Turbo-FSK, a physical layer for low-power wide-area networks: Analysis and optimization

Abstract

As the Internet-of-Things is becoming a reality, the need for a new Low-Power Wide-Area (LPWA) network emerged in the last few years. Numerous low-cost devices will be connected, and this requires an optimization of the link budget: the physical layer needs to be designed highly energy efficient. The combination of M-ary orthogonal Frequency-Shift-Keying (M-FSK) modulation and coding in the same process has been shown to be a promising candidate when associated with an iterative receiver (turbo principle). In this work, we study this new digital transmission scheme, called Turbo-FSK. An EXtrinsic Information Transfer (EXIT) chart analysis is realized. The influence of the packet length is investigated, and the scheme is shown to stay energy efficiency even with short packet sizes. Comparison with LPWA current technologies is performed, showing the potential of this technology.

Full text

C. R. Physique 18 (2017) 178–188
Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
Energy and radiosciences / Énergie et radiosciences
Turbo-FSK, a physical layer for low-power wide-area
networks: Analysis and optimization
Turbo-FSK, une couche physique pour les réseaux longue portée basse
consommation : optimisation et comparaison
Yoann Roth a,b,∗, Jean-Baptiste Doré a, Laurent Ros b, Vincent Berg a
aCEA, LETI, MINATEC Campus, 38054 Grenoble, France
bUniv. Grenoble Alpes, GIPSA-Lab, 38000 Grenoble, France
a r t i c l e i n f o a b s t r a c t
Article history:
Available online 13 December 2016
Keywords:
Turbo FSK
Low rate
Internet-of-Things (IoT)
Low-Power Wide-Area (LPWA)
As the Internet-of-Things is becoming a reality, the need for a new Low-Power Wide-
Area (LPWA) network emerged in the last few years. Numerous low-cost devices will be
connected, and this requires an optimization of the link budget: the physical layer needs
to be designed highly energy efficient. The combination of M-ary orthogonal Frequency-
Shift-Keying (M-FSK) modulation and coding in the same process has been shown to
be a promising candidate when associated with an iterative receiver (turbo principle). In
this work, we study this new digital transmission scheme, called Turbo-FSK. An EXtrinsic
Information Transfer (EXIT) chart analysis is realized. The influence of the packet length
is investigated, and the scheme is shown to stay energy efficiency even with short packet
sizes. Comparison with LPWA current technologies is performed, showing the potential of
this technology.
©2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access
article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
r é s u m é
L’Internet des objets devient une réalité, et depuis plusieurs années le besoin d’un nouveau
réseau à longue portée et basse consommation est apparu. Le but de ce réseau est de
connecter un grand nombre de nœuds à faible coût, tout en optimisant le bilan de
liaison. La couche physique doit alors être définie comme très efficace énergétiquement.
La combinaison de la modulation orthogonale de fréquence à Métats avec un codage
canal dans un processus conjoint, et non successif, à l’émission se révèle très efficace
lorsqu’un récepteur itératif est utilisé. Cet article étudie cette technique Turbo-FSK avec
l’outil d’analyse itérative EXIT (EXtrinsic Information Transfer, en anglais). La métrique est
adaptée au cas de la M-FSK, et l’influence de la taille du paquet est étudiée. On montre
alors que la technique reste performante, même lorsque que la taille de paquet est réduite.
*Corresponding author.
E-mail addresses: yoann.roth@cea.fr (Y. Roth), jean-baptiste.dore@cea.fr (J.-B. Doré), laurent.ros@gipsa-lab.grenoble-inp.fr (L. Ros), vincent.berg@cea.fr
(V. Berg).
http://dx.doi.org/10.1016/j.crhy.2016.11.005
1631-0705/©2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Y. Roth et al. / C. R. Physique 18 (2017) 178–188 179
La comparaison avec des techniques actuelles est réalisée, montrant le potentiel de la
technologie proposée.
©2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access
