system degrades with decreasing δ(Λ).
• Critically-sampled case (δ(Λ) = 1): It results in a
complete Gabor System. Bases exist, but they cannot
utilize well-localized prototype filters according to the
Balian-Low theorem [12]. This theorem states that there
is no well-localized function in both time and frequency
for a Gabor basis where δ(Λ) = 1. It dictates the
use of non-localized functions, e.g., rectangular and sinc
functions. A consequence of Balian-Low theorem can
also be observed when the dual filters are calculated as
in (7) and (8). If one attempts to utilize a well-localized
4
function, e.g., Gaussian, when δ(Λ) = 1, the Gram matrix
R in (6) becomes ill-conditioned. Hence, the calculation
of the dual pulse shape becomes difficult. The degree
of ill-conditioning can be measured via the condition
number of R. As stated in [18], the condition number
of R approaches to infinity for Gaussian pulses when
δ(Λ) → 1.
• Oversampled case (δ(Λ) > 1): It yields an overcomplete
set of functions. Gabor system cannot be a basis, but
it may be a frame1 with well-localized pulse shapes.
However, since the Gabor system is overcomplete, rep-
resentation of a signal might not be unique. Note that
non-unique representations do not always imply loss of
one-to-one mapping from modulation symbols to signal
constructed. For example, a finite number of modulation
symbols may be useful to preserve the one-to-one relation
between the signal space and the message space [24].
E. A Combined Approach: Lattice Staggering
It is possible to circumvent the restriction of Balian-Low
theorem on the filter design with lattice staggering2 [13], [15],
[32], [35], [36]. It is a methodology that generates inherent
orthogonality between the points in the lattice for real domain
through mandating symmetry conditions on the prototype
filter. Since the inherent orthogonality does not rely on the
cross-correlation between the filters, it relaxes the conditions
for the filter design. It is worth noting that the real domain
may be either the imaginary portion or the real portion of
the complex domain. Thus, in lattice staggering, the real and
imaginary parts of the scheme are treated separately. However,
processing in real domain does not imply that the real and
imaginary parts do not contaminate each other. Indeed, they
interfere, but the contamination is orthogonal to the desired
part.
The concept of lattice staggering is illustrated in detail
in Fig. 2. The lattices on real and imaginary parts of the
scheme are given in Fig. 2(a) and Fig. 2(b), respectively. They
also correspond to the lattices on in-phase and quadrature
branches in baseband. While the filled circles represent the
locations of the filters on the cosine plane, the empty circles
show the locations of the filters on the sine plane. Cosine
and sine planes indicate that the filters on those planes are
modulated with either cosine or sine functions. First, consider
the lattice given in the cosine plane of Fig. 2(a). According
to the Euler’s formula, a pulse modulated with a complex
exponential function includes components on both cosine
and sine planes. Hence, when the filters on this lattice are
modulated with complex exponential functions, there will be
same lattice on the cosine and sine planes, as illustrated in
Fig. 2(a) and Fig. 2(b). It is important to observe that the
1Frames are introduced in 1952 by Duffin and Schaeffer [33], as an
extension of the concept of a basis. They can include more than the required
elements to span a space. This issue corresponds to an overcomplete system,
which causes non-unique representations [6], [34].
2In the literature, this approach appears with different names, e.g. offset
quadrature amplitude modulation (OQAM), staggered modulation. Rather than
indicating a specific modulation, it is referred as lattice staggering throughout
the study.
Orthogonality based on
Nyquist filter design
Cosine Plane
Cosine Plane
Orthogonality
Orthogonality
based on
based on
symmetry
symmetry
Orthogonality based on
Nyquist filter design
5
Orthogonality
based on
symmetry
Sine Plane
Sine Plane
(a) Real part of the multicarrier signal (in-phase component).
(b) Imaginary part of the multicarrier signal (quadrature component)
(c) Illustration for the orthogonality based on even-symmetric filters. Inner product
of three functions is zero when τ0 is set to 1/2ν0.
Fig. 2. Lattice staggering.
cross-correlation among the points indicated by arrows in the
cosine plane of Fig. 2(a) is always zero, when the filters
are even-symmetric and the symbol spacing is selected as
τ0 = 1/2ν0. This is because of the fact that the integration of
a function which contains a cosine function multiplied with
a symmetric function about the cosine’s zero-crossings yields
zero, as illustrated in Fig. 2(c). From the communications point
of view, this structure allows to carry only one real symbol
without interference. Considering the same structure on the
imaginary part by staggering the same lattice, illustrated in
Fig. 2(b), another real symbol could be transmitted. Although
transmitting on the imaginary and real parts leads to contam-
inations on the sine planes, these contaminations are always
orthogonal to the corresponding cosine planes when the filter
is an even-symmetric function.
