Int. J. Communications, Network and System Sciences, 2010, 3, 256-265
doi:10.4236/ijcns.2010.33033 blished Online March 2010 (http://www.SciRP.org/journal/ijcns/).
Pu
On Cross-Layer Design of AMC Based on Rate Compatible
Punctured Turbo Codes
Fotis Foukalas, Evangelos Zervas
Departmen t of Inform a tics and Tel e communication s , Universit y of Ath ens, Athens, Greec e
Received December 24, 2009; revised January 27, 2010; accepted Febru a r y 28, 2010
Abstract
This paper extends the work on cross-layer design which combines adaptive modulation and coding at the
physical layer and hybrid automatic repeat request protocol at the data link layer. By contrast with previous
works on this topic, the present development and the performance analysis as well, is based on rate compati-
ble punctured turbo codes. Rate compatibility provides incremental redundancy in transmission of parity bits
for error correction at the data link layer. Turbo coding and iterative decoding gives lower packet error rate
values in low signal-to-noise ratio regions of the adaptive modulation and coding (AMC) schemes. Thus, the
applied cross-layer design results in AMC schemes can achieve better spectral efficiency than convolutional
one while it retains the QoS requirements at the application layer. Numerical results in terms of spectral effi-
ciency for both turbo and convolutional rate compatible punctured codes are presented. For a more compre-
hensive presentation, the performance of rate compatible LDPC is contrasted with turbo case as well as the
performance complexity is discussed for each of the above codes.
Keywords: Cross-Layer Design, Adaptive Modulation and Coding, Rate-Compatible Punctured Turbo Codes,
Hybrid ARQ, Codes Complexity
1. Introduction
The success of current standard such as 3GPP HSPA and
IEEE 802.11/16 in terms of high data rates prov ision and
quality of service (QoS) requirements satisfaction is prin-
cipally owed to Adaptive Modulation and Coding (AMC),
hybrid automatic repeat-request (HARQ) and fast sched-
uling [1,2]. The AMC realization uses different constel-
lation orders and coding rates according to the signal
strength [3]. By this way, when instantaneous channel
conditions are proper, link adaptation offers high data
rates at the physical layer. Th e proper usage of each con-
stellation order and coding rate, i.e., mode is specified by
the SNR regions in which each separate mode is active.
Enhancement of AMC performance can be achieved
by using different channel coding techniques. Particu-
larly, in case of turbo-coding implementation, an AMC
scheme can achieve the highest spectral efficiency even
if low SNR regions are met [4]. The original rate of a
turbo code could be 1/3; nevertheless by using punctur-
ing techniques greater code rates can be used for each
modulated symbol. Incorporating also rate compatibility
in punctured turbo codes, by which all of the code sym-
bols of a high rate punctured code are used by the lower
rate codes, an enhanced spectral efficiency is reached.
This gain is actually provided b y Rate-Compatible Punc-
tured Turbo (RCPT) codes [5]. RCPT codes have been
employed for HARQ implementations due to the fact
that no received information is discarded [6]. Such ARQ
schemes are well-known as Incremental Redundancy (IR)
HARQ schemes that improves the channel use efficiency
since parity bits for error correction are transmitted only
if this is required [7].
The aforementioned description is basically a cross-
layer combination of AMC at the physical layer and HARQ
at the data link layer for QoS provisioning in wireless
communication networks [8,9]. It has been shown that
such a cross-layer design outperforms in terms of spec-
tral efficiency, the case of AMC use only at the physical
layer or the combination of typical ARQ with a single
modulation and coding scheme [8]. Moreover, it has
been proved that IR HARQ based on convolutional
codes has much improvement in spectral efficiency than
that with type-I HARQ [9]. To this direction and since a
lot of research work has been done on turbo codes as
well as turbo coding and decoding is applied to all

