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Signal Processing Methods for Filtered Multitone
Receivers in Underwater Acoustic Communication
Systems
Luıs Pedro Pereira da Silva
Thesis to obtain the Master of Science Degree in
Electrical and Computer Engineering
Supervisor: Prof. Doctor Joao Pedro Castilho Pereira Santos Gomes
Examination Committee
Chairperson: Prof. Doctor Jose Eduardo Charters Ribeiro da Cunha Sanguino
Supervisor: Prof. Doctor Joao Pedro Castilho Pereira Santos Gomes
Members of the Committee: Prof. Doctor Rui Miguel Henriques Dias Morgado Dinis
May 2015
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Acknowledgments
During the year I spent working on this thesis a lot of people have helped, more than I can possible
mention here, to all of them my deepest thanks.
I begin by thanking to my adviser, Professor Joao Pedro Gomes, for all the trust he had in me and
in my autonomous work, always encouraging me to go further.
I thank to all my friends for the encouragement to push through the difficulties and also to my lab
colleagues for their support and the good work environment.
To all my family, that helped me keep going in the bad times and didn’t let me relax (too much) when
things were going well.
In particular I would like to thank to Beatriz Ferreira, Andre Matias and Vasco Ludovico for helping
with the hardest part of my thesis.
To Maria Raposo, with whom all the time spent was never wasted.
Finally, I thank my brother, Tiago, and my sister, Ana, for all times that they step in for me during
this year.
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Resumo
As comunicacoes sub-aquaticas tem sofrido uma grande evolucao nas ultimas decadas, atraindo cada
vez mais utilizadores que constantemente procuram maneira de ter maiores ritmos de transmissao.
Esta tese aborda o problema da escolha da modulacao que melhor se ajusta ao canal sub-aquatico. E
um facto bem sabido que o mar e um ambiente bastante adverso para comunicacao, pelo que so podem
ser usadas frequencias acusticas baixas o que limita os ritmos de transmissao possıveis. Isto significa
que o desempenho e a eficiencia espectral da modulacao utilizada sao determinantes para se aproximar
o maximo possıvel dos limites teoricos. Em particular sao comparadas a popular modulacao Orthogonal
Frequency Division Multicarrier (OFDM), com a modulacao mais robusta Filtered Multitone (FMT),
ainda que possa ter menos eficiencia espectral, e com a modulacao Single-Carrier. Varios algoritmos
sao deduzidos e implementados para os receptores das varias modulacoes, dando um especial enfase a
compensacao dos ecos e do efeito de Doppler. A esparsidade tıpica dos canais sub-aquaticos e levada em
consideracao para melhorar metodos ja existentes de estimacao e seguimento de canal.
Para o receptor de Filtered Multitone um novo metodo de treino e proposto. Para alem disto, um
conhecido metodo que combina informacao de diferentes sub-portadoras para melhorar as estimativas de
canal e reformulado.
Finalmente, todas as modulacoes sao comparadas em simulacao e atraves de dados experimentais.
Keywords: Comunicacoes sub-aquaticas, Orthogonal Frequency Division Multicarrier, Filtered Mul-
titone, Estimacao de resposta de impulsiva de canais esparsos, Treino de channel estimation based decision
feedback equalizers.
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Abstract
Underwater communications have seen a large increase in the last decades, with new developments
bringing more users who are always seeking for higher transmission rates.
This thesis focuses on the problem of choosing suitable modulation for underwater communications. It
is known that the ocean environment is very harsh on communications, so only low acoustic frequencies
can be used, which drastically limits the possible bit rates. This means that the performance and
spectral efficiency of the used modulation are crucial for approaching the theoretical limits. In particular
this work compares the popular modulation, Orthogonal Frequency Division Multicarrier (OFDM), the
more robust, yet possibly with worse spectral efficiency, Filtered Multitone (FMT) modulation and the
Single-carrier modulation. Several algorithms are derived and implemented for the receivers of these
modulations, with a special emphasis on compensating the multipath and Doppler effect. The sparsity
of the typical underwater channel is explored in order to improve already known methods for channel
estimation and tracking.
For the receiver of Filtered Multitone modulation, a new training method is proposed. Moreover, a
known method for combining information from different subcarriers to improve the channel estimates is
reformulated.
Finally, all the modulations are compared using experimental and simulated data.
Keywords: Underwater communications, Orthogonal Frequency Division Multicarrier, Filtered Mul-
titone, Sparse Channel estimation, Channel estimation based decision feedback equalizers training.
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Contents
1 Introduction 1
1.1 Characteristics of Underwater Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 OFDM, FMT and Single-Carrier Modulations . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Main Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Document Structure and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Orthogonal Frequency-Division Multiplexing 7
2.1 OFDM Modulation/Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Pre-Demodulation Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 OFDM Time Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 CP-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 ZP-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Channel Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Channel Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Multi-Receivers Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Filtered Multitone 23
3.1 FMT Modulation/Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Receiver Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Decision Feedback Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Channel-Estimation-Based DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Training and Initializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Channel Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 Fusion of Channel Estimates from Different Subcarriers . . . . . . . . . . . . . . . 37
3.3.3 Introducing Sparsity on the Fusion Process . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Multi-Receivers Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Multi-Receiver CHE-DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.2 Multi-Receiver Pre-Combiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Single-Carrier Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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4 Results 45
4.1 Results Relative to OFDM Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Results Relative to FMT Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Results Relative to Single-Carrier modulation . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Comparison Between OFDM and FMT Results . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Conclusion 61
5.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
References 63
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List of Figures
1.1 Example of underwater communications setup for multipath effect. . . . . . . . . . . . . . 2
1.2 Effect of sound speed variation with depth. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 OFDM pulses in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Use of a time guard to prevent inter-block interference in OFDM transmission . . . . . . . 12
2.3 Example of the threshold methods for channel sparsification. (a) Channel estimate without
noise, (b) and (d) Noisy channel estimates for different instants, (c) and (e) Application
of both methods to the correspondent noisy channel estimates. . . . . . . . . . . . . . . . 19
3.1 Raised-cosine pulses for different β values in (a) the time domain and (b) the frequency
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 FMT signaling pulses in (a) the time domain and (b) the frequency domain for β = 0.25. 25
3.3 Raised-cosine pulses, with β = 0.5, overlapping in the frequency domain. . . . . . . . . . . 25
3.4 Modulation and demodulation of FMT through the use of filter-banks. . . . . . . . . . . . 26
3.5 Structure of the basic DFE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Receiver structure for a DFE with a feedforward filter. . . . . . . . . . . . . . . . . . . . . 29
3.7 DFE with a feedforward filter and Doppler compensation. . . . . . . . . . . . . . . . . . . 30
3.8 CHE-DFE receiver with Doppler compensation. . . . . . . . . . . . . . . . . . . . . . . . . 32
3.9 Block diagram of the PLL implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.10 Application of the described training method assuming that the training sequence ends at
the 100th symbol of the packet and two repetitions are done. In (a) it is shown the original
packet and (b) is the artificially extended packet. . . . . . . . . . . . . . . . . . . . . . . . 34
3.11 CHE-DFE adaptation to a multireceiver scenario. . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 (a) Physical channel estimate for the 16 receivers and (b) the sound speed profile measured
on site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 MSE evolution as the Doppler effect increases for OFDM packets with (a) 128, (b) 256
and (c) 512 subcarriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Ratio of the MSEs between two OFDM packets throughout time. In the first packet, both
the transmitter and receiver are stationary, whereas in the second case the transmitter is
approaching the receiver at a constant speed of 2m/s. . . . . . . . . . . . . . . . . . . . . 47
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4.4 MSE of an OFDM packet with 128 subcarriers when the channel sparsity is explored with
(a) a fixed threshold and with (b) the LS-AT method. . . . . . . . . . . . . . . . . . . . . 48
4.5 MSE evolution as the number of receivers increases for an OFDM packet with 256 subcarriers. 48
4.6 Convergence of the (a) DFE, (b) DFE with feedforward filter and (c) CHE-DFE for the
first subcarrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.7 MSE for different initializations of matrix P from the RLSλ algorithm. . . . . . . . . . . . 49
4.8 Absolute values of matrix P when the RLSλ began by converging to (a) a local minimum
(δ = 10−5), (b) the absolute minimum (δ = 0.1) and (c) at the final of packet that initially
converged to a local minimum (δ = 10−5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.9 Possible MSE evolution for the different training methods as number of known symbols at
receiver increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.10 The two most common profiles of the MSE in experimental FMT1 packets as the fixed
thresholds increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.11 Evolution of the varying threshold for (a) a simulated and (b) an experimental FMT1 packet. 52
4.12 Channel estimates from all the subcarriers at the 100th symbol of a FMT1 packet for (a)
simple recursion method, (b) fixed threshold, (c) varying threshold and (d) minimization
of the L0-norm approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.13 CHE-DFE evolution through a FMT1 packet for (a) simple recursion method, (b) fixed
threshold, (c) varying threshold and (d) minimization of the L0-norm approximation. . . . 54
4.14 MSE evolution with increasing Doppler shifts for the different channel estimation methods
in packets of (a) FMT1 (b) FMT2 and (c) FMT3. . . . . . . . . . . . . . . . . . . . . . . 55
4.15 Results for the fusion process in a FMT1 packet: (a) Channel estimates, (b) High resolu-
tion channel estimate (c(n)), (c) CHE-DFE evolution and (d) constellation of the symbol
estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.16 MSE evolution for an increasing number of receiver (a) combining the signals in a single
CHE-DFE from an FMT1 packet, (b) from an FMT2 packet, (c) from an FMT3 packet
and (d) using the pre-combiner method for an FMT1 packet, (e) for an FMT2 packet and
(f) for an FMT3 packet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.17 Doppler compensation for a simulated FMT packet where the transmitter is approaching
the receiver at a constant speed of (a) 0.2 m/s, (b) 0.4 m/s, (c) 0.6 m/s and (d) 0.8 m/s. 58
4.18 Evolution of the CHE-DFE for an FMT1 packet where the receivers ambiguity destroyed
the RLSλ convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.19 Evolution of CHE-DFE on a single-carrier packet. . . . . . . . . . . . . . . . . . . . . . . 59
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List of Tables
4.1 Parameters for OFDM, FMT and single-carrier signals. . . . . . . . . . . . . . . . . . . . 45
4.2 MSE for λ = 0.995. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 MSE for λ = 0.998. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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List of Abbreviations
AUV Autonomous Underwater Vehicle
BER Bit Error Rate
CFO Carrier Frequency Ofset
CHE-DFE Channel Estimate Based Decision Feedback Equalizer
CP-OFDM Cyclic Prefix - Orthogonal Frequency-Division Multiplexing
DAB Digital Audio Broadcasting
DBV-T Digital Video Broadcasting-Terrestrial
DFT Discrete Fourier Transform
FBMC Filter-Bank Multicarrier
FMT Filtered Multitone
IBI Inter-Block Interference
ICI Inter-Carrier Interference
IDFT Inverse Discrete Fourier Transform
ISI Intersymbol Interference
LS Least-Squares
LS-AT Least-Squares Adaptive Thresholding
MSE Minimum Squared Error
OFDM Orthogonal Frequency-Division Multiplexing
OLA Overlap-Add
PLL Phase Locked Loop
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QPSK Quadrature Phase Shift Keying
RLS Recursive Least-Squares
SNR Signal to Noise Ratio
VDSL Very High-Speed Digital Subscriber Line
ZP-OFDM Zero padding - Orthogonal Frequency-Division Multiplexing
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Chapter 1
Introduction
Underwater communications have seen a large increase over the last decades, being important in a
number of areas that reach from marine research to offshore oil exploration and defense applications.
As the number of applications and needs increases, so does the technology improve and the number of
published articles sign of an increasing interest by part of the scientific community. The current inves-
tigations are focused on a diversity of subjects such as channel modeling, modulation schemes, receivers
structure and demodulation schemes, multi-receivers gain, networking, communication protocols, AUVs
communication and coding. In this work the goal is to establish a comparison between one of Filter-
Bank Multicarrier (FBMC) modulations, called Filtered Multitone (FMT), and the popular modulation
of Orthogonal Frequency-Division Multiplexing (OFDM).
1.1 Characteristics of Underwater Channels
Underwater communications are often confronted with time-varying frequency selective channels and
OFDM modulation is very sensible to such distortions, hence the motivation to search for alternatives.
Underwater high frequency signals suffer large attenuation, so the communications are done using sound.
Even with frequencies below 20-30 kHz the communication ranges do not usually go further than a few
kilometers, with the exception of ducts where sound can travel up to hundreds of kilometers [3]. Besides
the large attenuation, sea currents and sea surface variations produce a time-varying channel making it
more difficult to communicate. In deep waters, this has a low impact on transmitted signals, but on
shallow waters it is one the main causes for signal corruption. Another important phenomenon is the
constructive and destructive signal interference due to multipath. Once again, in deep waters this effect is
not as significant as in shallow waters, since on the second case there are usually more bottom and surface
reflections. A larger number of multi-paths will most likely lead to a channel frequency response where
many frequencies are attenuated or amplified. The multipath is also very common in indoors wireless
communications[5], so some solutions may be sought there, although there is a significant difference
between the intersymbol interference (ISI) in each case. In wi-fi systems, the possible paths are all much
similar among them so the echoes arrive with little or no interval whereas in underwater channel often
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there’s silence between them. Consider the communication setup in Fig. 1.1 where the transmitter and
receiver are both at a depth of 10m, the distance between them is 1,5km and the total depth is 160m.
For simplicity, the sound speed is 1500 m/s at all depths, which means that the direct ray will take 1s
to arrive, the first echo has a delay of 0.08ms and the second echo arrives about 26.5ms after the first
arrival. If each symbol has 4ms, the first echo is not distinguishable from the main arrival, although it
may have a destructive effect, but the second echo will interfere with the signal 27 symbols later. This
shows why the channel in underwater communications is almost always sparse, despite the several echoes.
160m 150m
10m 10m
150m
1,5km
750m
Figure 1.1: Example of underwater communications setup for multipath effect.
Contrary to this short example, the sound speed varies with the water density which usually increases
with the depth, although some warm or cold currents can cause other type of sound speed profile. The
Fig. 1.2(a) shows the various types of possible paths on a different topology and taking into account the
variation of the sound propagation speed in Fig. 1.2(b).
(a)
Sound speedDepth
0 m
(b)
Figure 1.2: Effect of sound speed variation with depth.
Note that as the water gets deeper the distance from the surface to the bottom becomes so large
that the attenuation does not allow for more than one reflection. Since the multipath and the time
variations are lighter in deep waters, the first underwater communications were done in that type of
channels. Although they allow for easier solutions, deep waters are usually further from the coastline.
