SIViP (2007) 1:239252

DOI 10.1007/s11760-007-0017-4
ORIGINAL PAPER
Turbo equalization of serially concatenated turbo codes
using a predictive DFE-based receiver
K. Vasudevan
Received: 24 June 2006 / Revised: 11 April 2007 / Accepted: 12 April 2007 / Published online: 15 May 2007
Springer-Verlag London Limited 2007
Abstract This paper investigates the performance of
various turbo receivers for serially concatenated turbo
codes transmitted through intersymbol interference (ISI)
channels. Both the inner and outer codes are assumed to be
recursive systematic convolutional (RSC) codes. The opti-
mum turbo receiver consists of an (inner) channel maximum
a posteriori (MAP) decoder and a MAP decoder for the outer
code. The channel MAP decoder operates on a supertrellis
which incorporates the channel trellis and the trellis for the
inner error-correcting code. This is referred to as the MAP
receiver employing a SuperTrellis (STMAP). Since the com-
plexity of the supertrellis in the STMAP receiver increases
exponentially with the channel length, we propose a simpler
but suboptimal receiver that employs the predictive decision
feedback equalizer (PDFE). The key idea in this paper is
to have the feedforward part of the PDFE outside the itera-
tive loop and incorporate only the feedback part inside the
loop. We refer to this receiver as the PDFE-STMAP. The
complexity of the supertrellis in the PDFE-STMAP receiver
depends on the inner code and the length of the feedback
part. Investigations with Proakis B, Proakis C (both chan-
nels have spectral nulls with all zeros on the unit circle and
hence cannot be converted to a minimumphase channel) and
a minimum phase channel reveal that at most two feedback
taps are sufcient to get the best performance. A reduced-
state STMAP (RS-STMAP) receiver is also derived which
employs a smaller supertrellis at the cost of performance.
Keywords Equalization Intersymbol interference
Turbo equalization Decoding
K. Vasudevan (B)
Department of Electrical Engg.,
Indian Institute of Technology, Kanpur, India
e-mail: vasu@iitk.ac.in
1 Introduction
The problemof intersymbol interference (ISI) in digital com-
munication is a classical one that has been addressed as early
as 1965 [1]. The maximum likelihood receiver for ISI chan-
nels was proposed by Forney [2]. With the introduction of
turbo codes [3, 4] in the last decade, it is now possible to
achieve very low error rates at signal-to-noise ratio (SNR)
close to 0 dBin additive white Gaussian noise (AWGN) chan-
nels. In the recent years, the method of turbo equalization for
mitigating the effects of ISI channels, has been an area of
active research. While turbo codes [57] achieve Shannons
capacity over additive white Gaussian noise (AWGN) chan-
nels, turbo equalization achieves the same over ISI channels.
Alow-complexity approach to turbo equalization is to use
a linear minimum mean squared error equalizer (LMMSE)
that utilizes a priori information about the data [811]. An
alternative approach is based on interference cancellation
[1216]. By exploiting the fact that the covariance between
the real and imaginary parts of the equalizer output is non-
zero, further improvement in performance over the LMMSE
equalizer is obtained in [17]. In [18] FFT is used to com-
pute the equalizer coefcients. Turbo equalization for mul-
tilevel signals is addressed in [1922]. In [23], it has been
extended to MIMO channels employing multilevel-coded
QAMsignalling. Yeap et al. [24] present a comparative study
of various turbo equalization schemes using convolutional,
convolutional turbo and block-turbo codes. It was found that
the convolutional turbo codes give the best performance. The
problem of iterative channel estimation has been discussed
in [2527]. Turbo equalization for magnetic recording chan-
nels has been studied in [28, 29]. A technique similar to
turbo equalization has been used in the multiuser detector
for CDMA systems in [30]. The concept of bit-interleaved
turbo equalization and the effect of constellation mapping
1 3
240 SIViP (2007) 1:239252
Fig. 1 Turbo equalization
using the predictive MAP
decoder for serially
concatenated codes
Channel
Deinterleaver
(for outer code)
MAP decoder
Interleaver
Outer code Inner code Interleaver
(QPSK symbols)
LMMSE
equalizer
Final decisions
Predictive
for inner
code
MAP
decoder
a priori
Initialize
AWGN
is
extrinsic info
extrinsic info
about
about
probabilities of
Receiver
Transmitter
(1, 1)
is described in [31]. The convergence behaviour of serially
concatenated systems using extrinsic information transfer
(EXIT) charts is studied in [32]. In [33] turbo decision feed-
back equalization using constrained delay a posteriori prob-
abilities is described.
Maximum likelihood detection of signals in the presence
of coloured noise has been studied earlier in [2, 3437] and
independently in [38, 39]. Optimal decoding of turbo coded
signals in the presence of coloured Gaussian noise has been
addressed in [40], where the predictive iterative decoder
(PID) was proposed. The PIDconsists of two predictive MAP
(maximum a posteriori) decoders that are interconnected via
an interleaver/deinterleaver pair. Detection of turbo coded
signals transmitted through ISI channels using an LMMSE
equalizer and the PID is discussed in [41]. A similar prob-
lem of detecting turbo coded signals in the presence of ISI is
discussed in [42]. An earlier work on turbo equalization of
serially concatenated codes is given in [43].
The channel MAP decoder in the STMAP receiver pro-
posed in this paper is analogous to decision-feedback
sequence estimation (DFSE) proposed earlier in the litera-
ture [4446], the only difference being that in the STMAP,
the MAP algorithm is used whereas in the latter, the Viterbi
algorithm (VA) or soft-output VA (SOVA) is used. Turbo
equalizationof trellis-coded8-PSKusingsoft-output sequen-
tial algorithm (SOSA) is described in [47]. It is also interest-
ing to compare our transmitter in Fig. 1 and scheme 2 (the
best scheme) in [48]. Note that we have used only one inter-
leaver in the transmitter, whereas in [48] two interleavers are
used.
This paper is organized as follows. In Sect. 2 we present
the system model. Maximum likelihood block decoding in
coloured Gaussian noise is described in Sect. 3. In Sect. 4
we derive the symbol-by-symbol predictive MAP decoder.
The implementation of the PDFE-STMAP receiver using the
BCJR algorithm is discussed in Sect. 5. The predictive MAP
decoder is a component of the PDFE-STMAP receiver. Other
receiver structures, which differ only in the implementation
of the inner (channel) MAP decoder are presented in Sect. 6.
In Sect. 7 we present the analytical and simulation results.
EXIT charts [4951] are used to estimate the BER perfor-
mance at a given SNR. Finally, in Sect. 8 we present our
conclusions and scope for future work.
2 System Model
In a typical digital communication system employing turbo
equalization (see Fig. 1), [5254], the input data a
k
, 0 k
L 1, is divided into frames of L bits each. Each frame is
encoded into N bits denoted by b
k
, 0 k N 1, using a
convolutional encoder (outer code). The interleaver scram-
bles b
k
to form c
k
. The output of the interleaver is fed to an
inner code (which could again be a recursive convolutional
code as proposed in this paper and also in [48]) or a recursive
precoder as in [11, 55, 56].
The objective of this paper is to propose new receiver
structures for such serially concatenated systems and to
compare their performance with the receivers reported ear-
lier in the literature. The simulation results indicate that the
1 3
SIViP (2007) 1:239252 241
performance of the proposed serially concatenated codes is
better than the classical turbo equalization approach in [11]
which uses an inner precoder. The reason can be attributed
to the fact that the precoder cannot correct errors. It can only
introduce dependencies in the data due to its innite mem-
ory, which enhances the performance of the channel MAP
decoder. Recall that a MAP decoder outputs a soft decision
about a symbol byobservinganentire frame of data. Since the
channel memory is nite, the precoding approach is expected
to perform better than the classical turbo equalization that
does not use precoding. However, our approach is expected
to do even better than the precoding approach, since we have
introduced error correcting capability, besides having innite
memory, in the inner code.
The channel with impulse response h
k
introduces ISI and
adds AWGN with zero mean and variance
2
v
. For conve-
nience, we assume that the channel has unit energy:
L
h