article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Energy is a major concern when it comes to embedded systems; some systems will harvest the surrounding sources of
energy, other will use low-power components to save it. In the context of wireless communication systems, the technique
used to transmit the information can be more or less energy efficient: for embedded systems, it becomes a critical aspect.
The Internet-of-Things (IoT) has emerged as one of the leading research areas for wireless communications as several bil-
lions connected devices are forecasted in years to come [1]. A significant part of communication transactions in the IoT
are expected to be done through Low-Power Wide-Area (LPWA) networks [2,3], for which requirements include low cost
and low energy consumption. A link budget improvement of 15/20 dB in comparison to existing cellular technologies is
expected [2], as reducing the costs to connect the devices to the wide-area network will encourage IoT deployment. To
meet these requirements, the current generation of LPWA technologies dramatically extended the sensitivity performance of
its receivers.
The sensitivity of a receiver is defined as the minimal power required to achieve an arbitrary level of error rate. It is
commonly expressed as
ρdBm =SNRdB
min +10 log10(BN0)(1)
where Bis the bandwidth of the signal (in Hz), SNRmin the minimum signal-to-noise ratio (SNR) to reach the level of error
rate, and N0the spectral density of the noise in mW/Hz (with 10 log10 (N0) –174 dBm/Hz at 20 ◦C). In Equation (1), ρ
is expressed in dBm, i.e. 10 times the log of the ratio of the power expressed in mW. The energy efficiency of a physical
layer can be estimated from the energy per bit to noise spectral density ratio, denoted by Eb/N0. This metric shows how
efficiently the energy is used: the lower is the required Eb/N0, the more energy efficient is the technology. The relationship
between the SNR and the energy efficiency is
SNR =R
B·Eb
N0=η·Eb
N0(2)
where Ris the binary data rate and η=R/Bis the spectral efficiency of the technology, expressed in bits/s/Hz. A different
expression of the sensitivity can then be given as
ρdBm =Eb
N0dB
min +10 log10(R)+10 log10 (N0)(3)
This expression clearly explains the current trend of LPWA networks towards low data rates: if the value of Ris reduced,
lower levels of sensitivity are required to guarantee the quality of service, and longer ranges of communication may be
provided by the system.
Reducing the data rate can be done by reducing the bandwidth Bfor a constant spectral efficiency, or by reducing the
spectral efficiency ηfor a constant bandwidth B. The first solution leads to narrow-band signaling, the option chosen by
the IoT company SigFox [4]. Dealing with narrow-band signals involves some technological issues, such as the necessity to
have precise oscillators. The second option, reducing the value of η, is often done by the use of the well-known spreading
factor, or repetition factor. Indeed, repeating by a factor λdivides both the values of ηand SNRmin by the same factor, thus
lowering the sensitivity level (when the bandwidth is fixed).
However, neither of these techniques change the energy efficiency, as the required Eb/N0is intrinsic to the modula-
tion used. It is furthermore bounded by the Shannon’s limit of the information theory [5], which defines the maximum
transmission rate with arbitrarily low bit-error probability, for a given SNR and bandwidth. A formulation of this limit can
be [6]
Eb
N0≥2η−1
η(4)
thus giving the minimum Eb/N0(i.e. maximum energy efficiency) for a reliable communication as an increasing function of
the spectral efficiency, with the ultimate limit Eb/N0=−1.59 dB when ηtends toward 0. As having a system’s performance
close to this ultimate limit would imply an optimal use of the energetic resource for a given spectral efficiency, it is clear
that decreasing the required Eb/N0should be a major concern.
An interesting option to combine both Eb/N0and ηreduction is the use of M-ary orthogonal modulations. Indeed, by
increasing the size of the alphabet M, the spectral efficiency (given by log2(M)/M) is reduced and the energy efficiency