So the current challenge is to develop methods that can deal with multipath in a time-varying channel
hence achieving a good performance in shallow waters or even in surf zones.
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Besides the important mentioned channel distortions, communications often occur between a base
station and AUV or between two AUVs. This means that Doppler effect has to be taken into account or
the communication will be limited to a fixed transmitter and receiver. Even if a transmitting or receiving
AUV isn’t moving, most likely it will be drifting which can also cause a non-negligible Doppler effect[17].
In a moving scenario, besides the Doppler effect, the channel time-variations will be much faster and
possibly abrupt, hence the method of fighting the multi-path effect must be very robust. It is important
to note that if the transmitter and receiver are moving with same speed and in the same direction there
will be no Doppler effect, but channel time-variations can still occur due to changes in the bottom of the
sea, motion of the sea surface, changes in the sound speed, etc.
1.2 OFDM, FMT and Single-Carrier Modulations
Multicarrier modulations have been long considered for many types of communications, including
non-submarine. Particularly, Chang[1] and Saltzberg[2], respectively in 1966 and 1967, published some
of the first developments on the Filter-Bank Multicarrier (FBMC) systems on the domain of telephone
communication on twisted pairs. Since then many developments were done both on FBMC and in general
multicarrier communications. On the 80’s and 90’s, some of these type of modulations were used for VDSL
applications, namely OFDM, and FMT [9],[12]. Later, OFDM was also considered for DAB and DBV-T
[7]. In most applications OFDM has the upper hand against FMT and other FBMC techniques since
it has a more efficient use of the spectrum. Still in some cases, e.g. where a significant Doppler effect
is present, OFDM performance decreases and more robust modulations, like FMT, may achieve better
results.
In order to mitigate all the mentioned channel distortions many techniques have been proposed and
implemented, especially in OFDM modulation. Some of these improvements lead to significant bit error
rate (BER) decrease, e.g., powerful coding and turbo equalization inspired in turbo coding [18], but won’t
be considered here because the goal is to compare the two modulations and it’s considered that those
type of techniques can be later adapted for FMT, or other modulations, producing similar performance
increases to the ones observed in OFDM. The focus of this work is on the basic receiver components that
allow to mitigate the channel distortions on both modulations.
In OFDM modulation, time guards will be considered to fight the multipath effect. To compensate
the Doppler effect a well known resampling approach will be employed, this is common to FMT and
single-carrier modulation, its the remaining residual Doppler that will be corrected differently for each
modulation. For OFDM the residual Doppler will be estimated as part of the channel tracking. Other
channel distortions, like ISI, are corrected by computing a channel estimate, where several methods will
be analyzed, some of which take into account the channel sparsity. To account for the channel variability
and correct the residual Doppler effect, channel tracking methods are implemented. To further improve
the overall performance the possible spacial diversity gain from using several receivers is investigated.
In FMT modulation a channel estimate based decision feedback equalizer (CHE-DFE) will be consid-
ered for the receiver. Several ways of computing the channel estimate will be considered. The CHE-DFE
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needs an initial training sequence which decreases the transmission efficiency. By exploring different
training methods and speeding up the convergence of the channel estimates, the CHE-DFE convergence
can be improved in its rate and in its quality, meaning in the number of correctly identified symbols. To
further improve the channel estimates, an algorithm for combining information from different subcarriers
will be proposed. Finally, a multireceiver scenario will also be considered.
A single-carrier modulation is also used, mainly to serve as a benchmark to OFDM and FMT. The
receiver for a single-carrier signal will be the same as in FMT modulation with slight adaptations.
To further enrich this work, the three types of modulation are compared through computer simulation
and from real data of an experiment realized in 2010 at Algarve’s shore, near Vilamoura, on the scope
of “Field Calibration and Communications Sea Trial” (CalCom’10). The signals from the simulation
and experimental setups have the same characteristics and include demodulations with several of the
developed and implemented methods, as well as variations of each modulation, i.e., on the number of
subcarriers.
1.3 Main Contributions of This Thesis
Throughout this thesis a special emphasis is given to the characteristics of the underwater channels so
the first contribution is the conclusions that come from that line of tought that allowed a careful adaption
of the used algorithms, highlighting better initializations or possible simplifications, and considering others
as unsuited for underwater communications.
Most of the methods used for the OFDM receiver have already been studied since this has been
applied to a number of communications systems, nonetheless some innovation is introduced by adapting
them to underwater communications, specifically on the channel estimation and tracking methods.
In the CHE-DFE, that will be used at the core of the FMT receiver, an innovative training method
is developed that allows for significant reduction of the number of symbols used for training and possible
improvements of the errors. Several ways for improving the channel estimates by taking into account
the underwater channel sparsity are compared for different situations, including Doppler effect. On the
multi-receiver scenario, a proposed method in the literature is tested concluding that it has significant
limitations, achieving similar or worse results than a simple extension of used CHE-DFE to this case.
Finally, a process of fusion that combines information from all subcarriers allows for even more accurate
channel estimates that further decrease the error. An adaption of this algorithm to a sparse channel
scenario is developed. This, together with the channel estimation algorithms, is the subject of the paper
“Sparse Channel Estimation and Equalization for Underwater Filtered Multitone Modulation”, that was
submitted to MTS/IEEE Oceans Conference of 2015.
Despite the fact that the important technical advances are done in the FMT receiver, the principal
goal of this thesis is the comparison between OFDM and FMT. By doing this, the main contribution is
the possibility of finding the best modulation between OFDM, FMT and single-carrier for underwater
communications given the specific characteristics of the underwater transmission channel.
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1.4 Document Structure and Notation
The OFDM and FMT modulations are analyzed and detailed descriptions of the receivers are given
in chapters 2 and 3, respectively. Due to its close relation with FMT modulation, the receiver used for
single-carrier is described in the last section of 3, which also includes a short theoretical comparison with
FMT. Simulation and experimental results are presented in chapter 4 that ends with a comparison of the
results for OFDM and FMT. The conclusions and possibilities of future improvements are presented in
chapter 5.
Throughout this work matrices and vectors will be denoted by uppercase and lowercase boldface
letters. The symbols T , ∗ and H will stand for transpose, conjugate and hermitian, and the convolution
will be identified as ∗.
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Chapter 2
Orthogonal Frequency-Division
Multiplexing
In this chapter the OFDM is analyzed beginning in section 2.1 by explaining the modulation/demo-
dulation process in the absence of a channel. Since, in underwater conditions the received signal can’t be
directly demodulated, a pre-demodulation processing is presented in section 2.2 and the use of a guard
interval is explored in section 2.3. Channel estimation techniques for compensating the remaining signal
distortion are explored in section 2.4. Finally, on section 2.5, the use of an array of receivers is considered.
2.1 OFDM Modulation/Demodulation
OFDM modulation arises from the goal of maximizing the efficiency of the use of the frequency
spectrum, something that doesn’t happen if a single-carrier signal is used occupying all the available
bandwidth. The basic idea is that each symbol, at each symbol instant, is modulated by an independent
subcarrier, meaning that each subcarrier power can be adjusted to the power of noise at its frequencies.
This allows for a closing on the channels capacity theoretical maximum. The choice of the used pulse will
be determinant on the possibility of retrieving the modulated symbols or not, and on the efficiency of the
spectrum use. In OFDM a set of complex symbols on the frequency domain, that code the information,
are loaded unto the subcarriers at consecutive time instants. Hence if there’s a sequence of K symbols
to transmit on M � K subcarriers, the symbols will be divided in blocks of M that will be transmitted
successively across the channel. In this work the OFDM block will be denoted by b, the subcarrier by m
and the continuous time by t. Taking this into account the transmitted signal on the time domain can
be written as
x(t) =∑b
M−1∑m=0
Sb(m)p(t− bT,m). (2.1)
Where the Sb(m) is the complex symbol transmitted on the m-th subcarrier at the b-th block, p(t−bT,m) is the pulse that will carry the b-th block symbol from the m-th carrier. Since the same type
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of pulse is used for all subcarriers and all time instants, just with the difference of a phase-shift and a
time-shift, the pulse can be expressed as:
p(t− bT,m) = p(t− bT )ej2πtfm (2.2)
Applying this to equation (2.1) yields:
x(t) =∑b
M−1∑m=0
Sb(m)p(t− bT )ej2πtfm (2.3)
Assuming an ideal channel, the symbols can be demodulated at the receiver if for a given decision
instant tbT and a given frequency fm all pulses have a zero value except for pm(t− bT ). In order for this
to happen every transmitted pulse must not interfere neither in the time domain, nor in the frequency
domain with any of the others at those time-frequency instants. In other words, the choice of p(t) must
be such that there is no inter-symbol interference (ISI) and no inter-carrier interference (ICI) in the
absence of channel distortions. These constraints are called the orthogonality condition as a parallelism
with orthogonal vectors whose inner product is zero. They can be mathematically summarized into a
single equation:
∫ ∞−∞
p(t− kT,m)p∗(t− nT, l)dt = δm,lδk,n (2.4)
δk,n is the Kronecker function, i.e., 1 if k is equal to n and zero in every other case. On FDM and
FBMC, the pulses are designed to be contained on a bandwidth B/M where B is the total bandwidth and
M is the total number of subcarriers. This simple approach to find a pulse that respects the orthogonal
condition, doesn’t lead to the best use of the spectrum like it was noted already in 1967 by Saltzberg[2].
Instead one should try to find a pulse with a bandwidth that exceeds the available B/M , but has a zero
value at all the other carriers center frequency. In OFDM, the chosen pulse is the sinc function because
not only it leads to a setup where the orthogonality condition is respected, but also allows for numerically
efficient implementation as it is shown next.
sinc(f) =sin(πf)
πf(2.5)
On the time domain, the sinc function corresponds to a rectangular pulse. This means that by using
simple rectangle pulses in the time domain, multiplied by complex exponentials with the subcarrier’s
frequency, one can respect the orthogonality condition and at the same time reach a greater efficiency in
spectrum use. Moreover, by using a rectangular pulse in (2.3) one gets:
x(t) =∑b
M−1∑m=0
Sb(m)rect(t− bTT
)ej2πtfm (2.6)
On the time and frequency domain the total energy of the pulse must be the same energy, so the
height of the rectangular window will be adjusted to 1/T and be defined as
rect(t) =
1 |t| ≤ 12
0 12 ≤ |t|
, (2.7)
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f
Figure 2.1: OFDM pulses in the frequency domain
such that
sb(t) = x(t)rect(t− bTT
). (2.8)
Noting that the second sum in equation (2.6) is now an inverse discrete Fourier transform (IDFT),
the equation can be rewritten as:
x(t) =∑b
sb(t) (2.9)
sb(t) =
M−1∑m=0
Sb(m)ej2πtfm (2.10)
This means that OFDM modulation can be realized with low complexity by implementing IDFTs
through very efficient fast Fourier transform (FFT) methods. The inverse process of obtaining the
symbols from the signal x(t) can also be done efficiently by using an DFT, after passing x(t) through a
rectangular window. Note that the symbols don’t overlap in the time domain, so a rectangular window
at time bT will filter only the time values corresponding to the symbols transmitted in the b-th block,
Sb(m),m = 0, 1, ...,M − 1.
Sb(m) =
M−1∑m=0
sb(t)e−j2πtfm (2.11)
Joining equations (2.8)) and (2.11), one can write the direct demodulation from the received signal
into the original symbols:
Sb(m) =
M−1∑m=0
x(t)rect(t− bTT
)e−j2πtfm =
M−1∑m=0
x(t)p∗(t− bT,m) (2.12)
Comparing this last equation with (2.1), one finds that the pulse used at the receiver should be the
conjugate of the one used at the transmitter. This was expected since at the receiver the multiplication
by ej2πtfm can be seen as applying a different phase shift to every symbol that should be undone when
one is trying to obtain the original symbol.
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Although this is the traditional setup for OFDM, it isn’t unique, for example in [30], the author
presents alternative filtered rectangular pulses whose frequency counterparts have lower side lobes than
the sinc functions. This increases the robustness to ICI at the cost of lowering the transmission bit rate or
introducing ISI. Such approaches are considered to be slight variations on the overall OFDM modulation,
hence aren’t pursued in this work.
2.2 Pre-Demodulation Processing
Considering a real scenario of shallow water communication where the received signal will be corrupted
and the orthogonality condition will no longer hold at most of the times, hence it will not be possible to
directly apply the DFT and some pre-processing will be required. The channel distortion can be divided
in amplitude attenuation and phase distortion that may be due to Doppler effect and bottom or surface
reflections, and additive noise. All of these depend on the time instant because the channel is time-
varying, and on the path crossed by the signal from the transmitter to the receiver since the multipath
effect is almost always present in underwater communication. Taking all this into consideration the
received signal can be described as
yd(t) =∑p
Ap(t)x(t)δ(t− τp(t)) + np(t), (2.13)
where yd(t) is the received signal, p denotes one of possible P paths, Ap(t) is the time-varying am-
plitude attenuation, τp(t) is the time-varying path delay and np(t) is additive noise. Furthermore, it’s
common to approximate τp(t) by a linear time-variation
τp(t) = τp + at, (2.14)
where first term, taup, is the relative path delay at the beginning of a block and the second term
is a Doppler scaling factor assuming that the Doppler effect only comes from the movement of the
transmitter and the receiver at relative constant velocities [21]. Finally, although the Doppler effect has
slight variations among the different paths, it can be considered approximately the same when the receiver
is stationary and the main motion comes from the receiver[33], which is the case for most of our data,
so from now on ap = a, p = 1, 2, ..P . It is important not to confuse the index that denotes the multipath
with the block number, to ease the reading and avoid confusion, the index for the path is always p, for
the block is always b and the math will be presented in such way that they won’t be used simultaneously.
A more practical interpretation is to consider that at the receiver there will be P copies of the original
OFDM signal x(t) usually separated in time, but approximately all with the same duration since they
will be time scaled versions of the original duration mainly due to the Doppler effect which is being
approximate by a to every path. This means that the different arrivals will produce ISI. Applying a
rectangular window to the received signal won’t remove the ISI like it did in the absence of a channel and
the orthogonality condition doesn’t hold anymore. Two main solutions have been proposed to solve this
problem. They consist on time extensions of the original signal, the first by the introduction of a cyclic
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prefix, CP-OFDM, and the other by a zero padding, ZP-OFDM [21]. Both of them are explored in the
next section, but independently of the one that is chosen, a Doppler compensation must be done before
to ensure that each received block has the correct time length.
