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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 1, January 2011
Performance of Iterative Concatenated Codes
with GMSK over Fading Channels
Labib Francis Gergis
Misr Academy for Engineering and Technology
Mansoura, Egypt
IACSIT Senior Member, IAENG Member
Abstract-Concatenated continuous phase achievable data rates obtained using codes
modulation (CCPM) facilitates powerful of a manageable decoding complexity.
error correction. CPM also has the However, a novel approach to error
advantage of being bandwidth efficient and control coding revolutionized the area of
compatible with non-linear amplifiers. coding theory. The so-called turbo codes
Bandwidth efficient concatenated coded [1,2], almost completely closed the gap
modulation schemes were designed for between the theoretical limit and the data
communication over Additive White rate obtained using practical
Gaussian noise (AWGN), and Rayleigh implementations. Turbo codes are based on
fading channels. An analytical bounds on concatenated codes (CC's) separated by
the performance of Serial (SCCC), and interleavers. The concatenated code can be
Parallel convolutional concatenated codes decoded using a low-complexity iterative
(PCCC) were derived as a base of decoding algorithm [3]. Given certain
comparison with the third category known conditions, the iterative decoding algorithm
as hybrid concatenated convolution codes performs close to the fundamental Shannon
scheme (HCCC). An upper bound to the capacity. In general, concatenated coding
soft-input, soft-output (SISO) maximum a provides longer codes yielding significant
posteriori (MAP) decoding algorithm performance improvements at reasonable
applied to CC's of the three schemes was complexity investments. The overall
obtained. Design rules for the parallel, decoding complexity of the iterative
outer, and inner codes that maximize the decoding algorithm for a concatenated code
interleaver's gain were discussed. Finally, a is lower than that required for a single code
low complexity iterative decoding algorithm of the corresponding performance
that yields a better performance is The parallel, serial, and hybrid
proposed. concatenation of codes are well established
as a practical means of achieving excellent
key words: Concatenated codes, continuous performance. Interest in code concatenation
phase modulation, GMSK, uniform has been renewed with the introduction of
interleaved coding, convolutional coding, turbo codes [4,5,6,7]. These codes perform
iterative decoding well and yet have a low overall decoding
complexity.
CPM is a form of constant-envelope
I. INTRODUCTION digital modulation and therefore of interest
for use with nonlinear and/ or fading
The channel capacity unfortunately only channels. The inherent bandwidth- and
states what data rate is theoretically possible energy efficiency makes CPM a very
to achieve, but it does not say what codes to attractive modulation scheme [8].
use in order to achieve an arbitrary low bit Furthermore, CPM signals have good
error rate (BER) for this data rate. spectral properties due to their phase
Therefore, there has traditionally been a continuity. Besides providing spectral
gap between the theoretical limit and the economy, CPM schemes exhibit a “coding
gain” when compared to PSK modulation.
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This “coding gain” is due to the memory where Eb, is the energy per symbol interval,
that is introduced by the phase-shaping T is the duration of the symbol interval, fc, is
filter and the decoder can exploit this. CPM the carrier frequency, and Ф(t+α) is the
modulation exhibits memory that resembles "phase function" responsible for mapping
in many ways how a convolutionally the input sequence to a corresponding phase
encoded data sequence exhibits memory - in wavefom.
both cases, a “trellis” can be used to display The term α = {αi} is the input sequence
the possible output signals (this is why taken from the M-ary alphabet ±l, ±3, . . . ,
convolutional encoders are used with CPM ± M - 1. For convenience the focus here will
in this paper. be on the binary case, αi є {±1}.