F. FOUKALAS ET AL. 257
known standards of wireless communications [2,10], we
extend this study by employing the aforementioned cross-
layer design (CLD) that combines AMC and HARQ
based on RCPT codes.
The rest of this paper is organized as follows. Section
2 presents the system model and its components in de-
tails. In Section 3, the cross-layer design of the system is
presented with its assumptions and constraints. In Sec-
tion 4, system performance is evaluated for both turbo
and convolutonal rate compatible codes and LDPC as
well. Besides, the complexity performance is evaluated
for each coded system. Finally, Section 5 provides the
concluding remarks and gives some directions for further
investigation in this topic.
2. System Model
The model of the adopted cross-layer design system is
illustrated in Figure 1. It shows the layer structure of the
system as well as the implementation details of the AMC
scheme (i.e. physical layer). In the following text, we
first describe concisely the functionality of each layer
and in sequel we go into details for each of layers’ com-
ponents.
2.1. Turbo Encoding and Decoding
First, confirm that you have the corr ect template for your
paper size. This template has been tailored for output on
the A4 paper size. If you are using US letter-sized paper,
please close this file and download the file for “MSW
US ltr format”. Turbo coding and decoding achieves per-
formances on error probability near to Shannon limit
[11]. In its main form, turbo coding is a channel coding
type that combines two simple convolutional codes in
parallel linked by an interleaver (i.e. Parallel Concate-
nated Convolutional Codes-PCCC) [12]. It had been
studied that recursive systematic convolutional codes
(RSC) are superior to nonrecursive counterparts for con-
catenated implementations [13]. The codewords of such
schemes consist of one information bit followed by two
parity check bits which both parallel encoders produce.
Thus, the rate code of a PCCC scheme with two RSC
constituent codes is 1/3
c
R

.
On the other side, the decoding process of concate-
nated codes is performed by a suboptimum decoding
scheme that uses a posteriori probability (APP) algo-
rithms instead of using Viterbi algorithms. Such a
scheme is constructed by “soft-in/soft-out” decoders that
exchange bit-by-bit or symbol-by-symbol APPs as soft
information that depends on the bit or symbol decoding
technique [14]. The input soft-information represents the
log-likelihoods of encoder input bits and code bits. This
is actually the input of the So ft-Input Soft-Outpu t (SISO)
“Maximum A Posteriori” (MAP) module presented in
[15]. The output soft-information of this module is up-
dated versions of input based on the information of the
constituent RSC of the turbo encoder.
More specific, turbo decoding based on a PCCC sche-
me is constructed by two SISO modules that linked with
Figure 1. Cross-layer system model combining AMC and HARQ based on RCPT codes.
C opyright © 2010 SciRes. IJCNS

F. FOUKALAS ET AL. 258
a deinterleaver (Figure 2). In addition to that, iterative
decoding is accomplished in order to improve the de-
coding performance. Henceforth, a feedback loop be-
tween the two constituent SISO decoders is established
that actually presents the turbo decoding principle [11].
This feedback loop appears an interleaver that gives in-
terleaved inputs to the first parallel decoder required for
the first iteration and so on. Multiple iterations between
these two decoders exchanging soft information give
near to Shannon limit results. Turbo codes and iterative
turbo decoders has been extensively studied for imple-
mentation purposes in current standards like 3GPP
HSPDA (High Speed Data packet Access) [16].
2.2. RCPT Codes
In general, a RCPT encoder consists of a turbo encoder
as described above followed by a puncturing block with
puncturing matrix P. The puncturing matrix P is known
as the puncturing rule or pattern and indicates the coded
bits that should be punctured [17]. Puncturing can be
applied both to information or/and parity bits. However,
the way of puncturing affects the coding scheme per-
formance and the coded-modulation scheme in general
[18]. Assuming only the impact of puncturing on turbo
coding scheme, one can realize that without puncturing
systematic bits, the code performance decrease is rea-
sonable. In addition to that, by puncturing periodically
the parity bits produced by two RSC codes, a better per-
formance of the coding scheme can be achieved.
The rate compatibility offered by a RCPT code has
been considered as the enabling technique for incre-
mental redundancy (IR) HARQ schemes [6]. IR HARQ
based protocols are major components of HSDPA offer-
ing rate matching capabilities [19]. During the rate
matching process, the transmitter sends only supplemen-
tal coded bits indicated by the aforementioned punctur-
ing rule. A representative example of IR HARQ scheme
for HSDPA with turbo encoder as mother code is pre-
sented in [20]. The RCPT encoder in particular is con-
structed by a turbo mother code with a rate code
resulted by 1/RM1
M

RSC encoders. The
puncturing matrix indicates the puncturing period
and actually the bits being punctured during the HARQ
scheme operation [6,18]. Therefore, the resultant family
of rate codes is:
P
,
P
RP