At this time it is convenient to note that by provoking time shortening or lengthening of the signal,
the Doppler effect is causing frequency shifts proportionally depending on the frequencies. This means
that each subcarrier will have suffered a different frequency shift and to compensate all the Doppler in
the frequency domain would take a large effort which translates into more receiver complexity. Instead
the Doppler can be easily compensated by resampling the signal, adjusting it’s duration to the correct
one. In the frequency domain, this will cause inverse frequency shifts to those produced by the Doppler
effect. Given that the ideal OFDM packet duration,T , is known, one can measure the time between the
beginning and end of OFDM packet markers, To, and resample yd(t) by a a factor of α = T/To. The
obtained signal is then an upper or down scaled version of the received signal with the same duration as
the original transmitted signal. The Doppler free signal is:
y(t) = yd(αt) (2.15)
It is important to note that even for small velocities, the Doppler effect will almost always cause
ICI because each subcarrier bandwidth B/M is as small as possible in order to maximize the number
of subcarriers. Any small error in the resampling factor will probably produce a signal y(t) where the
Doppler isn’t totally compensated, so often there will still be some frequency shift in each subcarrier.
In [21] this is called residual Doppler and the authors defend that it can be considered the same for all
subcarriers, since the frequency dependent shift was eliminated in the first step. The parameter that
measures this residual Doppler is called the carrier frequency offset (CFO) in a parallelism with radio
applications and is defined equally for all subcarriers as:
ε =a− α1 + α
fc (2.16)
Where fc is the center subcarrier frequency. The authors then proceed to find a cost function that
minimizes ε and suggest finding its minimum by a 1-D search algorithm. This approach isn’t followed in
this work, instead two methods are presented in section 2.3, in one the residual Doppler is corrected like
if it was a phase shift due to reflections and on the other an estimation of the residual Doppler effect is
computed in parallel with a channel estimate.
2.3 OFDM Time Extensions
After rescaling the OFDM signal, which eliminates most of the ICI, the main problem is the ISI
resulting from the multipath effect. While this isn’t dealt with there will not be orthogonality between
pulses transmitted at different times. The most common idea to solve this question is to introduce a time
guard interval, Tg, at the end of each OFDM symbol such that the echoes and pre-echoes are contained
in that interval and there’s no interference between blocks (IBI).
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To completely eliminate the IBI the Tg should be large enough to include the arrival of the rays that
have the largest delay, i.e., the ones which travel trough the slowest path. When this is the case there’s
a period of length at least Tu = T − Tg where the information arriving at the receiver from all paths
corresponds to the same block b. Note that in section 2.1 there was no time guard and consequently the
rectangular window length was the same as the total time length of each symbol. For the rest of this
chapter, to avoid misunderstands, T is the length of an OFDM block, including the time guard Tg, and
Tu is the total time length that corresponds to an OFDM block without the recently introduced time
guard and it also corresponds to the variable T that was used in the previous sections. Tu will be called
the summation period since it contains the time instants needed to obtain the symbols in the frequency
domain through an DFT. The use of the time guard to avoid the IBI can be seen below in Fig. 2.3 where
it is easy to understand how this technique works.
b-1
b-1
b
b b+1
b+1Main path
Largest delay path
Tg Tu
Shortest delay path bb-1 b+1
OFDM block
Figure 2.2: Use of a time guard to prevent inter-block interference in OFDM transmission
Supposing that the introduction of a Tg totally eliminates the IBI, there will still be interference
between the different time instants in each OFDM block. Sampling the transmitted and received signal
and moving on to discrete time, a relation can be established similar to the following:
yb = Hxb (2.17)
Where H modulates the channel effect and yb and xb are vectors containing a sampled version of
the transmitted and received signals. To solve this system or its equivalent in the frequency domain, a
matrix inversion of H is needed which in a general case has a complexity O(L3) where L is number of
columns of H. In order to avoid such a complex processing, the Tg can be used in a way that H as a
structure that is easily invertible. In the next two subsections is provided an analysis of the two already
mentioned ways of filling the Tg, the CP-OFDM and the ZP-OFDM. Both these methods are presented
supposing that a channel estimate is available, how this can be done is explored later in section 2.4.
2.3.1 CP-OFDM
In CP-OFDM the time-guard is filled with a repetition of the information at the end of the summation
period. To better understand this, consider s(n), n = 0, 1, ..., N − 1, to be the sampled version of the
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original transmitted b-th block with duration of Tu. Its cyclic extension will be sCP(n) = [s(N−Ng) s(N−Ng + 1) . . . s(N − 2) s(N − 1) s(0) s(1) . . . s(N − 1)] with total duration T . Note that the time length
Tg corresponds to Ng samples and the index CP has no relation with the block number which is being
omitted for notation simplicity and since there’s no IBI and each block is being treated individually
there’s no use in maintaining the block subscript. In this situation the received Doppler compensated
signal will be yCP(n) and if its first Ng entrances are discard the result is y(n).
Denoting the channel impulse response as h and considering that it has a length of L = Ng, the
relation between yCP(n) and sCP(n) can be defined as:
yCP(0)
...
yCP(Ng +N − 1)
=
h(0) 0 . . . . . . . . . . . . 0
h(1) h(0) 0. . .
. . .. . .
......
. . .. . .
. . .. . .
. . ....
h(L− 1). . .
. . .. . .
. . .. . .
...
0. . .
. . .. . .
. . .. . .
......
. . .. . .
. . .. . .
. . ....
0 . . . . . . 0 h(L− 1) . . . h(0)
sCP(0)
...
sCP(Ng +N − 1)
+
n(0)
...
n(Ng +N − 1)
(2.18)
Where n is additive noise. Since the first Ng entrances of yCP(n) are correspondent to the same
symbols as the last, they can be discarded and the equation becomes:
y(0)
...
y(N − 1)
=
h(L− 1) . . . h(0) 0 . . . . . . 0
0. . .
. . .. . .
. . .. . .
......
. . .. . . h(0)
. . .. . .
......
. . .. . .
. . . h(0). . .
......
. . .. . . h(L− 1)
. . .. . .
...
0 . . . . . . 0 h(L− 1) . . . h(0)
sCP(0)
...
sCP(Ng +N − 1)
+
n(Ng − 1)
...
n(Ng +N − 1)
(2.19)
Finally, the sCP last Ng entries are equal to the first, this means that the equation can be further
simplified to:
y(0)
...
y(N − 1)
=
h(0) 0 . . . 0 h(L− 1) . . . . . . h(1)
h(1) h(0) 0. . . 0 h(L− 1)
. . . h(2)...
. . .. . .
. . .. . .
. . .. . .
...
h(L− 1). . .
. . .. . .
. . .. . .
. . . 0
0. . .
. . .. . .
. . .. . .
. . ....
.... . .
. . .. . .
. . .. . .
. . . 0
0 . . . 0 h(L− 1) . . . . . . . . . h(0)
s(0)
...
s(N − 1)
+
n(Ng − 1)
...
n(N +Ng − 1)
(2.20)
Let the matrix on this last equation be denoted by H. H is a Toeplitz circulant matrix which means
that it can be decomposed as a product between three matrices, H = WΛWH , where W is unitary and
Λ is diagonal [14]. Using matrix notation, the last equation is equivalent to:
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y = WΛWHs + n (2.21)
Applying a Least-Squares (LS) minimization, one finds that the best estimate for s is:
s = WΛ−1WHy (2.22)
After obtaining an estimate for the original transmitted symbol it is possible to apply a DFT just like
in section 2.1 to obtain the original frequency symbols. Through this explanation it was assumed that
L = Ng, this doesn’t have to be true, in fact this method will always hold when L ≤ Ng which is already
assured since it is equivalent to have a Tg larger than the biggest channel delay.
2.3.2 ZP-OFDM
In CP-OFDM a significant amount of energy is spent in the transmission of repeated information, the
CP. ZP-OFDM is an alternative solution where the original signal is extended by padding Tg with zeros
so the total transmission power remains the same as in the case where there was no guard interval. In
this situation instead of transmitting a block s(n), the transmitted signal will be:
sZP(n) = [s(0) s(1) . . . s(N − 1) 0 0 . . . 0] (2.23)
At the receiver, after resampling, the signal y(n) is reduced to the original length through an overlap-
add (OLA) operation, i.e., the values corresponding to the time guard instants are added to the first
values:
yOLA(n) = [IN IZP] yZP(n) (2.24)
Where IN is the identity matrix with N columns, IZP is a repetition of the first M columns of IN and
yZP(n) is the received signal after resampling and Doppler compensation. The transmitted signal sZP(n)
and the received signal yZP(n) can be related by a channel matrix just like the one in equation (2.18).
Taking into the account that the last entries Ng = L entries of sZP(n) are zero and the remaining are
equal to s(n), the equation can be simplified to:
yZP (0)
...
yZP (Ng +N − 1)
=
h(0) 0 . . . . . . 0
h(1) h(0) 0. . .
......
. . .. . .
. . ....
h(L− 1). . .
. . .. . .
...
0 h(L− 1). . .
. . ....
.... . .
. . .. . . h(0)
.... . .
. . .. . .
...
0 . . . . . . . . . h(L− 1)
s(0)
...
s(N − 1)
+
n(0)
...
n(Ng +N − 1)
(2.25)
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Applying the overlap-add operation mentioned above, the equation becomes:
yZP (0) + yZP (N)...
yZP (Ng − 1) + yZP (Ng +N − 1)
yZP (Ng)...
yZP (N − 1)
=[H]
s(0)...
s(N − 1)
+
n(0) + n(N)...
n(Ng − 1) + n(Ng +N − 1)
n(Ng)...
n(N − 1)
(2.26)
Similarly to the case in CP-OFDM, this derivation leads to a system with the same Toeplitz circulant
matrix H and the same low complexity inversion can be performed.
In general, CP-OFDM and ZP-OFDM have an equal efficiency, since there’s no reason to have dif-
ferent lengths on the CP and the ZP extensions, and they achieve similar performances. More advanced
variations of CP and ZP have been presented in the literature, where ZP-OFDM does seem to slightly sur-
pass CP-OFDM [13], but these aren’t considered here because they add too much complexity. For some
applications, where the available energy is limited, the ZP method has a significant reduction relative to
CP.
2.4 Channel Estimation
In the previous section two methods were presented that allow the compensation of the IBI and also of
ISI in the same block provided that a channel estimate is available. The most intuitive way of computing
such estimate is by sending an OFDM block with known information at the receiver and solving the
system (2.17) in reverse to obtain H. This has two major drawbacks, the first is that a whole block is
spent in computing the channel estimate which decreases the overall effective bit rate. Note that, as the
time passes, the channel changes and the estimate progressively gets outdated, to update it another block
has to be wasted. The second, is that even for the first blocks received after the estimate is computed,
there’s no guarantee that it will produce good results since it’s based on another block. In [21] the
author proposes a different method where only some of the subcarriers, called pilot, are used to transmit
information known to the receiver. This section begins by an explanation of this method and is followed
by an extension based on tracking the channel estimate trough the OFDM packet.
Since the pilot subcarriers must be distinguished from the remaining active subcarriers the deductions
will be done in the frequency domain. The relationship between the signals DFTs is:
S(m) =Y (m)
H(m)(2.27)
It’s important to remember that in this equation H(m) is not the physical channel frequency response
because a large part of the Doppler effect and IBI due to multipath have already been eliminated, instead
it only includes the channel induced phase distortion, the residual Doppler and the amplitude attenuation.
Even more, S(m) is the DFT of s(n) and not the transmitted signal sZP(n) or sCP(n), meaning that
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it only comprises the samples correspondent to Tu. The same applies to y(n) and Y (m). Note that
the estimates of S(m) can be obtained by M complex divisions between the vectors Y (m) and H(m),
so as long as the process of obtaining the channel estimate has a good computational efficiency all the
demodulation process will be computationally efficient.
At the receiver, the initial processing associated with CP-OFDM or ZP-OFDM is done, but the final
system can’t be solved since not all the symbols are known to the receiver, instead one can consider just
the Mp pilot carriers and write the following smaller system[21]:
Y (p1)
...
Y (pMP)
=
S(p1)
. . .
S(pMP)
1 e−j2πp1K . . . e
−j2πp1LK
......
. . ....
1 e−j2πpMP
K . . . e−j2πpMP L
K
h(0)
...
h(L)
+
n(p1)
...
n(pP )
(2.28)
Where Y (pa) denotes the received corrupted symbol at the a-th pilot subcarrier or pa-th subcarrier,
S(pa) is the correct transmitted symbol, h(l) is the channel impulse response l − th coefficient on the
time domain and n(pa) is additive noise. The equation can be rewritten in matrix form as:
Yp = DpV h + np (2.29)
Applying a least squares (LS) criteria yields:
h = (VHDHp DpV)−1VHDH
p Yp (2.30)
h is used to denote the channel impulse response estimate which will usually differ from the true
channel impulse response h. If all the symbols in the constellation have unit amplitude, DHp Dp becomes
the identity matrix. Even more, if the pilot subcarriers are evenly spaced, VHV also becomes the identity
matrix. If one wants to estimate the channel response the ideal positioning of the pilot subcarriers would
be as spread out as possible which actually corresponds to have them equally spaced. Since both of these
constrictions are easily respectable, the channel estimate can be computed just as:
h = VHDHp Yp (2.31)
This important simplification allows to find h without applying a matrix inversion, hence maintaining
a low complexity receiver. The channel frequency response estimate, H(m), can then be obtained by
a L-point DFT and equation (2.27) can be solved giving S(m). The final step, is to decide on each
transmitted symbol for every block, Sb(m). For this, a simple minimum squared error (MSE) criteria
can be applied, meaning that each demodulated symbol is decided as the constellation symbol which is
closer. Denoting Db(m) as the decision of the received symbol from subcarrier m and block b, one gets:
Db(m) = decision[S(m), constellation symbols] (2.32)
The total number of pilot subcarriers should be adjusted according to the channel characteristics and
the needed MSE or transmission bit rate. This poses a difficult task, since to minimize the MSE one
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needs to have the best channel estimate, which corresponds to having many pilot subcarriers and the
useful transmission rate will be very low. So an equilibrium must be met between the useful bit rate and
the level of MSE.
2.4.1 Channel Sparsity
On chapter 1 a short analysis of the typical underwater communications scenario was presented,
including the fact that the different arrivals to the receiver due to multipath are usually spaced, which
translates into a sparse channel. Using the result in (2.31) from the LS minimization, doesn’t necessarily
yields a sparse estimate, on the contrary, most of the times, due to the noise present in the signal, channel
estimates will have some low coefficients have a negligible contribution to the model, not contributing
to the ISI mitigation, and would be best treated as zero. On the other hand, it’s common to have
coefficients with low magnitude on the channel response, for example, from distant echoes. So, if it’s
obvious that some values should be ignored in order to improve the ISI estimate, it’s not clear at all how
to choose those values. In [24] and [33] two methods are presented, the Fixed Threshold and LS Adaptive
Thresholding (LS-AT), like the names indicate they both consist in defining a threshold and reducing to
zero coefficients whose magnitudes are below the threshold.
