A Semi-Blind Frequency-Domain Concurrent Equalizer for OFDM Systems
Estevan M. Lopes
Department of Communications
National Institute of Telecommunications – INATEL
Santa Rita do Sapucai, Brazil
[email protected]
Abstract — This paper proposes a semi-blind algorithm for
frequency-domain (post-FFT) soft-decision concurrent equalization in OFDM systems. The objective is to improve system
performance by increasing data throughput or decreasing
power requirements, when compared with pilot based conventional channel estimation techniques. The Constant Modulus
Algorithm and the Soft Decision-Directed technique were
concurrently employed to adjust the coefficients of a post-FFT
equalizer bank. The algorithm works in a semi-blind mode
because it uses channel information, obtained from pilot subcarriers, to initialize and to supervise the equalizer bank when
pilots are present, otherwise remaining blind in the equalization process. To support such a concurrent equalization, the
system should provide pilot subcarriers only in the first symbol
of each OFDM super-frame, allowing algorithm initialization
when the receiver is turned on. In the remaining super-frame
symbols, pilot subcarriers can be suppressed to increase the
overall system throughput.
Keywords — OFDM equalization; Concurrent Algorithm;
Semi-blind equalization.
I.
INTRODUCTION
Conventional OFDM systems are conceived to basically
offer two types of protection against the degradations
caused by the channel response. The first protection is obtained by inserting a Cyclic Prefix (CP) in each OFDM
symbol in the time domain, with the purpose of preventing
Inter-Symbol Interference (ISI). The second protection is
obtained in the frequency domain, by exploiting pilot-based
channel estimation to compensate for amplitude attenuations
and phase rotations in each subcarrier. Both protections are
very effective in OFDM systems, but unfortunately inserting
cyclic prefix and pilot subcarriers may significantly reduce
system throughput.
Most existing equalization techniques for OFDM are
based on channel estimates which are first obtained from
pilot subcarriers and then extended for data subcarriers by
interpolating the pilot estimations. A review and comparative analysis of these techniques are presented in [1]. One
way of reducing or even eliminating the need for pilot subcarriers is to employ blind channel estimation and equalization techniques. Blind Channel Identification in OFDM
typically exploits the cyclostationarity induced by the cyclic
Fabbryccio A. C. M. Cardoso, Dalton S. Arantes
Department of Communications
School of Electrical and Computer Engineering
State University of Campinas - UNICAMP
Campinas, Brazil
(cardoso, dalton)@decom.fee.unicamp.br
prefix, as shown in [2]. However, the channel estimations
are performed in the time-domain with high computational
effort.
The present paper introduces a frequency-domain multicarrier extension of the single-carrier Constant Modulus
Algorithm with Soft Decision-Directed (CMA+SDD) equalizer, described in [3]. The equalization exploits the benefits
of the Concurrent Equalization (CEQ) [3][4], such as lowcomplexity, fast convergence and mod /2 phase recovery,
to improve both data throughput and Bit Error Rate (BER)
performance. The equalizer structure is configured as a filter
bank using a non-fractional one-tap modified CEQ for each
subcarrier. Every CEQ is concurrently adapted using the
blind LMS-like CMA and SDD, as described in [3]. Simulation results are presented for a digital television broadcasting scenario, both for a sufficient and for an insufficient
cyclic prefix.
This paper is organized to present the related work in
Section II, the proposed algorithm in Section III, to analyze
in Section IV the system throughput and the simulation
results and, finally, to present in Section V the conclusion of
the study.
II.
RELATED WORK
A previous application of the Concurrent Equalizer
(CEQ) to OFDM has been proposed in [5], in which a frequency-domain multi-carrier extension is obtained from the
original single-carrier CMA+DD algorithm [4]. The method
presented in [5] uses a two-taps and /2 fractionallyspaced CEQ in frequency domain, which requires
processing two FFTs per OFDM symbol. Two taps are
mandatory because this is the minimum filter length in a
/2 fractionally-spaced equalizer. Although the results in
[5] have been shown to offer a good BER performance, the
equalizer architecture can be further optimized. Simplifications on the architecture in [5] and the use of the
CMA+SDD, instead of the CMA+DD [4], are studied here.