180 Y. Roth et al. / C. R. Physique 18 (2017) 178–188
Fig. 1. (a) The Turbo-FSK transmitter with λstages. (b) The convolutional-FSK encoder.
is increased, eventually reaching Shannon’s limit for an infinite value of M[6]. However, since the spectral efficiency be-
comes close to 0, this solution is quite unrealistic. It is nonetheless purportedly used by a current commercial off-the-shelf
long-range solution, supported by the LoRa Alliance [7], as suggested by the patent [8] held by a company member of the
alliance. Another option to reduce both Eb/N0and ηis the use of channel coding [9]. In this area, the use of the turbo prin-
ciple [10] has been shown to be particularly efficient, but implies high consumption at the receiver’s side. Even though the
transmitter for this scheme has a low complexity, most of the current LPWA solutions rely on other Forward Error Correction
(FEC) codes, and a potential improvement can be achieved by introducing more sophisticated receiver algorithms.
The use of orthogonal alphabet and coding in the same transmit process combined with a turbo receiver was first pre-
sented for the case of the binary Hadamard code in [11]. In [12], we proposed the Turbo-FSK scheme, an adaptation of [11],
replacing the binary Hadamard codewords by the non-binary complex codewords of the orthogonal M-ary Frequency-Shift-
Keying modulation (M-FSK). This modulation is an interesting choice as its constant envelope property provides a power
efficient solution regarding the transmit power amplifier. The use of pure frequency waveforms also leads to robustness
through frequency-selective multipath channel. Demodulation can be performed using the Fast-Fourier Transform (FFT), as
in Orthogonal-Frequency-Division-Multiplexing (OFDM) receivers [13]. M-FSK is widely used for monitoring application, and
off-the-shelf optimized chips are available [14]. Limitations of the Turbo-Hadamard code from [11] have been studied in
[15], using the EXtrinsic Information Transfer (EXIT) chart analysis [16] and its extension to multi-dimensional codes [17].
The EXIT analysis is used to observe the exchanges of information inside the decoder, and to predict the “waterfall” region
of a turbo process, i.e. the region where the Bit-Error Rate (BER) curve drops significantly.
In this paper, we propose to extend the initial analysis of the Turbo-FSK done in [12]. After presenting the modulation
and demodulation procedures, we perform the EXIT analysis of the scheme, considering the modifications implied by the
use of the FSK alphabet. Using this analytical tool, we optimize the parameters of the scheme, by finding, for every set of
parameters, the best pinch-off value. This value is defined as the Eb/N0value for which the decoder can recover without
error the information word after a certain number of iterations. EXIT chart accurately predicts the convergence behavior of
the iterative decoder for large interleaving depth. Considering the LPWA context, where short packet lengths are expected,
we propose to perform extensive BER computations to confirm if the general trend obtained from the EXIT analysis is
confirmed when the packet length is shortened. The scheme is then compared to state-of-the art modulations, including
the Turbo Code standardized and used in the 3G and 4G standards [18]. This channel code is a natural candidate for the
next generation of cellular IoT network, the Narrow-Band IoT (NB-IoT) recently released by the 3GPP [19].
The paper is organized as follows: the system model is introduced in Section 2, while EXIT analysis and parameter opti-
mization are performed in Section 3. The performance of the scheme and comparison versus LPWA solutions are presented
in Section 4; Section 5concludes the paper.
2. System model
For the following parts, bold lowercase letters denote row vectors, e.g. x, and bold uppercase letters matrices, e.g. X.
The Discrete Fourier Transform (DFT) matrix of size 2r×2ris denoted Fr. jis the unit imaginary number,