On the first method a simple fixed threshold is proposed to be used throughout each OFDM packet
and instead of using h, the following sparse estimate is used:
ht = trunc(h, threshold) (2.33)
The function trunc denotes a truncation, i.e., it’s a function that cuts to zero all the coefficients
that are below the given threshold. The channel frequency response will be obtained by taking the DFT
of ht, instead of h. The threshold is defined as a percentage of the highest coefficient in the original
channel estimate and the author of this method presents results indicating that for threshold between
0 and 0.25 the final MSE will decrease, but for higher thresholds the MSE has an abrupt descent to
very low values. This behavior is easily understandable, since as the threshold rises, more and more
important coefficients are cut. Taking all this into account, a good input for trunc could be, for example,
0.2 max|h|1. Simplicity is the main advantage of this method, but it has several disadvantages, namely,
there’s not enough research to know if these thresholds will provide good results for different physical
channels and, even for the same location, the threshold may have to be lowered or raised depending
on the time-variations of the channel, including its noise. Setups where the threshold value is directly
dependent on the error are possible[24], but they will only produce changes a posteriori, meaning that the
necessary threshold corrections are only available for the following channel estimate. Instead one should
consider the second method where the threshold is adjusted for current estimate ensuring it has the
desired degree of sparseness. Besides the question on how to choose the threshold, this method doesn’t
distinguish between coefficients that have a physical meaning and those due to noise in the estimate.
On the alternative varying threshold method, a desired degree of sparseness can be defined a priori
and the authors propose that the threshold should be computed each time a channel estimate is done,
such that desired sparseness is achieved for all of them. How much a channel is sparse is measured as the
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longest sequence of coefficients that are zero. The LS-AT algorithm is described in detail below like it is
presented in [33], besides the desired degree of sparseness T0, a number of iterations or steps S should be
defined taking into account that the resolution of the final estimate is of 2−S .
Algorithm 1: LS-AT
1 Define S and T0
2 Initialize step=1, γ = 0.5, ∆γ = 0
3 while step ≤ S or (step > S and ∆γ > 0) do
4 hLS−AT = trunc(h, γ max |h|1 )
5 U0 ← Delay spread of hLS−AT
6 if U0 ≤ T0 then
7 ∆γ = 2−(step+1)
8 else
9 ∆γ = −2−(step+1)
10 end if
11 γ ← γ + ∆γ
12 step← step+ 1
13 end while
14 return hLS−AT
The chosen measure of sparseness may produce bad results for some channels, for example, if there
are two echoes of the main signal and the second has a delay double than the first. In this case, there’s
no guarantee that a large sequence of zeros isn’t discarding the first echo or that the channel estimate
has a large sequence of zeros after the first echo, but not a shorter one before. In any case, it does
take into account the variability of the channel and doesn’t require for an apriori specification of the
threshold. Choosing a desired delay or level of sparseness apriori may be easier, because the time varying
characteristics of the channel usually don’t include the level of amplitude, although it may happen in the
case where one or both the transmitter and the receiver are moving. This algorithm may also produce
better results than the first when it comes to cutting the coefficients corresponding to noise and leaving
most of those that have a real meaning, but it still can discard significant components of the channel
estimate, which it often does.
The example from Fig. 2.3 illustrates the use of both methods and points out how small differences
from the channels time-variations can lead to very distinct results in the case of the fixed threshold. These
two methods require low complexity from the receiver and are very intuitive, but the resulting channel
estimates can be very unnatural, often containing sharp transitions that probably don’t correspond to
the physical channel. Another disadvantage is that if two coefficients have the same value, they will both
be cut or none of them, independently of the adjacent coefficients. A more complex algorithm is applied
to the FMT modulation, yet it isn’t applied here because it’s recursive character makes it unsuitable for
OFDM where the symbols have a longer time duration so too many changes can occur on the physical
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0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Mag
time (n)
(a)
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (n)
Mag
Noisy channel estimateFixed threshold at 15% of the maximumVarying threshold for a 3 zeros sequence
(b)
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (n)
Mag
Channel estimate for fixed thresholdChannel estimate for varying threshold
(c)
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Mag
time (n)
Noisy channel estimateFixed threshold at 15% of the maximumVarying threshold for a 3 zeros sequence
(d)
0 2 4 6 8 10 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (n)
Mag
Channel estimate for fixed thresholdChannel estimate for varying threshold
(e)
Figure 2.3: Example of the threshold methods for channel sparsification. (a) Channel estimate without
noise, (b) and (d) Noisy channel estimates for different instants, (c) and (e) Application of both methods
to the correspondent noisy channel estimates.
channel for such a method to hold.
2.4.2 Channel Tracking
Channel tracking is the method of using the last channel estimates to compute a new updated estimate,
instead of just computing a new one. In this situation, the number of pilot subcarriers in the first block
is usually high too ensure that a high quality channel estimate is computed. Still, on the following blocks
the number of pilot subcarriers can be reduced because they just have to be enough to assure a good
tracking of the channel time-variations. In some cases the decisions from regular subcarriers are assumed
correct and used to track the channel, consequently zero or almost none pilot subcarriers are used. The
implemented method is based on the one presented in [33], although some variations are considered.
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Some steps of the algorithm require a method of channel estimation from the received signal assuming
knowledge of the transmitted symbols, for those situations the processing will be analogous to the pilot
method explained in the previous section. The channel estimate at time instant n will be denoted as
H(n), there will also be a channel estimate without the Doppler effect, H 0(n), and a Doppler estimate,
β0(n), such that if there were no errors H(n) = H0(n)ejβ0(n).
For the first symbol of each packet there’s no previous channel estimate, so it will be applied the
simple pilot method. For the remaining symbols there will be a need to compute a channel estimate from
decisions made and Y(n), at these times the pilot method will always be used, but instead of a few pilot
subcarriers, all the computed decisions will be considered correct.
For all the other symbols there’s an estimate of the channel from the previous symbol, H 0(n − 1),
and a Doppler prediction which is computed on the previous time instant, βp(n). The first step is to
make decisions on Y(n) using H 0(n− 1)ejβp(n). This set of decisions, D(n), is then used with Y(n) to
compute a new channel estimate H (n). The Doppler estimate will be done using the phase difference
from the two channel estimates, which is computed as:
∆β(n) = 6H (n)
H 0(n− 1)eβp(n)(2.34)
Since the Doppler effect has a linear dependence with the frequency, an estimate of Doppler between
this instant and the last, can be easily found by averaging the differences through the spectrum.
β(n) =1
M
M−1∑m=0
∆βm(n)
2πTfm(2.35)
Using this value the overall Doppler estimate, β0, can be update as
β0(n) = β0(n− 1) + 2πβ(n)T [f0 . . . fM ], (2.36)
and used to compute the new zero-Doppler channel estimate:
H 0(n) = λ H 0(n− 1) + (1− λ) H (n)e−jβ0(n). (2.37)
λ is a forgetting factor, similar to ones used in LS-algorithms. For the next time instant H 0(n) can
be used, since it’s not possible to know how the channel will vary. On the other hand, it’s expected that
the Doppler effect remains constant or has slow changes, so a prediction can be made using the current
estimates:
βp(n+ 1) = β0(n) + 2πβ(n)T [f0 . . . fM ] (2.38)
Before moving on to the next symbols, the decisions can be made again using H 0(n)e−jβ0(n). Most
of the times the improvement isn’t significant, but in very noisy or rapidly varying channels it might be
needed.
It was also verified that most of the times the main differences on the estimates are on the phase and
that the amplitude of the channel response is almost constant through an OFDM block and doesn’t need to
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be updated. Taking this into account this tracking algorithm can be simplified by using H 0(n) = H 0(0)
for all time instants, removing the step that correspond to equation (2.37). This approach could further
simplify the algorithm by computing only the phase estimate instead of a complete channel response
estimate for each time instant. Yet, no way of doing this was found, since the channel is estimated first
on the time domain.
For both methods, the number of pilot subcarriers is usually zero in all time instants, except for
the first symbol. Alternatively, a few pilots can be maintained, in this case their values will replace the
decisions in the respective subcarriers.
2.5 Multi-Receivers Gain
In underwater communications it’s very common to have such significant channel distortions and
noise that the use of more than one receiver becomes almost mandatory to achieve acceptable results.
For OFDM this can be easily done by combining the decisions made at each receiver. This idea is
presented in [21], it can be deduced starting from equation (2.27) and rewriting it as:
Y (m) = H(m)S(m) (2.39)
Considering a R-receivers scenario, each receiver will have a different signal and associated channel
estimate, but since the transmitted symbol is the same for all, one can write the following system:
Y1(m)
...
YR(m)
=
H1(m)
...
HR(m)
S(m) (2.40)
Where the subscripts denote the corresponding receiver. Considering Y(m) = [Y1(m) . . . YR(m)]T ,
H(m) = [H1(m) . . . HR(m)]T and applying a LS estimate for S(m) yields:
S(m) = (H(m)HH(m))−1H(m)HY(m) (2.41)
Note that the signals from each receiver are being treated completely independently. The symbol
estimates obtained for each one are compared and joined only on the last step before the final decision,
so everything else mentioned in the last sections must be repeatedly applied to each receiver.
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Chapter 3
Filtered Multitone
One of the biggest problems of the use of OFDM modulation in underwater channels is the large ICI
at the receiver. This happens, as mentioned before, due to the Doppler effect that shifts the subcarriers
from their original frequency (in a non-homogeneous way). Since for any small shift in the frequencies of
the closely packed sinc pulses their orthogonality is lost, a different pulse may have to be considered.
This is the guiding line that motivates leaving OFDM behind in search of an alternative modulation,
even if it implies moving back to a less efficient use of the spectrum. The most robust modulations to ICI
are those where the different subcarrier spectra don’t overlap at all and there’s a larger interval between
subcarriers. Such modulations have been well studied under the FBMC theory. Note that OFDM, having
overlapping subcarriers, isn’t usually though of as a FBMC modulation, yet that approach was considered
in [30] as a means of establishing a comparison between OFDM and FBMC techniques.
One of the most attractive FBMC modulations is FMT, which is presented and explained throughout
this chapter. On section 3.1 the basics of FMT modulation and demodulation and theoretical comparison
with OFDM are presented considering ideal conditions. The possible receiver structures for underwater
communications are explored in section 3.2, including setups for multiple receivers and an innovative
training method. Channel estimates are needed for some of the considered receivers, the methods for
doing this are developed in section 3.3. This section also includes a process of combining the estimates
from each sub-carrier. A multi-receiver scenario is considered in section 3.4. Finally, on section 3.5 an
adaptation of the receiver for Single-Carrier modulation and a short comparison between between both
modulations are presented.
3.1 FMT Modulation/Demodulation
In FMT modulation a popular choice for the signaling pulse is the raised cosine pulse, it has a very
tight spectral containment which means that it’s very robust to frequency shifts. The expression for
time and frequency domain of this pulse are shown below in equations (3.1) and (3.2), respectively. The
symbol T denotes the time interval between two consecutive symbols and β, which is called the roll-off
factor, controls the shape of the pulse. The minimum value for β is 0 and it corresponds to having a
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rectangular pulse in the frequency domain and a sinc function in the time domain, exactly the dual of
OFDM modulation. As β increases, the sharp edge of the rectangle becomes smoother and the side lobes
of the sinc function decrease. When the maximum value of 1 is reached, the frequency shape resembles
a cosine with a single period, raised above x axis hence the name for this pulse. It’s important to note
that as β increases the frequency duration of the pulse goes from 1/T to 2/T , so for smaller side lobes of
the time domain sinc function, the sub-carriers need to be more spaced. The pulse shapes for different β
values are shown in Fig. 3.1. The adjacent signaling pulse for the time and frequency domain are shown
in Fig. 3.2, where it’s easy to see that the orthogonality condition is respected, just like in OFDM, i.e.,
for a given time-frequency instant the only non-zero signal is the one from the corresponding symbol.
q(t) =
Ts 0 ≤ |t| ≤ 1−β
2Ts
Ts2 (1 + cos(πTsβ (|t| − 1−β
2Ts))) 1−β
2Ts≤ |t| ≤ 1+β
2Ts
0 1+β2Ts≤ |t|
(3.1)
Q(f) =cos(πβ f
Ts)
1− ( 2βfTs
)2
sin(π fTs
)
π fts
(3.2)
t
β=0β=0.25β=0.5β=0.75β=1
(a)
f
β=0β=0.25β=0.5β=0.75β=1
(b)
Figure 3.1: Raised-cosine pulses for different β values in (a) the time domain and (b) the frequency
domain.
The raised cosine pulse has several advantages comparing with the sinc pulse. First of all, it results in
a much more robust modulation regarding the Doppler effect, like it was already mentioned. Eventually,
for very large Doppler situations, the subcarriers frequency may be totally mismatched from the original
frequency, but note that in OFDM this happen for much lower Doppler values. Secondly, it’s possible to
adjust the pulse shape in case there’s more or less ICI or ISI, by choosing the best value for β.
The main downside of using FMT modulation is that time guards can no longer be used to fight ISI,
since the symbols aren’t disjoint in time, like it happened in OFDM. Several methods for compensating
ISI will be presented in the next sections, but it’s important to notice that by increasing the roll-off factor
the side lobes of the pulses on the time domain are decreased which reduces the magnitude of the possible
ISI.
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t
(a)
f
(b)
Figure 3.2: FMT signaling pulses in (a) the time domain and (b) the frequency domain for β = 0.25.
Another important aspect is the transmission bit rate that directly relates to the spectral efficiency.
For low roll-off factors, the sub-carriers can be much closer and at the limit of β = 0 the rectangle pulses
may be right next to each other. In the opposite situation the spacing between subcarriers would have to
be double to ensure an ICI free transmission in ideal conditions, which significantly affects the possible bit
rate. Instead, it’s common to allow for some overlap between the raised cosines to increase the spectral
efficiency[30], this is shown if Fig. 3.3. Actually, since the high values of the roll-off factor are better
for the situations where the Doppler effect is less significant, the small overlap in the frequency domain
doesn’t lead to a poorer performance.
f
Figure 3.3: Raised-cosine pulses, with β = 0.5, overlapping in the frequency domain.
At this point it’s easy to understand that the FMT modulation has a higher resistance to ICI than
OFDM at the cost of having lower robustness to ISI. As a direct consequence, for communication systems
where the Doppler effect isn’t present, the FMT doesn’t hold many advantages against OFDM ans
many times the latter was been chosen[7]. The hope that FMT can outperform OFDM in underwater
communications is based on the fact that significant frequency distortions, such like the Doppler effect,
occur very often, but that’s not all. In fact, it isn’t just about which modulation is more resilient to ICI,
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but which modulation can achieve a better equilibrium between robustness to ISI and ICI taking into
consideration how strong each of the effects is in underwater communications. To achieve this goal the
adjustable roll-off factor may give FMT an edge over OFDM, since it provides an extra degree of freedom
in finding that balance.
