Joint Iterative Demodulation and Decoding of Differential Frequency Hopping Signals

Abstract

In this paper, joint iterative demodulation and decoding of differential frequency hopping (DFH) signals based on the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is considered. The DFH is a nonlinear modulation with memory in the frequency sequence of the successive transmitted symbols, which can be represented by a trellis diagram. The proposed receiving scheme is primarily composed of an a posteriori probability (APP) decoder/filter and an APP demodulator, by which the extrinsic information on the coded bits extracted from the output of the APP filter can be utilized again by the demodulator as updated a priori information. Performance over AWGN and time-varying Rayleigh flat fading channels is investigated by Monte Carlo simulation. By comparison between the symbol-by-symbol maximum a posteriori probability (SBS-MAP) detection with soft decision Viterbi decoding and the proposed method, it can be shown that considerable improvement can be achieved for both static and time-varying fading channels

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JOINT ITERATIVE DEMODULATION AND DECODING OF
DIFFERENTIAL FREQUENCY HOPPING SIGNALS
Rui Zhang, Xiao-Ping Zhang
Department of Electrical
and Computer Engineering
Ryerson University
Toronto, Ontario, Canada
Kaihua Liu
School of Electronic
Information Engineering
Tianjin University
Tianjin, China
ABSTRACT
Inthispaper, jointiterative demodulation and decoding ofdif-
ferential frequency hopping (DFH) signals based on the Bahl-
Cocke-Jelinek-Raviv (BCJR) algorithm is considered. The
DFH is a nonlinear modulation with memory in the frequency
sequence of the successive transmitted symbols, which can
be represented by a trellis diagram. The proposed receiving
scheme is primarily composed of an a posteriori probabil-
ity (APP) decoder/filter and an APP demodulator, by which
the extrinsic information on the coded bits extracted from
the output of the APP filter can be utilized again by the de-
modulator as updated a priori information. Performance over
AWGN and time-varying Rayleigh flat fading channels is in-
vestigated by Monte Carlo simulation. By comparison be-
tween the Symbol-by-Symbol maximum a posteriori proba-
bility (SBS-MAP) detection with soft decision Viterbi decod-
ing and the proposed method, it can be shown that consider-
able improvement can be achieved for both static and time-
varying fading channels.
IndexTerms— Iterativeprocessing, differentialfrequency
hopping, MAP estimation, Rayleigh fading channels
1. INTRODUCTION
Differential frequency hopping (DFH) technique, proposed in
[1], is identified as a spread spectrum modulation with mem-
ory in the frequency sequence of the transmitted signals. It
has proved effective in the applications in which lower trans-
mitted power is more desirable than economical use of band-
width, such as military communications [2] and powerline
communications [3]. Recent study, [3, 4, 5], has been focused
on the design of DFH demodulator based on a trellis diagram.
Both maximum likelihood sequence estimation (MLSE) by
the Viterbi algorithm (VA) and symbol-by-symbol maximum
aposterioriprobability(SBS-MAP)detectionusingtheBCJR
algorithm are considered. By SBS-MAP detection, the de-
modulator in [3] and [5] can produce soft decision output
for coded bits and hence outperforms the system employing
MLSE detection. The demodulator, however, can not utilize
the information exploited by the decoder for further improve-
ment of the DFH system.
Iterative processing technique has been successfully ap-
plied to the decoding of serial concatenated codes (SCC) [6]
and the joint demodulation and decoding of signaling tech-
niques with recursive nature, such as DPSK and CPM signals,
and trellis codes [7], [8], and [9]. In this paper, the iterative
processing technique is applied to a new type of nonlinear
modulation with memory, i.e. DFH signals, in which the con-
catenation of an a posteriori probability (APP) decoder/filter,
defined in [8], and an APP demodulator by random interleav-
ing is employed. The fundamental principle is that by the
extraction of extrinsic information on the coded bits from the
output of the APP filter, which is obtained by both of the in-
put and the constraint of the code structure, the result can be
used as refined a priori information of the coded bits and fed
back to the demodulator to improve the performance of the
demodulation. The performance of the proposed scheme is
investigate by Monte Carlo simulation, in which both AWGN
and time-varying Rayleigh flat fading channels are consid-
ered. A rate-1/2 convolutional code with constraint length
3 and a random interleaver are used in the concatenation of
the channel encoder and the DFH modulator. Comparison
between DFH signals with SBS-MAP detection and iterative
processing is performed based on the simulation result.
