The Tenth International Symposium on Wireless Communication Systems 2013
Non-uniform FBMC - A Pragmatic Approach
Slađana Jošilo, Miloš Pejović, and Slobodan Nedić♦
University of Novi Sad, Trg Dositeja Obradovica 6, Novi Sad, Serbia
♦
Corresponding author: nedics@uns.ac.rs
Abstract
Outgoing from the known uniform FBMC (Filter-Bank Multi-Carrier) formats with Nyquist spectral shaped sub-channels and its
time-frequency dual Time-Limited Orthogonal (TLO) form, we introduce an orthogonal frequency division multiplex of nonuniformly spaced and unequal-width sub-channels. The goal is to attain potential gains in using relatively small number of subchannels, but still allow for frequency gaps needed for channelization with a small decrease in overall spectral efficiency. The
orthogonality conditions are evaluated through simulations using the extended OFDM framework, whereby the corresponding
referent filter-bank impulse responses are defined in frequency-domain by straightforward aggregations of the pertaining uniform
filter-bank sub-channels spectral shapes (for conventional FBMC with frequency limited sub-channels spectra), and by
transforming to frequency-domain of adequately aggregated time-limited referent impulse responses of uniform TLO
configuration. An analytical derivation, i.e. confirmation of the orthogonality conditions are also derived with reliance on the
uniform FBMC and TLO orthogonality conditions. Non-symmetrical spectral shaped sub-channels in the case of FBMC format
are also proposed.
Keywords- Filter-Bank Multi-Carrier (FBMC), uniform filter-bank, non-uniform filter-bank, orthogonal frequency division
multiplexing (OFDM), quadrature amplitude modulation (QAM), frequency-limited orthogonal (FLO), time-limited orthogonal
(TLO).
For the time being, we used the framework of extended
OFDM [5] for computer simulation based evaluation of
orthogonality conditions, which is based on frequencydomain implementation of the conventional overlap-andadd filtering method. We also experimented with
orthogonality for the unsymmetrical roll-off factors (in
frequency-domain), to be able to possibly better control
the latency inherent in the FBMC format.
After highlighting the time-frequency duality between
uniformly spaced FBMC and TLO formats in Section II,
a pragmatic method of aggregation of uniformly spaced
sub-channels and the related derivation of non-uniform
filter-bank orthogonality conditions are dealt with in
Section III, while in Section IV the asymmetrical subchannel spectral shaping is introduced. The similar
aggregation procedure, but in time-domain, is applied to
the TLO format in Section V, followed by the
conclusions part in Section VII.
I. INTRODUCTION
Although the FBMC formats have much better spectral
characteristics compared with the traditional CP-OFDM,
the longer referent impulse response needed therein has
turned into a certain disadvantage regarding the
additional latency introduced. To remedy this, the one
QAM interval (T-) long impulse response in the form of
cosine function in the range of its argument between –π/2
to π/2 (the Hanning window) proposed in [1, and its ref.
5], can be used. Since the staggered signaling formats, as
essentially real-domain ones, show great advantages in
case of the co-channel interference limited scenarios, as
shown in for example [2], the TLO format with the zero
roll-off factor in time-domain leads to a particular form
of Staggered, SCP-OFDM format, while 100% roll off
case gives OFDM of MSK signals.
In order to conciliate the advantages of using wider
sub-channels in terms of reduction of PAPR and increase
of spectral efficiency in situations when predetermined
power spectral density (PSD) masks have to be obeyed,
while at the same time being able to separate the adjacent
channels by relatively narrow frequency guard-intervals,
the need arises for a modification which would enable
utilization of sub-channels with differing widths within
scattered frequency bands (white-zones), and in particular
scarcely available frequency-gaps in the targeted Private
Mobile Radio (PMR) and Cognitive Radio applications.
This in principle can be done by adopting widely
explored and quite well studied non-uniform filter-bank
configurations known and used for source coding
applications, starting from some early proposals, as in[3].