zthe complex
conjugate of zand ZHis the Hermitian transpose of the matrix Z. Re(z)(resp. Im(z)) is the real part (resp. imaginary part)
of complex number z.
2.1. Transmitter
We denote by Dthe information bit matrix of size P×r, whose elements are {bp,n}, with p ∈{0, ..., P−1}and n ∈
{0, ...r−1}. We assume that the information bits are independent and equiprobable. The transmitter is shown in Fig. 1 (a).
It is composed of λstages, each one having for input an interleaved version of D.
The convolutional-FSK encoder is presented in Fig. 1 (b). The encoding process consists in adding one redundancy bit,
computed as the accumulated parity of the rinput bits. The r+1bits are then mapped to the FSK alphabet, which contains

Y. Roth et al. / C. R. Physique 18 (2017) 178–188 181
Fig. 2. Example of the mapping of the FSK codewords for the case M=4, and representation of the oversampled time waveform and the spectrum Safter
time-domain rectangular shaping.
2r+1codewords. Because of this encoding, each of the Pcodewords is linked to the previous one. An extra codeword is
appended to set back the memory to 0, leading to a total of P+1FSK codewords of length M. The alphabet Acan be
constructed as orthogonal or bi-orthogonal. In the orthogonal case, A=Frand r=log2(M) −1. In the bi-orthogonal case,
A=[Fr; −Fr]and r=log2(M). Each stage output S(k), k ∈{0, ...λ −1}, is then a (P+1) ×Mmatrix, and all the codewords
are concatenated in a serial way to be sent through the transmission channel, giving the (P+1)Mλelements of the vector x.
The code is entirely described by the parameters M, λand N. The spectral efficiency of the scheme is defined as
η=N
(N/r+1)Mλ≈r
Mλ(5)
where Nis the information block size with N=Pr. The approximation is valid when a large value of Nis considered.
For illustration purpose, a typical mapping of the FSK alphabet is represented in Fig. 2, along with the oversampled time
representation and the spectrum of each codeword.
2.2. AWGN channel and its parameters
The discrete vector xis transformed into an analog signal by using chip pulse at chip-time Tc=η/R=1/B. After up
transposition to carrier frequency, the RF real signal is transmitted over an AWGN channel with monolateral bandwidth
B =1/Tc, noise monolateral spectral density N0and useful signal average power Px=Ebη/B. At the receiver side, after
frequency down conversion, chip matched filter, and sampling at every Tc, the discrete model of the baseband signal can
be expressed in vector form as
y=x+n(6)
where nis a zero-mean circular complex white noise. All vectors are composed of (P+1)Mλchips. Each element of nis
independent and has a variance equal to σ2=N0B, and the power of the signal xis equal to Px.
2.3. Receiver
The Turbo-FSK receiver is depicted in Fig. 3 (a). After a serial-to-parallel conversion, the λvectors y(k)(with k ∈{0, ...λ})
of P+1 orthogonal codewords are fed to the FSK detector. This block consists in performing the projection of each
codewords in y(k)over the alphabet of all possible codewords, i.e. performing the matrix operation y(k)AH. Since a FSK
alphabet has been chosen, this step can be performed using the Fast-Fourier Transform (FFT) algorithm. The output Y(k)is
a (P+1)×2r+1matrix, which will be used as channel observations by its associated decoder. The turbo principle consists
in exchanging iteratively information between the decoders. After the appropriate deinterleaving, a decoder will use both
channel observation and extrinsic information from all the other decoders (the a priori information) to perform a new es-
timation of the information bits. The exchanged information between the decoders, denoted as Lin Fig. 3 (a), is the Log
Likelihood Ratio (LLR) of the information bits, defined as
L=L(bp,n)=log p(bp,n=0)
p(bp,n=1)p∈{0,...,P−1}
n∈{0,...,r−1}
(7)
where p(bp,n=1)(resp. p(bp,n=0)) is the probability that the bit bp,nequals 1 (resp. 0). The initial value for matrix Lis
all-zero, and does not include information about the trellis termination bits. While the sign of L(bp,n)gives the hard value

182 Y. Roth et al. / C. R. Physique 18 (2017) 178–188
Fig. 3. (a) The Turbo-FSK receiver with λstages. (b) Trellis of the Turbo-FSK, for an orthogonal alphabet and M=8.
of the bit, its magnitude gives the reliability. At the end of every iteration, the matrix Lincludes the information from all
the decoders, and a decision can be made on the information bits.
2.4. Probabilistic decoding
Each decoder performs an estimation of the information bits by computing the a posteriori probabilities (APP) of the
bits. We consider the received noisy FSK codeword yp, p ∈{0, ... P−1}, from the k-th stage. d={dn}n∈{0,...r−1}is the
decoded information word. ciis a codeword from the FSK alphabet A, and bi={bi
n}n∈{0,...r−1}its associated information
word (i ∈{0, ...2r+1−1}). The APP of a codeword is the probability of having the codeword knowing the observation, and
p(ci|yp)=p(yp|ci)p(ci)
p(yp)(8)
by applying Bayes’ theorem. p(ci)is the a priori probability of having the codeword ci, or that the decoded information
word dis bi. Introducing Yp
i, the i-th component of the DFT of yp, the APP becomes [12]
p(ci|yp)=Cexp 1
σ2Re Yp
i+
r−1

n=0
1−2bi
n
2L(dn)(9)
with Ca constant eliminated in further computations.
This expression conveniently uses the DFT of the received codeword, i.e. the channel observation, and a particular com-
bination of the LLR L(dn). The use of a memory in the encoding process, as described Fig. 1 (b), induces the construction
of a trellis. The Bahl, Cocke, Jelinek and Raviv (BCJR) [20] algorithm can be used to decode the trellis, as suggested in [11].
Denoting ci, i ∈{0, ..., 2r+1−1}, the codewords of the alphabet, the trellis for the special case of an orthogonal alpha-
bet and M=8is depicted in Fig. 3 (b). Each state transition is encoded by 2r−1codewords (i.e. 2 is this case). Due to
the orthogonality of the alphabet, every transition is orthogonal to each other. This orthogonality between the transitions
is a fundamental characteristic of the code. For one particular section of the trellis, the BCJR algorithm will update the
codewords probabilities with the knowledge of all the other sections. The updated probabilities are denoted P(ci|yp).
The LLR of the information bits can be computed with [21]
L(dn|yp)=log 
i∈Bn
0
P(ci|yp)−log 
i∈Bn
1
P(ci|yp)(10)
where Bn
0(resp. Bn
1) are the codeword indexes associated with an information word for which bit nequals 0 (resp. 1), with
n ∈{0, ...r−1}.
The complexity of the maximum a posteriori (MAP) algorithm presented here can be reduced using the max log approxi-
mation
log 
i
eximax
i(xi)(11)
This simplification will induce some performance loss, but reduces the computations to sum and max operations. Under
AWGN consideration, it also allows the system to work without any Channel State Information (CSI), as the term σ2from
(9) will be eliminated during the computations.