2. SYSTEM MODEL
The DFH modulator can be modeled as a first-order Markov
source, withthefrequenciesallowedtobeusedasthepossible
states. The number of branches starting from and ending in
each state of the trellis is determined by a parameter of DFH,
i.e. fanout, denoted by F. The relation between F and the
number of bits mapped onto each transmitted symbol is Fo=
2ν, where ν is the number of bits transmitted per hop. Hence,
the characterization of DFH signaling is similar to that of a
binary nonsystematic convolutional code, with the exception
that the associated transmitted symbol with each branch can
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be an M-ary symbol and is determined by the ending state of
that branch. Throughout the rest of this paper, it is assumed
that the size of the set of available frequencies is M.
The DFH signals can be modeled as M-ary FSK signals
with noncoherent demodulation [1]. An equivalent baseband
receiver is illustrated in Fig. 1.
Matched
Filter 1
Matched
Filter 2
Matched
.
Filter M
.
.
.
.
.
2||⋅
2||⋅
2||⋅
Detector
2
jm ft
π Δ
e
k
j
k a eφ
( )
r t
sample
γ
sample
sample
,
M k
γ
2,k
1,k
γ
1,k
Y
2,k
Y
,
M k
Y
DFH
modulator
Ck
PA(Ck)
APP demodulator
ˆ
()
k
C
Λ
( )
z t
Fig. 1. Equivalent baseband DFH receiver with an APP de-
modulator.
Given a frame of N successive transmitted symbols, as-
suming that the mth symbol was transmitted, the complex-
valued sampled output of the matched filter is
ri,k=
?
Esakejφkδ(i,m) + zi,k,
(1)
where i,m ∈ {1,2,...,M}. akis a Rayleigh random vari-
able. φkis a uniformly-distributed random variable ranging
within [0,2π]. zi,kis complex-valued Gaussian random vari-
ables. When transmitted through AWGN channels, akis a
constant equal to one. δ(i,m) is the Kronecker delta func-
tion. According to [3], Y1,k,Y2,k,...,YM,kare statistically
independent random variables. Furthermore, as for the cases
of i = m and i ?= m, Yi,k’s are respectively non-central chi-
square-dis-tributed random variable and central chi-square-
distributedrandomvariablewithtwodegreesoffreedom, whose
conditional probability density functions (PDF’s) are
p(Yi,k= yi,k|Xm,k= xm,k)
⎧
⎩
where Xm,kis the transmitted symbol at time k, and I0(•) is
the zeroth-order modified Bessel function of the first kind.
=
⎨
1
2σ2 exp
2σ2 exp?−yi,k
?
−a2
kEs+ym,k
2σ2
?
?
I0
?
ak√
Esym,k
σ2
?
,i = m;
1
2σ2
,i ?= m,
(2)
3. APP DEMODULATION OF DFH
SIGNALS WITH MODIFIED BCJR ALGORITHM
In this section, we assume that the signal has been coded be-
foreenteringthemodulator. WhenappliedtotheAPPdemod-
ulation of DFH signals, the branch metric at time k, denoted
by γk(s?,s), can be expressed as
?
γk(s?,s) =
Xk
P(Sk= s|Sk−1= s?) ·
P(Xk|Sk= s,Sk−1= s?)
M
?
i=1
P(Yi,k|Xk),(3)
where Skand Xkare the state and transmitted symbol at time
k, respectively.
Yk= {Y1,k,Y2,k,...,YM,k}Tis the input
vector of the detector at time k as shown in Fig. 1. The first
term of (3) is the joint a priori probability of coded bits input
to the modulator at one time. By sufficient interleaving, it can
be written as the product of marginal a priori probabilities,
− →
P(Sk= s|Sk−1= s?) =
ν?
p=1
PA(cp
k),
(4)
where ν is the number of bits modulated in each symbol in-
terval. This term will be updated by the feedback from the
APP filter in iterative processing. As for the second term of
(3), there is a zero-one relationship, such as (5), between a
transmitted symbol and a branch, which in turn represents a
state transition on the trellis.
