Real-domain SIC for MIMO with FBMC Waveforms
V. Stanivuk, S. Tomić, M. Narandžić and S. Nedić
Faculty of Technical Sciences
University of Novi Sad
Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia
orange@uns.ac.rs
Abstract—This paper proposes 2x2 MIMO OQAM/FBMC
receiver strategy based on successive interference cancellation
(SIC) as an extension of the SISO approach which uses entirely
real-valued formulation of the transmission system model]. Both
increase of throughput and SNR gain are observed through
appropriate maximum ratio combining (MRC) of the soft
symbols pertaining to the receiver-end diversity branches. The
2x1 multi-user detection (MUD) case has been also addressed to
certain extent. Additional gains, essentially related to SISO case,
coming from the cooperative use of in-phase (I) and quadrature
(Q) branches, primarily based on the quadrature noise
components’ independence, as well as from the explicit noise
cancellation within the SIC framework, have also been outlined.
Keywords— successive interference cancellation; staggered
multicarrier waveforms; FBMC; MIMO; diversity; noise prediction
I.
INTRODUCTION
Compensation of linear distortions introduced in the
subchannels of a FBMC signal with staggered, i.e. by T/2
offset, in-phase (I) and quadrature (Q) components of
transmitted QAM symbols was the first challenge towards
making it competitive to the traditional CP-OFDM
multicarrier waveform.. This issue has been further
pronounced in various forms of multiple-input multiple-output
(MIMO) configurations, where the so-called (complexdomain) intrinsic interference [1] has been considered as an
obstacle towards attaining the performance level comparable
with CP-OFDM. To address this within the context of nonlinear data detection methods, in [2] and [3] were proposed
combined (linear) minimum mean square error (MMSE) and
maximum likelihood sequence estimation (MLSE), and the
(entirely) maximum a posteriori probability (MAP)
approaches, respectively. In these cases, to be found also in
[7][9], while relying on the transmission system model in
complex-domain, the intrinsic interference is considered as an
“annoyance”, with a lot of effort/complexity invested in its
avoidance, i.e. partial elimination.
However, the orthogonality conditions for the FBMC
formats and single-input single-output (SISO) transmission are
determined entirely in the real-domain. For that reason in [4]
only the in-phase component of the subchannel signals was
used to provide the sufficient statistic for detection of the
transmitted signal in the framework of the successive
interference cancellation (SIC). Thereby, the subchannel
978-1-4799-5863-4/14/$31.00 ©2014 IEEE
impulse responses (one from the channel of interest and two
related to adjacent channels cross-talk) are estimated as realvalued samples and further used for reconstruction of the inphase component in an iterative process that is based on MAPlike “soft” estimated data symbols, with optional interaction
with the forward error code (FEC) decoder. In the real-domain
SIC formulation the intrinsic interference does not figure
explicitly and the MAP detection takes places with the halved
number of real-valued multiplications in the signal
reconstruction step, when compared with the complex-domain
formulation.
Consequently, the extension of the SIC SISO framework to
the MIMO case becomes free from the notion of the intrinsic
interference, in that it becomes included in the system-model
description. However, the problem of the MIMO/MISO
implicit MUD (multi-user detection) comes to the foreground
and needs to be addressed by appropriate detection strategy
that distinguishes impact of the residual interference and the
additive (Gaussian) noise to the overall estimation variance.
Since the FBMC subchannels signals I-component presents
sufficient statistics for the T/2 spaced real data samples, the
corresponding Q-component contains the same information
and, with the appropriately defined/estimated system-level
impulse response, it can be used for signal detection as well.
Moreover, since the noise quadrature components are
independent, the two components can be processed in such a
way to produce at least the AWGN diversity effect, and
potentially also the fading related one. Another possibility to
enhance the SIC performance for FBMC signal reception is
the fact that the analysis filter-bank (AFB) complex-valued
output noise samples are correlated at T/2 instants, enabling
the noise prediction and cancellation without the causality
constraints. This paper thus presents an attempt towards
allowing for the FBMC performance to significantly exceed
the CP-OFDM performance.
In the following, after presentation of general FBMC SISO
transmission system model and its extension to 2x2 MIMO
configuration in Section II, the proposed SIC-based receiver
structure and related flat fading channels simulation results are
presented in Section III, followed by the still generally
„problematic‟ 2x1 MUD configuration. The noise prediction
and the constructive use of the quadrature complex signal
branch are explored in Section IV. The drawn conclusions and
the planed future work are given in section V.
II.
