Joint Detection and Decoding over Multipath Channels: the Known Path

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p>Joint Detection and Decoding over Multipath Channels: the Known Path</p

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Joint Detection and Decoding over Multipath Channels: theJoint Detection and Decoding over Multipath Channels: the
Known PathKnown Path
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CC BY 4.0
SUBMISSION DATE / POSTED DATE
10-07-2023 / 17-07-2023
CITATION
Mirbadin, Anoush; Zaraki, Abolfazl (2023). Joint Detection and Decoding over Multipath Channels: the Known
Path. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.23651016.v1
DOI
10.36227/techrxiv.23651016.v1

1
Joint Detection and Decoding over Multipath
Channels: the Known Path
Anoush Mirbadin and Abolfazl Zaraki
Abstract—We use the sum-product algorithm (SPA) on factor
graphs to enable joint detection and decoding over multipath
channels in single-carrier modulation formats. We consider a
novel model for multipath channels at the receiver, obtained
by discretizing channel parameters, such as channel paths,
attenuations, and delays. We assign these parameters proper
random variables and consider suitable priors for them. Due
to the project’s complexity, we only evaluate the performance of
the benchmark algorithm called the known path in this article.
Index Terms—Detection, decoding, multipath, known path.
I. INTRODUCTION
DURING past decades, several approaches have been pro-
posed to reduce the effect of multipath impairment. The
proposed techniques are categorized into three main groups:
diversity, multicarrier transmission and their combinations.
Diversity considerably reduces the probability of simultaneous
fade on several replicas of the same information signal.
Diversity transmission techniques in fading environments were
extensively investigated in [1]– [5]. In multicarrier modulation
techniques such as orthogonal frequency division multiplexing
(OFDM), large symbol duration reduces intersymbol interfer-
ence (ISI), and the ISI can be obliterated by using cyclic
prefixes. OFDM and its application against ISI channels have
been well investigated in [6]. The multicarrier modulation
and diversity combination has also been well treated in the
literature, e.g., [7] and [8].
Lack of accuracy in the algorithms mentioned above was
our main motivation to simulate the channel impairments at
the receiver side as accurately as possible and better decide
on the transmitted symbols. In this project, we perform joint
detection and decoding over multipath channels using single
carrier modulation formats in a baseband communication
system. We consider a novel model for multipath channels at
the receiver. The proposed model is obtained by discretizing
channel paths and attenuation coefficients, then assigning a
Bernoulli random variable to each path, called path existence
(PE), and a uniform random variable to each attenuation,
called attenuation existence (AE). The prior probability of each
PE is set to 0.5while a uniform distribution considers the prior
of each AE. The delay profile is also modelled by the Weibull
random variable. We assumed that symbol duration is much
smaller than the coherent time of the channel; however similar
approach can be used for modelling the Doppler spread. Due
to the project’s complexity, in this article, we only evaluate the
performance of the benchmark algorithm, called known path,
Anoush Mirbadin is with SELTA Digital Platforms Group, Cadeo, Italy
(anoush.mirbadin@selta.com). Abolfazl Zaraki is with the Computer Science
department of the University of Hertfordshire, UK (a.zaraki@herts.ac.uk)
where the receiver has exact knowledge of the communication
channel. Given a vector of observations, we factorize the
joint posterior distribution of channel model parameters and
codeword bits. We apply the sum-product algorithm (SPA)
to the corresponding factor graph to compute the marginals
of desired random variables. The performance evaluations are
done through bit error rate (BER) curves.
II. SY S TE M DESIGN
A. Channel Model
Consider the general model for a L-Path multipath channel.
h(τ;t) =
L−1
X
i=0
αi(t) exp(j2πfcτi(t))δ(τ−τi(t)) (1)
in which fcis the carrier frequency, τi(t)is the propagation
delay of i-th path, and αi(t)is the gain of i-th path. Since we
consider a baseband communication system in which symbol
duration is much smaller than the coherent time of the channel,
the Doppler effect is negligible. We also assume that each
τi(t)is smaller than the symbol duration T. This means
that interferer paths can only affect the adjacent symbol. In
addition, we consider that channel remains constant during
each symbol period. This implies that the channel can change
symbol by symbol, i.e., Lchanges symbol by symbol. The
maximum number of paths per symbol is fixed for all paths in
our simulations. We also assume that τiis a Weibull random
variable. Let us ignore the channel attenuation for the moment.
