IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 2863
Near-Capacity
Irregular-Convolutional-Coding-Aided
Irregular Precoded Linear Dispersion Codes
Nan Wu and Lajos Hanzo, Fellow, IEEE
Abstract—In this paper, we propose novel serial concatenated
irregular convolutional-coded (IRCC) irregular precoded linear
dispersion codes (IR-PLDCs), which are capable of operating near
the multiple-input–multiple output (MIMO) channel’s capacity.
The irregular structure facilitates the proposed system’s near-
capacity operation across a wide range of SNRs, while maintaining
a vanishing bit error ratio (BER). Each coding block of the
proposed scheme and all the iterative decoding parameters are
designed with near-capacity operation in mind, using extrinsic
information transfer charts. By applying the irregular design prin-
ciple to both the inner and outer codes, the proposed IRCC-aided
IR-PLDC scheme becomes capable of operating as close as 0.9 dB
to the MIMO channel’s capacity for SNRs in excess of a certain
threshold.
Index Terms—EXIT-chart matching, EXIT-charts, irregular
channel coding, irregular space-time coding, linear dispersion
coding, near-capacity channel coding, wireless multiple-input–
multiple-output (MIMO) channel capacity.
I. INTRODUCTION
W
IRELESS communication systems using multiple an-
tennas at both the transmitter and receiver, which are
referred to as multiple-input–multiple-output (MIMO) systems
in the literature, have the potential of maintaining reliable
transmissions at high data rates [1]–[3]. The design of coding
schemes for MIMO systems operating at high SNRs involves
a tradeoff between the achievable rate at which the system’s
capacity increases and the rate at which the error probability de-
cays [4]. There is considerable interest in developing schemes
that provide different tradeoffs in terms of the achievable rate
and error probability, which are applicable for employment in a
broad range of antenna configurations.
The set of linear dispersion codes (LDCs), which was
first proposed by Hassibi and Hochwald [5], constitutes a
wide-ranging class of space-time codes exhibiting diverse
characteristics. Hence, this family encompasses numerous
Manuscript received October 23, 2007; revised August 23, 2008, October 21,
2008, and November 30, 2008. First published December 12, 2008; current
version published May 29, 2009. This work was supported in part by the
Engineering and Physical Sciences Research Council, U.K., and in part by the
European Union under the Optimix project nINFSO-ICT-214625. The review
of this paper was coordinated by Prof. E. Bonek.
The authors are with the School of Electronics and Computer Science,
University of Southampton, SO171BJ Southampton, U.K. (e-mail: lh@ecs.
soton.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2008.2010979
existing schemes [6], [7], providing a natural framework in
which such design problems can be posed. The joint design of
the space-time signal of several antennas using sphere-packing
modulation and iterative detection was proposed in [8] and
[9]. The revolutionary concept of LDCs [5], [10] invokes a
matrix-based linear modulation framework, where each space-
time transmission matrix is generated by a linear combination
of so-called “dispersion” matrices used to disperse or map
the symbols to the transmit antennas, where the weights of
the constituent matrices are determined by the transmitted
symbols. The set of dispersion matrices can be optimized
according to different objectives. The dispersion matrices
were originally [5] designed for maximizing the continuous-
input–continuous-output memoryless channel capacity [2] of
the MIMO system. However, the LDCs proposed in [5] did
not necessarily guarantee a low bit error ratio (BER) [10],
[11]. On the other hand, LDCs can also be optimized using the
determinant criterion [12] using the beneficial techniques of the
Golden codes [13], [14], where a nonvanishing determinant is
promised. By contrast, in this paper, we propose a novel method
of optimizing the set of dispersion matrices for the sake of
maximizing the discrete-input–continuous-output memoryless
channel (DCMC) capacity.
Serial concatenated codes (SCCs) are capable of attaining
a vanishing BER, while maintaining a manageable decoding
complexity [15], [16]. It has been demonstrated in [17] and [18]
that SCCs may operate near the MIMO channel’s capacity at
certain SNR values. However, the distance to the capacity is still
quite significant, particularly when higher order modulation
schemes are employed [17]. In [19] and [20], the authors
proposed to adopt irregular convolutional codes (IRCCs) as the
outer channel code of an SCC, since IRCCs exhibit flexible
extrinsic information transfer (EXIT) chart characteristics [20].
Unfortunately, these schemes may fail to approach the capacity
at different SNRs, where the inner and outer codes fail to
create an open EXIT tunnel. Since LDCs have the ability to
approach the MIMO channel’s capacity and to provide flexible
configurations, the novel contribution of this paper is the joint
design of irregular LDCs as the inner code and the IRCCs as
the outer code of an SCC scheme to approach the capacity for
a wide SNR range.
