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Iterative Frequency-Domain Equalization for
General Constellations
Rui Dinis(1,2,3), Paulo Montezuma (1,3), Nuno Souto(2,4) and Jo˜
ao Silva(2,4)
(1) Departamento de Engenharia Electrot´
ecnica, Faculdade de Ciˆ
encias e Tecnologia, FCT,
Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
(2) IT, Instituto de Telecomunicac¸˜
oes, Av. Rovisco Pais, Lisboa, Portugal.
(3) Uninova, Instituto de Desenvolvimento de Novas Tecnologias, Quinta da Torre, Caparica, Portugal.
(4) ISCTE, Av. das Forc¸as Armadas, Lisboa, Portugal.
Abstract - IB-DFE (Iterative Block DFE) is a promising
turbo equalization technique for SC-FDE schemes (Single-
Carrier with Frequency Domain Equalization). However,
typical IB-DFE implementations are tailored for a specific
constellation (usually QPSK).
In this paper we propose a general method for designing
IB-DFE receivers for any constellation. Our approach
relies on an analytical characterization of the mapping
rule were the constellation symbols are written as a
linear function of the transmitted bits. This method is
then employed in both uniform and non-uniform QAM
constellations.
I. INTRODUCTION
SC-FDE schemes (Single-Carrier with Frequency-Domain
Equalization) [1] are excellent candidates for future broadband
wireless systems since they can have good performance in
severely time-dispersive channels without requiring complex
receiver implementation [2], [3].
A promising IB-DFE (Iterative Block DFE) approach for
SC transmission was proposed in [4] and extended to diversity
scenarios [5], [6] and spacial multiplexing schemes [7]. With
IB-DFE schemes both the feedforward and the feedback
parts are implemented in the frequency domain. Since the
feedback loop takes into account not just the hard-decisions
for each block, but also the overall block reliability, the error
propagation problem is significantly reduced. Consequently,
the IB-DFE techniques offer much better performances than
the non-iterative methods [4], [5]. In fact, IB-DFE techniques
can be regarded as low complexity turbo equalization schemes
[8], since the feedback loop uses the equalizer outputs instead
of the channel decoder outputs.
Earlier IB-DFE implementations considered hard decisions
(weighted by the blockwise reliability) in the feedback loop.
To improve the performance and to allow truly turbo FDE
implementations IB-DFE schemes with soft decisions were
proposed [9], [10], usually only for QPSK constellations.
The extension to larger constellations leads to difficulties on
the computation of the reliability of each block, as well as
problems on the computation of the average symbol values
conditioned to the FDE and the channel decoder output.
In this paper we consider SC-DFE schemes with IB-DFE
receivers and we propose a general method for the computation
of the receiver parameters for any constellation. Our approach
relies on an analytical characterization of the mapping rule
were the constellation symbols are written as a linear function
of the transmitted bits. This method is then employed in both
uniform and non-uniform QAM constellations.
II. IB-DFE RECEIVERS
We will consider an SC-DFE scheme where the signal
associated to a given block is
s(t)=
N−1
n=−NG
snhT(t−nTS),(1)
with TSdenoting the symbol duration, NGdenoting the
number of samples at the cyclic prefix and hT(t)denot-
ing the adopted pulse shape. The transmitted symbols sn
belong to a given alphabet S(i.e., a given constellation)
with dimension M=#Sand are selected according to the
corresponding bits β(m)
n,m=1,2, ..., μ (μ= log2(M)), i.e.,
sn=f(b(1)
n,b
(2)
n, ..., b(μ)
n), with b(m)
n=2β(m)
n−1(throughout
this paper we assume that β(m)
nis the mth bit associated to the
nth symbol and b(m)
nis the corresponding polar representation,
i.e., β(m)
n=0or 1 and b(m)
n=−1or +1, respectively). As
with other cyclic-prefix-assisted block transmission schemes, it
is assumed that the time-domain block is periodic, with period
N, i.e., s(m)
−n=s(m)
N−n.
If we discard the samples associated to the cyclic prefix
at the receiver then there is no interference between blocks,
provided that the length of the cyclic prefix is higher than the
length of the overall channel impulse response. Moreover, the
linear convolution associated to the channel is equivalent to
a cyclic convolution relatively to the N-length, useful part of
the received block, {yn;n=0,1,...,N−1}. This means that
the corresponding frequency-domain block (i.e., the length-
NDFT (Discrete Fourier Transform) of the block {yn;n=
0,1,...,N −1})is{Yk;k=0,1,...,N −1}, where
Yk=SkHk+Nk,(2)
with Hkdenoting the channel frequency response for the kth
subcarrier and Nkthe corresponding channel noise. Clearly,
the impact of a time-dispersive channel reduces to a scaling
factor for each frequency.