FBMC TRANSMISSION SYSTEM MODEL
A. SISO to MIMO Extension
In continuous time representation, the OFDM/OQAM
modulated signal is defined by the relation
x(t )
j
k 2 2 M 1
k k1 n 0
k n
d (t ) .
k k
n n
essentially Lh 2 Lg 1 , where in the considered case Lg 3 .
Sets of the direct and the adjacent subchannels cross-talk
impulse responses for 100 flat fading realizations could be
seen in [8].
r
r'
-
(1)
nk (t ) g (t nT / 2)e j 2 k (t nT / 2) / T .
denoting the QAM signaling interval and its spectra occupies
bandwidth Bg (1 ) / T , where α designates the roll off
factor of subchannels‟ spectra.
After a transmission over the channel with the (complexvalued) impulse response c(t ) , the conventional maximum
likelihood (ML) receiver is obtained by considering (in
accordance with the FBMC orthogonality conditions) the real
part of the scalar product between the received signal
k
y(t ) c(t ) * x(t ) b' (t ) and the basis functions n (t ) :
LhT
y(nT / 2 t )nk (nT / 2 t )dt ,
σ2
rnk
d
Lh
k ' 1 n ' Lh
k k ' k ,k '
n n' n'
h
bnk .
(4)
interacting subchannels: hnk',0 is the impulse response in the
referent subchannel k, hnk',1 represents the interferences from
the upper subchannel with index k+1 and hnk', 1 interferences
from the lower subchannel with index k-1. It is worth noting
that while in the complex-domain formulation the subchannel
impulse response has just one sample, for the real-domain
system formulation the impulse response length is
In real-domain formulation noise samples are independent, E bnk bnkk 0 ,
while in the complex-domain formulation the are correlated
E b'kn b'nk k 0.5 (1 j ) .
d̂
h
Detection process in [4] comprises the estimation of the
individual impulse responses, successive interference
cancellation, matched filtering and FEC decoding, as shown in
Fig. 1. The very iterative procedure relies on the SIC block
Extrinsic probability in form of the conditional probability
P rnk | d nk exp ( f nk d nk E sk ) 2 /(2 k2 ) .
(5)
where the f nk is the matched filter (MF) output, calculated (in
simplified form [8]) as
f nk
r'
1
L
k ' 1
n ' L
k k '
n n'
hˆnk',k ' dˆ nk E sk .
(6)
In (6) the term r ' nk nk '' represents the residual of the received
signal sample(s) after subtracting influence of all previously
estimated symbols, d̂ nk , on the current bin, including itself:
r ' nk nk '' rnknk' '
while
dˆ
L
1
k " 1 n" L
k k ' k "
n n ' n"
hˆnk" k ',k " .
(7a)
d̂ nk is the “soft” estimate from the previous iteration
(d nk )
dˆnk E{d nk | r} d nk Pdapi
k
k
n
{d nk }
n
(7b)
and E sk represents the symbol energy per subchannel,
calculated in the following manner:
E sk
where hnk',k ' designates the overall impulse responses related to
1
L_app
SIC Detector
(3)
where the over-bar denotes complex conjugation. In general,
including the frequency-flat fading channel, the received
signal samples can be represented [4] as the output of a MISO
matrix filter with real-valued coefficients, superposed by the
AFB filtered white Gaussian noise samples1. The model of
transmission system that includes interference from adjacent
subchannels is given [4][8] by the following relation:
1
L_api
Fig. 1. SISO SIC Detector
(2)
The interfering subchannels take indices between k1 and k 2 .
The elementary signal g (t ) is a symmetric, real-valued,
square root Nyquist function which is normalized to have
unity power. It has finite time duration [ Lg T , Lg T ] , with T
LhT
Decoder
Deinter.
P(r|d=-1/sqrt(2)) P(r|d=+1/sqrt(2))
k
L_extr
Inter.
where the symbols d n are the real valued (Re{} and Im{}
parts of M-ary QAM) symbols that scale an orthogonal basis:
rnk Re
f
MF
hˆ
1
k '1
L
n ' L
k ,k '
n'
2
.
(7c)
The a priory probabilities used for for data samples estimates
in (7b) are produced, in presence of the FEC decoder, through
ext
Pdapi
(d nk1 ) Pextk d k (d nk1 ) Pcod
(d nk1 )
k
k
d k
n1
n
n1
n1
n1
(8)
where the right most term represents the Extrisnic probability
provided by FEC decoder, being used as the apriory
probability of the particular data symbol. Similarly, the
probability informations on paricular values of data bits are
passed to the FEC decoder from the SIC detector. If FEC
decoder is not included, this second multiplicand is omitted.