Y. Roth et al. / C. R. Physique 18 (2017) 178–188 183
3. EXIT chart analysis
The Turbo-FSK scheme offers two adjustable parameters, Mand λ. They can be optimized for better energy efficiency.
However, a theoretical approach must be done to find the best set of parameters. A useful tool for iterative process analysis
is the EXIT chart, or the tracking of the exchanges of information between the decoders. It was first introduced by ten Brink
[16], for the particular case of Parallel Concatenated Convolutional Codes (PCCC), or Turbo Codes (TC).
The idea of the EXIT chart is to compute the extrinsic information from the output of the decoder for a certain amount
of a priori information, which is simulated using a model. The metric used to measure the quantity of information is the
Mutual Information (MI). Considering two sources of information Xand Yand assuming that Xis a binary source with
equiprobable possible values +1or −1, the MI between these two sources is
I(X,Y)=1
2
x=±1y
fY|X(y|X=x). log 2fY|X(y|X=x)
f(y|x=+1)+f(y|x=−1)dy(12)
with fthe probability density functions. The value of the MI is between 0 (independence between the two sources) and 1
(complete correlation).
3.1. A priori model
The a priori information model chosen is the standard Gaussian model N(μA, σA), as suggested in [16]. The observation
of the exchanged LR in the decoder allows us to validate this hypothesis. Assuming the consistency of the LR, we have
μA=σ2
A/2, such that the model is now described by only one parameter. The a priori LR are expressed as
LA=μA·x+nA(13)
where xis the information (x =±1) and nAis a real white Gaussian noise with zero mean, with standard deviation σA. The
MI between the information source Xand the a priori source can be expressed with the function Jdefined in [16], giving
IA=J(σA)=1−+∞
−∞
1
√2πσA
exp ⎧
⎨
⎩−1
2σ2
Az−σ2
A
2x2⎫
⎬
⎭
log (1+e−z)dz(14)
3.2. Extrinsic information
The output of a decoder can be separated into three types of information: the a priori information LA, the channel
observation Lch and the extrinsic information LEgenerated by the decoder. The output LR is thus expressed as
L=LA+Lch +LE(15)
Extracting LEcan be realized by simply substracting the two other informations from the output LR L. The classic TC
construction of a codeword includes a systematic part, i.e. the uncoded information bits, and a parity part, or the redundancy
bits added by the encoder. Obtaining the channel observation of the systematic bits and computing the extrinsic LR is
then straightforward. However, in the Turbo-FSK case, the construction of the codewords does not include any systematic
part: the codewords are elements of the DFT matrix, composed of complex symbols. Channel observation Lch cannot be
retrieved: decoding only the FSK modulation (without decoding the trellis) would still include the extrinsic information of
the orthogonal code itself. This residual information from the channel must be taken into account for further interpretations.
3.3. Multi-dimensional EXIT chart
When two decoders are used, the exchange of information between the decoders during the successive iterations can
be represented in a two-dimension plot. The case where the iterative process includes more than two decoders has been
studied in [17]. If λdecoders are considered, the representation of the exchanges needs to be done in λdimensions.
Graphical interpretation becomes impossible for λ >3, but for the special case where all decoders are the same (which is
the case for the Turbo-FSK), a two-dimensional projection may be computed. If this projection does not intersect with the
line going from (0, 0)to (1, 1), it indicates that convergence toward the maximum of MI is possible. The formula of the
a priori MI must be changed [17] to
IA=J√λ−1·σA(16)
This formula can be interpreted as the fact that each decoder receives information from the λ −1other decoders.