II.
UNIFORMLY SPACED TLO-MC
AND FBMC
Difference between the standard FBMC, i.e.
OFDM/OQAM formats with frequency-domain shaping
(which could be termed as the FLO – Frequency-Limited
Orthogonal) and the TLO multi-carrier formats [1]
consists in the utilization of time-frequency dual impulse
responses, as illustrated in Fig. 1. It may suffice to only
note that in the case of zero-percent roll off case TLO,
the referent impulse response has rectangular form of
length T/2, while in the 100% roll-off case, it has the
ISBN 978-3-8007-3529-7
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
36
The Tenth International Symposium on Wireless Communication Systems 2013
roll-off factor (i.e. for their s.r.r.c – square-root raisedcosine Tx and Rx pertinent sub-channels‟ transfer
function) is 1/T, corresponding to QAM symbol rate. The
frequency shift between even and odd sub-channels
central frequencies is 1/2T. Each of the individual
spectral shapes shown here by triangles are represented
by 2K+1 equidistant samples. Their overlapping samples
are added at the KM-IFFT input, [5, Fig. 10], and the
signal samples at the KM-FFT outputs summed-up, [5,
Fig. 11], after previously having been weighted by the
same set of frequency domain samples.
If the KM-long vectors belonging to equidistantly
spaced (real and positive) sub-channels‟ samples, among
which there are only 2K+1 generally non-zero values, are
form of the half of the cosine function, and has length of
T. Also, since the OFDM requires use of the CyclicPrefix (CP), which now (if at all) has to be inserted at the
beginning of each of the T/2 long OFDM symbols, in
order for it to still be long enough to absorb the delay
spreads for particular usage scenarios, more spectral
inefficiency has to be allowed for, or the number of subchannels be doubled, with the consequent increased
impact of non-linear distortions and the related spectral
broadening caused by the use of High-Power Amplifiers
(HPA).
denoted by
Gˆ i , with i = 0 to KM-1, then the wider and
generally non-uniformly spaced sub-channels samples
can be defined by
iy x Ĝi .
2
For example, from the odd-spaced uniform filter-bank
the following arrangement of the non-equidistant subchannels can be made:
sCh1p= i70 Gˆ i , sCh2p= 11
ˆ i2 , sCh3p= 13
ˆ i2 ,
i 12 G
i 8 G
etc.
The sub-channels denoted sCh4p and sCh4m (p and m
are used respectively for marking positive and negative
frequencies) are thus defined by the original sub-channels
with roll-off factor 100%, so that the roll-offs of the
wider sub-channels become progressively reduced to
50%, 25% and 12.5% by the effective doubling of subchannels‟ bandwidths, along their signaling intervals
being correspondingly halved, with central frequencies
on the corresponding multiples of 1/2T, as can be
inferred from the illustration in Fig. 3a. (In sub-section A.
it will be shown that the orthogonality conditions
between sub-channels formed in this way reduce to the
ones of the uniform FBMC.)
The sum of the pertinent vectors of the non-uniform
sub-channels is fed to the input of the KM-size IFFT
[5, Fig 10.], and similarly reproduced at the output of
demodulator by appropriately weighting the KM-size
FFT output [5, Fig 11].
Sequencing of the QAM data samples parts happens at
the beginning and the half of the corresponding QAM
symbol intervals, and based on simulation experiments it
becomes determined by the odd and/or evenness of the
filter-bank arrangements produced by equidistant shifting
of the particular sub-channels: if particular sub-channel
produces an odd filter-bank, on it the signaling is always
by Re or j•Im, and if it produces an even filter-bank, then
interchangeably Re and j•Im are brought to KM-IFFT
block, so that the adjacent sub-channels at the same time
instant are purely real and purely imaginary, or the other
way around. For the nonuniform filter-bank arrangement
of Fig. 3a, the one of the two options for the QAM parts
signaling is shown in Fig. 3b. In all the figures Re part is
marked by a square, and the j•Im part by a diamond.