To cope with these channel effects we can employ a linear
FDE. However, the performance is much better if the linear
FDE is replaced by an IB-DFE, whose structure is depicted in
Fig. 1. For a given iteration, the output samples are given by
˜
Sk=FkYk−Bkˆ
Sk,(3)
where {Fk;k=0,1,...,N −1}and {Bk;k=0,1,...,N −
1}denote the feedforward and the feedback coefficients,
respectively. {ˆ
Sk;k=0,1,...,N −1}, is the DFT of the
block {ˆsn;n=0,1,...,N −1}, with ˆsndenoting the ”hard”
estimate of snfrom the previous FDE iteration.
The optimum coefficients Fkand Bkthat maximize the
overall SNR in the samples ˜
Skare given by [5]1
Fk=κH∗
k
α+(1−ρ2)|Hk|2,(4)
and
Bk=ρ(FkHk−1) ,(5)
respectively, where
α=E[|Nk|2]/E[|Sk|2](6)
and κis selected so as to ensure that
N−1
k=0
FkHk/N =1.(7)
^`
k
Y
X
^`
k
F
x
^`
k
B
IDFT
^`
k
S
DFT
+
-
^`
ˆ
n
s
^`
n
s
^
`
ˆ
k
S
Dec.
Fig. 1. Basic structure of an IB-DFE.
The correlation coefficient ρ, which can be regarded as the
blockwise reliability of the decisions used in the feedback loop
(from the previous iteration), is given by
ρ=E[ˆ
SkS∗
k]
E[|Sk|2]=E[ˆsns∗
n]
E[|sn|2].(8)
The IB-DFE structure described above is usually denoted
as ”IB-DFE with hard decisions”, although ”IB-DFE with
1It should be noted that, contrarily to [5], we are considering a normalized
feedforward filter.
blockwise soft decisions” would probably be more adequate,
as we will see in the following. In fact, (3) could be written
as
˜
Sk=FkYk−B
kSBlock
k,(9)
with ρB
k=Bkand SBlock
kdenoting the average of the
block of overall time-domain chips associated to a given
iteration, given by SBlock
k=ρˆ
Sk(as mentioned above, ρ
can be regarded as the blockwise reliability of the estimates
{ˆ
Sk;n=0,1,...,M −1}).
To improve the performances, we could replace the ”block-
wise averages” by ”symbol averages”, leading to what is
usually denoted as ”IB-DFE with soft decisions” [9], [10].
A simply way of achieving this is to replace the feedback
input {SBlock
k;k=0,1,...,N −1}by {SSymbol
k=Sk;k=
0,1,...,N−1}=DFT{sSymbol
n;n=0,1,...,N−1}, with
sSymbol
ndenoting the average symbol values conditioned to
the FDE output of the previous iteration ˜sn, with {˜sn=; n=
0,1,...,N −1}denoting the IDFT of the frequency-domain
block {Sk;k=0,1,...,N−1}. To simplify the notation, we
will use sninstead of sSymbol
nin the remaining of the paper.
For normalized QPSK constellations (i.e., sn=±1±j)
with Gray mapping it is easy to show that [10]
sn= tanh LI
n
2+jtanh LQ
n
2=
=ρI
nˆsI
n+jρQ
nˆsQ
n,(10)
with the LLRs (LogLikelihood Ratios) of the ”in-phase bit”
and the ”quadrature bit”, associated to sI
n=Re{sn}and sQ
n=
Im{sn}, respectively, given by
LI
n=2˜sI
n/σ2(11)
and
LQ
n=2˜sQ
n/σ2,(12)
respectively, where
σ2=1
2E[|sn−˜sn|2]≈1
2N
N−1
n=0 |ˆsn−˜sn|2.(13)
The hard decisions ˆsI
n=±1and ˆsQ
n=±1are defined
according to the signs of LI
nand LQ
n, respectively and ρI
n
and ρQ
ncan be regarded as the reliabilities associated to the
”in-phase” and ”quadrature” bits of the nth symbol, given by
ρI
n=E[sI
nˆsI
n]/E[|sI
n|2] = tanh |LI
n|/2(14)
and
ρI
n=E[sQ
nˆsQ
n]/E[|sQ
n|2] = tanh |LQ
n|/2(15)
(for the first iteration, ρI
n=ρQ
n=0and sn=0).
The feedforward coefficients are still obtained from (4), with
the blockwise reliability given by
ρ=1
2N
N−1
n=0
(ρI
n+ρQ
n).(16)
Therefore, the receiver with ”blockwise reliabilities” (hard
decisions), and the receiver with ”symbol reliabilities” (soft
decisions), employ the same feedforward coefficients; how-
ever, in the first the feedback loop uses the ”hard-decisions”
on each data block, weighted by a common reliability factor,
while in the second the reliability factor changes from bit
to bit. The receiver structure when we have soft decisions is
depicted in fig. 2, which is closely related to the IB-DFE with
hard decisions (fig. 1).