At the receiver side, we model the channel by
hm(τ) =
N−1
X
i=0
P Eiexp(j2πfcliTN)δ(τ−liTN)(2)
in which TN=T
N,Tis the symbol duration, and P Eiis the
Bernoulli random variable representing the path existence with
prior probability equals to 0.5.liTNis the delay of ith path
in which lifollows a discrete Weibull distribution. Its value
is greater than zero and less than a percentage of sample per
symbol for the line of sight (LOS) path (we considered that
the LOS path always exists). Therefore, for the rest paths, its
value is greater than l0and less than 50 Percent of samples
per symbol. If Ngoes to infinity and we know exactly about
path existences and delays, the error modelling goes to zero.
Now, we model the power delay profile in (1) and (2). In
this project, we model the power delay profile in (1) by the
exponential function as αi(t) = exp(−τi(t)a)in which ais
the attenuation coefficient. In (2), we use a similar channel
discretization strategy as in channel paths for power delay

2
profile, i.e., α0(liTN) = PN
0
−1
i0=1 exp −liTN
AEi0a
0(N
0)
1−AEi0a0(N0).
We discretize the attenuation coefficient into N0descritization
level and assign a uniform prior probability to each attenuation
existence, i.e., AEi0. If N0goes to infinity and we have the
exact knowledge of attenuation existence, the error modelling
goes to zero. It should be considered that channel attenuation
never becomes zero. For this reason AEi0never starts by zero.
The same strategy might be used for modelling the Doppler
shift. We left this for future work in this context. In this paper,
we should only evaluate the benchmark algorithm, which is
called the known path. We use the following channel model
at the receiver
hm(τ) =
N−1
X
i=0
N
0
−1
X
i0=0
P Eiexp −liTN
AEi0a0(N0)
1−AEi0a0(N0)!×
exp(j2πfcliTN)δ(τ−liTN)
(3)
B. System Model
Consider a modern baseband communication system. The
transmitter encodes data bits using a low-density parity-check
(LDPC) encoder. Pilot symbols are inserted to enhance the
performance of the receiver. This is followed by a BPSK
digital modulator. The sequence of coded modulation symbols
c= (c0, c1, ..., cK−1)are pulsed shaped using the raised
cosine filter then transmitted over an additive white Gaussian
noise (AWGN) channel impaired by multipath phenomena
according to equation (1).
C. Factor Graph
We consider the proposed channel model at the receiver in
(3). To this end, a maximum number of Npossible paths at
every time epoch which N−1of them can interfere with the
adjacent symbol and a maximum number of N0discretization
levels for power delay profiles are considered. For example, let
us consider a channel model in which N= 2. In this case, we
have three paths at every time epoch: the possible interferer
path of the previous symbol, the delayed version of the current
symbol, which contains portions of both the current symbol
and the previous one, and the line of sight signal. We pass
each of them through the receiver-matched filter to compute
the mean of conditional Gaussian observation (likelihoods).
To this end, we perform computations of the receiver-matched
filter for all possible paths offline and build a look-up table to
simplify the message-passing algorithm. Four numbers for the
above-mentioned signals are obtained at the receiver-matched
filter output. Let’s name them as βk,γk,ζk, and ψkin which
krepresents the current time epoch. From the system model,
observation at time kcan be computed by
rk=βk+γk+ζk+ψk +nk,(k= 0,1, ..., K −1) (4)
where rkis obtained after receiver matched filter. {nk}is
a sequence of independent and identically distributed (i.i.d.),
Gaussian noise components, i.e., nk∼ N (nk; 0, σ2).ψkis
obtained by matched filtering the line of sight signal. βkis
obtained by matched filtering a portion of the previous symbol
at time k−1, i.e., the delayed version of the symbol at time
k−1.γkis obtained by matched filtering a portion of the
previous symbol at time k.ζkis obtained by matched filtering
a portion of the current symbol at ktime epoch. Computations
of these signals by considering equation (4) are summarized
by
ψk= exp −l0TN
AE0a0(N0)
1−AE0a0(N0)!×
[ckp(n)]NTN
n=l0TN∗p∗(−n)|n=NTN
(5)
in which AE0and l0are detected parameters by the receiver
at epoch k.p(n)is the pulse shaping filter (here is the raised
cosine one). NTNis the symbol duration. l0TNis the delay
of LOS path. We assumed that LOS always exists.
βk= exp −li00 TN
AEi000 a0(N0)
1−AEi000 a0(N0)!×
[ck−1p(n)]NTN
n=NTN−li00 TN∗p∗(−n)|n=N TN
(6)
where AEi000 and li00 are detected channel parameters at time
epoch k−1.