2
Fig. 1. Time-frequency duality between the TLO and
FLO formats.
Implementation of these two forms of filter-banks,
both in the uniform and non-uniform configurations is
done through the extended OFDM approach [5],
essentially a frequency-domain based referent impulse
response design method, as described below.
III. AGGREGATING SUB-CHANNELS FOR
NU-FBMC
A direct extension of the uniform filter-bank towards
the non-uniform filter-bank configuration is to simply
aggregate the sub-channels of even, odd or combined
uniform filter-banks. In the following we construct
certain configurations and study the orthogonality
conditions in terms of the Re and j•Im signalling at T/2
spaced instants and their relationship, retaining the rolloff region spectral symmetry from the uniform filter-bank
arrangements shown in Fig. 2, with the frequency axis
normalized by fs/2π.
Fig. 2. The even (upper) and odd (lower) uniform
filter-banks, M=32.
The separation between the peaks of the stylized 100%
ISBN 978-3-8007-3529-7
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
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The Tenth International Symposium on Wireless Communication Systems 2013
A. Analytical derivation of orthogonality conditions
Conditions of intrinsic FLO system orthogonality
expressed in time domain provided by [1, i.e. ref. 4
therein] is given by
1
k l , n m (1)
*
Re gˆ n ,k (t ) gˆ m,l (t )dt
0
other
where the first and second indexes are for sub-channel
central frequency and T instant, respectively, and
2
Fig. 3. Non-uniform FBMC arrangement from odd
uniform FBMC, (a) and the example of sequencing the
QAM I and Q parts in odd-NU-FBMC.
gˆ n,k (t ) is given by
~
2k
nT
gˆ n,k (t ) f (t ) exp( j ~ t j n,k )
2
T
If the sCh1p and sCh1m would be merged (by the
square-root of the sum of their squares, as indicated
earlier), then the central (DC) sub-channel would
correspond to an even-spaced FBMC, illustrated as in
Fig. 4.
(2)
where
n k odd
n ,k 2
0 n k even
(3)
~
T is QAM symbol period, and f(t) is impulse response
of the square-root Nyquist filter with transfer function
F(ω). In [1], for derivation of the uniform FBMC-TLO
system it was used the fact that Gˆ ( ) , the Fourier
n,k
transform of gˆ (t ) can be used to express the FLO
n, k
orthogonality conditions in frequency-domain as
1 k l , n m
*
(4)
Re Gˆ n ,k ( )Gˆ m,l ( )d
0 other
where
~
nT
2k
(5)
j n ,k )
Gˆ n,k ( ) F ( ~ ) exp( j
2
T
By the straightforward designing of the non-uniformly
spaced sub-channels by aggregation of the uniform
FBMC sub-channels transfer functions, as shown again in
Fig. 6. as an example, it can be shown that the
orthogonality conditions of the former can be reduced to
conditions of the latter one.
Fig. 4. An example of the I/Q sequencing in NU-FBMC.
For derivation of a non-uniform FBMC configuration
by sub-channels aggregation it turns out that only noninteger ratios of the sub-channels bandwidth can be
produced. In order to keep the integer ratios, and thus to
have every NU-FBMC sub-channel belonging to either
the even or the odd extended uniform FBMC „category‟,
certain sub-channels have to be defined directly, or by
some combination of segments of the even and the odd
(referent) uniform FBMC arrangements. Fig. 5 illustrates
such a situation. Here, the sub-channels 2 and 3 contain
4.5 uniform sub-channels‟ lengths. Consequently, the
roll-off factors are 11.11%, 28.57 (sub-channels 2 and 3).
(Although the signaling intervals of sub-channel 4 are not
commensurate with the signaling intervals of the adjacent
sub-channels, the orthogonality conditions are satisfied
with the same QAM parts sequencing as in Fig. 4,
indicating that the non-integer signaling speeds are
possible, although it might be quite impractical for the
symbol synchronization.)