This paper is organized as follows. The "continuous phase" constraint in
Section II briefly describes continuous CPM requires that the phase function
phase modulation, using Gaussian minimum maintain a continuous amplitude. In general
shift keying GMSK, and how it can be the phase function is given by
separated into a finite-state machine and a
memoryless signal mapper. Section III N
describes in details the system model and Ф(t+α) = 2 π Σα n δn g( t – n T )
encoder structure of serial, parallel, and n=0
hybrid concatenated codes. (2)
Section IV derives analytical upper where δ is the modulation index, and g(t) is
bounds to the bit-error probability of the the phase pulse. The phase pulse g(t) is
three concatenated codes using the concept typically specified in terms of a normalized,
of uniform interleavers that decouples the time-limited frequency pulse f(t) of duration
output of the outer encoder from the input LT such that:
of the inner encoder, from the knowledge of
the input–output weight coefficients
(IOWC), Acw,h, for CC's. Acw,h is
represented related to the type of 0 ; if t < 0
concatenation. The choice of decoding t
algorithm and number of decoder g(t) = ∫ f(τ)dτ ; if 0 < t < LT
iterations is described in section V. 0
Factors that affect the performance are
discussed in section VI. Finally conclusion 1/2 ; if t > LT
results for some examples described in (3)
section IV have been considered in section
VII. The duration term (LT) is specified in
terms of the bit duration T, and identifies
the number of bit durations over which the
II. GMSK SYSTEM MODEL frequency pulse f (t) is non-zero, δ = 1/2, and
the frequency pulse is
Gaussian minimum-shift keying is a
special case of a more genet-ic class of
modulation schemes known as continuous f(t)= (1/2T) Q 2πB (t -τ/2)/ √ ln 2 -
phase modulation (CPM). In CPM schemes,
the signal envelope is kept constant and the
phase varies in a continuous manner. This Q 2πB (t+τ/2)/ √ ln 2
ensures that CPM signals do not have the
high-frequency components associated with (4)
sharp changes in the signal envelope and B is a parameter in GMSK which controls
allows for more compact spectra. CPM the amount of bandwidth used as well as the
signal s(t ) can be written [8,9] severity of the intersymbol interference, the
B parameter is expressed in terms of the
S(t) = (√2Eb/T) cos [2πfct + Ф(t+α) ] (1) inverse of the bit duration T.
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III. PERFORMANCE ANALYSIS OF It is clear from equation (8) that BER
COCATENATED CODES depends on major factors like signal-to-
noise ratio per bit, and the input–output
weight coefficients (IOWC), Acw,h for the
Consider a linear (n,k) code C with code code, Acw,h is represented related to the type
rate Rc = k/n and minimum distance hm. An of concatenation.
upper bound on the bit-error rate [BER] of The average IOWC for λ concatenated
the code C over memoryless binary-input codes with λ -1 interleavers can be obtained
channels, with coherent detection, using by averaging (5) over all over all possible
maximum likelihood decoding, can be interleavers. This average is obtained by
obtained as [4] replacing the actual ith interleaver (i = 1, 2,
… , λ-1), that performs a permutation of the
n k Ni input bits, with an abstract interleaver
BER ≤ Σ Σ (w/k) Acw,h D(Rc Eb / No , h) called uniform interleaver defined as a
h=dmin w=1 probabilistic device that maps a given input
(5) word of weight w into all distinct Ni
w
where Eb/No is the signal-to-noise ratio per permutations of it with equal probability
bit, and Acw,h for the code C represents the ψ = 1 / Ni .
number of codewords of the code with w
output weight h associated with an input
sequence of weight w. Acw,h is the input–
output weight coefficient (IOWC).The IV. DESIGN OF CONCATENATED
function D(.) represents the pairwise error CODES
probability which is a monotonic decreasing
function of the signal to noise ratio and the Concatenated codes represent a more
output weight h. For AWGN channels we recent development in the coding research
have D(Rc Eb / No , h) = Q( √ 2Rc h Eb / No ). field [1], which has risen a large interest in
For fading channels, assuming coherent the coding community.
detection, and perfect Channel State
information (CSI), the fading samples μi are IV. 1. Design of Parallel Concatenated
i.i.d. random variables with Rayleigh
Convolutional Codes ( PCCC )
density of the form f(μ)= 2μe-μ2. The
conditional pairwise error probability is
The first type of concatenated codes is
given by
parallel concatenated convolutional codes
h (PCCC) whose encoder is formed by two (or
D(Rc Eb No , h│μ) = Q (2Rc h Eb / No Σ μ2i )
more) constituent systematic encoders joined
i=1
through one or more interleavers. The input
information bits feed the first encoder and,
(6) after having been scrambled by the
where Q function can be defined as interleaver, they enter the second encoder.
A codeword of a parallel concatenated code
Q(x) ≤ (1/2) e –
x2 / 2
(7) consists of the input bits to the first encoder
followed by the parity check bits of both
By averaging the conditional bit error encoders. As shown in Fig. 1, the structure
rate over fading using (5), (6), and (7). The of PCCC consists of convolutional code C1
upper bound for BER is represented by with rate R1p = p/q1 , and convolutional code
n k C2 with rate R2p = p/q2, where the
BER ≤ 0.5 Σ Σ (w/k) A c
w,h . constituent code inputs are joined by an
h=hm w=1 interleaver of length N, generating a PCCC,
Cp, with total rate Rp. The output codeword
[1/(1+Rc Eb / No)]h length n = n1 + n2 [4].