Lattice staggering induces an important result for the filter
design: In order to obtain an orthogonal or biorthogonal
scheme using lattice staggering, the correlation of the transmit
filter and the receive filter should provide nulls in time and
frequency at the multiples of 2τ0 and 2ν0. In other words,
the unit area between the locations of the nulls in time-
frequency plane is 2τ0 × 2ν0, which is equal to 2 since τ0 is
set to 1/2ν0. This is because of the fact that even-symmetrical
filters provide inherent orthogonality between some of the
diagonal points in lattice when τ0 = 1/2ν0 even though the
filters do not satisfy Nyquist criterion, as illustrated in Fig. 2.
Also, this approach circumvents the Balian-Low theorem since
δ(Λ) = 1/τ0ν0 = 2. Hence, lattice staggering allows or-
thogonal and bi-orthogonal schemes with well-localized filters
while maintaining bandwidth efficiency as ǫ = β. From the
mathematical point of view, lattice staggering corresponds to
utilizing Wilson bases on each real and imaginary parts of the
scheme, which is equivalent to the linear combinations of two
Gabor systems where δ(Λ) = 2 [7], [9], [17], [18].
F. Summary
As a summary, the relations between the fundamental ele-
ments of a multicarrier scheme, i.e. symbols, lattice, and filter
are given in Fig. 3. One can determine these elements based on
Gabor theory, considering the needs of the communication sys-
tem. For example, let the signaling be an orthogonal scheme
which is based on a rectangular lattice geometry without lattice
staggering. Then, the localization of the filter is determined
according to the statements of Gabor theory. For instance,
equipping this system with well-localized filters yields an ill-
conditioned R when δ(Λ) = 1. Other inferences can also be
obtained by following Fig. 3.
III. MULTICARRIER SCHEMES
In this section, the concepts introduced in Section II are
harnessed and they are associated with known multicarrier
schemes. Rather than discussing the superiorities of schemes
to each other,
the relations between the orthogonal, bi-
orthogonal, and non-orthogonal schemes within the framework
of Gabor theory are emphasized. An interpretation of the
spreading operation in multicarrier systems (e.g., as in single
carrier frequency division multiple accessing (SC-FDMA)) is
also provided in the context of Gabor systems. Finally, mile-
stones for multicarrier schemes reviewed for completeness.
6
Filter
Lattice
Symbols
Multicarrier Schemes
Orthogonal
Non-orthogonal Bi-orthogonal
Rectangular
Hexagonal
Other
Real
Complex
Filters are
matched.
Non-localized
Filters
Filters are not
matched.
Localized
Filters
Finite Number
of Elements
Infinite Number
of Elements
Fig. 3. Multicarrier schemes based on lattices, filters, and symbols.
A. Orthogonal Schemes
The schemes that fall into this category have orthogonal
basis functions at both the transmitter and the receiver and
follow matched filtering approach. The phrase of orthogonal
frequency division multiplexing is often used for a specific
scheme that is based on rectangular filters. However, there
are other multicarrier schemes which provide orthogonality.
We begin by describing the orthogonal schemes which do not
consider lattice staggering:
• Plain & Zero Padded-OFDM (ZP-OFDM): Plain OFDM
is an orthogonal scheme which is equipped with rect-
angular filters at the transmitter and the receiver when
δ(Λ) = 1. In order to combat with multipath channel,
one can provide guard interval between OFDM symbols,
known as ZP-OFDM [37]. It corresponds to stretching
the lattice in time domain, which yields δ(Λ) < 1.
• Filtered multitone (FMT): FMT is an orthogonal scheme
where the filters do not overlap in frequency domain.
There is no specific filter associated with FMT. Instead
of the guard intervals in ZP-OFDM, guard bands between
the subcarriers can be utilized in order to obtain more
room for the filter localization in frequency domain.
Hence, it is based on a lattice where δ(Λ) ≤ 1. For more
details we refer the reader to the studies in [2], [38]–[42].
• Lattice-OFDM: Lattice-OFDM is the optimum orthog-
onal scheme for time- and frequency- dispersive chan-
nels in the sense of minimizing interference between
the symbols in the lattice [18]. It relies on different
lattice geometries and orthogonalized Gaussian pulses,
depending on the channel dispersion characteristics. For
more details, we refer the reader to Section VI-A.