Similarly to OFDM, the transmitted signal will be denoted as x(t), but now the symbols aren’t
separable by blocks, so the signal will be defined as:
x(t) =∑n
M−1∑m=0
sm(n)q(t− nT,m) (3.3)
Where q(t − nT,m) is the pulse signal q(t) shifted to the frequency of the m-th subcarrier, fm.
Although the symbols aren’t disjoint in the time domain, they are easily separable in the frequency
domain through a filter-bank. So in the modulation process the first step is to build the individual
subcarriers signals by passing the symbols through the filter-bank which is equivalent to multiplying the
successive pulses in every subcarrier by their respective symbols. Then the output of each filter is shifted
into the corresponding subcarrier frequency fm and all the signal are summed resulting in x(t).
At the demodulation the inverse procedure is applied to x(t). The signal is copied and each copy is
shifted to a different subcarrier frequency. After this, each subcarrier signal goes through a conjugated
filter-bank of the one at the modulation process. In ideal conditions, the outputs of the filter are the
original symbols from each subcarrier.
The modulation and demodulation of an FMT signal resorting to filter-banks is illustrated in Fig.
3.4 where an upsampling and downsampling operation are also included. By the sampling operations,
the polyphase components of the signals are available thus allowing for an efficiently implementation of
transmitter and receiver filter-banks using a DFT/IDFT[9]. The polyphase components can also be useful
to other receiver processes, this is explored in parallel with a multi-receiver scenario in section 3.2.1.
...
... Channel ...
...
Figure 3.4: Modulation and demodulation of FMT through the use of filter-banks.
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3.2 Receiver Structure
In real situations, the received signal y(t) will be different from x(t), but in most cases, at each
subcarrier frequency, fm, the signal still comes from just one subcarrier and the filter-bank processing can
be applied. This results in a set of symbols corrupted by ISI, phase shifts due to reflections and Doppler
effect, and additive noise. Since the signals of each subcarrier are already separated and sampled, each
sequence of corrupted symbols will be denoted as ym(n), where the subscript m indicates the subcarrier
and the index n denotes the time instant. The received signal is then:
ym(n) = hTm(n)[ sm(n− L2) . . . sm(n− L1) ]T + w(n), (3.4)
where hm(n) is the channel response that includes the filter-bank processing and the sampling of the
transmitted and received signals, its total length is L = L2 − L1 + 1 and it differs for each subcarrier,
sm(n) denotes the transmitted symbol at time instant n and w(n) is additive noise.
To figure out which symbol corresponds to the ym(n) values, one could try a similar strategy to
the one used in OFDM where known symbols are sent in each subcarrier and the channel estimate is
immediately computed by solving the equation system in (2.27). On FMT the number of sub-carriers
is typically much lower than in OFDM, this means that even if all the sub-carriers were transmitting
known information to the receiver, a channel estimate through the same method would require the use of
more than one time instant n. It’s clear that the useful bit rate would largely decrease, so this solution
isn’t attractive. Instead, the use of a decision feedback equalizer (DFE) and a variation that includes an
iterative channel estimate will be explored on the next sections. Note that, the FMT symbols are much
shorter in the time domain than the OFDM symbols, so it’s possible to compute a channel estimate from
consecutive symbols faster than the channel variations, which wasn’t possible before.
In this section and for the next, the algorithms and equation will be described as if the received signal
was a single-carrier and they must be repeated to each subcarrier. Since the initial processing doesn’t
combine the signals of the different subcarrier and is equal for all, the m subscript will be dropped until
is needed again.
3.2.1 Decision Feedback Equalizer
The basic DFE is shown in Fig. 3.2.1, it consists of a decision maker and a feedback loop that subtracts
a prediction of the error in the received symbol to the following. The idea is that although the channel
may vary in time it’s likely that consecutive symbols have similar or equal distortions. So, if the n-th
decision, d(n), is correct, then the error e(n) = d(n) − y(n) can be used to correct the next received
symbols: d(n+1) ' z(n+1) = y(n+1)+e(n). Of course that if one decision is incorrect, it will probably
cause another wrong in the following symbol. Including a feedback filter assures that several past errors
are taking into account which improves the error prediction.
To update the feedback filter coefficients the standard LS algorithm with a forgetting factor λ is a good
choice, since the oldest values are given less weight, without being totally disregarded. The forgetting
factor is very important, otherwise the estimate of the filter coefficients would progressively change less
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Decision+
-
d(n)z(n)
y(n)
e(n− L2) . . . e(n)
-+
BufferWFBe(n)
Figure 3.5: Structure of the basic DFE.
and less, which isn’t appropriate to the situation where the channel is time varying. To avoid doing all
the computations for each new symbol, the recursive version of this algorithm (RLSλ) will be used. The
implementation follows the one presented in [10] and it’s resumed in algorithm 2, the initial values for the
several parameters are analyzed in section 3.2.3. The algorithm is defined such that e(n) is the amount
of error in u(n)W(n) so that the RLSλ knows by how much W(n) needs to be corrected. In this simple
case the vector u(n) contains the last decisions, W(n) = WFB(n) and e(n) is the current error. P(n) is
the inverse of the correlation matrix that has no application outside the algorithm.
Algorithm 2: RLSλ
Input: u(n), e(n),W(n),P(n)
1: π(n) = uH(n)P(n)
2: K(n) = P(n)u(n)λ+π(n)u(n)
3: P(n+ 1) = 1λ (P(n)−K(n)π(n))
4: W(n+ 1) = W(n− 1) + K(n)e(n)∗
Output: W(n+ 1),P(n+ 1)
During the RLSλ convergence period it’s expected that many decision are done wrong, each one
resulting in a badly computed error that delays the convergence of RLSλ and leads to more wrong
decisions with increasing errors. When this happens the DFE is diverging, i.e., the values of z begin
to increase without any control and the remaining symbols of that FMT packet are most likely lost.
To avoid this the first symbols of each block are a priori known to the receiver which assures that the
first errors are computed correctly, this is called the training mode whereas in the rest of the packet the
decision mode is used. How the number of symbols in the training can be reduced is explored in section
3.2.3.
Since the channel impulse response has a total length of L symbols, one symbol transmitted at time
instant n will span over L symbols in the received signal. To better understand this, suppose that only
one symbol was transmitted, then the received signal would span over L samples:
[ym(n− L2) . . . ym(n− L1)]T = h(n)s(n) (3.5)
Note that L1 > 0 corresponds to the existence of a pre-echo, meaning an arrival of the signal that
is actually faster than the one that is considered as the main arrival due to its higher intensity. The
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equation (3.5) suggests that to include all the information relative to the n-th symbol, the receiver should
process a window of length L ranging from L2 to −L1. To process a window at a time a feedforward
filter is added to the DFE and the RLSλ algorithm will update the coefficients of the both filters. This
setup is shown in Fig. 3.6 and the defined RLSλ function will have as input
[WFF(n+ 1); WFB(n+ 1)] = RLSλ( [y(n); d(n)], e(n), [WFF(n); WFB(n)], P(n) ) (3.6)
where, for purpose of lighter notation, the vectors y(n) and d(n) were introduced corresponding to
[y(n− L2) . . . y(n− L1)]T and [d(n− L2) . . . d(n− 1)]T .
Decision+
d(n)z(n)
y(n)
e(n)
-
+
WFF (n)
RLSλ
WFB(n)
-
Bufferd(n)
-
Figure 3.6: Receiver structure for a DFE with a feedforward filter.
At this point is important to remember that in the demodulation process a downsampling operation
was included in such a way that the polyphase components of the signal are available at the receiver.
The number of polyphase components will denoted by φ which means that the n-th symbol occupies the
samples from φn to φn+φ−1. The processing will be very similar, but now the length of the feedforward
window is φ times longer. The RLSλ algorithm will update the feedforward filter coefficients such that
the ones that correspond to the polyphase component with better synchronization with the transmitter
will have higher values.
The so far presented receiver structure tries to correct the ISI and the other channel distortions, but
is doesn’t specifically aims to compensate the Doppler effect. For signals with constant phase shifts or
that vary very slowly the RLSλ algorithm can converge to a state where the filters coefficients successfully
correct the received signal and evolve accordingly, but not if significant Doppler effect is present. Note
that the subcarriers are separated by the filter-bank despite the Doppler induced frequency shift, but the
Doppler effect also causes the received symbols to have a continuous phase shift, i.e., the constellation
spins in time as the transmitter and receiver get further apart or closer. The phase shift is typically much
faster than the remaining channel time-variations, this means that the RLSλ algorithm can’t track the
channel variations along with the faster Doppler effect and a different solution must be found.
There are two possible approaches to compensate the Doppler effect, the first is to effectively correct
the received the signal by introducing a phase correction and the second is to rotate the decision constel-
lation adapting it to the new incoming symbols. The first step to both of them is finding an estimate of
the Doppler effect, which will be denoted as θ(n). Once this estimate is available, equation (3.7) or (3.8)
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should be applied depending on the chosen method. It’s important to remember that a different estimate
θ(n) must be derived for each subcarrier, so this process has to be repeated M times.
yθ(n) = y(n)e−jθ(n) (3.7)
constellation(n) = constellation(0) ejθ(n) (3.8)
To compute the phase correction needed for instant n a phase locked loop (PLL) will be used driven
by the error of the previous decision. Since the goal is to minimize the squared error, MSE, one must
take the derivative of e(n)e∗(n) with respect to θ(n)
e(n) = d(n)− z(n) = d(n)− (y(n)ejθ(n) − yFB(n))TWFF(n) (3.9)
∂e2(n)
∂θ(n)= 2 Re
{∂e(n)
∂θ(n)e∗(n)
}= −2 Im
{WFF(n)Hy(n)ejθ(n)e∗(n)
}(3.10)
The output of the PLL at time instant n will be θ(n+ 1), that can be used for any of the techniques
mentioned before to compensate the Doppler effect. Due to reasons that will be explained in section 3.4,
the chosen solution is to use the Doppler estimate to correct the received signal. More details about the
PLL implementation and initialization are provided in section 3.2.3. This DFE configuration is shown if
Fig. 3.7.
Decision+
d(n)z(n)
y(n)
e(n)
-
+
WFF (n)
RLSλ
WFB(n)
-
Bufferd(n)
yθ(n)
PLL
x
e−jθ(n)
Figure 3.7: DFE with a feedforward filter and Doppler compensation.
Using independent algorithms for the Doppler estimate and updating the filter coefficients, both using
the same error as the input, is a big risk since the whole receiver becomes ambiguous. Supposing that
the phase corrections have an error of ∆θ, but the filters coefficients are correctly estimated, then e(n)
will indicate that a correction must be made in the prediction. The problem is that both the RLSλ and
the PLL will change their predictions for the next symbol, when only the PLL needed a change. This
can eventually cause the DFE to diverge, similarly to the earlier mentioned situation. Yet this isn’t very
common since once the RLSλ has converged it will produce small changes, so it diverges slower than the
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PLL converges. The main problem would be that the deficient correction of the ISI or the Doppler effect
probably would cause some wrong decisions.
There’s a worse scenario where the ambiguity can be more problematic, which is when the phase
corrections have an error of ∆θ and the filter coefficients have an error of −∆θ. In this case z(n) and
d(n) will be the same as if no error had been made in the compensations, although neither of them is
being correctly made. If there’s no Doppler, then this doesn’t have any consequence in the decisions made
throughout the transmission. When the Doppler is significant it’s hard to explain in which situations this
ambiguity will cause bad results. It’s supposed that if the Doppler is constant through an FMT packet,
the quick phase variation will be immediately tracked by the PLL, resulting in a very different evolution
rate of the PLL and the RLSλ algorithm estimates so it’s not likely that any problems should arise. The
worst case is if the Doppler effect only appears in the middle of an FMT packet. In this situation, if
the compensation are being done wrong, when the symbols start to shift the PLL will quickly change
its prediction, leaving an unbalanced RLSλ algorithm that had converged to bad estimates and will take
a long time to converge to the correct values of the filter coefficients. During this transition period it’s
likely that many of the decisions are done wrong, extending the duration of the DFE convergence or
causing it to diverge. Since this last situation isn’t very common, the ambiguity is tolerated as it has also
been in the literature.
3.2.2 Channel-Estimation-Based DFE
The last DFE configuration has already a complex structure, but most of its components don’t have
an explicit physical meaning, except for the Doppler phase compensation. In [5] a different interpretation
of the DFE is presented, the author proposes that from the past symbols and decisions a channel estimate
can be computed, instead of an error prediction or a feedback filter, and can be used to estimate the
ISI for future symbols. In this new configuration the feedforward filter will compensate eventual pre-
echoes of future symbols and capture future echoes of the current symbol whereas the feedback loop will
subtract echoes from the past symbols, computed by the channel estimate from the previous decisions.
This structure is called the channel estimate based DFE (CHE-DFE) and is depicted in Fig. 3.2.2. The
RLSλ algorithm will update the filter coefficients and the Doppler is still estimated using a PLL. How to
compute a channel estimate is explored in section 3.3.
The fact that now the receiver components have a direct physical meaning allows for some changes
and optimizations. Remembering that the channel estimate as a total length L = L2 − L1, for every
decision the channel estimate should produce an ISI estimate for L symbols, this will be described as:
yISI(n) = h(n− 1)d(n− 1) + yISI(n) (3.11)
The second term of the sum assures that in yISI(n) is included the interference from all the symbols
from n − L2 up to n − 1. The ISI is removed by subtracting yISI(n) from the received signal and the
resulting signal goes through the feedforward filter. Since, ideally, all the past interference was removed
from yFF(n), the feedforward filter doesn’t need to look at the past received symbols, instead its window
should range from n to n−L1. This will improve the resulting error and lead to a significant complexity
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Decision+d(n)
z(n)yθ(n)
e(n) -
+
WFF (n)
RLSλ
-
Buffer
d(n)
yISI(n)
PLL
e−jθ(n)
y(n)
Channel estimatorh(n)
x
Figure 3.8: CHE-DFE receiver with Doppler compensation.
reduction in this receiver component because, almost always, L2 � −L1, which means that a large
decrease of the filter window is made. The RLSλ input will be:
WFF(n+ 1) = RLSλ( yFF(n), e(n), WFF(n), P(n) ) (3.12)
3.2.3 Training and Initializations
On the last two sections several receiver alternatives where explored assuming that convergence of
the algorithms was guaranteed for the majority of times. In this section the training and initialization
problems are considered, searching for the configuration that allows for better results and, if possible,
faster convergence.