k=0
h
2
k
= 1. (1)
The receiver consists of a channel MAP decoder (commonly
referred to as the MAP equalizer in the literature) and a MAP
decoder for the outer code (we henceforth refrain from using
the term MAP equalizer, since according to the classi-
cal denition, an equalizer inverts the channel, whereas a
MAP decoder reconstructs the noise-free signal at the chan-
nel output). The two MAP decoders are interconnected via
the interleaver/deinterleaver. Though this approach is opti-
mum in principle, it may be difcult to implement in prac-
tice when the channel length L
h
is large, since the channel
MAP decoder would have to operate on a trellis with 2
L
h
states, for BPSK signalling. Note that when the inner code is
a recursive precoder, the trellis complexity does not increase
[11, 55, 56].
Throughout this article we assume that both the outer and
inner codes are rate-1/2 recursive systematic convolutional
codes (RSC) having the generator matrix:
G(D) =
_
1
1+D
2
1+D+D
2
_
(2)
Thus N = 2L. The output of the inner code is mapped onto a
QPSK constellation so that the overall rate of the transmitter
is 1/2 (2L coded QPSK symbols for every L uncoded data
bits). We assume that b
2k
= a
k
and b
2k+1
is the parity bit
corresponding to a
k
.
The important feature of the PDFE-STMAP receiver pro-
posedinthis article is that the LMMSEequalizer (the feedfor-
ward part) is outside the iterative loop, whereas the feedback
part (the prediction lter) is incorporated into the trellis of
the inner RSC to form a super-trellis [44, 45]. The predictive
MAP decoder operates on the super-trellis. The super-trellis
enables the feedback part to process all valid combinations
of data separately and is thus expected alleviate the problem
of error propagation that is usually associated with decision
feedback equalizers. Note that it is essential to maintain a
small number of feedback taps, since the complexity of the
super-trellis grows exponentially with the length of the feed-
back lter.
3 ML decoding in coloured Gaussian noise
Assume that N symbols have been transmitted (recall that
there are N = 2L symbols in a frame). The symbols are
taken from an M-ary alphabet. The received signal can be
written as [38]:
r = S
(i )
+ w (3)
where r is an N 1 column vector of the received samples,
S
(i )
is an N 1 vector of the i th possible symbol sequence
(assuming that the symbols are independent 0 i M
N

1) and w is an N 1 column vector of correlated Gaussian

noise samples with zero-mean and variance
2
w
. In general
we denote complex quantities by a tilde, e.g. x, though we
prefer not to use a tilde when denoting complex symbols.
Thus the symbol estimates are denoted by

S
k
instead of

S
k
.
Real quantities are denoted without a tilde, e.g. x. Boldface
letters denote vectors or matrices.
The ML detector maximizes the joint conditional pdf:
max
j
p
_
r|S
( j )
_
for 0 j M
N
1 (4)
which is equivalent to:
max
j
1
(2)
N
det
_

R
_
(5)
exp
_

1
2
_
r S
( j )
_
H

R
1
_
r S
( j )
_
_
where the covariance matrix, conditioned on the j th possible
symbol sequence, is given by