 0,1,...,( 1)
M
P (1)
An example of RCPT encoder dedicated to ARQ
mechanisms with M = 3 and P = 3 is illustrated in Figure 3
which is constructed from two constituent RSC encoders
with rate 1/2 and offers a family of RCPT code rates
12
...
M
Rc RcRc with the following decreasing
order {(1), (3/4), (3/5), (1/2), (3/7), (3/8), (1/3)}.
Figure 2. PCCC (i.e. turbo) decoding with SISO modules.
Figure 3. RCPT-ARQ encoder.
2.3. RCPT-ARQ Protocol
By puncturing the bits that will be transmitted in the
current and future transmission attempts, the HARQ
scheme (i.e. RCPT-ARQ) brakes the packet unit with
size into blocks of bits with size . The number of
transmitted bits of the RCPT-ARQ protocol at the
transmission attempt can be expressed as
Li L
i
L
ith [10]
1
1
1
()
11
()
i
ii
LRc
LLRcRc 
















(2)
1,
1
i
iC


where i with Rc 1,..,iC

denotes the rates pro-
duced by the RCPT encoder. Going into further details,
we assume a single stop-and-wait ARQ strategy of
RCPT-ARQ protocol (i.e. hybrid ARQ) described by the
following step-by-step functionality:
C
 Depending on previous channel cond ition the adap-
tive scheme operates on mode n.
 The L-long packet size is encoded by the turbo
mother code. The coded packet is stored at the transmit-
ter and is broken into blocks with size of {1 /}
iic
LLR

bits with 1, .., C


. Bits selection is performed for each
transmission attempt according to the puncturing rule.
 Constituent blocks’ transmission with size i L is
initialized according to the puncturing matrix.
 At the receiver, iterative decoding is performed for
each separate transmission block i L. If decoding is not

F. FOUKALAS ET AL. 259
successful after the number of maximum transmissions
max
t
N is reached, then a NAK is sent to the transmitter
and the adaptive scheme updates to the corresponding
mode according to the channel condition.
 Otherwise, an ACK is sent to the transmitter and
the adaptive scheme continues to the current mode n.
3. Cross-layer Design
The cross-layer system structure described above is re-
lied on the following assumptions:
 Channel SNR estimation is perfect and in conse-
quence the channel state information (CSI) that is avail-
able at the receiver as well, although the impact of errors
in SNR estimation on adaptive modulation is negligible
[21]. In our implementation, the channel estimator is
implemented using the Error Vector Magnitude (EVM)
algorithm for AWGN channel [22].
 Feedback channel dedicated to mode selection
process is error free and without latency. The mode se-
lection is performed in a packet-by-packet basis i.e. the
AMC scheme is updated after max
t
N transmission at-
tempts. Alternative update policies with e.g. updates for
every transmission attempt (i.e. block-by-block basis),
will be left for further investigation [10].
 System updates are based on received SNR denoted
as

that is actually the estimated channel SNR at the
receiver. It is assumed that the received SNR

values
per packet is described statistically as i.i.d random vari-
ables with a Rayleigh probability density function (pdf):
1
() exp p










(3)
where {}E

  is the avera ge received SNR.
3.1. Cross-Layer Design of AMC and HARQ
A cross-layer design approach that combines the AMC at
the physical layer with a hybrid ARQ at the data link
layer could follow the procedure presented both in [9]
and [10]. Applying this method, the following constraints
must be imposed in order to keep a particular QoS level
at the application layer:
Constraint1 (C1): The maximum allowable number of
transmissions per inform a tion packet is .
max
t
N
Constraint2 (C2): The probability of unsuccessful re-
ception after transmissions is no greater than
.
max
t
N
Prloss
C1 is calculated by dividing the maximum allowable
delay at the application layer and the round trip time re-
quired for each transmission at the physical layer. For
example, assuming the QoS concept of 3GPP, the audio
and video media streams for MPEG-4 video payload
allows a maximum delay value equal to 400 ms [23]. In
addition to that the round trip delay between the terminal
and the Node B for retransmissions in case of HSDPA
could be approximated less than 100 ms [23]. Thereafter,
in such a context, the should be 4. On the other
hand, C2 is related to the bit-error rate (BER) at the
physical layer and the packet size at the data link layer.
Hence, if the BER imposed by the QoS requirements at
the application layer is equal to and the informa-
tion packet size is L = 1000 then the should be
max
t
N
6
10
Prloss
3
10