(8)
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input data nM
C1 n1
Acw,h ≈ Σ ( Nj / j! p! ) Acw,h,j
j=1
to Modulator
interleaver
(12)
Length = N Inserting (12) into (9), we obtain the
input–output weight coefficients (IOWC),
C2 n2 Acpw,h, for PCCC as [4]
N/p N /p
Fig. 1. Parallel Concatenated nmax nmax n1 n2
Convolutional Code ( PCCC ) A cp
w,h ≈ Σ Σ N
: n1=1 n2 =1 w
The input–output weight coefficients c1 c2
(IOWC), Acpw,h, for PCCC can be defined as .A w,h,n1 A w,h,n2
[3]
nmax nmax
A c1
w,h1 x A c2
w,h2 ≈ Σ Σ w!
Acpw,h = Σ Acpw,h1,h2 = Σ n1=1 n2 =1
N pn1+n2 n1! . n2 !
h1,h2 : h1,h2 : w
h1+h2=h h1+h2=h
(9 )
. Nn1+n2-w . Ac w,h,n Ac w,h,n 1
1
2
2
(13)
where Acpw,h1,h2 is the number of codeword
of the PCCC with output weights h1, and h2
associated with an input sequence of weight
w. IV. 2. Design of Serially Concatenated
Let Acw,h.j be the IOWC given that the Convolutional Codes ( SCCC )
convolutional code generates j error events
with total input w, and output weight h. The Another equally powerful code
Acw,h.j actually represents the number of configuration with comparable performance
sequences of weight h, input weight w, and to parallel concatenated codes is serially
the number of concatenated error events j concatenated convolutional codes (SCCC).
without any gap between them, starting at The structure of a serially concatenated
the beginning of the code. For N much convolutional code (SCCC) is shown in Fig.
larger than the memory of the convolutional 2. It refers to the case of two convolutional
code, the coefficient of the equivalent code CCs, the outer code Co with rate Roc = k/p,
can be approximated by and the inner code Ci with rate Ric = p/n,
joined by an interleaver length N bits,
nM N/p generating an SCCC with rate Rc = k/n.
A w,h ≈
c
Σ j Acw,h,j
j=1 outer Code interleaver inner Code
(10) Ci length = N Co
where nM, the largest number of error
events concatenated in a codeword of weight Fig. 2. Serially Concatenated
h and generated by a weight w input Convolutional Code ( SCCC )
sequence, is a function of h and w that
depends on the decoder. From the knowledge of the IOWC of
outer and inner codes, which called Aco(w,L)
N and Aci(l,H). Exploit the properties of the
j ≈ Nj / j! (11) uniform interleaver, which transforms a
codeword of weight l at the output of the
Substitution of this approximation in (10) outer encoder into all its distinct N
Yields permutations. l
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As a consequence, each codeword of the
outer code Co of weight l, through the action
of the uniform interleaver, enters the inner
encoder generating N codewords of Interleaver π1 Parallel n1
the inner code Ci. l length = N1 Encoder
Thus, the IOWC of the SCCC scheme,
TO CHANNEL
Acsw,h ,of codewords with weight h associated
with an input word of weight w is given by Outer Interleaver π2 Inner n2
Encoder length = N2 Encoder
N i/p
Acsw,h = Σ Acow,l Χ Acil,h Fig. 3. Hybrid Concatenated
l=0
Convolutional Code ( HCCC )
N
l (14)
Using the previous result of (12) with
j=ni for the inner code, and the analogus
one, j=no, for the outer code.
composed of three concatenated codes, the
parallel code Cp with rate Rpc = kp/np, the
no M N/p outer code Co with rate Roc = ko/po, and the
Acow,l ≈ Σ
o
no Aow,l,no inner code Ci with rate Ric = pi/ni , with two
interleavers N1 and N2 bits long. Generating
n =1
(15) an HCCC CH with overall rate RH. For
Substituting (15) into (14) defines Acsw,h for simplicity, assuming kp = ko and po = pi = p,
SCCC in the form then RH = ko / ( np + ni ).