R =
1
2
E
_
w w
H
_
. (6)
The diagonal elements of

Ris a constant equal to
2
w
, which is
equal to the variance of coloured noise. Performing Cholesky
factorization on

R, we get [38, 39, 57]:

R
1
=

G
H
D
1

G (7)
where

G is an N N lower triangular matrix given by:

G

=
_
_
_
_
_
1 0 . . . 0
g
1, 1
1 . . . 0
.
.
.
.
.
.
.
.
.
.
.
.
g
N1, N1
g
N1, N2
. . . 1
_

_
(8)
1 3
242 SIViP (2007) 1:239252
where g
k, p
denotes the pth coefcient of the optimal
kth-order prediction lter. The N N matrix D is a diagonal
matrix of the prediction error variance denoted by:
D

=
_
_
_

2
0
. . . 0
.
.
.
.
.
. 0
0 . . .
2
N1
_

_. (9)
Observe that
2
j
denotes the prediction error variance for the
optimal j th-order predictor (0 j N 1). Substituting
(7) into (5) and noting that det
_

R
_
is independent of the
symbol sequence j , we get:
max
j
exp
_

1
2
_
r S
( j )
_
H

G
H
D
1

G
_
r S
( j )
_
_
. (10)
which is equivalent to:
max
j
N1

k=0
exp
_
_
_

z
( j )
k

2
2
2
k
_

_ (11)
where the prediction error at time k for the j th symbol
sequence, z
( j )
k
, is an element of z
( j )
and is given by:
_
_
_
_
_
_
z
( j )
0
z
( j )
1
.
.
.
z
( j )
N1
_

= z
( j )
=

G
_
r S
( j )
_
. (12)
Note that the prediction error variance is computed using
j = i as follows:

2
k

=
1
2
E
_
z
(i )
k
_
z
(i )
k
_

_
. (13)
When w consists of samples from a Pth-order AR process,
then it is easy to see that a Pth-order prediction lter is suf-
cient to completely decorrelate the elements of w. In this
situation we have

2
k
=
2
P
for k P. (14)
In practical situations, we would in any case use a nite order
prediction lter.
4 The predictive MAP decoder
The block diagramof the systemincorporating the predictive
MAP decoder is depicted in Fig. 1. Observe that the two bits
at the output of the inner code at the transmitter are combined
to forma QPSKsymbol S
k
= d
k
+j e
k
. Here d
k
= c
k
denotes
the systematic bit and e
k
denotes the parity bit. The output
of the LMMSE equalizer at the receiver is S
k
plus coloured
Gaussian noise (we assume that the noise is Gaussian even
though it may not be strictly true).
Table 1 Super trellis for the inner MAP decoder in the PDFE-STMAP
receiver, for a rst-order (P = 1) prediction lter and the rate-1/2
encoder given in (2)
Present state (time n) Input Next state (time n +1)
0 0 0
0 1 3
1 0 0
1 1 3
2 0 6
2 1 5
3 0 6
3 1 5
4 0 2
4 1 1
5 0 2
5 1 1
6 0 4
6 1 7
7 0 4
7 1 7
Before we proceed to describe the predictive MAP
decoder, note that the trellis for the inner code must be mod-
ied to account for the additional memory due to the predic-
tion lter. In fact, we obtain a super-trellis [44, 45] containing
S
ST
= S
E
2
P
(15)
states where P is the order of the prediction lter and S
E
is
the number of states in the encoder trellis. We assume for
simplicity that a Pth-order prediction lter completely dec-
orrelates the noise. This is illustrated in Table 1 for P = 1
and the encoder is given by (2). Observe that
Supertrellis state in decimal notation
= 2
P
encoder state in decimal
+prediction lter state in decimal. (16)
The output sample is given by (32).
Assume that a received block of N = 2L symbols needs
to be decoded. The predictive MAP decoder computes the a
posteriori probabilities:
P (c
k
= +1| r) and P (c
k
= 1| r) (17)
for 0 k N 1, where c
k
{1} is the systematic bit
input to the inner code at the kth time instant and r is a 1N
vector denoted by:
r =
_
r
0
. . . r
N1
_
. (18)
In the above equation
r
k
= S
k
+ w
k
(19)
1 3
SIViP (2007) 1:239252 243
where S
k
is a QPSK symbol, w
k
denotes samples of ACGN
with zero-mean and variance
2
w
.
Now (for 0 k N 1)
P (c
k
= +1| r) =
p ( r|c
k
= +1) P (c
k
= +1)
p( r)
(20)
P (c
k
= 1| r) =
p ( r|c
k
= 1) P (c
k
= 1)
p( r)
where p() denotes the probability density function and
P(c
k
= +1) denotes the a priori probability that c
k
= +1.
Noting that p( r) is a constant and can be ignored, the above
equation can be written as:
P (c
k
= +1| r) = C
k+
P (c
k
= +1)
(21)
P (c
k
= 1| r) = C
k
P (c
k
= 1)
where
C
k+
=
2
N1

j =1
p
_
r|c
( j )
k+
_
P
( j )

k
(22)
C
k
=
2
N1

j =1
p
_
r|c
( j )
k
_
P
( j )

k
where the 1 N vectors
c
( j )
k+
=
_
c
( j )
0
. . . +1 . . . c
( j )
N1
_
(23)
c
( j )
k
=
_
c
( j )
0
. . . 1 . . . c
( j )
N1
_
are constrained such that the kth symbol is +1 or 1, respec-
tively, for all j . Assuming that c
( j )
k
are independent
P
( j )

k
=
N1

i =0
i =k
P
_
c
( j )
i
_
. (24)
Expanding the conditional pdf in (22) we get
C
k+
=
2
N1

j =1
N1

i =0

( j )
i, k+
P
( j )

k
(25)
C
k
=
2
N1

j =1
N1

i =0

( j )
i, k
P
( j )

k
where
( j )
i, k+
and
( j )
i, k
are calculated as (from Sect. 3):