Well
Well
Ill
Well
Well
Well
Condition Number of
(a) Plain OFDM.
(c) SMT.
(b) FMT.
(d) CMT.
Fig. 4.
Illustrations of various orthogonal multicarrier schemes.
real or imaginary part of the modulation symbols. The
main difference between the SMT and the CMT is the
modulation type. While SMT uses quadrature amplitude
modulation (QAM) type signals, CMT is dedicated to
vestigial side-band modulation (VSB). Yet, SMT and
CMT are structurally identical; it is possible to synthesize
one from another by applying a frequency shift operation
and proper symbol placement [36].
Illustrations for Plain/ZP-OFDM, FMT, SMT, and CMT in
time and frequency are provided in Fig. 4.
The orthogonal schemes which consider lattice staggering
are given as follow:
B. Bi-orthogonal Schemes
• Staggered multitone (SMT) and Cosine-modulated multi-
tone (CMT): Both schemes exploit the lattice staggering
approach to obtain flexibility on the filter design when
ǫ = β [2], [36], where ǫ is the bandwidth efficiency
and β is the bit per volume, introduced in (15). In these
schemes, the symbols are real numbers due to the lattice
staggering, however, as a special case, they are either
These schemes do not follow matched filtering approach
and do not have to contain orthogonal basis functions at the
transmitter and the receiver. However, transmit and receive
filters are mutually orthogonal to each other.
• Cyclic Prefix-OFDM (CP-OFDM): Plain OFDM is often
utilized with cyclic prefix (CP) to combat with the
multipath channels. CP induces a lattice where δ(Λ) > 1.
the same time,
At
it results in a longer rectangular
filter at the transmitter, compared to one at the receiver.
Therefore, CP-OFDM does not follow matched filtering
and constructs a bi-orthogonal scheme [3], [21], [43]. Yet,
it provides many benefits, e.g. single-tap equalization and
simple synchronization.
• Windowed-OFDM: OFDM has high out-of-band radiation
(OOB) due to the rectangular filter. In order mitigate
the out-of-band radiation, one may consider to smooth
the transition between OFDM symbols. This operation
smooths the edges of rectangular filter, and commonly
referred as windowing. If the windowing is performed
with an additional guard period, a bi-orthogonal scheme
where δ(Λ) > 1 is obtained.
• Bi-orthogonal frequency division multiplexing (BFDM):
In [16], it is stated that extending the rectangular filter as
in CP-OFDM is likely to be a suboptimal solution under
doubly dispersive channels, since this approach does not
treat the time and frequency dispersions equally. As an
alternative to CP-OFDM, by allowing different filters at
the transmitter and the receiver, BFDM with properly
designed filters can reduce the interference contribution
from other symbols in doubly dispersive channels. In
[16],
the design is given based on a prototype filter
constructed with Hermite-Gaussian function family and
a rectangular lattice geometry when δ(Λ) = 1/2. In
order to maintain the bandwidth efficiency, one can utilize
BFDM with lattice staggering, as investigated in [35],
[44].
• Signaling over Weyl-Heisenberg Frames: Main motiva-
tion is to exploit overcomplete Gabor frames with well-
localized pulses and finite number of symbols for digital
signal transmission. It is a unique approach that allows a
scheme where δ(Λ) > 1 with the perfect reconstruction
property, which exploits subspace classifications [24].
It is interesting to examine bi-orthogonal schemes from
the point of equalizers. We refer the reader to the related
discussion provided in Section VI-B.
C. Non-orthogonal Schemes
The schemes that fall into this category do not contain
orthogonal basis functions at the transmitter or the receiver.
Also, there is no bi-orthogonal relation between the filters at
the transmitter and the filters at the receiver.
• Generalized Frequency Division Multiplexing: General-
ized frequency division multiplexing (GFDM) is a non-
orthogonal scheme which allows correlation between the
points in the lattice in order to be able to utilize well-
localized filters when δ(Λ) = 1 [45]. Also, it utilizes
complex symbols and CP along with tail biting in the
pulse shape. At the receiver side, successive interference
cancellation is applied to remove the interference between
the symbols [46].