^`
k
Y
X
^`
k
F
x
^`
k
B
IDFT
^`
k
S
DFT
+
-
^
`
n
s
^`
n
s
^`
k
S
LLR
comput.
^`
()m
n
O
Dec.
Average’s
comput.
Fig. 2. IB-DFE with soft decisions.
III. ANALYTICAL CHARACTERIZATION OF MAPPING
RULES
A. General Mapping
It can be shown [11] that the constellation symbols can be
expressed as function os the corresponding bits as follows2:
sn=g0+g1b(1)
n+g2b(2)
n+g3b(1)
nb(2)
n+g4b(3)
n+...
=
M−1
i=0
gi
μ−1
m=0 b(m)
nγm,i ,(17)
for each sn∈S, where (γμ−1,i γμ−2,i ... γ1,i γ0,i)is the
binary representation of i. Since we have Mconstellation
symbols in Sand Mcoefficients gi, (17) is a system of
Mequations that can be used to obtain the coefficients gi,
i=0,1, ..., μ −1. Writing (17) in matrix format we have
s=Wg with s=[s1s2... sM]T,g=[g0g1... gμ−1]Tand
Wis a Hadamard matrix with dimensions M×M. Clearly, the
vector of constellation points sis the Hadamard transform of
the vector of coefficients g. Therefore, for a given constellation
we can obtain the corresponding coefficients gifrom the
inverse Hadamard transform of the vector of constellation
points.
For an uniform M-PAM constellation with S=
{±1,±3, ..., ±(M−1)}and a natural binary mapping the
only non-zero coefficients are g1,g
2,g
4, ..., gM/2(i.e., the
coefficients g2i,i=0,1, .., μ −1). Moreover, g2i=2
μ−i
which means that
sn=
μ−1
m=0
2mb(m)
n(18)
2It should be noted that in this subsection sndenotes the nth constellation
point but in the previous section sndenotes the nth transmitted symbol;
the same applies to b(m)
n(or β(m)
n) that here denotes the mth bit of the n
constellation point (instead of the mth bit of the nth transmitted symbol).
If we have
sn=
μ−1
m=0
2m−1
μ
m=μ−m+1
b(m)
n
(19)
then the mapping is Gray.
For uniform 4-PAM constellations we have
sn=b(1)
n+2b(2)
n(20)
for a natural binary mapping and
sn=2b(2)
n+b(1)
nb(2)
n(21)
for a Gray mapping. For uniform 8-PAM constellations we
have
sn=b(1)
n+2b(2)
n+4b(3)
n(22)
for a natural binary mapping and
sn=4b(3)
n+2b(3)
nb(2)
n+b(3)
nb(2)
nb(1)
n(23)
for a Gray mapping.
The same approach can be employed for non-uniform
hierarchical constellations such as the ones adopted in
multi-resolution schemes [12]. In fact, from (18) and (19),
an M-PAM constellation with either natural binary mapping
or Gray mapping can be regarded as the sum of μbinary
sub-constellations, each one with twice the amplitude of
the previous one. By reducing the amplitude of successive
sub-constellations we obtain hierarchical constellations with
different error protections.
By combining two PAM constellations, each with dimension
√M, one for the in-phase (real) component and the other for
the quadrature (imaginary) component we obtain a M-QAM
constellation.
IV. COMPUTATION OF RECEIVER PARAMETERS
An IB-DFE receiver with soft decisions (as described in sec.
II) has the following constellation-dependent tasks (see fig. 2):
•De-mapping the time-domain samples at the output of the
FDE, ˜sn, into the corresponding bits. This can be imple-
mented by computing the log-likelihood ratios associated
to each bit of each transmitted symbol.
•Computation of the average symbol values conditioned
to the FDE output of the previous iteration ˜sn, denoted
sn.
•Computation of the blockwise reliability ρ, required for
obtaining the feedforward coefficients (see (4)).