184 Y. Roth et al. / C. R. Physique 18 (2017) 178–188
Fig. 4. The EXIT chart computation.
Fig. 5. (a) EXIT charts of the Turbo-FSK decoder with parameters M=128, λ =4, for various values of Eb/N0. The information block size was set to N=
100,000. (b) EXIT chart and BER performance of the Turbo-FSK scheme with parameters M=128, λ =4. The EXIT chart is computed for Eb/N0=−1.12 dB
and the information block size was set to N=100,000. BER performance is computed using N=100,000 and the MAP algorithm.
3.4. EXIT chart computation
The process to compute the EXIT chart is depicted in Fig. 4. As previously mentioned, the code is not systematic, implying
the impossibility to extract the channel observation and the necessity to convert the a priori LR to codewords probabilities.
The subtraction after the decoder will give an extrinsic information that still contains the channel observation. The extrinsic
MI block computes the integrals summation from (12). The a priori MI is computed using the function Jfrom (14) and the
correction of (16). For a specific value of σch, the extrinsic MI is computed for several values of σA, so that the a priori MI
spans between 0 and 1. The EXIT chart computation does not depend on the interleaving function, and needs to be done
using very large block sizes, to ensure a good statistic for the density probability functions estimations (this assumption also
implies statistical independence of all incoming messages into the decoder).
The EXIT chart computation results for three values of Eb/N0are depicted in Fig. 5 (a), along with the diagonal (0, 0)
to (1, 1). The Turbo-FSK scheme with parameters M=128, λ =4is selected, using an orthogonal alphabet, an information
block size of N=100,000 and the MAP algorithm. For Eb/N0=−1.6dB, the EXIT chart intersects the diagonal around
IA0.1. It can be concluded that whatever the number of iterations, the decoding process cannot converge towards the
error-free information word for this Eb/N0value. However, for Eb/N0values of 0 and 1dB, there is no intersection with
the diagonal: the process will then, after a certain number of iterations, successfully retrieve the exact information word.
An exhaustive search using these parameters shows that the smallest value of Eb/N0for which the EXIT chart does not
intersect with the middle line, or the “pinch-off” value, is Eb/N0=−1.12 dB. The EXIT chart for this value is illustrated
in Fig. 5 (b), along with the BER performance of the Turbo-FSK scheme with the same parameters, and 100 decoding

Y. Roth et al. / C. R. Physique 18 (2017) 178–188 185
Tabl e 1
Pinchoff values in Eb/N0for every Mand λtested. EXIT charts are computed using an information block size N=100,000. Best value for each Mis
denoted with a symbol ∗, and the best couple of parameters is in bold.
λM
16 32 64 128 256 512 1024 2048
22.23 1.61 1.15 0 .86 0.53 0.30 −0.13 −0.61
30.33 −0.17 −0.59 −0.94 −1.23∗−1.30∗−1.19∗−1.07∗
40.02 −0.46 −0.84 −1.12∗−1.20 −1.04 −0.83 −0.59
5−0.06∗−0.51∗−0.86∗−1.09 −1.03 −0.77 −0.47 −0.18
6−0.06∗−0.51∗−0.84 −1.02 −0.84 −0.52 −0.14 0.19
Fig. 6. Values from Tabl e 1 (EXIT chart analysis with N=100,000) and BER simulation for N=1,000 and N=100 in the plot normalized spectral efficiency
versus energy efficiency.
iterations. The EXIT Chart predicts accurately the beginning of the “waterfall” region, i.e. the Eb/N0region where the BER
drops significantly. Indeed, after the pinch-off value, the BER decreases to reach a value of 10−5at Eb/N0=−1.03 dB.
3.5. Parameters optimization
The performance of the Turbo-FSK scheme directly depends on the value of parameters M, λand N, the information
block size. Since the EXIT chart predicts the waterfall region, minimizing the pinch-off value will give the most efficient
scheme under the assumed hypothesis. An exhaustive search of the pinch-off value is done for every set of parameters
(M, λ), using an orthogonal alphabet. Coherent detection of the FSK is assumed and a MAP algorithm without any ap-
proximation is used. The results are presented in Table 1. For each value of M, we denoted with a ∗symbol the lowest
pinch-off value depending on λ, and the overall lowest pinch-off is in bold. The results show that Shannon’s ultimate ca-
pacity (Eb/N0−1.56 dB for these spectral efficiencies) can be approached by only 0.29 dB, using the set of parameters
(512, 3). A similar result was shown in [15], where a Hadamard systematic bi-orthogonal alphabet was used. The perfor-
mance depending on the value of λis not predictable, justifying the exhaustive approach for the optimization.
These results show the good asymptotic performance of the code. However, for the considered LPWA scenario, the size
of the payload should be limited to a few bytes, up to a few hundreds of bytes. Small information block sizes are expected,
but the EXIT chart analysis is inaccurate when the value of Nis small [16]. We can nonetheless perform BER computations
to evaluate the performance depending on the set of parameters. For each set, we choose N=1,000 and N=100 bits and
search the Eb/N0value for which the BER reaches 10−4. Ten decoder iterations and a random interleaver are used for the
simulations. To depict the results, the normalized spectral efficiency of the scheme using the different sets of parameters,
as defined in (5), is proposed. The results can be compared to the ultimate Shannon limit as defined by (4), and are
depicted in Fig. 6. For each alphabet size, λvalues from 3to 6are represented. Since increasing λlowers the normalized
spectral efficiency, the point with the highest ηis λ =3, then 4, and so on. The figure clearly shows the optimum value
of the parameters, for both EXIT and BER analysis. The general trend obtained with the EXIT chart is confirmed even
when the block size is shortened. Reducing the block size to N=1,000 bits (125 bytes) induces a performance loss of
approximately 1dB (on average), implying a minimum gap to Shannon’s limit equals to 1.3 5 dB. For a block size of N=100,
the performance loss compared to the asymptotic pinch-off is 3dB, with the best set of parameters (128, 4)being 3.42 dB
away from Shannon’s limit.