Fractionally-spaced equalizers are appropriated for nonminimum phase channels. However, one can take advantage
of the OFDM guard interval (cyclic prefix) and the number
of orthogonal subcarriers to model each narrow-band sub-
carrier channel as being flat. In this case, the fractional sampling is not necessary and its removal can simplify considerably the equalization solution. Actually, frequency domain equalization with CEQ is required only to compensate
for amplitude attenuations and for phase rotations.
To further simplify the receiver, the de-spinning solution
proposed in [5], to solve the mod /2 phase ambiguity, was
not adopted here because it demands a high computational
effort. Rather than using a de-spinning approach, the equalizer bank is initialized with an initial estimate of the channel. This initialization reduces the initial phase range of the
combined channel-equalizer response, avoiding the need of
an additional de-spinning module.
III.
CMA
ns(n)
s(2n)
2
y(n)
r(n)
h(n)
2
y(2n)
wd (n)
SDD
Figure 1. Baseband communication model for concurrent equalization.
(Source: Authors).
CMA
ns (n)
SEMI-BLIND CONCURRENT EQUALIZER
The Constant Modulus Algorithm (CMA) [4][6] and the
Decision-Direct (DD) are the most effective and widely
used blind equalization techniques. These two methods are
usually combined by switching from the CMA to the DD
mode, after partial convergence. This strategy aims at improving the Mean Square Error (MSE) performance of
CMA in communication systems employing high order
Quadrature Amplitude Modulation (QAM). However, as
stated in [3], the DD requires a sufficiently low level of
MSE, a condition which may not be always achievable by
the CMA.
On the other hand, rather than switching to DD when detecting CMA convergence, the concurrent approach can use
both DD and CMA simultaneously. The CEQ [3] uses two
equalizers in a parallel master-slave configuration, as shown
in Figure 1. The master equalizer is initialized with a flat
response, i.e., with a single spike initialization and is
adapted using the CMA. The slave equalizer is initialized to
produce a zero response with all taps set to zero. This
second equalizer is updated using the DD output, but conditioned to a non-linear function which measures the reliability of the CMA adaptation. Such reliability is evaluated by
comparing the equalizer output decisions before and after
the CMA adjustment. If the decided outputs are the same,
the decision is probably the right one.
Among all the subsequent developments [3][7][8][9] that
followed the original concurrent approach, probably the
most important has been presented in [3], which proposes a
CMA and Soft Decision-Directed (CMA+SDD) concurrent
equalizer with about the same complexity of the original
CMA+DD scheme [4], but with a still faster convergence.
Additionally, the non-linear master-slave link between the
CMA and the SDD equalizers has been removed in [3]. The
link is really not necessary because SDD naturally deals
with decision uncertainties within a given four symbol constellation region. Simultaneous updates of both equalizers
are then enabled without worrying with error propagation
due to incorrect adaptations.
wc(n)
s(2 n)
2
h(n)
r(n)
y(n)
w(n)
y( 2n)
2
SDD
Figure 2. Baseband communication model for the single filter concurrent
version. (Source: Authors.)
Figure 3. Baseband communication model of the OFDM system with equalizer bank #$ %
. ( Source: Authors.)
The semi-blind approach for concurrent equalizers in
OFDM systems is presented in this section. The original
single-carrier approach is reviewed in the first part with the
inclusion of a compact single filter version. The modified
OFDM CEQ extension is then presented in the second part
of this section.
A. CMA+SDD Concurrent Equalizer
The CEQ is depicted in Figure 1. It was originally designed for fractionally-spaced blind equalization of high
order QAM modulation and single-carrier transmission
models [4]. As developed in [3], the CEQ with CMA+SDD
algorithm are given by the following set of equations
= + ,
2 + 1
= 2 + 1
= 2
+ 2
∗ 2
,
2
= 2
Δ − |2
| ,
Δ = E|
| !⁄E|
| !,
(1)
2 + 1
= 2 + 1
= 2
+ &'()*+ , 2
,
&
TABLE I.