Fig. 6. Spectral aggregation example: (a) uniform
configuration (b) corresponding non-uniform
configuration
Here, sub-channels i and j are described by (6) and
(7) respectively
2
0
Gˆ s ,r ( ) xi Gˆ n x,k x ( )
(6)
2
j
(7)
Gˆ p ,q ( ) y 0 Gˆ m y ,l y ( )
By placing the newly generated transfer functions into
(4), it follows
Fig. 5. Non-uniform FBMC arrangement derived from
even form FBMC.
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© VDE VERLAG GMBH · Berlin · Offenbach, Germany
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The Tenth International Symposium on Wireless Communication Systems 2013
Re Gˆ s ,r ( )Gˆ p ,q ( )d
*
V. AGGREGATING REFERENT IRS FOR
THE NU-TLO
2
2
0
j
Re
xi Gˆ n x ,k x ( ) ( y 0 Gˆ m y ,l y ( ) ) d
Since the complex conjugation (*) can be brought
under the square-root, as well as under the squaring
operation, and since (c1 + c2)* = c1*+ c2*, the following
expression is produced
*
Similarly as the frequency domain representations of
the sub-channels spectra were produced by aggregation
of the uniform frequency-domain description, for the
non-uniform TLO formats, the aggregation is performed
at corresponding (time-domain) referent impulse
responses. Their frequency-domain representations are
subsequently produced, and appropriately positioned as
in case of the NU-FBMC counterpart. Spectrum of the
uniform TLO FBMC format for the even filter banks
arrangements is shown in Fig. 8a, and spectrum of the
non-uniform TLO FBMC format from the odd filter
banks arrangements is shown in Fig. 8b.
2
2
2
2
Re Gˆ ni ,k i ( ) Gˆ n(i1),k (i1) ( ) Gˆ n1,k 1( ) Gˆ n,k ( )
*2
*2
*2
*2
Gˆ m, l ( ) Gˆ m1,l 1( ) Gˆ m( j 1),l ( j 1) ( ) Gˆ m j ,l j ( )d
Due to spectral confinement to 1/T widths, from all of
the products under the root-square operation only
2
*2
Gˆ n , k ( ) Gˆ m, l ( ) is different from zero, so that the
orthogonality conditions are reduced to those of the
uniform FLO FBMC format, i.e.
*
*
Re Gˆ s ,r ( )Gˆ p, q ( )d Re Gˆ n,k ( )Gˆ m,l ( )d
1
0
k l, n m
other
(8)
~
( T is used above for interval T for the reason of
generality.)
IV.
USING
CHANNELS FOR
ASYMMETRICAL
(a)
SUB-
NU-FBMC (NU-FLO)
The relatively low roll-off factors resulting from the
uni-form FBMC aggregation might be unfavorable for at
least two reasons: increased transmission delay and
increased PAPR (Peak-to-Average Power Ratio). To
remedy these problems it would be of interest to keep the
NU-FBMC sub-channels‟ roll-off factors as large as
possible, while retaining high flexibility in assigning the
sub-channel bandwidths. This turns out to be possible by
directly defining the sub-channels frequency-domain
samples (shapes) for example, as illustrated in Fig. 7b.
(b)
Fig. 8. TLO FBMC arrangement: (a), even uniform,
M = 16, K =4; (b), odd non-uniform, M = 64, K = 4.
As was the case with non-uniform FLO formats, for
non-uniform TLO formats the orthogonality conditions
can also be shown to reduce to those of the uniformly
spaced case. Since in [1] the orthogonality conditions of
the uniform TLO FBMC format is derived by applying
the time-frequency duality, formally be replacing
~
symbols Gˆ , , F , T from (5) by g, t, w and 2π/T‟
respectively, (5) gets converted to
g n,k (t ) w(t kT ) exp( j
(a)
As a dual form,
n
t j n ,k )
T
g n, k (t )
(9)
also satisfies the
orthogonality condition given by (10) and thus forms the
uniform TLO orthogonal base.