( j )
i, k+
= exp
_
_
_

z
( j )
i, k+

2
2
2
Q
_

_
(26)

( j )
i, k
= exp
_
_
_

z
( j )
i, k

2
2
2
Q
_

_
where
2
Q
is the noise variance at the output of the optimal
Qth-order predictor and
Q = min(i, P)
z
( j )
i, k+
=
Q

l=0
g
Q, l
_
r
i l
S
( j )
i l, k+
_
(27)
z
( j )
i, k
=
Q

l=0
g
Q, l
_
r
i l
S
( j )
i l, k
_
where g
Q, l
is the lth coefcient of the optimal Qth-order pre-
dictor. The terms S
( j )
i, k+
and S
( j )
i, k
denote the (QPSK) symbol
at time instant i , generated by c
( j )
k+
and c
( j )
k
, respectively.
Finally, the extrinsic information that is to be fed to the
MAP decoder for the outer code, is computed as (for 0
k N 1):
E(c
k
= +1) = C
k+
/(C
k+
+C
k
)
(28)
E(c
k
= 1) = C
k
/(C
k+
+C
k
).
It is clear that a straightforward computation of C
k+
and
C
k
as given in (25) would be prohibitively expensive. Let
us now turn our attention to the efcient computation of C
k+
and C
k
using the BCJR algorithm [58].
5 The PDFE-STMAP receiver using the BCJR
algorithm
In this section we describe the BCJR algorithm for the
receiver shown in Fig. 1 which is referred to as the predic-
tive DFE-MAP receiver employing the SuperTrellis (ST), in
short the PDFE-STMAP receiver. The number of supertrellis
states is given by (15). Throughout this paper, it is under-
stood that a MAP decoder operating on the encoder trellis is
employed on the outer code. The receiver discussed in this
section and the other receivers discussed later differ only in
the implementation of the inner MAP decoder.
5.1 BCJR for the inner code
Let D
n
denote the set of states that diverge from state n. For
example
D
0
= {0, 3} (29)
implies that states 0 and 3 can be reached from state 0. Sim-
ilarly, let C
n
denote the set of states that converge to state n.
Let
i, n
denote the forward sum-of-products (SOP) at time
i (0 i N 2) at state n (0 n S
ST
1). Then the
1 3
244 SIViP (2007) 1:239252
forward SOP can be recursively computed as follows [58]:

i +1, n
=

mC
n

i, m

i, m, n
P
_
c
i, m, n
_

0, n
= 1 for 0 n S
ST
1 (30)

i +1, n
=

i +1, n
_
_
S
ST
1

n=0

i +1, n
_
where P(c
i, m, n
) denotes the a priori probability of the sys-
tematic bit corresponding to the transition from superstate m
to superstate n, at time i (this is set to 0.5 at the beginning of
the rst iteration) and

i, m, n
= exp
_

z
i, m, n

2
2
2
Q
_
(31)
where
z
i, m, n
=
Q

l=0
g
Q, l
_
r
i l
S
l, m, n
_
. (32)
The term S
0, m, n
denotes the input symbol corresponding
to the transition from superstate m to n. The other symbols
S
l, m, n
for 1 l Q, are extracted from superstate m
(more precisely, from the memory of the prediction lter).
The normalization step in the last equation of (30) is done
to prevent numerical instabilities. Equation (32) constitutes
the key step in the predictive MAP decoder using the BCJR
algorithm.
Similarly, let
i, m
denote the backward SOP at time i
(1 i N 1) at state m (0 m S
ST
1). Then the
recursion for the backward SOP can be written as:

i, m
=

nD
m

i +1, n

i, m, n
P
_
c
i, m, n
_

N, m
= 1 for 0 m S
ST
1 (33)

i, m
=

i, m
_
_
S
ST
1

m=0

i, m
_
.
Once again, the normalization step in the last equation of
(33) is done to prevent numerical instabilities.
Let
+
(n) denote the state that is reached from state n
when the input symbol is +1. Similarly let

(n) denote the

state that can be reached from state n when the input symbol
is 1. Then for 0 k N 1 we have
C
norm, k+
=
S
ST
1

n=0

k, n

k, n,
+
(n)

k+1,
+
(n)
(34)
C
norm, k
=
S
ST
1

n=0

k, n

k, n,

(n)

k+1,

(n)
.
Note that due to the normalization step in (30) and (33)
C
norm, k+
= A
k
C
k+
(35)
C
norm, k
= A
k
C
k
where C
k+
and C
k
are given in (25), and A
k
is a constant. It
can be veried that in the absence of the normalization step
in (30) and (33), A
k
= 1 for all k.
Finally, the extrinsic information that is to be fed to the
MAP decoder for the outer code, is computed as:
E(c
k
= +1) = C
norm, k+
/(C
norm, k+
+C
norm, k
)
(36)
E(c
k
= 1) = C
norm, k
/(C
norm, k+
+C
norm, k
)
which is identical to (28). Equations (30), (33) and (34) con-
stitute the BCJR recursions for the predictive MAP decoder.
5.2 BCJR for the outer code (common to all receivers)
Let
i, n
denote the forward SOP at time i (0 i L 2,
N = 2L) at state n (0 n S
E
1). Then the forward SOP
is recursively computed as follows:

i +1, n
=

mC
n

i, m

sys, i, m, n

par, i, m, n
P
_
a
i, m, n
_

0, n
= 1 for 0 n S
E
1 (37)

i +1, n
=

i +1, n
_
_
S
E
1

n=0

i +1, n
_
where P(a
i, m, n
) denotes the a priori probability of the sys-
tematic bit corresponding to the transition from state m to
state n, at time i (in the absence of any other information,
this is xed at 0.5 [11]) and

sys, i, m, n
=
_
E(c
(2i )
= +1) if condition H
1
E(c
(2i )
= 1) if condition H
2
.