The PLL implementation can be best explained by a block diagram of the equivalent filter, this is
shown in Fig. 3.2.3 where f(n) is the result of equation (3.10) and g(n) is the estimated phase error
relative to the current symbol such that θ(n+ 1) = θ(n) + g(n).
g(n)f(n)b0
b1−a1
z−1
Figure 3.9: Block diagram of the PLL implementation.
The equivalent transfer function is written in (3.13) and to assure a good and quick convergence its
zero and pole must be placed correctly. A common choice is to place the zero such that 10b0 = 9b1 [24]
and consider a1 = −1 so that the pole is at the origin. On the PLL initialization it is considered that
θ(0) = 0.
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H(z) =b0 + b1z
−1
1 + a1z−1(3.13)
The RLSλ algorithm has a more complex adjustment, for a correct explanation it’s important to
notice that at time instant n the recursive update is equivalent to find the LS estimate for W(n) that
minimizes the following expression:
d(1)λn−1
...
d(n)
−
yTFF(1)λn−1
...
yTFF(n)
W1(n)...
WL(n)
(3.14)
where it is assumed that the values of y(n) for n ≤ 0 are 0 and that the receiver is a CHE-DFE.
The most commonly used forgetting factors produce good results, i.e, λ= 0.998 or 0.995. On the [10]
is suggested that W should be initialized as vector of zeros and the matrix P as δ−1I, where I is the
identity matrix. The proposed criteria for choosing δ is that it should be smaller than 0.01σ2, where σ2
corresponds to the variance of the algorithms input signal, in this last case yFF. Yet this was found to
cause a bad initial convergence. For the first iterations the rank of matrix P isn’t complete, note that
P is an L × L square matrix. Sometimes, it was verified that during these instants the estimate of the
filter coefficients W converged to a local minimum. When n→ L a process of re-convergence happened,
i.e., the estimates diverged for few symbols, usually less than 20, and then converged to the absolute
minimum. The ambiguity in the receiver structure caused the PLL to interpret this as a sign that the
Doppler estimates where being done wrong, resulting on a second convergence period. Usually this didn’t
cause the DFE or CHE-DFE to completely diverge, but caused a significant reduction of the achieved
MSE. This problem is explored in the results section for several initializations of matrix P and it was
found that the initializing with δ ∈ [0.1; 1] eliminated this effect.
Besides the suited initializations, a training for the initial part of each packet was considered. During
the training mode the correct decisions for each symbol are known to the receiver, this ensures that the
input error for the PLL and RLSλ algorithms is correct which increases the convergence rate. Some of
the FMT packets have so few symbols that the convergence time is still too large, consequently the useful
bit rate becomes too low unless the training is reduce which can lead to poorer MSE. This motivated
a new training method that consists in an artificial extension of the packet by repeating the symbols
corresponding to the training period, this process is illustrated in Fig. 3.2.3. The number of repetitions
can be arbitrarily large, but it should be an even number like it is justified below. One of the most
significant aspects of this method is to ensure that recursive procedures related to the feedforward filter,
the Doppler estimate and the channel estimate are adapted correctly in the time instants where the
processing order is reversed.
The RLSλ algorithm looks for the coefficients of W that minimize e(n) = d(n)− yTISI(n)W(n) taking
into consideration all the past instants with a decaying importance controlled by the parameter λ. To
understand how the training works it will be considered the non-recursive version of the algorithm, note
that the results are equivalent, but it allows for an easier understanding of this method. Considering
the example shown in Fig. 3.2.3, at the end of the training, i.e., the second time the 100th symbol is
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1 2 100 101
Symbol processing order
99
Original FMT packet
(a)
991 2 100 99 99 100 10122 1
Symbol processing order
Artificially extended FMT packet
(b)
Figure 3.10: Application of the described training method assuming that the training sequence ends at
the 100th symbol of the packet and two repetitions are done. In (a) it is shown the original packet and
(b) is the artificially extended packet.
considered, and if no errors were made, the algorithm is trying to minimize the following vector difference:
d(1)λ297
...
d(99)λ199
d(100)λ198
d(99)λ197
...
d(2)λ100
d(1)λ99
d(2)λ98
...
d(99)λ1
d(100)
−
yTFF(1)λ297
...
yTFF(99)λ199
yTFF(100)λ198
yTFF(99)λ197
...
yTFF(2)λ100
yTFF(1)λ99
yTFF(2)λ98
...
yTFF(99)λ1
yTFF(100)
W1(n)
...
WL(n)
(3.15)
Although the application of the forgetting factor makes all vector entrances different, it’s clear that
some values are redundant. Thus the expression can be reduced to almost a third of the size considering
only the first 100 symbols and this is equivalent to the case where the regular training method is applied.
Yet during this training time the PLL, the channel estimate and the filter coefficients are still converging
to good values, so, even if the decisions are all correct, the values of the ISI estimate will vary from
one sweep to the next. This small difference is enough to allow the RLSλ algorithm to have a better
convergence than just with the regular training method. If the algorithm has already converged after the
first sweep, then the others will be redundant and can be avoided. It’s also easy to see that after a few
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sweeps the gain stops and it’s even possible that too many sweeps result in a over fitting of the algorithm
producing bad results on the first symbols processed after the training, since a new convergence time
would happen.
At each each time that the symbols processing order is inverted it is important to note that:
• The Doppler effect is now inverted so θ(n)← −θ(n).
• When the ISI from one symbol is re-computed, the vector of interferences yISI should be update by
new estimate and removing the values from the old estimate that was done in a previous sweep.
• A window from n − L2 to n − L1 should be interpreted as referring to the original symbol order
this insures that there’s no need to change the equations used so far or to reverse the order of the
coefficients in the feedfoward filter and in the channel estimate.
If the number of sweeps is even, i.e., the train process ends at the first symbol, it isn’t possible to
proceed to the 101-th symbol with the same feedfoward filter and channel estimate, unless is assumed
that the channel has no time variability. Taking into account all the constraints, the number of sweeps
is usually three.
3.3 Channel Estimation
Typically in shallow underwater communication there are a lot of echoes coming from alternative paths
with bottom and/or surface reflections. It has already been concluded that this will lead to multipath
effect and, due to the significantly different propagation times, this will correspond to a long and sparse
channel impulse response. The goal, like in OFDM, is to find a good estimate of the channel and include
sparsity in that estimate in order to approximate it as much as possible to the real channel response. In
OFDM, the estimate was done in a single instant and adapted to the following ones to account for the
channel time variations. On FMT the symbols are much shorter in time, so not only one estimate can be
used to the following instant as it is possible to compute a channel estimate recursively from one symbol
to the next.
A simple method for estimating the channel impulse response is presented below. In sections 3.3.1
and 3.3.2 complementary methods that explore the channel sparsity and combine information for the
different subcarriers are explored.
Basic Channel Estimation Algorithm
The basic channel estimation algorithm is adopted from [24]. Referring to (3.4) it’s possible to write
h(n) = E{y(n)d∗(n)
}. (3.16)
Note that outside the training, the actual symbols aren’t known, so the made decisions have to be used.
By stochastic approximation, the channel estimate at time n can be obtained as
h(n) = (1− λ)
n∑i=0
λn−iy(i)d∗(i), (3.17)
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which is easily transformed into a recursive process
h(n) = λh(n− 1) + (1− λ)y(n)d∗(n). (3.18)
The resulting estimate can be directly used in equation (3.11) completing the CHE-DFE receiver as is
depicted in Fig. 3.2.2. Besides its simplicity, this method of estimating the channel estimate can also be
used to track throughout the packet without having to send any known symbols to the receiver, except
the ones from the train. Still, it is clear that this way of estimating the channel response can take some
time to converge to a good estimate, especially under low SNR. Since the channel is sparse, it is possible
to modify this algorithm and speed up its convergence to a good channel estimate. In the case where
the polyphase components are being considered a different channel estimate should be computed for each
since different ISI estimates are needed.
3.3.1 Channel Sparsity
The most simple methods of introducing sparsity into a channel estimate are based in thresholds, in
this work two algorithms of this kind have already been presented in section 2.4.1. They can be directly
applied to the CHE-DFE by changing the equation (3.11) to:
yISI(n) = trunc(h(n), threshold)d(n) + yISI(n) (3.19)
Where the function trunc performs a truncation by cutting all the coefficient of the estimate that
are below the threshold, like it was defined before in section 2.4.1. Contrary to the case in OFDM, the
channel estimates are computed recursively so the sparse version can only be used to compute the ISI,
the uncut version must be kept to be used in the recursion (3.18).
Channel Estimate Through a L0-norm Minimization
The methods used before where focused on cutting the lowest values of the channel estimate, without
looking at the overall result. In this section is presented a method whose goal is to maximize the number
of zero entries in h(n) while still providing a good match to (3.16) [34]. The original idea is to design
a cost function that quantifies how good the estimate is by taking into account both the error e(n) and
the sparsity of the channel. The best way to measure the sparsity of a vector is through its L0 norm
since it provides a count of how many of its entries are non-zero. However, the L0 norm is neither a
convex function nor differentiable, and its exact minimization is a computationally intensive combinatorial
problem. In [34] an alternative is presented. First, the true L0 norm is approximated as
||h(n)||0 'L−1∑k=0
1− e−η|hk(n)|1 , η > 0. (3.20)
The index k denotes the entry of the vector h(n). After this, the authors suggest the following cost
function for channel estimation
J(n) =∑||e(n)||2 + δr(n)HP(n− 1)r(n) + ζ
L−1∑k=0
1− e−η|hk(n)|1 , (3.21)
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where e(n) = [e(n−N) . . . e(n)] and the second term is a Riemannian distance between h(n) and h(n−1),
which encourages the vector of channel coefficients to vary parsimoniously over time, according to an
appropriate metric. The stationarity condition ∇r(n)J(n) = 0 yields a set of time-recursive equations
that are presented below and will be used as an alternative to (3.18) [34]
A(n) = G(n− 1)S(n) (3.22)
B(n) = (S(n)HA(n) + δ−1I)−1 (3.23)
C(n) = A(n)B(n) (3.24)
D(n) = G(n− 1)−C(n)A(n)H (3.25)
vk(n) = e−η|hk(n)|1csgn(hk(n)), k = 0, ..., L− 1 (3.26)
h(n) = h(n− 1) + µC(n)e(n)∗ − µζη
2δD(n)v(n− 1) (3.27)
Above, csgn corresponds to the complex signum function. The matrix S(n) contains the transmitted
symbols when in training mode or the last decisions, in decision-directed mode, such that its i-th column
is equal to [s(n− i) . . . s(n− i−N)]T . G(n) is a diagonal matrix whose l-th diagonal entry is given by
gl(n) =1− τ2L
+ (1 + τ)|hl(n)|1
2 ‖h(n)‖1 + ε. (3.28)
The above expressions require some parameters to be set, namely, µ ∈]0; 1] which is a step size, δ > 0,
η > 0, ζ ' 10−5 > 0, ε ' 10−5 and τ ∈ [−1; 1]. The most important parameter is τ , which controls the
sparsity of the channel estimate and in this work lies between 0 and 0.5.
The most appealing features of this algorithm are the smoothness of its sparse channel estimates, as
opposed to sparsification by hard thresholding, and its natural time-recursive structure. On the downside,
it is significantly more complex than other sparsification methods. Typically L � N , where N is the
number of past decisions and errors that are taken into account in the cost function (often less than 10).
Then, the number of required algebraic operations scales as O(L2) [34].
3.3.2 Fusion of Channel Estimates from Different Subcarriers
So far, the channel estimate was improved by taking advantage of the fact that the physical channel is
sparse. In this section is explored the possibility of viewing all subcarriers as signals that where distorted
by the same physical channel, instead of processing them independently. This approach was considered
in [26] where a method that computes a global channel estimate by combining lower-quality estimates
across subcarriers is presented. The original algorithm has doesn’t take sparsity into account, so it will
be redesigned resorting to process of L0-norm minimization that was presented in the previous section.
Remembering equation (3.3) and dividing the signal into subcarriers, the following relation for the channel
responses at each subcarrier can be written
hm(n) = q(n) ∗ c(n)e−j2πnfm . (3.29)
The subscript m identifies the subcarrier, q(n) is the time-domain signaling pulse, c(n) denotes the
real channel impulse response with total length K = K2 + K1 which is common to all subcarriers. For
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a situation where an array with R receivers is used, the equation is replicated R times. Note that it
is possible to include information from all subcarriers into the same channel estimate, but not from all
the receivers, this is explained in section 3.4. The convolution in the last equation is transformed into a
matrix multiplication, this results in:
hm(n) = Qmc(n), (3.30)
where
Qm =
q(K1 − L1)e−j2πfmK1 . . . q(K2 − L1)e−j2πfmK2
.... . .
...
q(K1 − L2)e−j2πfmK1 . . . q(K2 − L2)e−j2πfmK2
(3.31)
Since c(n) is the same for all subcarriers, the individual equation systems can be joined in a single large
system where the vector hcat(n) is a concatenation of the channel estimates from each subcarrier.
Q(n)c(n) = hcat(n), (3.32)
Q =
Q0
...
QM−1
, (3.33)
hcat(n) =
h0(n)
...
hM−1(n)
. (3.34)
Assuming that Q(n) has a full column rank, the solution for system (3.32) is simply
c(n) = (Q(n)HQ(n))−1Q(n)H hcat(n). (3.35)
To obtain the new channel estimates, one has only to multiply c(n) by Q(n):
hfuse(n) = Q(n)(Q(n)HQ(n))−1Q(n)H hcat(n). (3.36)
Separating hfuse(n) yields the new channel estimates for each subcarrier which can be used to compute
the ISI as in (3.11). After the next decision each channel estimate will be updated using (3.18) followed
by the fusion method resulting in hfuse(n + 1). While the matrix Q(n) is usually tall, it was found to
be nearly column-rank deficient due to the lack of information about the channel frequency response for
the transitions from one sub-band to the next. Poor conditioning of the matrix can lead to erroneous
or meaningless estimates of the channel response. Combining this algorithm with the previous L0-norm
minimization technique will have the double benefit of ensuring that the channel estimate is sparse and
at the same time avoiding a badly conditioned system.