• Concentric Toroidal Pulses: By exploiting the orthog-
onality between Hermite-Gaussian functions, concentric
toroidal pulses are introduced to increase the bandwidth
efficiency of the transmission [47]. Four Hermite pulses
7
are combined on each point
in the lattice and each
Hermite pulse carries one symbol. Although Gaussian-
Hermite functions are orthogonal among each other, the
pulses between neighboring points are not orthogonal.
• Faster-than-Nyquist & Partial Response Signaling: When
δ(Λ) > 1, certain conditions may yield the recon-
struction of the transmitted symbols. This issue firstly
is investigated by Mazo in 1975 as faster-than-Nyquist
by addressing the following question: to what extent
can the symbols be packed more than the Nyquist rate
without loss in bit error rate (BER) performance? It
is shown that the symbol spacing can be reduced to
0.802T without suffering any loss in minimum Euclidean
distance between the synthesized signals for binary mod-
ulation symbols and sinc pulse [31]. In other words, BER
performance is still achievable with optimal receivers
even when the symbols are transmitted at a rate greater
than the Nyquist rate. The minimum symbol spacing
that keeps the minimum Euclidean distance is later on
referred as the Mazo limit
in the literature. By gen-
eralizing faster-than-Nyquist approach to other pulses,
various Mazo limits are obtained for root-raised cosine
(RRC) pulses with different roll-off factors in [48]. For
example, when roll-off is set to 0, 0.1, 0.2, and, 0.3,
Mazo limits are derived as 0.802, 0.779, 0.738, and,
0.703 respectively. The faster-than-Nyquist approach is
extended to multicarrier schemes by allowing interference
in time and frequency in [49]–[51], which show that two
dimensional signaling is more bandwidth efficient than
one dimensional signaling. Another way of developing a
scheme where δ(Λ) > 1 is to transmit correlated symbols.
This approach corresponds to partial-response signaling
and introduced in [30]. Similar to the faster-than-Nyquist
signaling and partial-response signaling, in [23], Weyl-
Heisenberg frames (δ(Λ) > 1) is examined considering
a hexagonal lattice geometry and sequence detector is
employed at the receiver for symbol detection.
D. Multicarrier Schemes with Spreading Approaches
Spreading operation is commonly used to reduce the peak-
to-average-power ratio (PAPR) in multicarrier schemes,
in
which modulation symbols are mapped to the multiple points
in the lattice. One way to interpret and generalize the spreading
operation in multicarrier systems (e.g., as in SC-FDMA [52]
and filter-bank-spread-filter-bank multicarrier (FB-S-FBMC)
[53]) is to consider another Gabor system that spreads the
energy of the modulation symbols into multiple subcarriers.
In other words, as opposed to using single Gabor system at
the transmitter and receiver, two Gabor systems combined with
serial-to-parallel conversions are employed at the transmitter
and the receiver. For example, it can be said that SC-FDMA,
which allows better PAPR characteristics and frequency do-
main equalization (FDE) along with CP utilization [54]–[57],
employs an extra Gabor system equipped with a rectangular
filter and δ(Λ) = 1 to spread the modulation symbols at
the transmitter (i.e., discrete Fourier transformation (DFT))
and de-spread them at the receiver (i.e., inverse DFT). On
the contrary, in OFDM, since there is no spreading of the
modulation symbols in frequency domain, employed prototype
filter for spreading is a Dirac function.
E. Milestones for Orthogonal Schemes
Having discussed the different variations of multicarrier
systems in the earlier subsections, this subsection provides a
brief history on the development of aforementioned multicar-
rier systems. Earlier works related to orthogonal multicarrier
schemes actually date back to 1960s [58], [59], which uti-
lize a bank of filters for parallel data transmission. In [58],
Chang presented the orthogonality condition for the multi-
carrier scheme schemes considering band-limited filters. This
condition basically indicates that the subcarriers can be spaced
half of the symbol rate apart without any interference. This
scheme has then been re-visited by Saltzberg in 1967 [59] by
showing the fact that Chang’s condition is also true when the
time and frequency axes are interchanged, based on OQAM.
Indeed, Chang and Saltzberg exploits the lattice staggering
for their multicarrier schemes which includes the basics of
CMT and SMT. However, the idea of parallel transmission
suggested in [58] and [59] were unreasonably expensive and
complex for large number of data channels at that time. In [60],
through the use of DFTs, Weinstein and Ebert eliminated the
banks of subcarrier oscillators to allow simpler implementation
of the multicarrier schemes. This approach has been later
named as OFDM, and it has become more and more popular
after 1980s due to its efficient implementation through fast
Fourier transformation (FFT) techniques and FDE along with
CP utilization [61] compared to other multicarrier schemes.