= + .
Initialize each equalizer k:
#N %
= interp 7%, O%
, OP%
8.
&'()*+ , 2
=
&
9
∑=
3:=;< ∑4:9;< exp 1−
9
∑=
3:=;< ∑4:9;< exp 1−
Calculate the output for each subcarrier k:
$ %
= #$ %
D$ %
2%
− 34 2
6 734 − %
8
25
2%
− 34 2
6
25
∗ 2
For each data subcarrier k, adapt the coefficients using the CMA
and SDD:
CMA:
(2)
where 34 spans the four symbols of a QAM region >=9 . The
QAM constellation can be formed by tiling these four symbol regions. Indexes ?, @
are found by searching the constellation region which comprises the equalizer soft output.
For more details see [3]. Function f?B_DEF?G∙
, to be
presented later in Table I, compares the I and Q equalizer
outputs to the region boundaries to obtain such indexes
?, @
.
The CEQ is basically characterized by the adaptation of
two independent FIR filters. However, it is easily shown
that the same algorithm can be realized as only one FIR
filter concurrently adapted by the CMA and SDD algorithms. The resulting CEQ can then be implemented as
depicted in Figure 2 and according to the following equations
= ,
2 + 1
= 2 + 1
= 2
+ 2
∗ 2
,
2
= 2
Δ − |2
| ,
Δ = I|
| !/I|
| !,
2 + 1
= 2 + 1
= 2
+ CONCURRENT EQUALIZATION ALGORITHM FOR OFDM
SYSTEMS1.
(3)
&'()*+ , 2
.
&2
This compact (single filter) version of the concurrent algorithm was developed here and adopted in this work for
saving memory and operations.
B. Concurrent Equalization Applied to OFDM System
Our proposal here is to design, in the frequency domain, a
concurrent blind CMA+SDD equalization algorithm which
should be capable of regenerating the amplitude and phase
information of each subcarrier, without using pilots as references. The resulting OFDM receiver, illustrated in Figure 3,
was conceived to use a non-fractional frequency-domain
version of (3) with an equalizer bank using only one coefficient per subcarrier.
The equalization model in Figure 3 assumes that for
#$Q< %
= #$ %
+ $ %
$ %
D$∗ %
$ %
= ∆ − |$ %
|
∆ =
SDD [3]:
I|$ %
| !
I|$ %
| !
#$Q< %
← #$Q< %
+ ?, @
= find_region$ %
&'()*+ #, $ %
&#
&'()*+ #, $ %
=
&#
9
∑=
3:=;< ∑4:9;< exp 1−
2$ %
− 34 2
6 734 − $ %
8
25
2$ %
− 34 2
9
∑=
6
3:=;< ∑4:9;< exp 1−
25
D$∗ %
1. The algorithm presented here is the blind concurrent. The semi-blind
concurrent uses the same algorithm, except for every first symbol of a superframe. In this case, the algorithm switches to an LMS-like for both pilot and
data subcarriers in the first symbol. In the pilot subcarriers, the error is
calculated using the knowledge of the pilots as reference so that the update
equation is given by #$Q< %
= #$ %
+ TO%
− OP%
U D$∗ %
. On the
other hand, for data subcarriers the error is calculated using, as reference, the
decision output over the linear-interpolation estimates. In this case, the
update equation is given by #$Q< %
= #$ %
+ Tdecision\$ %
−
$ %
U D$∗ %
, where \$ %
= D$ %
× interp`%, Oa9 /OPa9 , Oab /
OPab !.
OFDM systems the communication channel can be
represented by J parallel and orthogonal narrow-band
channels, without mutual interference between them. The
power spectrum of each subcarrier can then be considered
plain, which means that their bandwidth is much smaller
than the channel coherence bandwidth. This model is guaranteed if cyclic prefix is sufficient to avoid ISI. Therefore,
there is no need to use a fractionally-spaced equalizer in
such scenario since each channel is considered flat.
In this article, K$ %
is the coefficient vector L#$ %
M
representing the equalizer bank at the symbol instant ,
where % is the subcarrier index of the equalizer. The total
number of subcarriers is J. The vectors collecting the equalizer-bank inputs and outputs are represented respectively
TABLE II.