3.3.3 Introducing Sparsity on the Fusion Process
Applying the described method of L0-norm approximation and minimization to (3.32), one obtains
the following cost function
J2(n) = ||eh(n)||2 + δ2r2(n)HZ2(n)r2(n) + ζ
L−1∑k=0
(1− e−η|ck(n)|1), (3.37)
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where eh(n) = hcat(n) − Q(n)c(n) and the second term now corresponds to the Riemannian distance
between c(n) and c(n − 1). Following similar derivations to those of [34], results in the following set of
recursive equations
A(n) = G(n− 1)QH(n) (3.38)
B(n) = (Q(n)HA(n) + δ−1I)−1 (3.39)
C(n) = A(n)B(n) (3.40)
D(n) = G(n− 1)−C(n)A(n)H (3.41)
vk(n) = e−η|ck(n)|1csgn(ck(n)), k = 0, ..., L− 1 (3.42)
c(n+ 1) = c(n) + µC(n)eh(n)∗ − µζη
2δD(n)v(n). (3.43)
Finally, the sparse channel estimate for each subcarrier can be obtained by equation (3.44) and disaggre-
gating the vector hfuse(n) as in the original method
hfuse(n+ 1) = Q(n)c(n+ 1). (3.44)
Two approaches for computing the channel estimate have been described, namely, the recursion in (3.18)
and the set of equations (3.22)–(3.27). After one of these procedures the process of fusion just described
is applied to obtain a better channel estimate. The improved channel estimates are then filtered and fed
to the CHE-DFE in each subcarrier for ISI compensation. Note, however, that the (pre-fusion) channel
estimation procedure in each subcarrier must continue to be done through (3.16).
3.4 Multi-Receivers Gain
In this section two methods for combining information from several receivers are analyzed. The first
part of this section is dedicated to an adaptation of the CHE-DFE to a multi-receiver scenario. On the
second part is presented a structure that combines the signals from the different receivers before the
CHE-DFE. Both methods process each subcarrier independently, so the subscript m will be dropped,
instead the letter r denoting signals from different receivers will be used.
Applying the combining method presented in section 2.5 for OFDM is not an option for the CHE-DFE
receiver structure. In OFDM, the estimate of each decision was obtained by multiplying the received
signal by the correspondent channel estimated gain. If individual CHE-DFEs were used for the different
receivers, the estimated symbols, z(n), would be obtained by a filtering operation that comprise several
instants of the received signals. So it is not possible to have an equation system equivalent to (2.39) to
the FMT receiver.
3.4.1 Multi-Receiver CHE-DFE
This first option is simply an adaption of the already described CHE-DFE to process several signals at
once. It’s not possible to do this for signals from different subcarriers because, for the same time instant,
they correspond to different symbols and a single decision is made in each CHE-DFE. The basic idea is
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Decision+d(n)
z(n)yθ(n)
e(n)
-
+
WFF (n)
RLSλ
-
d(n)
yISI(n)
PLL
e−jθ(n)
yr(n)
Channelh(n)
xestimator
d(n)
z(n)e∗(n)
y1(n)
yR(n)
Figure 3.11: CHE-DFE adaptation to a multireceiver scenario.
to extend the feedforward filter to have RL coefficients, where R is the total number of receivers, and the
RLSλ algorithm will give more higher/lower values to the coefficients from the receivers with more/less
significance. Although one filter is enough for all receivers, different ISI estimates must be computed for
each one. To do this the equation (3.18) has to be repeated for each yr(n) resulting in R distinct channel
estimates. The process of introducing sparsity must also be repeated for each channel estimate, because
the arrivals of the different receivers can have echoes arriving at different times. This receiver structure
is shown in Fig. 3.4.1.
Although it isn’t explicit in the figure, the yISI from all receivers are joined in one single signal
yISI,T = [yISI,1 . . . yISI,R]T and the feedforward filter can be thought of as a concatenation of the filters
for each receiver, WFF = [WFF, 1 . . .WFF,R]T . The RLSλ block implements the following:
WFF(n+ 1) = RLSλ( yISI,T (n), e(n), WFF(n), P (n) ) (3.45)
The main problem of this structure is that the large increase in the size of the filter will probably
lead to a decrease in the convergence rate. Of course the resulting gains will compensate and the overall
outcome is a positive gain, but this suggests that it could exist a better way of combining the signals
from the different receivers.
In section 3.2.1 a decision was made on which Doppler compensation method should be used justified
by the fact that a rotating constellation for symbol decision would be incompatible with the multi-receiver
processing. Suppose for a moment that R = 2, then the feedforward filter input would be two signals, each
one affected by slightly different Doppler effect. The filter output value will consist in a combination of the
values from both receivers that has a phase distortion that doesn’t correspond to none of the estimated
Doppler values. Hence it’s impossible to know by how much the constellation should be shifted. The
two possible solutions would be to pass the Doppler estimates through the feedforward filter and use
the output to correct the constellation or have a different constellation for each receiver. The first case
is the same as multiplying each signal by e−jθm,r(n) which corresponds to the other proposed Doppler
compensation method. The second case is only possible if a different CHE-DFE is used for each receiver,
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meaning that their information would only be combined on the last step, like in OFDM. Since such an
approach has already been discarded, the Doppler compensation must be done using equation (3.7).
3.4.2 Multi-Receiver Pre-Combiner
This second method was proposed in [24], the author suggests that the signals from all the receivers can
be combined into a few signals, no more than 2 or 3, and those are enough to achieve the same total gain
with a structure like the one described in the previous section. This means that the feedforward coefficients
decrease significantly speeding up even more the convergence rate and increasing the performance of the
whole system.
The element that joins the signals from all receivers is called a pre-combiner and it is applied before
the phase compensation. The pre-combiner can be described by a matrix C with R×R′ entries, where R′
corresponds to the number of signals that will go into the CHE-DFE. Mathematically this is equivalent
to:
y’(n− L2 : n− L1) = C(n)y(n− L2 : n− L1) (3.46)
where y’ is a vector that contains the R′ signals that go from the pre-combiner to the multireceiver
CHE-DFE and y is a vector that contains the signals from all R receivers. The authors suggests that C(n)
should be updated through a second RLSλ algorithm that would function in parallel with the already
existing and the PLL, driven by that same error e(n). This means an obvious increase in the ambiguity
of the receiver algorithms, so although the performance may improve in most cases, the risk of divergence
is higher and more FMT packets may be lost due to CHE-DFE divergence.
Another possible limitation of this setup is that the signals from each receiver can have significantly
different associated channel impulse responses. If this is the case, combining the different signals without
performing Doppler compensation or subtracting the ISI, can lead to poor results. The method of fusion
channel estimates is very sensitive to the fact of the signals coming from very similar channels, that
is why it can’t be used to combine information from different receivers. Since the pre-combiner isn’t
directly related to the computation of the channel estimates it may be more robust to these differences,
but testing is needed to confirm whether or not it will hold in real situations.
It’s interesting to note that the multi-receiver gain is based on the differences between received signals.
This means that the possible gain increases with the distance between the receivers, although this is
conditioned by the wave length [36]. If it’s confirmed that the pre-combiner has poor results when the
received signals are too different, then the possible gain of this method will be maximum at an equilibrium
where the signals are different, but still very similar. This means that, depending on the physical channel,
the maximum possible gain could be much lower than in the first case where the performance is expected
to increase proportionally to the theoretical gain. This poses serious difficulties to the success of this
method against the multireceiver CHE-DFE.
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3.5 Single-Carrier Modulation
Single-carrier modulation is the simplest of the considered modulations and also the one that has been
more studied over the years. This section begins with a brief a comparison with generic modulations,
with a special focus on FMT, followed by a description of the used receiver whose basic element is a
CHE-DFE and is very similar to the one used in the case of FMT modulation. FMT modulation can be
seen as sending multiple signals of single-carrier modulation each one at a different frequency, all with
the same bandwidth and pulse shape. This relation, which is further explained below, allows for an easy
adaption of the receiver components from one modulation to the other.
Comparison of Single-Carrier and Multicarrier Modulations
There are two main reasons for including single-carrier signals in this work. The first is to serve as
benchmark to OFDM and FMT, if none of the multicarrier modulations could achieve better results than
single-carrier, then this would be the best option. The second is to highlight some of the advantages of
the multicarrier modulations and why they should be preferred.
One advantage of single-carrier is that there’s no ICI, but the Doppler effect still shifts the carrier
frequency. This is a very well known problem, for example from received radio waves in a moving car,
which usually is solved with a PLL, just like it was done for the individual subcarriers of FMT. As
a direct consequence of having just one carrier, the signal processing has lower complexity. Although
single-carrier modulation has some advantages, it has several limitations that make it unsuitable for most
of the communications systems.
First of all, one carrier only allows for setups where a user at a base station is transmitting the same
information for all the users or a scheme with two users where they takes turns into transmitting and
receiving. By using several subcarriers the spectra is divided such that every user can have its channel. A
second disadvantage is that for a high bit rate, the receiver must have accordingly fast electronics which
means that the monetary costs increase. The same bit rate in a multicarrier system will be divided for
the M subcarriers, so the electronics can be up to M times slower and the receiver can become cheaper
than in single-carrier case.
To compare a single-carrier modulation with FMT, it is considered that available bandwidth is B for
both systems, which means that each FMT subcarrier will have an available bandwidth of B/M . The first
thing to consider is the spectral efficiency. Since FMT has no guards times or bands, the same efficiency
can be achieve in both cases if the same type of pulse is used. Given that the single-carrier signal were
generated raised cosine pulses both modulation have the same spectral efficiency. The main difference
comes from the equalization schemes used in the receivers. To address this question, it’s important to
note that the shortest possible time duration of a symbol in a single-carrier system will be 1/B where
as in FMT the shortest time duration of a symbol will be M/B. Thus in a channel where two arrivals
differ from a number of ms, the ISI will span more symbols in the case of the single-carrier modulation
which means that the processing windows of feedforward filter and of the channel estimator must be
longer. In ideal conditions, to accommodate the echo from the same physical channel the windows of a
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single-carrier receiver should be M times longer than for the an equivalent FMT receiver. Finding the
optimal window for the filter is a complicated process that involves trial and error approach, so apriori
the FMT receiver should be easier to parametrize than the single-carrier. Note that a shorter window
also means that the convergence will be faster, but since the symbols are longer it will correspond to
about the same time. In other words, if the packets have the same time length for both modulations, the
convergence period will correspond to approximately the same percentage of the packet duration. This
reasoning was done considering that the complexity of the RLSλ algorithm grows linearly in the number
of iterations as the number of the filter coefficients increases. While the complexity of basic versions of
RLS scales quadratically with the number of parameters, linear-complexity versions are available[10].
Receiver Structure
The receiver used for processing the single-carrier signal is a simple adaption of the CHE-DFE con-
sidered in 3.2. Considering single-carrier modulations, where the signal comes from an array of receivers
and is oversampled, i.e., the signal polyphase components are used, the receiver structure can be similar
to the one in section 3.4. The main difference is that there’s no possibility of fusion since there’s only one
carrier. The channel estimates can be computed in the same manner as in FMT. It’s important to notice
that the channel estimate windows are longer than in FMT so the convergence of the channel estimates
could be slower. This means that the sparsity methods have an increased importance.
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Chapter 4
Results
In this section the results for three modulations are presented. The signals used in simulation and real
experiments were similar and a summary of their characteristics is available in Table 4.1. The section 4.1
contains the results relative to OFDM. The results for FMT are presented in section 4.2 and the results
for single-carrier modulation are in section 4.3, where a short comparison with FMT results is included.
Based on the presented results, OFDM and FMT are compared in section 4.4.
The experimental results are from data collected off the coast of Vilamoura – Algarve, Portugal,
during the CalCom’10 sea trial in June 2010. A physical channel estimate is shown in Fig. 4.1a where
the channel differences for each receiver are clear, mainly the arrival times. The bottom depth was 140 m
and the distance between the transmitter and the receiver is about 3 km. The transmitter was set at 10
m depth and the 16-receiver array spanned 6 m to 66 m depth, with 4 m spacing between each receiver.
Simulation results were generated with the Bellhop ray tracer, whose configuration approximates the
experimental conditions of CalCom’10. In all cases the bandwidth was 4.5 kHz, the carrier frequency was
5.5 kHz and the packets lasted approximately 2.5 s.
Some algorithm parameters are the same for all the results in this section, namely, the number of
steps for the LS-AT algorithm presented in section 2.4.1 was always 6 and the desired delay spread was
20ms. For the FMT CHE-DFE receiver, the filter window spanned 7 symbols and the forgetting factor
Table 4.1: Parameters for OFDM, FMT and single-carrier signals.
OFDM1 OFDM2 OFDM3 FMT1 FMT2 FMT3 Single-carrier
Bandwidth [kHz] 4.5 4.5 4.5 4.5 4.5 4.5 4.5
Number of subcarriers 128 256 512 8 16 32 1
Carrier spacing [Hz] 35.2 17.6 8.8 562.5 281.25 140.63 -
Symbol interval [ms] 58.4 86.9 143.8 3.56 7.1 14.2 0.44
Number of symbols 47 31 18 703 352 176 5625
Bit rate [bps] 4312.5 5625 6649 4500 4500 4500 4500
Constellation QPSK QPSK QPSK QPSK QPSK QPSK QPSK
45
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−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
10
20
30
40
50
60
0.2
0.4
0.6
0.8
1
t (s)
Depth (m)
Mag
(a)
1506 1508 1510 1512 1514
0
20
40
60
80
100
120
Dep
th (
m)
Sound Speed (m/s)
(b)
Figure 4.1: (a) Physical channel estimate for the 16 receivers and (b) the sound speed profile measured
on site.
for the RLS algorithm was 0.998. The parameters for L0-norm minimization and fusion algorithms were
the following: µ = 0.1, δ = 0.1, ζ = 10−5, β = 0.25, ε = 10−4, η = 10.
4.1 Results Relative to OFDM Modulation
In OFDM, three methods were proposed for the channel estimation and tracking in section 2.4 and
their performances for varying Doppler values are presented in fig. 4.2. The pilot subcarriers where 25%
of the total for the basic pilot method, in the two other cases the same amount of subcarriers was used for
the first channel estimate, but during the tracking is was reduced to 12.5% . It’s clear that using the pilot
method yields the worst results. Tracking the channel, either fully or just the phase evolution, achieves
significant improvements of more than 10 dB in some cases. Unfortunately both the tracking methods
revealed themselves very sensible to the Doppler effect. Note that the first method is much less affected
by the Doppler effect and it actually outperforms the others for the more severe case. This is probably
due to the fact that as the transmitter approaches the receiver, significant channel variations can occur,
for example, the attenuation gets progressively smaller and some echoes might appear or disappear. If
this happens, the magnitude of the channel response estimate changes rather quickly and the tracking
might not be able to keep up. This is supported by Fig. 4.3, which depicts the ratio of the MSE of
two OFDM packets. This first is transmitted in a Doppler free situation whereas the in the second the
transmitter was moving towards the receiver at a constant speed of 2m/s. It’s clear that the quality of
the channel estimate decreases faster in the second case where the physical channel varies much more.