P(Xk|Sk= s,Sk−1= s?) =
Finally, the third term is the conditional probability of receiv-
ing Yk, which is a symbol corrupted by additive Gaussian
noise and fading, assuming Xkwas transmitted, as in (2). By
substituting (2), (4), and (5) into (3), the branch metric can be
calculated by
?ν
?
?1,
if Xktransmitted;
otherwise
0,
(5)
γk(s?,s) =
p=1PA(cp
2Mσ2M
−a2
k)
·
exp
kEs+?M
i=1yi,k
2σ2
?
I0
?
ak
?Esym,k
σ2
?
. (6)
The log-likelihood ratio (LLR) of the output of the APP
demodulator can be expressed as
?
?
where Λcj
cj
priori LLR’s of the jth bit of the group of coded bits input
to the modulator at time k. αk(s) and βk(s) are forward and
backward state metrics, and
Λcj
k=Λa
cj
k+ ln
{s?,s}∈Cj,1
k
αk−1(s?)Ak(s?,s)βk(s)
αk−1(s?)Ak(s?,s)βk(s), (7)
{s?,s}∈Cj,0
kare respectively the a posteriori and a
k
kand Λa
Ak(s?,s) =
ν?
p?=j
p=1
PA(cp
k)
M
?
i=1
P(Yi,k|Xk).
(8)
By decomposing the output of the APP demodulator as (7),
the APP demodulator is suitable for iterative processing, as
shown in the next section.
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4. JOINT ITERATIVE DEMODULATION AND
DECODING OF DFH SIGNALS
Convolutional
encoder
Bit
interleaver
DFH
Modulator
Fading
Additive noise
APP
dmodulator
Bit
deinterleaver
APP
decoder/filter
Bit
interleaver
_
+
+
_
ˆ
()
k
C
Λ
ˆn d
n
d
n c
k c
k
X
k
Y
k a
k z
ˆ
()
n
C
Λ
()
a
k
C
Λ
Fig. 2. Block diagram of the DFH system with joint iterative
demodulation and decoding.
We shall use the following notation for expositional con-
venience. Λ(q)
bit, i.e. the output on an information bit by the APP decoder.
Λa(q)
the input on an information bit of the APP decoder. Λ(q)
represents the a posteriori LLR on a coded bit by the APP
filter. Λi(q)
coder/filter. Λ(q)
APP demodulator. Finally, Λa(q)
of a coded bit, i.e. the input LLR on a coded bit of the APP
demodulator. The superscript q represents the qth iteration.
As illustrated in Fig. 2, in a DFH system employing itera-
tive processing, the channel encoder and modulator is serially
concatenated at the transmitting end, which is due to the fol-
lowing facts. A basic principle of the iterative processing is
that the the information exploited by a certain stage in the
previous iteration must not be feedback to itself as the input
of the current iteration. To satisfy this condition, it must be
guaranteed that the input to the current stage is separable from
the output of that stage as in (7). Therefore, a bit interleaver
is employed to connect the channel encoder and the DFH
modulator and to make the input to both the APP demodu-
lator and APP decoder as uncorrelated as possible. At the
receiving end, the APP demodulator calculates the Λ(q)
for each coded bit and by subtracting the a priori informa-
tion Λa(q)
result will be used by the APP decoder/filter after deinterleav-
ing.
dec(ˆdn) is the a posteriori LLR of an information
dec(dn) denotes the a priori LLR of an information bit, i.e.
fil(ˆ cn)
fil(cn) is the input LLR on a coded bit of APP de-
dem(ˆ ck) is a posteriori LLR on a coded bit by
dem(ck) denotes a priori LLR
dem(ˆ ck)
dem(ck) from the a posteriori information as in (9), the
Λi(q)
fil(ck) = Λ(q)
dem(ˆ ck) − Λa(q)
dem(ck) = Λ(q)
ck− Λa(q)
ck,
(9)
where Λi(q)
ation, the a priori information Λa(q)
updated by the extrinsic information extracted from output of
the APP filter in the last iteration, i.e. Λ(q−1)
leaving. The APP /decoder filter operates in a similar way as
fil(ck) is to be deinterleaved. Since the second iter-
ck
of the coded bits will be
fil
(ˆ cn), after inter-
the APP demodulator with the difference lying in that the a
priori LLR of the information bits, Λa(q)
to zero. By subtracting Λi(q)
calculated by the APP decoder/filter, the extrinsic information
Λe(q)
cn
on each coded bit can be computed by (10) and used as
theaprioriinformationofthenextiterationafterinterleaving.
dec(dn), are always set
fil(cn) from the a posteriori LLR
Λa(q+1)
cn
= Λe(q)
cn
= Λ(q)
fil(ˆ cn) − Λi(q)
fil(cn).