B. SISO to MIMO Extension
In the following we consider 2T2R configuration in the
SDM (space-division multiplexing) mode illustrated in Fig. 2.
h1,1
Tx1
Rx1
h2,1
h1,2
Tx
Tx2
Rx2
h2,2
Rx
III.
PROPOSED DETECTOR STRUCTURE
By previously estimated real-valued impulse responses
pertaining to direct and cross-antenna transmission channels,
the iterative interference cancellation framework becomes
rather straightforwardly applicable, with involvement of
certain “strategic” and “tactical” measures in terms of signals
ordering and overcoming of non-Gaussian statistics of the
residual interference during the iterations.
The iterative procedure starts with the signal that has the
highest energy and temporarily treating the weaker signal as a
‟noise‟. By using the LLR (log-likelihood ratio) formulation,
the soft symbols corresponding to the stronger signal, denoted
with dˆ аpi in Fig. 3, are determined for each receive antenna
j
Fig. 2. 2x2 MIMO configuration.
.
tanh ( Lextr i , j n Lapp j ) / 2
according to expression, in lieu of (7b)
dˆ j n api
Starting from (4), by simple extension, the received signal into
one of received antennas has the form
ri n
k
1
d1n n'hi,1n'
k k '
Lh
k ,k '
k ' 1 n ' Lh
1
k k '
k ,k '
k
d2n n'hi,2n' bi n . (9)
Lh
k ' 1 n ' Lh
where the first and the second index of h correspond to
receive and transmit antenna, respectively, while indices 1 and
2 of d differentiate between data symbols sent from the
antennas 1 and 2, respectively.
k
signals,
passing through independent channels and
term for the i-th receive antenna chain.
n
2
(12)
uses two different log-likelihood ratios (LLRs): extrinsic
Lextr i , j and a posteriori Lapp j . The extrinsic LLR is
determined at output of each „SIC detector‟ block, starting
from (10), as
k
received from different transmit antennas after
bi kn
k
The division by two is used to adjust the saturation level of
tanh function (1) to the transmitted constellation
d *j 1 / 2 in MIMO subchannels2. The proposed strategy
The first two summands in (9) represent the (real-valued)
ri , j kn ,
k
Lextr i , j n
is the noise
P(ri , j k d j k 1 / 2)
n
n
.
ln
P(ri , j k d j k 1 / 2)
n
n
(13)
k
The term Lapp j is calculated as average of two LLR values:
n
Before proceeding with detection, the impulse responses
hi , j kn ,'k ' pertinent to the four MIMO sub-channels are estimated
sequentially by the same procedure as for the SISO case
[4][8].
the first one is obtained after „weighting‟ block, Lextr j , and the
second one at the output of decoder, Lapi j . In initial iteration
when output of decoder is still not available, Lapp j k is set to
n
For the considered 2x2 MIMO system we have to calculate
four different expressions for conditional probability, inside
each of the “SIC detector” block, see Fig. 3. As in (5) the
conditional probabilities are calculated as:
( f k d k E k )2
i, j n
j n i, j s
P ri , j k | d j k exp
n
n
2 i , j 2
k
.
1
k ' 1
hˆ k ,k '
n ' L i , j n '
L
.
rˆi , m n n ' r
k k'
(10)
where Ei, j k represents the symbol energy and along the
s
k ,k '
adequate impulse response, hˆi , j n ' is calculated as:
Ei, j ks
zero. Then, in accordance with Fig. 3, which describes the
iterative cancellation of MIMO inter-stream and SISO selfinterference, the residuals
2
(11)
k k '
i n n'
dˆ
1
L
k " 1 n" L
k k ' k "
m n n ' n"
k ,k '
api hˆi, m n ' .
(14)
are interchangeably sent to the „SIC detector‟ blocks. Here,
m designates opposite antenna or signal from the set m 1, 2 .
Now, four Lextr i , j values corresponding to „SIC detector‟
outputs, are combined by MRC-like weighting to get
Lextr j values:
Lextr j
Calculation of the individual MF outputs, f i, j k , to be used in
n
(10), will be described in the next section along with
description and explanation of the proposed receiver structure.
2
E1, j Lextr1, j E 2, j Lextr 2, j
E1, j E 2, j
.
(15)
In this way total power of MIMO signal remains the same as in SISO case.
Lextr j are then passed through deinterleaver and sent to
decoder. The outputs from decoder are interleaved to get
adequately ordered Lapi j values for use within SIC detectors.