γk= exp −liTN
AEi0a0(N0)
1−AEi0a0(N0)!×
[ck−1p(n)]NTN−liTN
n=0 ∗p∗(−n)|n=NTN
(7)
ζk= exp −liTN
AEi0a0(N0)
1−AEi0a0(N0)!×
[ckp(n)]NTN
NTN−liTN∗p∗(−n)|N TN
(8)
The receiver can be implemented through factorizing the
joint posterior distribution of data bits and channel parameters,
given the observation samples after the matched filter and
applying the SPA to the corresponding factor graph.
p(b,P E,AE ,l|r)∝P(b)P(P E )P(AE)P(l)p(r|b,P)∝
K−1
Y
k=0
prk|ck, ck−1,P Ek,P E k−1,AEk,AEk−1,lk,lk−1
N−1
Y
k0=0
P(P Ek
k0)
N
0
−1
Y
k00 =0
P(AEk
0
k00 )
N−1
Y
k000 =1
P(P Ek−1
k000 )
N
0
−1
Y
k0000 =1
P(AEk
00
k0000 )P(lk)P(lk−1)T[c=ηc(b)]
(9)
where T[.]is the code indicator function that is equal to 1if c
is the codeword corresponding to band to zero, otherwise. It
should be noted that unlike AE and P E, which are detected by
the receiver, the receiver tracks lthrough a semi Bahl-Cocke-
Jelinek-Raviv (BCJR) algorithm (Just like phase tracking in
[9]) since there is a markovianity between their successive
samples as follows.
P(lk) = P(l0)
N−1
Y
i=1
P(li|li−1)(10)

3
Code Check Nodes
ck−1ck
Pdk−1
pk−1
uk−1(ck−1)Pk
dk(ck)
Pk−1
dk(ck−1)
Pk
uk−1(ck−1)
prk−1|ck−1, ck−2,PE k−1,P Ek−2,AE k−1,AEk−2,lk−1,lk−2prk|ck, ck−1,P Ek,P E k−1,AEk,AEk−1,lk,lk−1
... ...
PE k−1
0PE k−1
1PE k−1
N−1PE k
0
PE k
1
PE k
N−1
P(PE k−1
0)P(PE k−1
1)P(PE k−1
N−1)P(PE k
0)P(PE k
1)
P(PE k
N−1)
......
pk−1
uk−1(PE k−1
1)
pk
dk−1(PE k−1
1)
...
...
...
AEN−1
0AEN−1
1AEN−1
N0
−1
AE0
0AE0
1AE0
N0
−1...
...
... AE0
0
AE0
1
AE0
N0
−1
AEN−1
0
AEN−1
1
AEN−1
N0
−1
P(AE0
0)P(AE0
1)
P(AE0
N0
−1)
P(AEN−1
0)P(AEN−1
1)
P(AEN−1
N0
−1)
P(AEN−1
0)P(AEN−1
1)P(AEN−1
N0
−1)
P(AE0
0)P(AE0
1)P(AE0
N0
−1)
pk−1
uk−1(AEN−1
0)
pk
dk−1(AEN−1
0)
...
pk
dk−1(lk−1
N−1)
pk−1
uk−1(lk−1
N−1)
lk
0
P(lk
0)P(lk
1|lk
0)P(lk
N−1)
lk
1lk
N−1
...
lk−1
0
P(lk−1
0)P(lk−1
1|lk−1
0)P(lk−1
N−1)
lk−1
1lk−1
N−1
Fig. 1. Factor graph representation of joint detection and decoding over multipath channels.
where P(l0)is a uniform distribution in the range of 3.125
up to 12.5percentage of samples per symbol. P(lk|lk−1)is a
transition probability matrix.
The corresponding factor graph is depicted in Fig. 1. From
equation (6), one can conclude by considering the mentioned
example.
prk|ck, ck−1,P Ek,P E k−1,AEk,AEk−1,lk,lk−1∝
exp (−(rk−(βk+γk+ζk))2
2σ2)
(11)
III. KNOW N PATH
The iterative receiver consists of three algorithms to detect
PEs first, then detect AEs and at the final step, track the delay
profile of the channel. This is followed by hard decisions on
variable nodes. The detection may preferably be performed
bit by bit. This detection procedure is iterative. After some
iterations, LDPC decoding is performed. The entire process is
iterated a few times. Then, computations of the error rate are
performed at the receiver. In this article, we only evaluate the
performance on the known path algorithm, i.e., the benchmark
for the main algorithm.