Figure 4. Pilot subcarriers in super-frame for equalizer bank. (Source: Authors.)
by c$ %
and d$ %
. Notice in Figure 3 that the output is
calculated by
DATA THROUGHPUT IN MBITS/S FOR THE SIMULATED
SYSTEMS
Transmitter Known
Parameter
Channel
Blind and Semi- Linear InterpolaBlind Concurrent
tion
CP = 1/32
CP = 1/64
43.36
44.03
43.64
44.31
TABLE III.
34.91
35.45
ITU BRAZIL A CHANNEL PROFILE
Coefficient ceil
(Delay/Ts)+1
1
3
20
26
49
50
(4)
Delay (µs)
0
0.15
2.22
3.05
5.86
5.93
Phase recovery in CEQ is obtained by decisions over a
QAM square constellation, resulting in a mod /2 phase
ambiguity. Typically, OFDM transmissions in channel profiles like ITU Brazil-A [12] may result in signal phases that
vary widely throughout the subcarriers and some of them
may present rotations larger than 45º, resulting in wrong
phase convergence. Such scenario should be avoided because wrong phase recovery in even a few subcarriers can
substantially degrade the overall BER performance of the
system. It may be argued that sync-words, differential encoding and other countermeasures could be applied to individual subcarriers to resolve 90º ambiguities in phase rotations, but we are avoiding the use of these techniques in
favor of power efficiency.
The solution proposed here to deal with phase ambiguities consists in initializing the equalizer bank with an initial
channel estimate. This solution impacts the system if, for
example, pilot subcarriers are inserted to allow channel
estimation at the receiver. However, in the present case the
system can use pilot symbols with a much lower rate compared to the traditional OFDM equalizers, reducing the
impact on the overall throughput. As shown in Figure 4, our
proposal is to use pilot subcarriers only in the first symbol
of each super-frame. Moreover, in the first symbol the pilots
are interlaced with data subcarriers with the purpose of
increasing the overall throughput. Initialization for data
subcarrier is then obtained by interpolating the initializations from pilot subcarriers.
In this work, once initialized, the equalizer bank can be
concurrently adapted by the CMA+SDD, either blindly or
semi-blindly. We have tested both algorithms. The blind
concurrent uses the pilot symbols only once for initialization, while the semi-blind concurrent switches to an LMSlike algorithm in the first symbol of every super-frame (see
details in Table I). Pilots are repeatedly inserted in each first
symbol of the super-frame. Such strategy is necessary to
start the equalizer when the receiver is turned on or when
the equalization is lost.
The channel estimate used to initialize the equalizer bank,
considering pilot and data subcarriers, is given by #N %
=
O%
/OP%
for
pilot
subcarriers
and #N %
=
Gain (dB)
0
-13.8
-16.2
-14.9
-13.6
-16.4
$ %
= #$ %
D$ %
for % = 1, 2, ⋯ , J.
interp f%,
+gh +gi ,
j,
+Pgh +Pgi where interp∙! is a linear interpo-
lation function taken at data subcarrier k considering the
knowledge of channel estimates at the left (a9 ) and right
(ab ) neighboring pilot subcarriers; O%
is the transmitted
pilot associated with the k-th subcarrier; OP%
is the corresponding received pilot; and #N %
is the initial value of the
k-th one-tap equalizer of the equalizer bank.
Another important aspect that appears in the application
of the CEQ without the non-linear master-slave link in
OFDM systems is the possibility of antagonistic forces
between the CMA and SDD. This can occur when a subcarrier is highly attenuated by the channel. Empirically, we
have observed that for attenuations greater than 20 dB, the
SDD adaptation harms the CMA performance.
Fortunately, the same initialization solution adopted for
the problem of phase ambiguity can also solve the problem
of antagonistic forces in the concurrent process involving
the CMA and the SDD. In fact, initializing the equalizer
taps from the channel estimate can provide a favorable initial condition for the equalization in OFDM systems.