par, i, m, n
=
_
E(c
(2i +1)
= +1) if condition H
3
E(c
(2i +1)
= 1) if condition H
4
.
(38)
where
H
1
: systematic bit from state/superstate m to n is +1
H
2
: systematic bit from state/superstate m to n is 1
(39)
H
3
: parity bit from state/superstate m to n is +1
H
4
: parity bit from state/superstate m to n is 1
and (2i ) and (2i + 1) denotes the interleaver map at
time 2i and 2i + 1 respectively. Note that E(c
k
= +1) and
E(c
k
= 1) have already been computed as in (36).
Similarly, let
i, m
denote the backward SOP at time i
(1 i L 1) at state m (0 m S
E
1). Then the
1 3
SIViP (2007) 1:239252 245
recursion for the backward SOP can be written as:

i, m
=

nD
m

i +1, n

sys, i, m, n

par, i, m, n
P
_
a
i, m, n
_

L, m
= 1 for 0 n S
E
1 (40)

i, m
=

i, m
_
_
S
E
1

m=0

i, m
_
where again P(a
i, m, n
) denotes the a priori probability of the
systematic bit corresponding to the transition from state m
to state n, at time i (this is again xed at 0.5 [11]).
Let
+
(n) denote the state that is reached from state n
when the systematic bit is +1. Similarly let

(n) denote
the state that can be reached from state n when the input
(systematic) bit is 1. For 0 k L 1 dene:
B
2k+
=
S
E
1

n=0

k, n

par, k, n,
+
(n)

k+1,
+
(n)
(41)
B
2k
=
S
E
1

n=0

k, n

par, k, n,

(n)

k+1,

(n)
.
Similarly, let
+
(n) and

(n) denote the states that are

reached fromstate n when the parity bit is +1 and 1, respec-
tively. For 0 k L 1 dene:
B
2k+1+
=
S
E
1

n=0

k, n

sys, k, n,
+
(n)

k+1,
+
(n)
(42)
B
2k+1
=
S
E
1

n=0

k, n

sys, k, n,

(n)

k+1,

(n)
.
The extrinsic information that is fed as a priori probabilities
to the inner decoder is given by (for 0 k N 1):
E(b
k
= +1) = B
k+
/(B
k+
+ B
k
)
(43)
E(b
k
= 1) = B
k
/(B
k+
+ B
k
).
In other words, P(c
i, m, n
) in (30) and (33) is given by (for
0 i N 1):
P(c
i, m, n
) =
_
E(b

1
(i )
= +1) if condition H
1
E(b

1
(i )
= 1) if condition H
2
(44)
6 Other receiver structures
6.1 The STMAP receiver
Here the equalizer is not used. The inner MAP decoder
directly receives the input y
k
from the channel and operates
on a supertrellis of size S
ST
= S
E
2
L
h
states, where the
length of the channel is L
h
+1. This is the optimum receiver
which gives the lowest BER performance. The BCJR algo-
rithmfor the inner and outer MAP decoders is again identical
to that discussed in Sects. 5.1 and 5.2 excepting that z
i, m, n
in (31) is given by
z
i, m, n
= y
i

L
h

l=0

h
l
S
l, m, n
(45)
where

h
l
denotes the channel coefcients and S
0, m, n
denotes
the input QPSK symbol corresponding to the transition from
superstate m to n. The other QPSK symbols for 1 l L
h
are extracted from the superstate. Note also that
2
Q
in (31)
must be replaced by the channel noise variance
2
v
.
6.2 The reduced state STMAP (RS-STMAP) receiver
Here the number of supertrellis states is
S
ST
= S
E
2
L
r
(46)
where L
r
< L
h
. The term z
i, m, n
in (31) must be replaced
by
z
( j )
i, m, n
= y
i

L
r

l=0

h
l
S
l, m, n

L
h

l=L
r
+1

h
l
S
( j )
lL
r
, m
(47)
where

h
l
denotes the channel coefcients and S
0, m, n
denotes
the input QPSK symbol corresponding to the transition from
superstate m to n. The QPSK symbols for 1 l L
r
are
extracted fromthe superstate. The remaining QPSKsymbols
S
( j )
lL
r
, m
are extracted from the path history leading to state
m. Since there is more than one path that leads to state m, the
superscript j refers to the j th such path.
Similarly,
i, m, n
in (31) must be replaced by
( j )
i, m, n
and
the forward SOP in (30) must be modied to:

i +1, n
=

mC
n

i, m

( j )
i, m, n
P
_
c
i, m, n
_
. (48)
The backward SOP in (33) must be simularly modied.
Finally
2
Q
in (31) must be replaced by the channel noise
variance
2
v
.
6.3 Classical Turbo Equalization (TEQ)[11]
In the case of classical TEQ without precoding, there is no
inner code. Therefore, the inner channel MAP decoder oper-
ates on a trellis with just 2
L
h
states. Moreover, the transmitted
symbols are BPSK. Hence z
i, m, n
in (31) is given by
z
i, m, n
= y
i