In a known path scenario, the receiver has all the channel
impairments information (not the transmitted symbols). Thus,
the algorithm is more complex than its counterparts in other
scenarios (especially when the interferer paths corrupt the
adjacent symbol and other successive symbols). Therefore,
in the known path algorithm, detector part of the factor
graph does not exist. We should apply forward and backward
recursions to compute bit log-likelihood ratios (bitLLRs). To
this end, we do not have the backward messages in the forward
recursion. And we assume that we do not have the forward
ones in backward recursion. We perform one shot forward
and one shot backward recursions. Then, LDPC decoding is
performed. The overall process is iterated for a small number
of iterations. Forward and backward recursions are explained
mathematically as follows.
pk
uk(ck) = X
ck−1
Pdk−1pk−1
uk−1(ck−1)×
prk|ck, ck−1,P Ek,P E k−1,AEk,AEk−1,lk,lk−1
(12)
pk+1
uk(ck) = X
ck+1
Pdk+1 pk+1
uk+2 (ck+1)×
prk+1|ck+1 , ck,P Ek+1,P E k,AEk+1,AE k,lk+1,lk
(13)
where, we do hard decisions on Pdk−1and Pdk+1 to determine
the symbols ck−1and ck+1, respectively. Finaly, compute
bitLLRs from
puk(ck) = pk+1
uk(ck)pk
uk(ck)(14)
IV. SIMULATION RESULTS
To evaluate the performance of the known path algorithm
in multipath scenarios, computer simulations were carried out.
To this end, we draw the BER versus the signal-to-noise
ratio (SNR) curves. We consider a (3,6)-regular LDPC code

4
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Eb/N0
10-4
10-3
10-2
10-1
BER
LDPC
0.33-2 Known Path
0.465-3 Known Path
0.52-4 Known Path
Fig. 2. Simulation results for known path algorithm in joint detection and
decoding over multipath channels.
of length 4000, the employed code in [10] and [11]. The
modulation format is BPSK. We set the pilot interval to 4000
(1pilot at first position on every packet and 1pilot at last). In
a baseband communication system, pulse shaping was carried
out using raised cosine filter with a filter span per symbol
equal to 10 and samples per symbol equal to 64, i.e., dividing
the symbol duration to 64 (64 possible PEs). We assumed
that symbol duration is much smaller than the coherent time
of the channel; thus, no Doppler effect. We also considered
the transmitter and receiver filter delays in our simulations.
Channel was simulated according to the following parameters.
We fixed the number of paths and attenuation parameters, i.e.,
a= 0.33/(1 −0.33) for L= 2,a= 0.465/(1 −0.465)
for L= 3,a= 0.52/(1 −0.52) for L= 4. We assume
that LOS always exists and its delay is 0. It is important to
mention that this assumption should not be considered in the
main algorithm where synchronization is also considered. So
for the main algorithm, we consider that delay and attenuation
of the LOS path are random and not zero. The delay profile
of the channel follows a Weibull distribution with a scale
parameter equal to 1and a shape parameter equal to 0.4.
Attenuation coefficients and the number of paths are chosen
such that energy per bit of the above-mentioned scenarios is
similar at the receiver at every point. So, a fair comparison of
their performance could be carried out (the signal’s energy
at the receiver was computed during the simulations). We
consider code rate in the computation of noise variance at
the receiver. We also set the minimum error packets to 100,
detector iterations to 1, the maximum number of decoder
iterations to 200 and global iterations of the receiver to 20.
We used a lookup table to compute the means of Gaussian
likelihoods according to the explained instruction in this paper.
Figure 2 shows the simulation results. The curve labelled by
”LDPC” was obtained through pulse shaping, matched filtering
and LDPC decoding in an AWGN channel. The name of the
rest curves is based on their attenuation coefficients and their
multipath numbers, e.g., a= 0.33-L= 2 Known Path. The
simulation results prove that, at the same energy per bit, the
BER curve shows lower performance by increasing the number
of paths and shifts to the right side of the SNR axis. One may
conclude that the two-path model is almost the shifted version
of the LDPC curve by the amount of 1dB while, in successive
models, by adding an extra path, the performance degradation
is about 0.2dB.
V. CONCLUSION AND FUTURE WORK
The performance of the known path was evaluated in joint
detection and decoding over multipath channels, which utilizes
a channel discretization strategy and assigning proper priors
to the desired random variables. We will evaluate the main
detection algorithm under different schedules for future work.
This will be followed by complexity analysis. Moreover,
one may propose an information theoretic criterion for the
proposed channel discretization strategy to find the optimum
channel discretization levels.
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