[23].
It is obvious that the aforementioned cross-layer de-
sign (CLD) dictates the code rates that will be used for
each transmission at the data link layer and therefore
specifies the AMC switching thresholds at the physical
layer. Moreover, the proposed CLD scheme will be af-
fected by constituent encoders (i.e. RSC encoders) of
turbo code as well the puncturing rules [6,18]. However,
in current investigation, we present the results derived
using one of the optimal RCPT code and puncturing rule
presented in [6], and we will present the RCPT codes and
puncturing impact on our CLD in our future work.
3.2. AMC Schemes
The design of AMC schemes is the process by which the
switching thresholds are specified. The switching thresh-
olds of an AMC scheme at the physical layer are speci-
fied by a given target BER () [3,4]. The switch-
ing thresholds are boundary points of the total SNR
range denoted as

argtet
BER
1
0
N
nn



specifying nonoverlapping
consecutive intervals1 [, )
nn



. Afterwards, each
mode is selected in accordance to the switching thresh-
olds derived from the.
argett
BER
However, in a combined system in which the unit of
interest is the packet at the data link layer, the AMC de-
sign follows the value. More specific, in order
to satisfy the aforementioned constraints of the proposed
combination, the switching thresholds should be derived
from the following inequality:
argtet
PER
arg
PrPr:Pr
t
Nlosstet
 (4)
where is the packet error probability (i.e. packet
error rate) after transmissions at the data link layer.
In the following paragraphs, we derive the boundary
points
Pr t
N

N
nn

t
N
1
0


for each modulation and coding scheme
(MCS).
The packet error probability can be expressed in rela-
C opyright © 2010 SciRes. IJCNS

F. FOUKALAS ET AL. 260
tion to BER by the following equation
Pr1 (1)
L
BER  (5)
only if each demodulated and decoded bit inside the
packet has the same BER and bit-errors are uncorrelated
[9,10]. On the other hand, known closed form expres-
sions for the PER1 and BER is not available in the litera-
ture and closed-form expressions for the BER of turbo-
coded modulations in AWGN channel is not available
either [8]. All the same, one can use the union bound for
turbo-coded modulation system using the bounding tech-
nique introduced in [24]. However, this technique is ap-
plied for 16QAM system and indeed needs more inves-
tigation in case of turbo-coded AMC schemes with mul-
tiple modulation modes. Thereafter and since further
investigation on union bounds of turbo-coded modula-
tion is not the aim of our current work, we take BER and
PER values through simulations. Finally, the simulated
PER values are compared with those derived from fitting
the curves and those derived from Equation (5).
Figure 4 shows the PER values versus received SNR
of each mode in coding step with, where
i
Rc 1, 2, 3i

number of transmissions. We use the 1/ 2 QPS K, 3/ 4 QP SK,
1/2 16QAM and 3/4 16QAM modulations with RCPT
code rates 1, 1/2 and 1/3 for each transmission respec-
tively. The packet size is length and the
puncturing follows the optimal rules according to [6]
(Table 1). The constitu ent RSC encoders of PCCC turbo
codec is the optimal encoder B proposed by [6] with rate
1/2, memory
1536L
4

 (i.e. 16 states) and generator matrix
(1, /
ba
g
g), where the generator polynomial a
g
and
b
g
have octal representations and
respectively. The number of iterations is 8. The figures
depict the simulated PER, the fitting curves and the val-
ues derived from Equation (5).
(15)octal (13)octal
In order to have a more clear view on RCPT perform-
ance combining with AMC, we should compare it with
the other types of rate compatible codes. To this end, we
implement also the aforementioned CLD first using
RCPC (Rate Compatible Convolutional Code) and sec-
ond using RC-LDPC (Rate Compatible Low-density
Parity-check codes). We use the same rates for both two
RC codes. Specifically, the RCPC is a convolutional
encoder with rate 1/2, generator polynomial (171, 133)
Table 1. The block size and the puncturing matrix (in oc-
toctal) of applied rcpt codes.
Family of Code Rates
Block size 1 2/3 1/2
L = 1536 17
00
00
17
01
01
17
05
05
and constraint length 7 [9]. For LDPC, we employ the
same codes as in [25] with rate 1/2 (1008,504) and a
variable node degree equal to 3. The corresponding per-
formance of these modulation and coding schemes (MCS)
is depicted separately for each code in Figure 4.
-2 0 2 46 81012
10
-2
10
-1
10
0
SNR(dB)
PER
RCPT
MCS1
MCS2
MCS3
MCS4
-4 -20 2 4 6810 121416
10
-4
10
-3
10
-2
10
-1
10
0
SNR(dB)
PER
RCP C
MCS1
MCS2
MCS3
MCS4
-2 0246810 1214
10
-4
10
-3
10
-2
10
-1
10
0
SNR(d B )
PER
RC-LDP C
MCS1
MCS2
MCS3
MCS4
Figure 4. PER simulation performance of 4 AMC modes
using RCPT, RCPC and RC-LDPC.
1For the rest of this document the packet error probability Pr will be
replaced with PER notation.