L
h

l=0

h
l
c
l, m, n
(49)
where

h
l
denotes the channel coefcients and c
0, m, n
denotes
the input BPSK symbol (1) corresponding to the transition
from state m to n. The other BPSK symbols for 1 l L
h
1 3
246 SIViP (2007) 1:239252
Table 2 Complexity of the inner MAP decoder for various receivers
Inner MAP decoder No of trellis states
PDFE-STMAP S
E
2
P
STMAP S
E
2
L
h
RS-STMAP S
E
2
Lr
Classical TEQ 2
L
h
TEQ+precoding 2
L
h
are extracted from the state. Again
2
Q
in (31) must be
replaced by
2
v
.
6.4 Turbo equalization with precoding [11]
When precoding is used, the number of trellis states remains
unchanged (2
L
h
) and (49) must be modied to:
z
i, m, n
= y
i

L
h

l=0

h
l
c

l, m, n
(50)
where
c

0, m, n
= c
0, m, n
c

1, m, n
(51)
where (note that multiplication corresponds to the XORoper-
ation when bit 0 maps to +1 and bit 1 maps to 1).
The complexity of the inner MAP decoder for various
receivers is illustrated in Table 2.
7 Simulation results
In this section we compare the BER performance of various
receivers for three different channel characteristics namely,
Proakis B and Proakis C and a minimum phase channel.
For the classical TEQapproach [11], the average SNRper
bit (E
b
/N
0
) in dB is given by:
E
b
/N
0
= 10 log
10
(1/
2
v
) (52)
since E
b
= 2 (there are two code bits in one uncoded bit
duration) and N
0
/2 =
2
v
. However in our approach, QPSK
symbols taken from the constellation 1 j are transmitted.
Hence, the SNR per bit is given by [59]:
E
b
/N
0
= 10 log
10
(2/
2
v
) (53)
since E
b
= 4, (two QPSK symbols in one uncoded bit dura-
tion). Note that
1
2
E
_
| v
k
|
2
_
=
2
v
. (54)
Moreover in classical TEQ [11], the generator matrix for
the outer (nonsystematic) code is given by:
G(D) =
_
1 + D
2
1 + D + D
2
_
(55)
whereas in our approach the generator matrix for both the
inner and outer (systematic) codes is given by (2).
The other parameters for simulation are as follows. Unless
otherwise specied, the dataframesize is L = 10
3
bits and
the simulations were carried out over 10
5
frames. Random
interleaving is used and the interleaver pattern is identical for
all receivers. For all the receiver structures, the MAP decoder
for the outer code has four states. In the case of the PDFE-
STMAP receiver, the optimum nite-length predictive DFE
[59, 60] with feedforward part having forty coefcients was
used. The length of the feedback part was varied between
one and two.
The impulse response of Proakis B channel (normalized
to unit energy) is given by:
h
n
=
1

6
(
n
+2
n1
+
n2
) . (56)
The simulation results for Proakis B channel, with ten iter-
ations is given in Fig. 2. As expected, the STMAP receiver
with a super-trellis of 4 2
2
= 16 states for the inner MAP
decoder has the best performance. This is followed by the
PDFE-STMAP receiver. In the case of the PDFE-STMAP
receiver, a rst-order predictor is used. Hence the number of
supertrellis states for the inner MAP decoder is 4 2
1
= 8
states. Moreover, the PDFE-STMAP is better than classical
TEQ with precoding and the RS-STMAP by about 1.5 dB
at a BER of 10
5
, but is worse than the STMAP receiver
by only 0.2 dB. Note that the classical TEQ approach (with
and without precoding) requires only four states for the inner
MAP decoder whereas the RS-STMAP requires eight states.
The Outer code in AWGN curve in Figs. 2 and 6 is for the
code in (55) in AWGN with soft-decision Viterbi decoding.
The SCTC in AWGN curve in Figs. 2, 6 and 8 denotes
the performance of the proposed serially concatenated turbo
code in AWGN.
The EXIT chart [4951] for various receivers for Proakis
B channel is presented in Fig. 3 for E
b
/N
0
= 2 dB. Here the
termCCMAP refers to the MAP decoder for the outer con-
volutional code. Note that the EXIT chart for (2) and (55) are
very nearly identical. The procedure for obtaining the EXIT
chart for the inner and outer MAP decoders is described in
the Appendix.
Figure 3 reveals that the STMAP and the PDFE-STMAP
receivers must theoretically have zero BER, since the xed
point is at unity (I
D
o
= 1). In practice however the BER
is nite due to the nite length of the interleaver. In fact
I
D
o
= 1 implies that an arbitrarily low BER can be achieved
byincreasingthe interleaver size. Table 3shows the estimated
and simulated BER for the STMAP and PDFE-STMAP
receivers for data framesize of L = 10
3
and L = 10
4
bits
at E
b
/N
0
= 2 dB. We nd that the STMAP and PDFE-
STMAP receivers require a data framesize of 10
4
bits to
attain BER < 10
8
.
The EXIT chart for the classical TEQ approach (with and
without precoding) for various SNRs is depicted in Fig. 4.
1 3
SIViP (2007) 1:239252 247
Fig. 2 Simulation results using
Proakis B channel with ten
iterations
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 1 2 3 4 5 6
B
i
t