N/p N/p Since the HCCC has two outputs, the
N nmax nmax n1 n2 upper bounds on the bit error probability in
(8) can be modified to
Acs
w,h ≈ Σ Σ Σ N
:l=dof no=1 ni=1 l n1 n2 k
o i
BER ≤ Σ Σ Σ (w/k) . ACHw,h1,h2
. A w,l,no A l,h,ni h=h p
h= h i
w=wm
(16)
where dof is the free distance of the outer
. Q [ √ 2RH (h1 + h2 ) (Eb / No) ]
code. Inserting the approximation (11) in
(16) yields
(18)
N nom nim
where ACHw,h1,h2 for the HCCC code
Acsw,h ≈ Σ Σ Σ l! represents the number of codewords with
l=dof no=1 ni=1 output weight h1 for the parallel code and
pno+ni . no ! . ni ! output weight h2 for the inner code,
associated with an input sequence of weight
. Nno+ni . Aow,l,n Ail,h,n o i w, ACHw,h1,h2 is the IOWC for HCCC, wm is the
(17) minimum weight of an input sequence
generating the error events of the parallel
code and the outer code, hp is the minimum
IV. 3. Design of Hybrid Concatenated weight of the codewords of Cp, and hi is the
minimum weight of the codewords of Ci .
Convolutional Codes ( HCCC )
With knowledge of the IOWC ACpw,h1 for
the constituent parallel code, the IOWC
ACow,l for the constituent outer code, and the
The structure of a hybrid concatenated
IOWC ACil,h2 for the constituent inner code,
convolutional code is shown in Fig. 3. It is
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using the concept of the uniform interleaver, and Rayleigh fading channels with GMSK
the ACHw,h1,h2 for HCCC can be obtained. modulation scheme, using interleaver of
lengths N = 100, 1000, and 2000 bits.
N2 Acpw,h1 x Acow,l x Acil,h2
ACHw,h1,h2 = Σ 1.e-1
l=0 N1 N1
1.e-2
w l
1.e-3
(19) 1.e-4
1.e-5
ACHw,h can be obtained by summing
ACHw,h1,h2 overall h1, and h2 such that 1.e-6
BER
Co
h1+h2=h. A w,l is the number of codewords 1.e-7
of Co of weight l given by the input 1.e-8
sequences of weight w. Analogous
1.e-9
definitions apply for Acpw,h1 and Acil,h2.
1.e-10 N = 100
1.e-11 N = 1000
1.e-12 N = 2000
V. EXAMPLES CONFIRMING THE
DESIGN OF CONCATENATED 1.e-13
1 2 3 4 5 6 7 8 9
CODES RULES
Eb / No dB
To obtain the design rules obtained Fig. 4. Analytical bounds for PCCC with
asymptotically, for different signal-to-noise GMSK Modulation Scheme through different
ratios and large interleaver lengths, N, the interleaver lengths
upper bounds for (8) to BER for several
types of the concatenated codes were
evaluated, with different interleaver lengths, 2. Serially Concatenated
and compare their performances with those Convolutional Codes (SCCC)
predicted by the design guidelines.
Consider a rate 1/3 SCCC using as outer
1. Parallel Concatenated code a convolutional encoder Co with rate
Convolutional Codes (PCCC) Roc = 1/2, and the inner code Ci with rate Ric
= 2/3, joined by an uniform interleaver of
length N = 100, 1000, and 2000 bits, as
Consider a PCCC with overall rate = 1/3, shown in Fig. 2. Using the previously
formed by two convolutional codes, C1, and analysis for SCCC that defined in (8), and
C2, have equal rate = 1/2, linked through an (17), we obtained the bit-error probability
uniform interleaver with length N, and bounds illustrated in Fig. 5. The
whose encoder is shown in Fig. 1. We have performance was obtained over AWGN,
constructed different PCCCs through and Rayleigh fading channels with GMSK
interleavers of various lengths, and passed modulation scheme.
through the previous steps to evaluate their
performance with GMSK modulator.