Table I summarizes the CEQ for use in OFDM systems
with the modifications suggested in this article. It is
important to emphasize that in the simulations, as presented
in the next section, the CMA and SDD versions were
normalized by the average power of the input signal, as in
the normalized LMS. The power normalization factor was
omitted from Table I for sake of simplicity. The parameters
used in the simulations were empirically chosen as =
0.1, = 0.0001 and 5 = 0.7.
IV.
PERFORMANCE EVALUATION
Blind and semi-blind CEQ performance are evaluated
here in terms of BER versus Im /nN . The performance results are also obtained for known channel and linear interpolation receivers, which are used here as reference models to
represent a lower bound and an upper bound on bit error
rate, respectively. The known-channel scenario uses the
channel knowledge on all subcarriers o$ %
, with % =
1, 2, ⋯ , J, to adjust the channel amplitude and phase response by a factor of 1/o$ %
for each subcarrier. Knowing
the channel response is obviously the ideal scenario and
represents a lower bound on bit error rate.
On the other hand, the one-dimensional linear interpolation receiver uses pilot tones equally spaced by data subcarriers in all OFDM symbols. The channel estimates obtained
for pilot subcarriers are then interpolated to estimate the
channel behavior on the data subcarriers. The onedimensional linear interpolator is well known as one of the
simplest method of interpolation with reasonable performance. It is important to emphasize that linear interpolation
is also used here to initialize both the blind and the semiblind CEQ and to calculate the supervision reference for the
semi-blind CEQ when pilots are present. The results presented next show that the performance of the concurrent
algorithm significantly improves over the linear interpolation receiver. In fact, the present results are shown to be
quite close to the optimum known-channel receiver. Therefore, in the present case there is no need to use
more sophisticated interpolation schemes for comparison,
such as a non-linear one, for example.
The simulation results were obtained for an OFDM system with 2048 subcarriers. OFDM symbols, including cyclic-prefix, are sampled at 8.127 MHz which corresponds to a
sample time of = 63/512 × 10s ~ 123.05 . To
format the transmission spectrum, 158 null subcarriers are
padded to 1890 data and pilot subcarriers, resulting in a total
of 2048 subcarriers. Data subcarriers are modulated with 64QAM while pilots are modulated with BPSK.
When the channel response is assumed to be known, the
system model is configured to operate without pilot subcarriers, providing a system throughput uvwx , in bits/s, given
by the inverse of × 2048 × 1 + oO
/1890 × 6
. For
linear interpolation-based channel estimation, the system
model is configured with one pilot added to every other 4
subcarriers, resulting in 378 pilots and 1512 data subcarriers
per OFDM symbol. In this case, the system throughput
u|w}~€ is given by the inverse of × 2048 ×
1 + oO
/1512 × 6
. Considering the blind and the semiblind concurrent equalizers, pilots are used only in the first
symbol of each super-frame with the same configuration
used for the linear interpolation. In the other symbols of the
super-frame, all 1890 available subcarriers are used for data
transmission. The super-frame contains 32 symbols and the
system throughput u‚w is given by the weighted mean
u‚w = 1 × u|w}~€ + 31 × u‚wƒ /32.
In this work, the algorithm was tested for values of cyclic
prefix equal to 1/32 and 1/64. Table II summarizes the values of system throughput for the receivers considered here.
It is worth emphasizing that no channel encoding scheme
was used since the objective was to only evaluate the equalization performance. Observe in Table II that the use of the
CEQ increases data throughput by about 8 Mbits/s when
compared with the linear interpolation.
The channel profile used in all simulations is the ITU
Brazil A [12], described in Table III. The ITU has standardized channel profiles as a function of delay and gain for
multipath components. The channel was then digitized by
rounding the delay profile to multiple values of the sample
time Ts. In Table III we show only the nonzero coefficients
of the digitized channel profile.
The CP duration considered here is given by 2048 ×
oO × . For oO = 1/32 and 1/64, it leads to durations of
7.875 and 3.9375 , respectively. Therefore, as the
maximum channel delay for Brazil A is about 5.93 , then
the oO = 1/32 is enough to avoid ISI, while CP = 1/64
cannot prevent it, degrading system performance. In this
work, the BER performance is evaluated for both scenarios.