e
r
r
o
r

r
a
t
e
SNR per bit (dB)
TEQ + precoding, 4-states
Classical TEQ, 4-states
STMAP 16-states
PDFE-STMAP 8-states
RS-STMAP 8-states
Outer code in AWGN
SCTC in AWGN
Fig. 3 EXIT charts for various
receivers using Proakis B
channel at E
b
/N
0
= 2 dB
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
CC MAP
Classical TEQ, 4-states
TEQ + precoding, 4-states
STMAP decoder, 16-states
PDFE-STMAP decoder, 8-states
Fig. 4 EXIT charts for classical
turbo equalization using Proakis
B channel at various SNRs
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
CC MAP
No precoding, Eb/N0 = 2 dB
With precoding, Eb/N0 = 2 dB
With precoding, Eb/N0 = 3 dB
No precoding, Eb/N0 = 3 dB
1 3
248 SIViP (2007) 1:239252
Table 3 The estimated and simulated BER using Proakis B channel at
E
b
/N
0
= 2 dB
Simulated BER
Receiver Estimated BER L = 10
3
L = 10
4
10
5
frames 10
4
frames
STMAP 0 2.7 10
4
< 10
8
PDFE-STAMP 0 7.6 10
4
< 10
8
Table 4 The estimated and simulated BER for classical turbo equal-
ization using Proakis B channel at various SNRs
Receiver E
b
/No (dB) Estimated BER Simulated BER
No precoding 2 3.1 10
2
3.4 10
2
No precoding 3 5.1 10
2
5.3 10
3
Precoding 2 1.2 10
1
1.8 10
1
Precoding 3 0 2.9 10
3
(< 10
8
)
The number in bracket denotes the BER for L = 10
4
bits
The estimated and simulated BER given in Table 4 shows
the accuracy of our results. In fact, we can conclude that the
EXIT chart predicts the BER quite accurately when the xed
point is less than unity. It is interesting to note that at 2 dB
SNR, I
D
o
= 0.2 for precoding whereas I
D
o
= 0.7 without
precoding. This shows that precoding may not always give a
better BER performance, especially at low SNR.
The trajectory of the PDFE-STMAP receiver for the Proa-
kis Bchannel for various framesizes is depicted in Fig. 5. The
deviation between the actual trajectory and the EXIT chart
can be explained as follows. With successive iterations, the
pdfs of the LLRs L
b, k
in (63) are not Gaussian distributed
and neither is the variance of the LLRs twice that of the mean.
Since these LLRs are fed back as a priori probabilities to the
inner decoder, the Gaussian assumption for the a priori prob-
abilities gets violated. On the other hand, the EXIT charts are
drawn by assuming that the LLRs of the a priori probabilities
to the inner decoder are Gaussian distributed, that is, the pdf
of
L
b, k
= ln
_
(k, +1)
(k, 1)
_
(57)
where (k, c) is given by (62), is Gaussian distributed with
variance equal to twice the mean.
The impulse response of Proakis C channel, normalized
to unit energy is given by:
h
n
=
1

19
(
n
+2
n1
+3
n2
+2
n3
+
n4
) . (58)
The simulation results for Proakis C channel with ten iter-
ations is given in Fig. 6. The results are for L = 10
3
. We
nd that the PDFE-STMAP receiver with two feedback taps
(P = 2), is inferior to the STMAP receiver by only about 0.3
Table 5 The estimated and simulated BER using Proakis C channel at
E
b
/N
0
= 4 dB
Simulated BER
Receiver Estimated BER L = 10
3
L = 10
4
10
5
frames 10
4
frames
STMAP 0 3.9 10
6
< 10
8
PDFE-STAMP 0 1.4 10
5
< 10
8
Classical TEQ 1.2 10
1
2.4 10
1
2.4 10
1
(no precoding)
TEQ (with precoding) 1.7 10
1
2.1 10
1
2.1 10
1
dB and superior to classical TEQ with precoding by 2 dB.
Interestingly, here classical TEQ with precoding is inferior
to the case without precoding upto an SNR of about 6 dB.
The EXIT charts for various receivers for the Proakis C
channel is shown in Fig. 7. The estimated and simulated BER
for various receivers for the Proakis Cchannel at an SNRof 4
dB is shown in Table 5. Once again, observe the accuracy of
our predictions when the xed point is less than unity. Note
that increasing the framesize for the classical TEQ approach
does not improve the BER, as expected.
Note that the STMAP receiver, which has the best perfor-
mance has an inner MAP decoder with 64 states. However,
the PDFE-STMAP receiver with P = 2 and the classical
TEQ receiver (with and without precoding) have only 16
states for the inner MAP decoder. Thus we have attained
the performance of the STMAP receiver (which is the ideal
receiver) at a much reduced complexity. The RS-STMAP
receiver with 16 states gives a very poor performance. Note
that Proakis B and C channels cannot be converted to mini-
mum phase channels since all the zeros lie on the unit circle.
Finally, simulations were carried out for a minimumphase
channel whose impulse response is given by:
h
n
=

0.45
n
+

0.25
n1
+

0.15
n2
(59)
+

0.1
n3
+

0.05
n4
.
The results are shown in Fig. 8. Here it is worth noting that the
performance of the RS-STMAP is much better than the Proa-
kis channels, but still inferior to the proposed PDFE-STMAP
receiver by about 1.5 dB. The optimumSTMAPis better than
the PDFE-STMAP by only 0.3 dB and its supertrellis com-
plexity (64 states) is four times that of the PDFE-STMAP
(16 states).
Note that both the STMAP and the PDFE-STMAP are
better than scheme 2A (the best scheme) in Fig. 11 in [48].
Whereas the PDFE-STMAP is better than scheme 2A by
0.4 dB, the STMAP is better by 0.6 dB, at a BER of 10
5
.
Observe that the STMAP and PDFE-STMAP require only
ten iterations, while scheme 2A [48] requires 12 iterations
to attain the given performance. It is also worth noting that
1 3
SIViP (2007) 1:239252 249
Fig. 5 The trajectory of the
PDFE-STMAP receiver for
various framesizes at
E
b
/N
0
= 2 dB, for Proakis B
channel
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
CC MAP
PDFE-STMAP
Traj. of PDFE-STMAP, L=10
5
Traj. of PDFE-STMAP, L=10
4
Traj. of PDFE-STMAP, L=10
3
Fig. 6 Simulation results using
Proakis C channel with ten
iterations
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 1 2 3 4 5 6 7
B
i
t