186 Y. Roth et al. / C. R. Physique 18 (2017) 178–188
4. Performance
As the Turbo-FSK parameters have been optimized to allow high energy efficiency, our interest goes into comparing
the performance of the scheme against current published LPWA solutions. Comparisons are done versus three other state-
of-the-art Machine-to-Machine (M2M) transmission schemes: the IEEE 802.15.4k standard [22], the LoRa physical layer, as
described in the patent [8], and the serial concatenation of a TC [18,23] with a M-FSK modulation.
4.1. Proposed solutions
For each solution, the physical layer is first described. Since mostly the transmitter side is discussed in the literature,
we assume for each scheme the receiver to be the dual receiver version of the transmitter, with soft decoding (MAP) and
demodulation when possible. The parameter λis introduced, which will allow the spectral efficiency to be modulated for
comparison purpose.
4.1.1. IEEE 802.15.4k
The IEEE 802.15.4k is a standard designed for local and metropolitan area networks. It is part of the Low-Rate Wireless
Personal Area Networks (LR-WPANs). This standard supports three PHY modes: DSSS with BPSK or Offset-Quadrature Phase
Shift Keying (O-QPSK), or 2-FSK. Since DSSS with BPSK modulation is the more robust scheme, we select this configuration.
The standard specifications allow the use of a spreading factor value λfrom 16 to 32,768. In addition to the modulation
and the spreading factor, the standard uses FEC, a convolutional code of rate 1/2, with generators polynomial [171 133]
(in octal) and constraint length k =7. An interleaving operation is considered between the steps of channel coding and
modulation. The spectral efficiency of the scheme is defined as
η1=1
2λ(17)
Soft decoding using the well-known soft-input soft-output Viterbi decoder is considered.
4.1.2. OSSS with block code
Non-linear modulations based on an alphabet of M-ary orthogonal waveforms are known to be energy efficient, and to
reach channel capacity for infinite size of alphabet [6], as discussed in the introduction of this paper. This property can
be extended to Orthogonal Sequence Spread Spectrum (OSSS), and is purportedly used by the LoRa technology [8] with a
Hamming block code in addition. For this study, we consider a Hamming block code of rate 4/7. The parity bits can be
punctured to increase the rate. We denote by pthe number of parity bits (p ∈{0, 1, ...3}). After encoding, the transmitter
consists in an interleaver and an orthogonal modulation of size 2λ. The spectral efficiency is equal to
η2=4
4+p
λ
2λ(18)
The receiver side consists in a soft demodulation, and soft Hamming decoding.
4.1.3. The UMTS/LTE TC
The recent 4G standard specification includes a TC [18]. Turbo codes are known to be very effective FEC schemes, and its
use in the Cellular-IoT context is considered. They consist in the parallel concatenation of two identical recursive systematic
convolutional codes. In the specification, each code is rate 1/2, of constraint length k =4with feedback and generator
polynomials [13 15](in octal). The overall rate of the TC is approximately 1/3(extra few bits are required to close the
trellis). With this FEC, we consider an orthogonal modulation of size M, and a spreading factor λ. The spectral efficiency
can be expressed
η3=log2(M)
3λM(19)
The receiver side consists in a turbo receiver [10,23]. The use of soft iterative decoding implies high consumption at the
receiver side, but complexity at the transmitter side is rather low. This scheme, denoted as FSK +TC in the sequel, is an
interesting element of comparison against our proposed Turbo-FSK scheme. Indeed, it also implies the use of a powerful
channel code and an orthogonal modulation, but in two successive steps, instead of the joint process used in the Turbo-FSK
scheme.
4.2. Performance comparison
To compare the different schemes, we compute simulations for the AWGN channel, with coherent reception and ideal
synchronization. Random interleavers are used, except for the TC UMTS where the interleaver specified by the 3G standard
is used. The packet size is set to N=1,000. For the iterative receivers, 10 iterations are performed (TC and Turbo-FSK),
and the MAP algorithm without any approximation is used. We choose, for the Turbo-FSK, the parameters M=128 and