Simulation results were obtained for each point of Im /nN
by the estimate of the mean for BER in † = 20 outcomes
with a Confidence Interval (CI) of 90%.
The performance curves of ‡Iu × Im /nN are plotted here
with solid lines for the mean estimate of ˆˆˆˆˆˆ
‡Iu and with
dotted lines for the lower and upper bounds of the confidence interval.
Figure 5 shows the simulation results for the scenario
with CP = 1/32 , which is sufficient to prevent ISI. The
results for linear interpolation saturate near a BER of
5 × 10; due to imprecision on channel estimates for data
subcarriers. On the other hand, with the blind and semiblind concurrent no saturation was observed for BER. Such
results follow the behavior of the known-channel result with
a loss of only about 0.5 dB for the blind and semi-blind
concurrent in the evaluated points of Im /nN ≥ 15 B‡. The
results of the blind and semi-blind are practically the same.
The fast convergence of CEQ associated with the static
behavior of the channel profile explains the similarity of
blind and semi-blind results.
The results obtained with CP = 1/64, which is insufficient to avoid ISI, are presented in Figure 6. In this scenario,
inter-carrier interference (ICI) is introduced due to the orthogonality loss in the subcarriers. In this case, the assumption that every subcarrier is independently attenuated and
rotated by a single channel coefficient is not strictly valid.
However, the objective of this study is also to evaluate the
algorithm robustness when the assumption of orthogonality
is not valid. The results have shown very similar performance for both the blind and semi-blind concurrent and
better performance than the linear interpolation even when
the latter is used with sufficient cyclic prefix. This means
that the proposed concurrent equalization is able to successfully improve the initial channel estimate and outperform
the linear interpolation results.
The results presented here were obtained for a case study
which uses the standard Brazil-A channel profile on static
0
0
10
10
-1
10
-1
10
-2
10
-3
BER
BER
10
-4
-2
10
10
concurrent
semi-blind conc
linear interp.
known ch
CI 90%
-5
10
-6
10
-7
10
0
5
10
15
-4
20
E b / N0
25
30
35
40
Figure 5. Performance of BER versus Im /nN for the scenario with oO =
1/32 (Source; authors).
digital TV broadcast environment with sufficient and insufficient OFDM cyclic prefix. The objective was to show that
the proposed receiver would successfully operate on a typical Brazilian broadcast scenario. Further studies are still
required to investigate the algorithm behavior on dynamic
environments and to confirm its good performance also on
adverse scenarios with co-channel interference and with
non-line-of-sight weak signal reception.
V.
concurrent
semi-blind conc
linear interp.
known ch
CI 90%
-3
10
CONCLUSION
Concurrent equalization methods were investigated in this
work for operation with OFDM systems. The motivation
was the low complexity, fast convergence, blindness property and phase-recovery features of the CEQ to improve the
overall throughput without compromising BER performance.
This study has shown that the CEQ can drastically reduce
the number of pilot subcarriers. Besides, it was shown that it
is not necessary to use the two concurrent filters as originally described in [4], but only a single filter concurrently
adapted by the CMA and SDD. It has also been shown
that the CEQ does not need to be fractionally spaced, since
the channel model is supposed to be flat in each subcarrier.
To initialize and to add partial supervision capability to
the CEQ bank, pilot subcarriers were inserted in the first
symbol of each super-frame. The proposed blind concurrent
algorithm uses the pilot information only to initialize the
equalizer bank. On the other hand, the proposed semi-blind
algorithm can supervise the equalizer bank in the first symbol of each super-frame, when pilot subcarriers are available.
Results have shown that in scenarios with sufficient sizes
of cyclic prefix, the system performance follows the known
channel results with a loss of only 0.5 dB. However, when
cyclic prefix is insufficient to prevent ISI, the BER performance saturates, but to a level still better than the linear
interpolation with sufficient CP.
10
0
5
10
15
20
Eb / N0
25
30
35
40
Figure 6. Performance of BER versus Im /nN for the scenario with oO =
1/64 (Source: authors).
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