e
r
r
o
r

r
a
t
e
SNR per bit (dB)
TEQ + precode, 16-states
Classical TEQ, 16-states
PDFE-STMAP, 8-states
PDFE-STMAP, 16-states
STMAP 64-states
RS-STMAP, 16-states
Outer code in AWGN
SCTC in AWGN
Fig. 7 EXIT charts for various
receivers using Proakis C
channel at E
b
/N
0
= 4 dB
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
CC MAP
Classical TEQ, 16-states
TEQ + precoding, 16-states
STMAP decoder, 64-states
PDFE-STMAP decoder, 16-states
1 3
250 SIViP (2007) 1:239252
Fig. 8 Simulation results for a
minimum phase channel with
ten iterations
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
B
i
t

e
r
r
o
r

r
a
t
e
SNR per bit (dB)
PDFE-STMAP, 8-states
PDFE-STMAP, 16-states
RS-STMAP 16-states
STMAP, 64-states
SCTC in AWGN
scheme 2A [48] requires two interleavers at the transmitter
whereas our approach (Fig. 1) requires only one interleaver.
8 Conclusions and future work
This paper describes turbo equalization of serially concate-
nated turbo codes using a receiver based on the predictive
decision feedback equalizer (PDFE). The key feature of this
receiver is that the feedforward part of the PDFE is outside
the iterative loop and the feedback part is incorporated into
the trellis of the inner code to forma supertrellis. The number
of states in the supertrellis depends on the number of states
in the trellis of the inner code and the number of feedback
taps in the PDFE. This receiver is referred to as the PDFE
MAP receiver employing the supertrellis (PDFE-STMAP).
The optimum receiver, referred to as the STMAP, uses a
supertrellis whose number of states depends on the trellis of
the inner code and the channel memory. Simulation results
with Proakis B, C and a minimum phase channel show that
the performance of the PDFE-STMAP receiver is as good as
the STMAP receiver at a much less complexity.
The BERof various receivers is accurately predicted using
EXIT charts and the simulated performance is found to be
close to the theoretical performance.
It would be interesting to apply the PDFE-STMAP
receiver to time-varying channels. Though we have devel-
oped our theory using the exact MAP algorithm, it is straight-
forward to convert to the log-MAP or max-log-MAP
algorithm. In practical implementations, one would use the
conventional DFE instead of the predictive DFE [59, 60].
Finally, it would be interesting to investigate the performance
of reduced-state PDFE-STMAP receivers which have only a
portion of the feedback part inside the supertrellis.
Acknowledgments The author would like to thank anonymous
reviewers for their helpful suggestions and drawing his attention to
[48].
Appendix
The EXIT chart for the CCMAP (MAP decoder for the outer
code) is plotted as follows. For 0 k N 1 let
y
1, k
= b
k
+w
1, k
(60)
where w
1, k
is N(0,
2
1
). Next we set (for 0 i L 1)

sys, i, m, n
=
_
(2i, +1) if condition H
1
(2i, 1) if condition H
2
.
(61)

par, i, m, n
=
_
(2i +1, +1) if condition H
3
(2i +1, 1) if condition H
4
.
where
(k, c) =
1

2
exp
_

(y
1, k
c)
2
2
2
1
_
(62)
Next, the log-likelihood ratio (LLR) of the extrinsic infor-
mation at the CC MAP decoder output is obtained as (for
0 k N 1):
L
b, k
= ln
_
E(b
k
= +1)
E(b
k
= 1)
_
(63)
where E(b
k
= +1) and E(b
k
= 1) is obtained from (43).
The input mutual information is estimated according to (11)
in [51] for 0 <
2
1
< . For every
2
1
, the mean and vari-
ance of L
b, k
is estimated which is used in the computation
of the output mutual information.
The EXIT chart for the inner MAP decoder is plotted as
follows. Firstly,
i, m, n
is computed according to (31) with
1 3
SIViP (2007) 1:239252 251
z
i, m, n
depending on which receiver is used. However, the a
priori probabilities are set to (for 0 i N 1):
P(c
i, m, n
) =
_

1
(i ), +1
_
if condition H
1

1
(i ), 1
_
if condition H
2
(64)
Finally, the LLR of the extrinsic information at the output of
the inner MAP decoder is computed as (for 0 i N 1):
L
c, i
= ln
_
E(c
i
= +1)
E(c
i
= 1)
_
(65)
where E(c
i
= +1) and E(c
i
= 1) are given in (36). Once
again, the input and output mutual information is plotted as
0 <
2
1
< .
The BER is estimated from the xed point (the point of
intersection between the CC MAP and the other types of
inner decoders) in the EXIT chart as follows [51].
BER =
1
2
erfc
_

D
2

2
_
(66)
where

2
D
=
2
E
+
2
A
+
2
Z
(67)
where
2
E
denotes the variance of L
b, k
corresponding to the
xed point,
2
A
= 0 (since the a priori probabilities to the
CC MAP decoder is xed at 0.5, see also (37) and (40)) and

2
Z
= 4/
2
1
at the xed point.
References
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