Upper bounds to the error probability
based on the union bound described in(8),
and (13) present a divergence at low values
of signal-to-noise ratio. Fig. 4. shows the bit
error probability of a PCCC over AWGN,
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1.e-1
1.e-1
1.e-2 1.e-2
1.e-3 1.e-3
1.e-4
1.e-4
1.e-5
1.e-5 1.e-6
1.e-7
1.e-6
1.e-8
1.e-7 1.e-9
BER
1.e-10
1.e-8
BER
1.e-11
1.e-9 1.e-12
1.e-13
1.e-10
1.e-14
1.e-11 1.e-15
1.e-16
1.e-12 N = 100
1.e-17
1.e-13 N = 1000 1.e-18 N = 100
1.e-14 1.e-19 N = 1000
N = 2000 1.e-20 N = 2000
1.e-15 1.e-21
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Eb / No dB Eb / No dB
Fig. 5. Analytical bounds for SCCC with Fig. 6. Analytical bounds for HCCC with
GMSK Modulation Scheme through GMSK Modulation Scheme through
different interleaver lengths different interleaver lengths
3. Hybrid Concatenated VI. ITERATIVE DECODING OF
Convolutional Codes (HCCC) CONCATENATED CODES
Maximum-likelihood (ML) decoding of SCCC,
Consider a rate 1/4 HCCC formed by a PCCC, and HCCC with large N is an almost
parallel systematic convolutional code with rate complex and impossible achievement. To acquire
1/2, an outer four convolutional code with rate a practical significance, an iterative algorithm
1/2, and an inner convolutional code with rate consists of a soft-input, soft-output (SISO)
2/3, joined by two uniform interleavers of maximum a posteriori (MAP) decoding algorithm
length N1 = N and N2 = 2N, where N=100, 1000, applied to CC's [10], [11], and [12]. A functional
and 2000 bits, as shown in Fig. 3. Using (8), and diagram of the iterative decoding algorithm for
(18) we have obtained the bit error probability PCCC, SCCC, and HCTC are illustrated in
curves over AWGN, and Rayleigh fading figures 7, 8, and 9, respectively.
channels with GMSK modulation scheme,
shown in Fig. 6. from Demodulator
MAP not used
SISO
C1 π MAP
SISO
C2
π-1
Fig. 7. Iterative decoding algorithm for
PCCC
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from Demodulator
1.e-1
MAP not used to Decision 1.e-2 N = 1000
SISO 1.e-3
inner Ci π-1 MAP 1.e-4
SISO 1.e-5
outer Co 1.e-6
1.e-7
1.e-8
1.e-9
1.e-10
BER
π 1.e-11
1.e-12
Fig. 8. Iterative decoding algorithm for 1.e-13
SCCC 1.e-14
1.e-15 No iteration
from Demodulator 1.e-16 iteration = 2
1.e-17 iteration = 3
1.e-18 iteration = 4
MAP
SISO 1.e-19 iteration = 5
inner π2-1 1.e-20
MAP 1 2 3 4 5 6 7 8
SISO Eb / No dB
outer
Fig. 10. Analysis of Iterative decoding algorithm
π2 for 1/3 PCCC with different No of
to decision
iterations
1.e-1
MAP 1.e-2
N = 1000
SISO 1.e-3
π1 parallel π1-1 1.e-4
1.e-5
1.e-6
1.e-7
1.e-8
Fig. 9. Iterative decoding algorithm for 1.e-9
1.e-10
HCCC 1.e-11
1.e-12
BER
1.e-13
VI . THE EFFECT OF VARIOUS 1.e-14
PARAMETERS ON THE 1.e-15
1.e-16
PERFORMANCE OF 1.e-17
CONCATENATED CODES 1.e-18
No iteration
1.e-19
1.e-20 iteration = 2
1.e-21
The performances of concatenated codes 1.e-22
iteration = 3
were evaluated and analyzed in the previous 1.e-23 iteration = 4
sections. There are many parameters which 1.e-24 iteration = 5
1.e-25
affect the performance of CC's when 1 2 3 4 5 6 7 8
decoded with iterative decoder over AWGN
Eb / No dB
and Rayleigh fading channels. It is shown,
briefly, the effect of the interleavers lengths,
and the number of decoding iterations. Fig. 11. Analysis of Iterative decoding algorithm
for 1/3 SCCC with different No of
iterations
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1.e-1
1.e-2
1.e-3 N = 1000 VII . CONCLUSIONS
1.e-4
1.e-5
1.e-6
1.e-7 A construction of concatenated codes
1.e-8
1.e-9 CC's have presented and constructed in this
1.e-10 paper with three main basic schemes:
1.e-11
1.e-12 PCCC, SCCC, and HCCC, over AWGN
1.e-13 and Rayleigh fading channels. The effects of
BER
1.e-14
1.e-15 various parameters on the performance of
1.e-16 CC's, using an upper bound to the soft-
1.e-17 input, soft-output (SISO) maximum a
1.e-18
1.e-19 posteriori (MAP) decoding algorithm are
1.e-20 investigated. These parameters are : the
1.e-21 No iteration
1.e-22 iteration = 2
interleaver length, and the number of
1.e-23 iterations. The analytical results showed
1.e-24 iteration = 3
1.e-25 iteration = 4 that coding gain was improved by
1.e-26 iteration = 5 increasing the interleaver length, and the
1.e-27
number of iterations.
1 2 3 4 5 6 7 8
Eb / No dB
Fig. 12. Analysis of Iterative decoding algorithm
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85 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
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