More explicitly, motivated by the aforementioned flexibility
of the irregular outer code design philosophy, in this paper,
we circumvent the IRCC-related outer code limitations by
proposing irregular precoded LDCs (IR-PLDCs) as the in-
ner code of the SCCs and serially concatenate the resultant
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2864 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009
Fig. 1. Schema of the IRCC-coded IR-PLDC using iterative decoding.
IR-PLDCs with the outer IRCCs to operate close to the MIMO
channel’s capacity across a wide SNR region, while main-
taining a vanishing BER. The rationale of using a unity-rate
precoder in an SCC scheme [21] is that it allows us to create
an infinite impulse response system at the cost of a low im-
plementation complexity, which has the benefit of improving
the extrinsic information exchange among the component codes
at the receiver. The novelty of the proposed IRCC-coded IR-
PLDC scheme is described in the following.
• It is capable of operating close to the MIMO channel’s
capacity across a wide range of SNRs.
• The irregular design principle applied is to both the inner
code using IR-PLDCs and the outer employing IRCCs.
• The constituent LDCs of the inner IR-PLDC scheme are
optimized by maximizing the DCMC capacity.
• The maximum achievable rate of the IRCC-aided IR-
PLDC scheme is determined with the aid of EXIT
charts [22], [23] when using minimum-mean-square-error
(MMSE) detectors.
• The proposed scheme can be designed for an arbitrary
number of transmit and receive antennas combined with
arbitrary modulation schemes.
• IR-PLDCs may be reconfigured by activating different
dispersion matrices for different IRCC component codes.
We commence our discourse by providing a detailed descrip-
tion of the proposed IRCC-coded IR-PLDC scheme’s structure
in Section II. In Section III, we demonstrate how the proposed
system and all the iterative decoding parameters are optimized
from a capacity maximization perspective. Our simulation re-
sults are discussed in Section IV. Finally, we conclude our
discourse in Section V.
II. S
YSTEM DESCRIPTION
We consider a MIMO system employing M transmit and
N receive antennas. Fig. 1 portrays the system model of the
proposed serially concatenated IRCC-coded IR-PLDC scheme.
A total of P = P
out
+ P
in
irregular components are employed
in the proposed system, where the outer IRCC scheme contains
P
out
, while the inner IR-PLDC scheme has P
in
number of
component codes.
At the transmitter in Fig. 1, a frame of information bits u
1
is
encoded by an IRCC encoder. The particular fraction of the bit-
stream u
1
, which will be fed into each IRCC component code,
is controlled by the “irregular partitioner” in Fig. 1 based on the
weighting coefficient vector γ =[γ
1
,...,γ
P
out
]. The aggregate
rate of the outer IRCC scheme is denoted as R
out
, and the
rate of its component codes is R
i,IRCC
, where i =1,...,P
out
.
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WU AND HANZO: NEAR-CAPACITY IRCC-AIDED IRREGULAR PRECODED LINEAR DISPERSION CODES 2865
Then, the encoded bits c
1
are interleaved by a random bit
interleaver, yielding the outer encoded bits u
2
.Again,theIR-
PLDC’s “irregular partitioner” feeds the appropriately selected
fraction of u
2
into the various precoded LDC (PLDC) compo-
nent codes, governed by the weighting coefficient vector λ =
[λ
1
,...,λ
P
in
]. Memory-1 unity-rate precoders were employed
for all the PLDC components. We refer the readers to [21] for
the encoding/decoding structure, as well as the calculation of
the extrinsic information of the unity-rate precoders. Within
each PLDC encoder, the resultant precoded bits c
2
are inter-
leaved by a second interleaver, yielding the interleaved bits u
3
,
which are fed into the bit-to-symbol mapper of the LDC block
in Fig. 1. After modulation, the information-bearing vector
K =[s
1
,...,s
Q
]
T
containing QL-PSK symbols is dispersed
according to the space-time block matrix S of (1) by the
“ST mapper,” spanning M spatial and T temporal dimensions.
More explicitly, S is given by
S =
Q
q =1
A
q
s
q
,q=1,...,Q (1)
where A
q
having a size of (M × T) elements represents the
qth dispersion matrix. Hence, S is transmitted over the uncor-
related Rayleigh fading channel contaminated by AWGN at
each receive antenna. Since unity-rate precoders are employed,
aPLDC(MNTQ) scheme’s rate is the same as that of an
LDC(MNTQ) scheme, which is defined as R
LDC
= Q/T
(sym/slot).