F. FOUKALAS ET AL.
261
tra af
i n
4. Performance Analysis and Numerical
Results
4.1. System Performance
In case of a general type-II HARQ that uses punctured
codes, the probability of unsuccessful reception after t
N
nsmissions represents the event of decoding failure
with code i Rc ter i transmissions [10]. In case of
limited transmissions, the packet error probability of this
using AMC mode nN under channel states
is given by [2 6]
1,...,
()
n

}
(1)
{ ,..,t
N i
nn
  (2) ,.
n

1
()( )
t
Ni
n
i
PER PER



 (6)
By using (6) over for each retransmission and
for each mode the packe t error ra tes after
transmissions are resulted. The
i
PER
...,1,nNi
i
n

denotes the region
boundaries for each MCS and obtained as follows
()
1
()
arg
()
1
0,
1ln(),2,..., ,
i
in
ntet
ni
i
an
gPER







N (7)
The is reached using the corresponding
decoder when the imposed transmission at-
tempts is reached either. Assuming
argtet
i
PER
i
Rc t N
max 3
t
N

and
, the derived switching thresholds are
listed in Table 2. Tab le 2 includes also the parameters of
MCSs for convolutional and LDPC codes.
2
10 
loss
PER
We next evaluate the system performance in terms of
spectral efficiency when the AMC scheme is combined
with type II HARQ (i.e. IR HARQ). In each transmi-
ssion attempt, the number of transmitted bits is speci-
fied according to RCP T code rates
i
i
L
...
12
M
Rc RcRc
i
Rn
as
mentioned in Section 2. In addition to that, when mode
is used, each transmitted symbol carry n

information bits where
2
log ()
i
Mn Rc n
M
derived
from
M
ary
1/
modulation scheme. As in [9], we as-
sume a Nyquist pulse shaping filter with bandwidth
s
BT, where
s
T
S
is the symbol rate. Afterwards, the
spectral efficiency gives the bit rate in bits per
symbol that can be transmitted per unit bandwidth and is
given by
e
eL
SL  (8)
In (7), where is the input information packet size
and
L
L is the average of transmitted symbols in order to
Table 2. the parameters of each rc-coded modulation at the
physical layer.
RCPT MCS1 MCS2 MCS3 MCS4
Modulation QPSK QPSK 16-QAM 16-QAM
Coding rate Rc 1/2 3/4 1/2 3/4
Rate
(bits/symbol) 1.0 1.5 2.0 3.0
an 4.4111 4.0152 3.9111 3.7011
gn 5.4355 3.3311 2.7151 1.9461
(dB) (1)
n

3.0103 5.9989 9.9663 11.9224
(dB) (2)
n

-1.92301.7144 4.9422 7.11145
(dB) (3)
n

-3.0312-0.3161 3.1221 5.3121
RCPC MCS1 MCS2 MCS3 MCS4
Modulation QPSK QPSK 16-QAM 16-QAM
Coding rate Rc 1/2 3/4 1/2 3/4
Rate
(bits/symbol) 1.0 1.5 2.0 3.0
an 0.3351 0.2197 0.2081 0.1936
gn 3.2543 1.5244 0.6250 0.3484
(dB) (1)
n

4.3617 7.4442 11.2882 13.7883
(dB) (2)
n

0.1150 2.8227 6.6132 9.0399
(dB) (3)
n

-1.65320.7272 4.4662 6.8206
RC-LDPC MCS1 MCS2 MCS3 MCS4
Modulation QPSK QPSK 16-QAM 16-QAM
Coding rate Rc 1/2 3/4 1/2 3/4
Rate
(bits/symbol) 1.0 1.5 2.0 3.0
an 4.1352 3.9164 3.7071 3.5916
gn 5.1441 3.1233 2.5152 1.7373
(dB) (1)
n