On the other hand, Weinstein’s DFT method in [60] limits
the flexibility on different baseband filter utilization while
modulating or demodulating the subcarriers, but instead used a
time windowing technique to cope with the spectral leakage.
In [62], by extending Weinstein’s method, Hirosaki showed
that different baseband filters may also be digitally imple-
mented through DFT processing by using a polyphase network
(PPN) [63], [64]. Several other developments over the last
two decades have demonstrated low complexity and efficient
implementations of lattice staggering, paving the way for its
consideration in the next generation wireless standards (see
e.g., [2], [15], [65], and the references listed therein).
IV. FILTER DESIGN
In a multicarrier scheme, a prototype filter determines the
correlation between the symbols and the robustness of the
scheme against dispersive channels. This issue induces to
design prototype filters which are suitable for communications
in time-selective and frequency-selective channels. The goal of
this section is to review the filters available in the literature.
In order to reveal the connections between the filters, we
categorized the filters based on their design criteria: 1) energy
concentration [18], [66]–[76], 2) rapid-decay [77]–[80], 3)
spectrum-nulling, and 4) channel characteristics and hardware.
Analytical expressions of the investigated filters are given in
TABLE I. For more detailed discussions on the discussed
filters, we refer the reader to the review papers [2], [81], [82]
and the books [34], [42].
8
A. Design Criterion: Energy Concentration
In practice, limiting a pulse shape in time decreases the
computational complexity and reduces the communications
latency, which are inversely proportional to the filter length.
However, using shorter or truncated filter may cause high
sidelobes in the frequency domain. Prolate spheroidal wave
functions (PSWFs) address this energy-concentration trade-off
problem through obtaining a time-limited pulse with minimum
out-of-band leakage or a band-limited pulse with maximal
concentration within given interval. There are severals ways
to characterize PSWFs [75]. A convenient definition for the
prototype filter design is that PSWFs, {ψn,τ,σ (t)}, is a family
that includes the orthogonal functions which are optimal in
terms of the energy concentration of a σ-bandlimited function
on the interval [−τ, τ ], where n is the function order. In the
family, ψ0,τ,σ (t) is the most concentrated pulse and the con-
centration of the functions decreases with the function order. In
other words, ψn,τ,σ (t) is the most concentrated function after
ψn−1,τ,σ (t) and it is also orthogonal to ψn−1,τ,σ (t). Hence,
if one provides the filter length and the bandwidth (where the
pulse should be concentrated) as the design constraints, the
optimum pulse becomes ψ0,τ,σ (t) constructed based on these
constraints.
PSWFs have many appealing properties [67], [71]. For
example, they are the eigenfunctions of the operation of first-
truncate-then-limit-the-bandwidth. Therefore, these functions
can pass through this operation without any distortion or
filtering effect excluding the scaling with a real coefficient,
i.e, eigenvalue, which also corresponds to the energy after
this operation. Assuming that
the energy of the pulse is
1, eigenvalues will always be less than 1. Also, PSWFs
correspond to an important family when τ = σ → ∞, known
as Hermite-Gaussian functions which are the eigenfunctions
of Fourier transformation. Hermite-Gaussian functions provide
optimum concentration in time and frequency at the same time.
Hence, they are able to give isotropic (same) responses in
time and frequency. We also refer the reader to the detailed
discussions on the properties of PSWFs in [66], [68]–[70],
[75], [76]
In the following subsections, the prototype filters that target
time-frequency concentration are discussed. Their characteris-
tics are inherently related with the PSWFs.
1) Prolate Window: Prolate window addresses the energy
concentration in frequency for a given filter length and band-
width. In time domain, its expression corresponds to ψ0,τ,σ (f )
or 0th order Slepian sequence in time for the discrete case
[70]. This issue is explained as a sidelobe minimization
problem in [2], as shown in TABLE I. The time and frequency
characteristics of prolate window are given in Fig. 5.
2) Kaiser Function: An efficient solution for a filter with
finite length is proposed by Jim Kaiser by employing Bessel
functions to achieve an approximation to the prolate window
[72], [81]. It offers a suboptimal solution for the out-of-band
leakage. A favorable property of Kaiser filter is its flexibility
to control the sidelobes and stop-band attenuation, through a
single design parameter β with a closed-form expression. The
expression is given in TABLE I where I0(x) denotes the zeroth
order modified Bessel function of the first kind.
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