The weighting coefficient vectors λ and γ areassumedtobe
known at the receiver. The received signal matrix Y is related
to S by
Y =
ρ
M
HS + V (2)
where the channel impulse response (CIR) matrix H models an
independent identically distributed flat Rayleigh fading chan-
nel. The channel’s envelope and phase are assumed to be con-
stant during T symbol periods, but they are faded independently
from one space-time matrix to the next. The CIR matrix H is
assumed to be known to the receiver but not to the transmitter.
The noise matrix V is assumed to have independent samples
of a zero-mean unit-variance complex-valued Gaussian random
process, and ρ denotes the SNR.
It is desirable to rewrite the input–output matrix relationship
of (2) in an equivalent vectorial form. Define the vec() oper-
ation as the vertical stacking of the columns of an arbitrary
matrix. Subjecting both sides of (2) to the vec() operation gives
the equivalent system matrix [10]
¯
Y =
ρ
M
¯
HχK +
¯
V (3)
where
¯
Y ∈ ζ
NT×1
denotes the equivalent matrix of complex-
valued received signals, and
¯
V ∈ ζ
NT×1
represents the cor-
responding complex-valued noise matrix. More explicitly,
χ having a size of (MT × Q) is referred to as the dispersion
character matrix (DCM), which is given by
χ =[vec(A
1
),vec(A
2
),...,vec(A
Q
)] (4)
while
¯
H in(3)isgivenby
¯
H = I
T
⊗ H (5)
where ⊗ denotes the Kronecker product, and I
T
is the identity
matrix having a size of (T × T ). Note that the DCM χ uniquely
and unambiguously determines a particular LDC(MNTQ).
Again, the IR-PLDC decoder’s “irregular partitioner” in
Fig. 1 determines the specific portion of the received signal, as
well as the aprioriinformation I
A
(u
2
) to be detected by each
PLDC component decoder, according to the weighting coeffi-
cient vector λ. Similarly, the IRCC decoder’s “irregular parti-
tioner” operates according to the weighting coefficient vector
¯
γ =[¯γ
1
,...,¯γ
P
out
]. Note that the IRCC encoder’s weighting
coefficient vector γ quantifies the specific fraction of input bits
encoded by each component at the transmitter side, whereas
¯
γ determines the fraction of the log-likelihood ratios (LLRs)
fed into each individual decoder at the IRCC’s decoder, and the
components of γ and
¯
γ are related by
γ
i
=
¯γ
i
× R
i,IRCC
R
out
,i=1,...,P
out
. (6)
Then, an iterative decoding structure is employed, where
extrinsic information is exchanged between the three soft-in–
soft-out modules, namely, the MMSE detector [24], the pre-
coder, and the outer IRCC decoder, in a number of consecutive
iterations. To be specific, in Fig. 1, I
A
() denotes the apriori
information represented in terms of LLRs, while I
E
() denotes
the extrinsic information also expressed in terms of LLRs. Note
that the intermediate rate-1 precoder processes two aprioriin-
puts, namely, those arriving from the MMSE detector and those
from the outer decoder, and generates two extrinsic outputs.
More detailed discussions on the iterative decoding process are
provided in [15]. It is worth mentioning that the activation of
different IR-PLDC components is implemented by employing
different dispersion matrices, and the associated hardware cost
is modest, since it does not require the implementation of P
separate component codes at the transmitter and receiver. By
contrast, the component codes of the IRCC scheme are obtained
by puncturing of and/or adding extra generator polynomials to
a selected mother RSC code; hence, the associated complexity
may become potentially high.
III. E
XIT-CHART-BASED DESIGN OF
IRCC-CODED IR-PLDCS
In our forthcoming EXIT chart analysis, the unity-rate pre-
coder’s decoder and the MMSE decoder are considered as a
single “inner” decoding block, constituted by the IR-PLDC’s
decoder seen in Fig. 1. The advantage of this representation is
that the IR-PLDC block’s extrinsic information output I
E
(u
2
)
is entirely determined by the received signal matrix Y and
the aprioriinput I
A
(u
2
); hence, it remains unaffected by the
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2866 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009
extrinsic information exchange between the precoder’s decoder
and the MMSE detector. Thus, we can project the three-stage
system into a two-stage system, and hence, the traditional 2-D
EXIT charts [23], [25] are applicable.