The soft a posteriori estimates of data symbols are obtained as
tanh ( Lapi j Lextr j n ) / 2
dˆ j n app
k
k
k
n
2
.
(16)
and their impact is subtracted from received signal, in the
same manner as dˆ аpi is applied in (14):
j
rˆi , m n n ' r
k k'
k k '
i n n'
k k ', k "
k k ' k "
dˆ1n n'n"app hˆm,1n' .
L
1
k " 1 n" L
(17)
In Fig. 3 the dashed (blue and red) lines are feeding the
estimates of the symbols d 1 kn , and the solid lines carry the
estimates of the symbols d 2 kn .In Fig. 3 solid green line
represents column vector of estimates of the symbols d 1 kn and
d 2 kn . From resulting estimates rˆi , j nk nk '' ,
produced as:
r 'i , m n n ' rˆi , m n n '
k k '
k k '
dˆ
1
L
k " 1 n" L
k k ' k "
2 n n ' n"
the residuals are
k k ', k "
app hˆm, 2 n"
.(18)
Under assumption that all symbols are properly estimated the
residuals from (18) correspond to noise term, the effect of
which is partially suppressed by the MF. The output of MF is
added to properly scaled symbol estimation to form a new
decision variable:
f i, j kn
r'
1
k '1
L
n ' L
k k ' ˆ k ,k '
i , j n n' hi , j n'
dˆ
k
k
j n appEi , j s
i , j k2
1 P
P 1
r'
1
k'
L
n ' L
k k '
i , j n n '
k ,k '
hˆi , j n '
.
2
(20)
where P represents number of T/2 instants during the
transmission block of data samples.
The newly estimated quantities in (19) and (20) enable a
new estimation of conditional probability in (10) and LLR
value in (13). In this way, we are getting the novel information
about reliability of the estimated soft symbols and repeating
the process iteratively. The message passing within SIC and
FEC decoder, as well as within the two, proceeds similarly to
the SISO case.
The described detector structure is obtained as extension of
our approach to develop the 2x1 MUD system. In that system
we still have the problem to overcame the similar or too
different levels of interfering and signal of interest. In
proposed detector this problem is overcome by including the
second antenna at receiver side.
A. Simulation results
As most challenging for the MUD setup, the primarily
simulated channel model was Rayleigh flat fading where, due
to absence of time-frequency bins independence on signal
block level, there is no contribution from interleaving and
FEC decoding. The channel parameters are estimated with a
training period long enough (60 T intervals) to ensure a
reasonably well LS-type channel estimation. The system uses
R=1/2 (133,171) convolutional code followed by an
interleaver of length 56. For this purpose we used system with
six out of eight active subchannels. The results presented in
Fig. 4 are produced after seven iterations.
. (19)
Fig. 4. BER comparasion: MIMO vs. SISO.
The remaining term from (10), variance i , j 2 is calculated as
k
Fig. 3. Receiver structure for MIMO 2x2 system.
As shown on Fig. 4, the increase of signal-to-noise ratio
(SNR) brings additional reduction of overall BER for MIMO
transmission. The receiver diversity effect is achieved through
the appropriate (MRC) combining of the soft data symbols
pertaining to signals received from the two receive antennas
for each of the two transmitted data streams, without
noticeable effect of their generally disparate received energy
levels and effects of residual inter-stream cross-talk.
IV.
ADVANCED SISO PROCESSING - I/Q DIVERSITY AND/OR
NOISE PREDICTION AND CANCELLATION
Rather than considering the “intrinsic interference” as a
problem, its constructive use appears to be possible, primarily
through independence of the noise components present on the
in-phase and quadrature branches, with likely further
benefitting from at least certain independence among the inphase and quadrature branches of fading channels impulse
responses. As in (3), the quadrature branch system model then
becomes
LgT
rnk imag Im y( nT / 2t )( nT / 2t )dt .
LgT
(21)
The block diagram of such a configuration, with
incorporation of the prediction and cancellation of the noise
part colored by the receiver-end filter-bank is shown in Fig. 5.
Fig. 6. BER in function of SNR for the case with and without exploation of
imaginary branch
Since the joint (bivariate) distribution of two independent
Gaussian processes, is equivalent to product of individual
distributions, the effective extrinsic probability would be a
product of extrinsic probabilities of Re{} and Im{} branches,
Pexteff Pext[Re] Pext[Im ]
( f k [Re] d nk Esk [ Re] )2 ( f nk [Im] d nk Esk [Im] )2
1
exp n
2
2
2 k
2 k
Fig. 5. Block diagram of noise prediction and cancellation in SIC framework.