Y. Roth et al. / C. R. Physique 18 (2017) 178–188 187
Tabl e 2
Parameters used for comparison.
PHY-layer 802.15.4k OSSS FSK +TC Turbo-FSK
Modulation DBPSK 512-Orthog 128-FSK 128-FSK
FEC CC [171 133] Hamming TC [13 15] Turbo-FSK
Binary code-rate 1/24/6 1/3–
λ43 9 24
η(·10−2)1.163 1.172 0.907 1.170
Fig. 7. (a) BER and (b) PER performance comparison versus Eb/N0, using the parameters given in Tab le 2. The packet size is set to 1,000 bits.
λ =4, i.e. the best couple of parameters for this size of alphabet. The spectral efficiency is equal to η=1.17 ·10−2for
these parameters. For each scheme, we can choose a different value of λin order to equalize the spectral efficiency of the
schemes. The selected parameters are summarized in Table 2.
Fig. 7 (a) depicts the BER performance versus Eb/N0for the selected parameters. The OSSS scheme uses the Hamming
code, which is less powerful than the convolutional code of the IEEE 802.15.4k standard, but offers better performance when
combined with a relatively large size of orthogonal alphabet (512). The gain using the turbo principle is illustrated by the
performance of both the FSK +TC and the Turbo-FSK. The scheme FSK +TC reaches a BER of 10−5for value of Eb/N0of
2.91 dB, and the Turbo-FSK outperforms all the other scheme, showing a 4.8 dB gain versus the OSSS +Hamming scheme
for the same BER. The gain offered by the Turbo-FSK scheme versus the scheme FSK +TC also shows the benefit of jointly
optimizing the modulation and the channel coding instead of treating them separately.
The Packet Error Rate (PER) versus Eb/N0is depicted in Fig. 7 (b). For a PER of 10−3, the Turbo-FSK with these param-
eters outperforms the scheme OSSS +Hamming code by 5.5 dB. This level of PER is reached for Eb/N0=0.12 dB. Using
Equation (2), with the spectral efficiency given in Table 2, the equivalent SNR is equal to −19.2dB, demonstrating the
ability of the system to work at low levels of SNR.
These two figures show the real benefit of turbo processing on the sensitivity gain. Because all the spectral efficiencies
are normalized, the Eb/N0gain can be interpreted as the sensitivity gain between two schemes, when the same bandwidth
is considered (see Equation (3)). The 5.5 dB gain between Turbo-FSK versus the LoRa based scheme for a PER or 10−3
means that the sensitivity level will be lower using our scheme. This can be interpreted as a potential reduction of the
transmit power by a factor 3.5 while ensuring the same level of performance, or a distance increase by a factor 1.8(under
the approximation of a free path loss exponent equal to 2). Comparison with other schemes could be considered; the use
of a turbo code with a linear modulation also offers a good tradeoff between performance and spectral efficiency.
The sensitivity performance improvements are done at the expense of an increase of complexity at the receiver side. As
we focus on the complexity at the node level, having a more complex receiver at the base-station is acceptable. However, a
recent study showed that the Turbo-FSK physical layer can be implemented on off-the-shelf components [24], demonstrating
that the complexity increase can be handled by components with low computation capacity.
5. Conclusion
While global LPWA solutions aim at lowering the sensitivity, improving the energy efficiency should remain a major
concern. The Turbo-FSK scheme proves to be a serious candidate for the PHY layer of LPWA networks, offering high energy
efficiency with reasonable spectral efficiency. The optimization realized using the EXIT analysis and the performance versus
State-of-the-Art solutions show that very interesting gain in sensitivity can be achieved. The best set of parameters is
shown to be only 1.3 5 dB away for Shannon’s limit for a packet size of 128 bytes, a fair size in the M2M context. The
Turbo-FSK demonstrates the positive impact of mixing orthogonal modulations and turbo processing, and achieves promising
performance while ensuring low complexity and a constant envelope at the transmitter side. Further studies should however

188 Y. Roth et al. / C. R. Physique 18 (2017) 178–188
be considered about the complexity increase, and the synchronization, which can reveal itself arduous for the considered
levels of SNR.
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