Following the approach of [26], we now carry out the EXIT
chart analysis of the proposed IR-PLDC scheme. Thus, the
corresponding EXIT transfer function is
I
E
(u
2
)=Γ
in
[I
A
(u
2
),ρ] . (7)
The employment of irregular codes was proposed by Tüchler
and Hagenauer [19], [20], who used IRCCs as an outer chan-
nel code. In [19], the authors have shown that the aggregate
EXIT function of an irregular code can be obtained from the
linear combination of that of its component codes, under the
assumption that the probability density function of the LLRs is
symmetric and continuous. More explicitly, the EXIT function
of the proposed inner IR-PLDC scheme is given by
Γ
in
=
P
in
i=1
λ
i
Γ
i
(I
in
,ρ) (8)
where Γ
i
denotes the EXIT function of the ith PLDC compo-
nent. Similarly, the EXIT function of the outer IRCC scheme
becomes
Γ
out
=
P
out
i=1
¯γ
i
¯
Γ
i
(I
in
) (9)
where
¯
Γ
i
denotes the EXIT function of the ith IRCC compo-
nent, and it is independent of the operating SNR ρ.
In the following sections, we will characterize each block
of the IRCC-coded IR-PLDC scheme in Fig. 1 using various
parameters, which are optimized from a capacity maximization
perspective with the aid of EXIT charts.
A. Generating Component Codes for the IR-PLDC
In this section, we demonstrate how to generate an in-
ner IR-PLDC coding scheme containing P
in
=11 compo-
nents for a MIMO configuration having M =2transmit and
N =2 receive antennas when quadrature phase shift keying
(QPSK) modulation was employed. It has been shown in [10]
that the maximum achievable spatial diversity order of an
LDC(MNTQ) is N · min(M, T). We commence by setting T
equal to M for the first component. Hence, the resultant scheme
has the potential of achieving the maximum diversity order of
D =4. By setting Q =1, we create an LDC(2221) scheme that
is suitable for this parameter combination. Hence, we are able
to search for the specific DCM χ that maximizes the DCMC
capacity of the LDC(2221) scheme using [27]
C
ML
LDC
=
1
T
max
p(K
1
),...,p(K
F
)
F
f=1
∞
−∞
···
∞
−∞
p
¯
Y|K
f
p(K
f
)
· log
2
p
¯
Y|K
f
F
g =1
p
¯
Y|K
g
p(K
g
)
d
¯
Y bits/sym/Hz (10)
TABLE I
P
in
=11COMPONENT CODES OF THE IR-PLDC SCHEME IN FIG.1
G
ENERATED FOR A MIMO SYSTEM HAV IN G M = 2 AND N = 2
R
ECEIVE ANTENNAS AND EMPLOYING QPSK MODULATION
where K
f
denotes a candidate vector. Since each transmitted
symbol vector contains Q symbols of an L-PSK constellation,
we have a total of F = L
Q
number of possibilities. The normal-
ization factor 1/T corresponds to the fact that an LDC scheme
spans T channel uses.
Consequently, we can obtain more components by gradually
increasing the value of Q to increase the rate. We impose the
limit of Q ≤ MT for the sake of maintaining a low complexity,
although employing a higher value of Q is feasible. Hence, by
increasing the value of T and maximizing the corresponding
DCMC capacity of each LDC(MNTQ), we can generate a set
of beneficial LDCs. Naturally, low Q and T values are desirable
for the sake of maintaining a low complexity. The resultant
P
in
=11component codes designed for our IR-PLDC scheme
are listed in Table I. Hence, an inner PLDC(MNTQ) scheme
can be directly obtained by combining a memory-1 unity-rate
precoder [21] with an LDC having the parameter (MNTQ).
The complexity of each PLDC component code is jointly
determined by the precoder’s memory, the MMSE detector’s
complexity, and the number of inner iterations j. To quantify
the complexity in a unified manner, we count the number of
addition and multiplication operations required to calculate a
single LLR value in the logarithmic domain. Since the number
of addition and multiplication operations can be quantified in
terms of the so-called add–compare–select (ACS) arithmetic
operations, the complexity of each PLDC component is quan-
tified by the ACS operations per LLR computation. Observe
in Table I that when the value of T is fixed, the complexity is
increased by increasing the value of Q. Furthermore, increasing
the value of T typically resulted in substantially increased
complexity.