(10)
In the last iteration, the APP decoder will calculate the a pos-
teriori LLR Λ(q)
tion bits accordingly.
dec(ˆdn) and make hard decision on the informa-
5. SIMULATION RESULT
The bit error rate (BER) performance of DFH system with
iterative processing under both AWGN and rayleigh flat fad-
ing channels are investigated using Monte Carlo simulation.
A rate-1/2 convolutional code with generator [7,3] is used
as the channel coding scheme. A random interleaver is em-
ployed. The case of ν=1 is considered. The APP demodula-
tion and decoding/filtering are performed in a frame-by-frame
manner. The state of the channel encoder is forced to return to
zero, whereas the DFH modulator is left open. Noncoherent
demodulation is used in the APP demodulator because of the
difficulty in the acquisition of the phase of the received fast
frequency hopping signals such as DFH.
Results of AWGN channel are shown in Fig. 3. It has
been concluded in [5] that the performance of SBS-MAP de-
tection with soft-decision Viterbi decoding of convolutional
coded DFH signals is better than that of MLSE detection of
DFH signals with the same channel coding. Therefore, we
only draw the comparison between the iterative processing
and the SBS-MAP detection with soft-decision Viterbi decod-
ing of DFH signals. Because the input a priori information on
the information bits of the APP decoder is always zero, based
on which the MAP criterion and ML criterion are equivalent
to each other, the performance of the first iteration is simi-
lar to that of the system without iterative processing. Since
the second iteration, the performance of the APP demodu-
lator is improved by using the extrinsic information, which
in turn augments the signal-to-noise ratio (SNR) for the de-
coder. Therefore, better BER performance can be achieved
as shown in Fig. 3. It can be observed that about 2.5dB of
gain is achieved by the joint demodulation and decoding at
the BER of 10−3after 5 iterations relative to the SBS-MAP
detection with soft-decision decoding. In addition, the BER
performance begins to converge after the 5th iteration.
Results over Rayleigh flat fading channel are shown in
Fig. 4. A fading rate of 0.05R, i.e. fDmaxTs = 0.05, has
been selected. The comparison between SBS-MAP detection
with soft-decision Viterbi decoding and the iterative process-
ing is made again. The similarity between the performance
of SBS-MAP detection with soft-decision Viterbi decoding
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456789
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Eb/No
BER


SBSMAP+SD
1st iteration
2nd iteration
3rd iteration
4th iteration
5th iteration
Fig. 3. BER performance of joint iterative demodulation and
decoding of DFH signals over AWGN channel, compared
with the SBS-MAP detection with soft-decision Viterbi de-
coding of DFH signals.
and the first iteration of the iterative processing is due to the
same reason as explained in the case of AWGN channels. At
the BER of 10−3, we can achieve 3dB of gain after 5 itera-
tions relative to the other method simulated. The performance
tends to converge after the 5th iteration.
6. CONCLUSION
A DFH system with joint iterative demodulation and decod-
ingisproposed. TheperformanceisinvestigatedunderAWGN
and Rayleigh flat fading channels. It has been shown that
the DFH modulator can be represented by a trellis diagram,
making the iterative processing an effective solution to ex-
ploit the system performance as much as possible. Simula-
tion results show that by joint iterative demodulation and de-
coding, the BER performance can be improved considerably
over the systems without employing iterative processing. Fu-
ture work would be further investigation on the system per-
formance over frequency selective fading channels, a typical
channel model as a platform to test the communication algo-
rithm for powerline communications.
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56789 1011
10
−5
10
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10
−2
10
−1
10
0
Eb/No
BER


SBS−MAP+SD
1st iteration
2nd iteration
3rd iteration
4th iteration
5th iteration
Fig. 4. BER performance of joint iterative demodulation and
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