2.9311 5.9222 9.6772 11.4211
(dB) (2)
n

-1.51401.4237 5.0131 7.33188
(dB) (3)
n

-3.0312-0.6161 3.0331 5.1235
transmit an information packet. The average of transmit-
ted symbols for each mode is given by n
1
(1)
1
2
()( )
t
i
Ni
i
nnn
i
nn
L L
L
RR



 
P
(9)
For cross-layer designed AMC schemes with n = 1,..,N
modes, the average spectral efficiency needs to be calcu-
lated in order to evaluate system performance. By aver-
aging the n
L values in the range of for
over all
() (1)
( ,...,)
t
N

1,...,nN

modes, the average number of
transmitted symbols in order to transmit an information
packet is
C opyright © 2010 SciRes. IJCNS

F. FOUKALAS ET AL. 262
1() ()
1
1
()
n
n
Nii
nn
ni
L
Lp
Rn


d





1(1)() ()
1
21
()()
tn
n
NNii
nnn n
in i
LPERp d
Rn


i




 (10)
Figure 5 depicts th e average spect ral efficiency of the
combination of AMC and type-II HARQ relied on con-
straints and. In this figure, it is
shown the performance of AMC at physical layer when
3
t
N0.01
loss
PER 
05 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
average SNR(dB)
average S pectral E ffi cienc y
RCPT
Nt= 1
Nt= 2
Nt= 3
0510 15 20 25 30
0
0.5
1
1.5
2
2.5
3
average SNR(dB)
average Spect ral E fficiency
RCPC
Nt=1
Nt=2
Nt=3
0510 15 20 25 30
0
0.5
1
1.5
2
2.5
3
average SNR(dB )
aver age Spec tral Efficiency
RC-LDP C
Nt=1
Nt=2
Nt=3
Figure 5. The average spectral efficiency of each RC-coded
modulation based on CLD design.
rate compatible punctured codes are employed under the
constraints of the previous described cross-layer design.
The parameters of each MCS are those listed in Table 2
considering a channel with Rayleigh fading phenomena
as described above.
In Figure 6, we make contrast of the average spectral
efficiency derived for each rate compatible punctured
code. We illustrate the values of third transmission (i.e.
Nt = 3). Figure 6 shows the performance merit of RCPT
against RCPC. This corroborates the benefit of turbo
scheme against convolutional one in terms of communi-
cation performance as it is well known. Indeed, this per-
formance benefit is more evident in low regions of aver-
age SNR than in high regions. Moreover, it is obvious
that RC-LDPC achieves performance close to RCPT
code. This is a useful outcome considering these two
families of codes since LDPC codes are used in several
standards and especially in space communications. The
fact that turbo and LDPC codes show identical perform-
ance has also concluded both in [27] and [28]. [27] has
focused on performance in terms of PER values at the
physical layer both in AWGN and multipath Rayleigh
fading channel. [28] has proposed the PEG (Progressive
Edge-Growth) construction method for LDPC codes and
has concluded that turbo coding is identical of LDPC in
terms of bit-level performance. To this direction, we
evaluate the system performance under the aforemen-
tioned cross-layer design and we have also concluded in
the same result.
4.2. Comparison Complexity
However, the comparison between different codes should
not be considered only in terms of performance related to
communication efficiency. It should be also studied in
terms of complexity even when the achieved system
performance is identical between different codes (e.g.
turbo and LDPC). Most of code complexity issues are
051015 20 25 30
0
0.5
1
1.5
2
2.5
3
average SNR(dB)
average S pectral Eff i ciency
RCPT
RCPC
RC-LDPC
Figure 6. Comparison of RC codes in terms of average spec-
tral efficiency under the constraints of CLD design.