To elaborate further on the LDC’s ability to support arbi-
trary (MNTQ) parameter combinations, as well as support
arbitrary modulation schemes, in Table II, we characterized
another group of LDCs designed for a MIMO system having
M =3 and N =2 antennas and using binary phase shift
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WU AND HANZO: NEAR-CAPACITY IRCC-AIDED IRREGULAR PRECODED LINEAR DISPERSION CODES 2867
TABLE II
P
in
=6COMPONENT CODES OF THE IR-PLDC SCHEME IN FIG.1
G
ENERATED FOR A MIMO SYSTEM HAV IN G M = 3 AND N = 2
A
NTENNAS AND EMPLOYING BPSK MODULATION
keying (BPSK) modulation. Again, we set T =3to achieve the
maximum possible diversity gain at a low decoding complexity.
It is certainly feasible to choose higher T values to increase the
integrity at the price of higher complexity.
B. Number of Inner Iterations j
The number of “inner” iterations between the MMSE de-
tector and the precoder required for each PLDC component in
Table I is also optimized for the sake of achieving the maximum
capacity. In other words, we propose to answer the question of
how many inner iterations (j) per outer iteration are necessary
for approaching the capacity. Ideally, a small value of j is
desirable in the interest of minimizing the decoding complexity.
The so-called area property [25], [28] of EXIT charts may
be formulated by stating that the area under the “outer” IRCC
curve is approximately equal to its code rate R
out
.Thus,ifwe
assume that the area under the EXIT curve of an outer code
can be perfectly matched to the area under the inner code’s
EXIT curve at any SNR, then it is possible to approximate the
maximum achievable rate of a serial concatenated scheme by
evaluating the area under the EXIT curves, given the rate of the
“inner” IR-PLDC code R
in
, which is expressed as
C(ρ) = log
2
(L) · R
in
· R
out
bits/sym/Hz (11)
when L-PSK modulation is used.
Fig. 2 quantifies the maximum achievable rates for the dif-
ferent PLDC component codes in Table I employing different
number of inner iterations j. For each set of comparisons,
the DCMC capacity of the LDC using (10) is plotted as a
benchmark. For the rate R
3,2224
=2PLDC(2224) scheme, we
observe a clear gap between the DCMC capacity and the corre-
sponding maximum rate, when the number of inner iterations is
j =0. However, when we have j =1, the aforementioned rate
loss is eliminated, and a further increase of the number of inner
iterations j has only a modest additional rate improvement. In
fact, the maximum achievable rate loss is less than 1%, when
we have j =1. For the PLDC(2222) scheme in Table I having
arateofR
1,2222
=1, we observe in Fig. 2 that although the
aforementioned maximum achievable rate loss compared with
the DCMC capacity is still present, when employing j =0
inner iterations, the associated discrepancy is narrower than that
seen for the PLDC(2224) scheme. Observe in Fig. 2 for the
Fig. 2. Comparison of the maximum achievable rates for three PLDC schemes
in Table I having j =0, 1, and 2 inner iterations when using QPSK modulation
in conjunction with an MMSE detector.
PLDC(2241) scheme having a rate of R
8,2241
=0.25 that there
is no maximum achievable rate loss even in the absence of inner
iterations.
The above observations are related to the EXIT characteris-
tics of the LDC MMSE decoding block. When a single symbol
is transmitted, the resultant EXIT curve is a horizontal line,
which is a property of the Gray labeling employed [16]. There-
fore, regardless of the number of inner iterations employed, the
MMSE detector in Fig. 1 always outputs the same extrinsic
information. When Q is increased, the resultant EXIT curve
becomes more steep; therefore, higher extrinsic information can
be obtained upon increasing the aprioriinformation by using
a higher number of inner iterations. Therefore, the resultant
maximum achievable rate observed in Fig. 2 has an increasing
discrepancy with respect to the DCMC capacity, when a higher
number of symbols Q is transmitted by each LDC block. We
observe that for the component PLDCs in Table I, where we
have Q>1, employing j =1 inner iteration will enable the
system to attain 99% of the DCMC capacity. Similarly, we are
capable of determining the number of necessary inner iterations
required for the group of LDCs listed in Table II designed
for a MIMO system having M =3antennas. Again, observe
in Table II that employing a single inner iteration is adequate
for the LDC schemes having Q>1 to approach the associated
DCMC capacity.
C. Optimizing the Weighing Coefficient Vector
Apart from the specific shape of the component EXIT curves,
the aggregate inner IR-PLDC and outer IRCC schemes’ EXIT
curves characterized in (8) and (9) are also affected by the
weighting coefficient vectors λ and
¯
γ. Since λ quantifies the
specific fraction of information bits fed into the IR-PLDC
encoder/decoder in Fig. 1, λ =[λ
1
,...,λ
P
in
] has to satisfy
1=
P
in
i=1
λ
i
,λ
i
∈ [0, 1]. (12)
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