The log-likelihood ratio of the mth bit of the nth transmitted
symbol is given by
λ(m)
n= log Pr(β(m)
n=1|˜sn)
Pr(β(m)
n=0|˜sn)
= log ⎛
⎝s∈Ψ(m)
1exp −|˜sn−s|2
2σ2
s∈Ψ(m)
0exp −|˜sn−s|2
2σ2⎞
⎠,(24)
where Ψ(m)
1and Ψ(m)
0are the subsets of Swhere β(m)
n=
1or 0, respectively (clearly, Ψ(m)
1Ψ(m)
0=Sand
Ψ(m)
1Ψ(m)
0=∅). For obtaining the average symbol values
conditioned to the FDE output, sn, we need to obtain the
average bit values conditioned to the FDE output, b(m)
n. These
are related to the corresponding log-likelihood ratio as follows:
b(m)
n= tanh λ(m)
n
2.(25)
By taking advantage of the analytical characterization of
the mapping rules (17) and assuming uncorrelated bits (e.g.,
thanks to de adoption of suitable interleaving), we have
sn=
M−1
i=0
gi
μ−1
m=0 b(m)
nγm,i .(26)
Finally, the reliability of the estimates to be used in the
feedback loop is given by
ρ=E[ˆsns∗
n]
E[|sn|2]=M−1
i=0 |gi|2μ−1
m=0 ρ(m)
nγm,i
M−1
i=0 |gi|2,(27)
where ρ(m)
nis the reliability of the mth bit of the nth
transmitted symbol, given by
ρ(m)
n= tanh |λ(m)
n|
2.(28)
For QPSK constellations these results reduce to the ones
presented in sec. II.
V. N UMERICAL RESULTS
In this section we present a set of performance results con-
cerning IB-DFE receivers with soft decisions for generalized
constellations. The blocks have N= 256 symbols, plus an
appropriate cyclic prefix and we consider a severely time-
dispersive channel with perfect synchronization and channel
estimation at the receiver.
Let us first consider an uniform 64-QAM an uniform 64-
QAM constellation with Gray mapping based on 2 8-PAM
constellations characterized by g7/g6=g6/g4=0.5.Fig.
3 shows the BER performance for the IB-DFE receivers
described in this paper. Clearly, the performance improves
significantly with the iterations: when compared with a con-
ventional linear FDE, we have about 7dB gain for BER=10−4
after 4 iterations.
Let us consider now a non-uniform 64-QAM constellation
based on 2 8-PAM constellations characterized by g1/g2=
g2/g4=0.4, for natural binary mapping, and by g7/g6=
g6/g4=0.4, for Gray mapping. These constellations alow
bits with three different error protections, denoted LPB (Least
Protected Bits), IPB (Intermediate Protected Bits) and MPB
(Most Protected Bits), respectively.
Figs. 4 and 5 show the evolution of the LLR as a function
of the FDE output, for the different bits for SNR=10dB and
SNR=0dB, respectively. As expected, the LLR grows faster
for MPB.
15 20 25 30 35
10−4
10−3
10−2
10−1
100
Es/N0(dB)
BER
− − − − : Linear FDE
⋅ ⋅ ⋅ ⋅ : Iter. 2
− ⋅ − ⋅: Iter. 4
_____ : Iter. 5
Fig. 3. BER for uniform 64-QAM with Gray mapping.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−5
−4
−3
−2
−1
0
1
2
3
4
5x 1010
Re{˜sn}or Im{˜sn}
MPB
IPB
LPB
(*): Gray
(o): Binary
⋅ ⋅ ⋅ : LPB (m=1)
− − − : IPB (m=2)
____ : MPB (m=3)
Fig. 4. Evolution of the LLR of the different bits for a non-uniform 8-PAM
constellation (or the real and imaginary part of a 64-QAM constellation) and
SNR=10dB.
Figs. 6 and 7 show the BER performance for the differ-
ent type of bits when we have Gray mapping and natural
binary mapping, respectively. These results are expressed as
a function of Es/N0, with Esdenoting the average symbol
energy and N0the noise power spectral density. In both cases
the performance improves significantly with the number of
iterations, outperforming significantly the linear FDE. This
improvement is higher for LPB, since they suffer more from
the residual ISI that is inherent to a linear FDE optimized
under the MMSE criterion. The performance is similar for
both for Gray and binary mappings, with exception of the
region of very low SNR.
VI. CONCLUSIONS
In this paper we proposed a general method for designing
IB-DFE receivers for any constellation. For this purpose, we
defined an analytical characterization of the mapping rule were
the constellation symbols are written as a linear function of
−2 −1.5 −1 −0.5 00.5 11.5 2
−5
−4
−3
−2
−1
0
1
2
3
4
5
Re{˜sn}or Im{˜sn}
λ
(m)
n
(*): Gray
(o): Binary
⋅ ⋅ ⋅ : LPB (m=1)
− − − : IPB (m=2)
____ : MPB (m=3)
Fig. 5. As in fig. 4, but for SNR=0dB.
10 15 20 25 30 35 40
10−4
10−3
10−2
10−1
100
Es/N0(dB)
BER
* MPB
x IPB
o LPB
− − − : Linear FDE
⋅ ⋅ ⋅ : Iter. 2
____ : Iter. 4
Fig. 6. BER for the different bits of a non-uniform 64-QAM modulation
with Gray mapping.
the transmitted bits. This method was then employed in both
uniform and non-uniform QAM constellation.
ACKNOWLEDGMENTS
This work was supported in part by FCT (pluri-annual
founding and project U-BOAT).
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