F. FOUKALAS ET AL. 263
related to computational complexity measuring the addi-
tional operations required by each code. Another impor-
tant aspect of code complexity relies on architectural
issues introduced by code design. [29] studies the com-
plexity of decoding algorithms that is measured in terms
of computational operations such as multiplications, di-
visions and additions. In Table 3 is listed the number of
operations (i.e. additions, divisions, etc.) needed for each
decoding procedure using the max-log MAP (Maximum
A Posteriori) algorithm and the Viterbi algorithm in case
of turbo and convolutional decoder respectively. These
are actually the decoding algorithms that we have im-
plemented in the RCPT and RCPC decoding procedure.
In Table 3, M is the constraint length used by each en-
coder at the transmitter side.
Figure 7 shows the complexity of each decoding pro-
cedure (i.e. turbo and convolutional) in terms of number
of operations vs. the number of iterations and code con-
straint length respectively. It is obvious from this figure
that the decoding complexity in case of convolutional
scheme is noticeably less than turbo case. In our case, the
convolutional decoding procedure uses Viterbi decoder
with constraint length equal to seven. On the other hand,
turbo decoding uses max-log MAP with iterations equal
to eight. The declension of turbo decoding complexity is
close to two times the complexity of convolutional one
since convolutional decoding scheme exhibits 1200
number of operations while turbo one exhibits approxi-
mately 2400 number of operations .
Table 3. Complexity of turbo and convolusional decoders.
Number of Equivalent Operations
Turbo (Max-l og
MAP algorithm ) 28×2M–3
Convolutional
(Viterbi algorithm) 10×2M–3
1 2 3 4 5 6 7 8 9
0
1000
2000
3000
4000
5000
6000
Iterations (Turbo) / Constraint length (Convolutional)
Complexity (num ber of operat i ons)
Comparison compl exity
Convolut i onal , M =7
Turbo, M=4, max-log-MAP
Figure 7. Complexity comparison between Turbo and Con-
volutional decoders.
On the other hand, performance comparisons between
turbo and LDPC codes in terms of decoding complexity
have shown that when both codes achieve an identical
performance then the decoding complexity remains ap-
proximately the same. For instance, [28] have claimed
that 80 iterations using the belief propagation algorithm
produces the same decoding complexity as a turbo code
does with 12 iterations using the BCJR decoding algo-
rithm. [27] has studied the performance comparison be-
tween turbo and LDPC codes in more details cons idering
computational complexity. The authors have measured
the computational complexity in terms of number of op-
erations per iteration per information bit that they could
be additions or comparisons. Table 4 shows the compu-
tational complexity per information bit of the sub-opti-
mum decoding algorithm for code rate R = 1/3. The com-
plexity is expressed in relation to number of iterations
and it is illustrated in Figure 8.
itr
N
Assuming the same configuration as in [27] the turbo
decoding with 8 iterations when a max-log-MAP algo-
rithm is used exhibits approximately the same complex-
ity in terms of number of additions with the LDPC de-
coding scheme that uses the BP algorithm. In our com-
parative study, we use the decoding schemes from [28]
that consist of a turbo decoder with max-log-MAP plus 8
iterations and LDPC decoder with PEG decoding graphs
plus 80 iterations. Henceforth, it could be claimed that
both turbo and LDPC decoders show the same computa-
tional complexity.
Table 4. Complexity of turbo and LDPC decoding algorthms.
Number of Oper a ti o n s p er
information bi t
Turbo (max-log
MAP algorithm ) itr
N106
LDPC (Belief
Propagation algorithm)itr
N26.9
510 15 20 25 30 35 40
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Number of Iterations
Complexity (Num be r o f Additions)
Comparison com plex i t y of Turbo and LDPC
Turbo(max -Log-M A P )
LDPC(BP)
Figure 8. Complexity comparison between Turbo and
LDPC decoders.
C opyright © 2010 SciRes. IJCNS

F. FOUKALAS ET AL. 264
5. Conclusions and Future Work
In this paper, we have extended the cross-layer design
combining AMC with HARQ using RCPT codes. To this
end, a hybrid FEC/ARQ based on RCPT codes has been
assumed. In previous works, the proposed CLD was in-
troduced with uncoded modulations, convolutional and
rate-compatible convolutional coded modulations dedi-
cated to AMC schemes. In addition to that, we have im-
plemented a CLD approach using puncturing techniques
for rate compatibility purposes. The system performance
has been evaluated for type-II hybrid ARQ mechanism.
Moreover, we have illustrated comparative results of
system performance of other rate compatible codes as
convolutional and LDPC as well. In order to have a more
comprehensive view of coding and decoding schemes we
also discuss the computational complexity of each code
separately, in terms of the required number of operations
either in each iteration attempt or for each memory len-
gth. However, since turbo coding and indeed punctured
turbo codes are able to accomplish better performance
with different RSC encoders and puncturing rules name-
ly optimal encoding and puncturing [26], a future work
should be the performance evaluation of AMC and
HARQ combination implementing different encoders
and puncturing rules using RCPT-A RQ.
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