The vertical dashed line shows demarcation between the complex- and realdomain processing.
It might be worth nothing that while the utilization of
redundancy contained within the I/Q staggered formats had
been something normal for offset QAM (OQAM) in satellite
communications [5], for FBMC application - with the
essentially same modulation in subchannels - it has only
recently been made the related proposal in [6]. The
preliminary simulation results shown in Fig. 6 have indicated
the 3dB of SNR gain in the AWGN case, but some more work
is to be done to produce a comprehensive set of compelling
enough simulation results, in particular regarding the absence
of the gain in the coded case. While difference between the
real- and complex-domain MAP, that is SIC framework in
terms of implementation complexity might be practically nonexistent due to the longer impulse responses in the former
case, its advantage can actually be sought in the context of the
I and Q redundancy based gain.
(22)
The superscripts are added to differentiate between the two
branches that have the same form as probability in (5).
This becomes conceptually quite appealing, in that the
product of the two probability density functions (with the
same average values and the same variances) reveals narrowed
shape compared with any one of them, implying (by the
product of two numbers smaller than 1 is smaller than either
one of them) the reduction of the overall variance, that is the
effective SNR, which lies in the essence of the maximal-ratio
combining of the two (diversity) branches. The given
simulation results are, however, related to the equivalently
performing configuration that is based on averaging the
residuals of real and imaginary branches:
r 'kn avr
r 'kn real r 'kn imag
.
2
(23)
(The calculation of the term r 'kn imag requires the previous
estimation of impulse responses for imaginary branch, which
is performed using the same procedure as for the real branch.)
Another possibility of making the FBMC performance to
even be exceeding those of the CP-OFDM ones, rather than
just striving to attain them, is based on reduction and
cancellation of the noise part which has been colored by
passage of the AWGN through the receiver-end filter-bank
subchannels. Namely, the subchannel noise in complexdomain becomes quite strongly correlated with T/2 lag, by
about factor 0.5 in case of 100% subchannels‟ roll-off factor.
In order to produce the candidate noise samples for prediction,
the complex received signal has to be reconstructed, whereby
the SIC framework applied at the signal-block (frame) level
eliminates the problem of causality present in conventional
noise prediction configurations.
The complex-domain FBMC signal reconstruction, in line
with (1), is based on soft data samples estimates and has the
form
~
ynk
j
k2 2 M 1
k k1 m0
k m
d nknk (t ) .
The predicted noise samples in complex-domain,
V.
(24)
{ ynk ~
ynk ) ,
are produced based on the difference between the actual and
interpolated complex receive samples,
further explored and a remedy be possibly be found in
introducing more inertness in the iterating process.
ynk and ~ynk .
For the FBMC referent impulse response
g [0 0.0008 - 0.0260 0.0245 0.9996 0.9992 0.0243 - 0.0259 0.0008 0]
that is designed by the procedure proposed in [10], the
prediction coefficients are calculated by the spectral
decomposition, e.g. using the Levinson-Durbin algorithm:
p [-0.9691 0.8869 - 0.8383 0.6716 - 0.5250 0.3750 - 0.2650 0.1268 ] .
Simulation results for noise prediction and cancellation for
an AWGN channel (without fading) are shown below in Fig.
7.
In this work the novel MIMO 2x2 scheme for successive
iterative interference cancellation is proposed. The SNR gain
is observed as result of the described iterative procedure.
Future contribution based on proposed receiver structure will
be extended to multi-tap channels, where the strategy for
estimation of soft symbols should be devised separately for
each subchannel.
Performance enhancements based on redundant in-phase
and quadrature components and cancellation of the predictable
noise power spectral density, although partly demonstrated,
need further elaboration. Also, the 2x1 MISO multi-access
configuration remains to be tackled based on insights gained
from the 2x2 MIMO, where 2x1 MISO is contained as sub-set.
ACKNOWLEDGMENT
This work has been supported by the FP7 project 318362
“ICT-EMPhAtiC”
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http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=476088
[1]
Fig.7. Simulation results for the cases without and with noise suppression for
the AWGN channel.
While in the both cases the effect of noise suppression and
gain of utilization of imaginary branch are clearly visible,
when realistically estimated noise samples are used in
prediction only the uncoded BER performance becomes
significantly improved, for even more that 3dB, while the
coded performance remains unchanged with respect to the
case without noise suppression. This behavior needs to be
CONCLUSION AND FUTURE WORKS
Vladimir Stanivuk
Slobodan Nedic