1
k l, n m
(10)
Re g n,k (t ) g*m,l (t )dt
0
other
Based on the time-frequency duality between FLO and
TLO, expression (8) for non-uniform FLO applies as well
to the non-uniform TLO FBMC case, with appropriately
exchanged variables. Derivation is given below only for
the simplest case, where one sub-channel has the doubly
(b)
Fig. 7. NU-FBMC arrangement with symmetrical spectra
(a), and asymmetrical spectra (b).
ISBN 978-3-8007-3529-7
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
39
The Tenth International Symposium on Wireless Communication Systems 2013
longer signaling interval compared to its adjacent one, as
it is illustrated in Fig. 9.
VII.
We presented rather pragmatic approach towards
extending the uniformly spaced FBMC formats in the
forms of FLO and TLO to non-uniform ones. The
analysis and elaboration of non-uniform filter-bank, with
derivation of their orthogonality conditions, may provide
an useful introduction for further elaborations and its
practical use as an element of a flexible channelization,
and in particular for the realization of the individual
user‟s communication channel. In down-link, the
asymmetrical spectral shaping can avoid need for
separation of adjacent users‟ channels realized by
uniform FBMC with differing sub-channel widths. For
UL directions, by deploying the NU-FBMC individual
user‟s channels can be separated by a negligible loss in
spectral efficiency, to accommodate constraints of
exploitation, as the PSD mask, unsynchronized symbol
timings, and the carrier frequency offset uncertainties.
Fig. 9. Uniform configuration (a), and its corresponding
non-uniform configuration for two sub-channels (b).
If the referent impulse response of sub-channels
marked by 1, 0 and 2 are g n1,k 1 (t ), g n,k (t ) and
g m,l (t ), and the impulse response of the doubly wide sub-
channel 1 is
g s , r (t ), the process of time-domain
aggregation is expressed through
g s , r (t ) g n21,k 1(t ) g n2,k (t )
(11)
Going from (1) and by placing the newly generated
impulse responses (11) into the above expression, it
follows
Re g s ,r (t ) g*m,l (t )dt Re g n21,k 1(t ) g n2,k (t ) g*m,l (t )dt
Re g 2n1,k 1(t ) g*m2,l (t ) g 2n,k (t ) g*m2,l (t )dt
ACKNOWLEDGMENT
This work was supported by an FP7 grant, project
EMPhAtiC (http://www.ict-emphatic.eu/).
The last author thanks Mirjana Maksimović for her
earlier collaboration on this subject.
[1]
Because of absence of overlapping among the uniform
TLO sub-channels‟ impulse responses separated by (at
least) one shortest impulse response, the product
g2
g *2
n 1, k 1 (t ) m, l (t ) is equal to 0, so that the orthogonality
[2]
criterion reduces to the one corresponding to the uniform
FBMC case , i.e.
Re g s ,r (t ) g*m,l (t )dt Re g n,k (t ) g*m.l (t )dt
1
k l, n m
(12)
other
0
since the same expression is produced as in equation
(10), from which we started.
[3]
[4]
VI. COMPARISON WITH
CONVENTIONAL APPROACHES
[5]
The field of the non-uniform filter-banks analysis for
the source coding applications has been very well
developed, and two basic configurations have emerged –
modifications of the uniform filter banks implementation
by combination of (I)FFT and Polyphase Networks (PN)
by appropriately optimizing the referent low-pass filter
impulse response [7][8], and the application of
quadrature-mirror filters based branching, or tree
architectures [9]. While in the first case the roll-off factors
remain relatively high, the tree-branching method keeps
the same roll-off factor for all sub-channels, with maximal
value inversely proportional to number of non-uniform
sub-channels. While these two structures may have
similar complexity of implementation, the one considered
here appears to be able to provide higher flexibility.
[6]
[7]
[8]
[9]
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CONCLUSIONS
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Slobodan Nedic