It is also interesting to note that as the number of subcarriers increases there is an increase in the
performance if there’s no Doppler effect. This is explained by the fact that packets with more subcarriers
46
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have less symbols in each carrier, this translates in less updates of the channel tracking algorithm,
hence the channel estimate quality doesn’t degrade so much. Another important factor, is that more
subcarriers in the same global bandwidth, means that each individual bandwidth decreases, hence the
Doppler compensation is harder which explains why for higher velocities the performance worsens instead
of improving with the number of subcarriers.
0 0.5 1 1.5 2 2.5 3−30
−25
−20
−15
−10
−5
0
MS
E [d
B]
||vtx
− vrx
|| [m/s]
Pilot methodChannel tracking methodPhase tracking method
(a)
0 0.5 1 1.5 2 2.5 3−30
−25
−20
−15
−10
−5
0
MS
E [d
B]
||vtx
− vrx
|| [m/s]
Pilot methodChannel tracking methodPhase tracking method
(b)
0 0.5 1 1.5 2 2.5 3−35
−30
−25
−20
−15
−10
−5
0
MS
E [d
B]
||vtx
− vrx
|| [m/s]
Pilot methodChannel tracking methodPhase tracking method
(c)
Figure 4.2: MSE evolution as the Doppler effect increases for OFDM packets with (a) 128, (b) 256 and
(c) 512 subcarriers.
0 10 20 30 40 50
0.2
0.25
0.3
0.35
0.4
0.45
0.5
MS
Es
ratio
symbol index (n)
Figure 4.3: Ratio of the MSEs between two OFDM packets throughout time. In the first packet, both
the transmitter and receiver are stationary, whereas in the second case the transmitter is approaching
the receiver at a constant speed of 2m/s.
The results relative to the sparsity introduction in channel estimates, in the context of the OFDM
receiver, were very poor. In fig.4.4(a) is shown the very common case where the fixed threshold method
doesn’t yield any gain. Resorting to the LS-AT algorithm leads to slightly better results with small
gains being possible, note that fig. 4.4(b) isn’t from the situation as fig. 4.4(a). Even so, for most of
the possible values for the desired delay the outcome of using this algorithm is negative. The interval
between two OFDM symbols is between 58.4ms and 143.8ms depending on the type of packet, given that
the channel estimated delay is 20ms, it’s easy to understand that in this case there’s no sparsity to be
explored by the algorithms. Besides this, the used guard times can not prevent IBI which should also be
responsible for a significant degradation of the results.
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0 0.2 0.4 0.6 0.8 1−16
−14
−12
−10
−8
−6
−4
−2
MS
E [d
B]
Threshold
(a)
5 10 15 20 25 30 35 40 45 50−11
−10.5
−10
−9.5
−9
−8.5
MS
E [d
B]
Length of the largest sequence of zeros
(b)
Figure 4.4: MSE of an OFDM packet with 128 subcarriers when the channel sparsity is explored with
(a) a fixed threshold and with (b) the LS-AT method.
The final step in OFDM demodulation is the combining of the decisions from different receivers, the
result for this can be seen in fig.4.1, to different numbers of receivers, note that they were chosen in
order, i.e., the case of r refers to receivers 1, 2, . . . , r. Three different situations are presented and for
all of them it is clear that the biggest gain happens when going from 1 up to 4 receivers. For the channel
tracking algorithm the improvement from using several receivers is about 10dB more than in the case
of the simple pilot method. A signal distorted by a small Doppler effect was also included achieving a
similar performance to the first case. Still, it suffers a significant performance decrease when the number
of receivers is too high. Note that this also happens to the other cases, yet with less amplitude. There’s
no obvious explanation for this, since theoretically the gain should increase with the number of receivers.
0 2 4 6 8 10 12 14 16−22
−20
−18
−16
−14
−12
−10
−8
−6
−4
MS
E [d
B]
Number of receivers
Pilot MethodChannel Tracking MethodPilot Method with Doppler Effect
(a)
Figure 4.5: MSE evolution as the number of receivers increases for an OFDM packet with 256 subcarriers.
48
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4.2 Results Relative to FMT Modulation
For FMT, the CHE-DFE was chosen over the DFE as a receiver, because is achieves significantly
better results. In Fig. 4.6 is shown the typical receiver when a DFE and a CHE-DFE are considered for
an experimental FMT packet. Note that the channel estimate was obtained with the basic algorithm from
recursion (3.18) without exploring the sparsity. With so many possible improvements in the CHE-DFE
receiver and so little in the DFE it is clear that the first should be used. The feedforward filter window
goes from n − 5 up to n + 1 in both cases, the feedback filter window goes from n − 5 up to n − 1 and
the channel estimate ranges from to n− 10 up until n+ 10.
0 100 200 300 400 500 600 700 800−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re{
z 1(n)}
symbol index (n)
(a)
0 100 200 300 400 500 600 700 800−2
−1.5
−1
−0.5
0
0.5
1
1.5
2R
e{z 1(n
)}
symbol index (n)
(b)
0 100 200 300 400 500 600 700 800−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re{
z 1(n)}
symbol index (n)
(c)
Figure 4.6: Convergence of the (a) DFE, (b) DFE with feedforward filter and (c) CHE-DFE for the first
subcarrier.
10−6
10−4
10−2
100
102
−8
−7
−6
−5
−4
−3
−2
−1
δ
MS
E [d
B]
FMT1FMT2FMT3
Figure 4.7: MSE for different initializations of matrix P from the RLSλ algorithm.
In Fig. 4.7 is shown the effect of choosing different values of δ for the initialization of matrix P from
the RLSλ algorithm. To emphasize how much the convergence period can be improved, the MSE was
taken considering only the first 150 symbols of each packet. The values come from the experimental data
and it’s clear the best value is δ = 0.1, although the same evolution was observed for simulated data so
that value was chosen to generate all remaining results involving the CHE-DFE. Note that the variance
for both simulated and experimental received symbols is typically around 0.1, which would indicate that
δ should be smaller than 10−3[10]. Like it was explained in section 3.2.3 the choice of wrong δ causes the
49
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RLSλ to converge to a local minimum, which then leads to a second convergence period. In Fig. 4.8(a)
is shown a case of the convergence to a local minimum for δ = 10−5. The matrix P was observed at
time instant 56 for a CHE-DFE receiver where 4 receiver and 2 polyphase components were considered
and the filter window spanned over 7 symbols, so the rank of P is 56. In Fig. 4.8(b) is shown a matrix
P for the same circumstances, but δ is 0.1 so the algorithm converged to the absolute minimum. At the
end of the packet, where the algorithm has converged, the P matrix is almost diagonal, this can be seen
in Fig. 4.8(c). This final matrix is very similar in all the packets for both simulated and experimental
data and for any value of δ. The particular matrix shown in Fig. 4.8(c) was obtained at the end of same
packet from where the matrix in Fig. 4.8(a) was taken. The absolute of the diagonal average value is
1.045 which is between the inverse of the two values of δ that minimized the MSE in Fig. 4.7, 0.1 and 1.
This explains why the algorithm converges immediately to the absolute minimum when P is initialized
close to 1.045I, but has an initial bad convergence if δ is too far from this value.
(a) (b) (c)
Figure 4.8: Absolute values of matrix P when the RLSλ began by converging to (a) a local minimum
(δ = 10−5), (b) the absolute minimum (δ = 0.1) and (c) at the final of packet that initially converged to
a local minimum (δ = 10−5).
All the other results of this chapter were obtained using δ = 0.1. The forgetting factor the RLSλ
algorithm, was found to influence very little the rate of convergence or the final MSE. Although λ = 0.998
usually produces best results, the typical difference was almost always less than 1dB, still this value was
used for generating the remaining results. As example, some MSE values are provided in Tables 4.2 and
4.3 for different signals in similar simulated and real conditions.
The training methods where compared by increasing the length of the training between 12 and 200
symbols for packets with 700 symbols and the results are presented in Fig. 4.2 for three sweeps of the
new training method. It’s important to note if training is too short, then the RLSλ algorithm doesn’t
converge. When training is long enough for ensure convergence, the final MSE won’t increase by further
extending the training period, since the vast majority of the decisions made are already correct. This
effect is very clear in Fig. 4.2 for regular training. In the new training method the average training length
is about 10 symbols whereas typically regular training needs at least 20 symbols. More important, the
new training can lead to a better convergence of the CHE-DFE, consequently significantly decreasing
the MSE. Fig. 4.2(a) depicts a remarkably good case where the error decreases by about 15 dB, even
with shorter training. Unfortunately, this happens very rarely, usually on packets with small channel
50
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Table 4.2: MSE for λ = 0.995.
FMT1 FMT2 FMT3
Simulation [dB] -12.46 -14.36 -11.42
Experimental [dB] -8.33 -6.90 -5.76
Table 4.3: MSE for λ = 0.998.
FMT1 FMT2 FMT3
Simulation [dB] -12.37 -15.58 -12.02
Experimental [dB] -8.08 -6.92 -5.86
distortion, in this particular case the packet corresponded to a simulated FMT1 packet in a Doppler free
scenario. In the majority of the simulated packets the final MSE improves at most by 4 dB, although
there’s a big variance of the results. Another problem of this method is that in some cases, spikes in
the MSE occur for some training lengths like it’s shown in Fig. 4.2(b) with no explanation found so far.
Finally, in Fig. 4.2(c) is shown a case where the new training method yields worse results, which happens
more frequently than the case in Fig. 4.2(a). In most experimental packets the final MSE is the same for
both trainings, although the training length can usually be decreased when the new method is employed.
0 50 100 150 200−35
−30
−25
−20
−15
−10
−5
0
MS
E [d
B]
Train length
Training with 1 sweepTraining with 3 sweeps
(a)
0 50 100 150 200−25
−20
−15
−10
−5
0
MS
E [d
B]
Train length
Training with 1 sweepTraining with 3 sweeps
(b)
0 50 100 150 200−12
−10
−8
−6
−4
−2
0
MS
E [d
B]
Train length
Training with 1 sweepTraining with 3 sweeps
(c)
Figure 4.9: Possible MSE evolution for the different training methods as number of known symbols at
receiver increases.
Contrary to the case of OFDM modulation, the threshold methods for achieving sparse channel
estimates achieve good results. Still the fixed threshold method revealed to be least efficient, with results
most of the times below the varying threshold method. Fig. 4.10 shows the mean-square error (MSE)
values for two FMT1 packets. On the first, the result is similar to the one in [24], but in the second, the
MSE is almost constant for every threshold. This situation happened many times, and may be caused
by channel time variations that render a fixed-threshold approach too restrictive. It can also be due to
the fact that channel estimates take some time to converge and during the transient period it would be
better to have a high threshold to avoid discarding possibly relevant arrivals, whereas upon convergence it
would be best to set a low threshold. This hypothesis is supported by the analysis of the threshold values
on the varying threshold method, which is typically high in the beginning of the packet, but afterwards
decreases to close to zero. Fig. 4.11 shows the variation of the threshold for the same packet of Fig. 4.10b,
51
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0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
Threshold
−M
SE
(a)
0 0.2 0.4 0.6 0.8 17.85
7.9
7.95
8
8.05
8.1
8.15
8.2
8.25
−M
SE
Thresholds
(b)
Figure 4.10: The two most common profiles of the MSE in experimental FMT1 packets as the fixed
thresholds increases.
0 100 200 300 400 500 600 7000
0.05
0.1
0.15
0.2
0.25
time (n)
Thr
esho
ld
(a)
0 100 200 300 400 500 600 7000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Thr
esho
ld
time (n)
(b)
Figure 4.11: Evolution of the varying threshold for (a) a simulated and (b) an experimental FMT1 packet.
which is representative of the FMT data set. From this last figure, it can also be seen that in a time
varying channel, the threshold typically raises again, possibly because the channel estimate is getting
outdated at the end of the packet.
Fig. 4.12 shows several channel estimates computed independently in each subcarrier, i.e., without
fusion process, all from the same transmitted packet of type FMT1. The first estimate, which has no
sparsity, was obtained by using just the simple recursion in (3.18). At the 100th symbol, the varying
threshold is already very close to zero, which explains the similarity of Figs. 4.12a and 4.12c. The last
estimate, obtained by minimizing the L0−norm approximation of the channel estimate, is much smoother
and in line with what one would expect of a real noiseless underwater channel. Another important fact is
that in all cases the estimates from different subcarriers are almost overlapping, which supports the idea
that joint estimation may enhance the performance. Moreover, it is clear that channel estimates from
different receivers, shown in Fig. 4.1, are much more dissimilar than the ones from different subcarriers.
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−10 −5 0 5 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Mag
nitu
de
Time [symbols intervals]
(a)
−10 −5 0 5 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Mag
nitu
de
Time [symbols intervals]
(b)
−10 −5 0 5 100
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Mag
nitu
de
Time [symbols intervals]
(c)
−10 −5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Mag
nitu
de
Time [symbols intervals]
(d)
Figure 4.12: Channel estimates from all the subcarriers at the 100th symbol of a FMT1 packet for (a)
simple recursion method, (b) fixed threshold, (c) varying threshold and (d) minimization of the L0-norm
approximation.
This is why we can group all the different subcarriers estimates into the same equation system (3.32),
but must have a different equation system for each receiver.
Fig. 4.13 shows the evolution of the output of CHE-DFE for one of the FMT subcarriers in a simulated
packet where the transmitter moves towards a stationary receiver at 2 kn. MSE values as a function of
relative speed are given in Fig. 4.14, making it clear that sparsification through L0-norm approximation
outperforms the other methods even in the presence of significant Doppler shifts. It can also be seen that
for fewer subcarriers the Doppler effect is better supported, this comes directly from the fact the these
correspond to the case where the subcarriers have larger bandwidth so they are more robust to frequency
shifts.
Results for the fusion process are present in Fig. 4.14 to provide a comparison with other channel
sparsifying methods, but also in Fig. 4.15. The resulting channel estimate is similar to the ones obtained
by independently minimizing the approximated L0-norm for each subcarrier, as before, but the main
difference is in the evolution of the CHE-DFE. Previously, the quickest convergence was attained after
about 150 symbol intervals for the L0-norm minimization, whereas here the fusion process decreases
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0 100 200 300 400 500 600 700 800−1.5
−1
−0.5
0
0.5
1
1.5
symbol index (n)
Re{
z 1(n)}
(a)
0 100 200 300 400 500 600 700 800−1.5
−1
−0.5
0
0.5
1
1.5
symbol index (n)
Re{
z 1(n)}
(b)
0 100 200 300 400 500 600 700 800−1.5
−1
−0.5
0
0.5
1
1.5
symbol index (n)
Re{
z 1(n)}
(c)
0 100 200 300 400 500 600 700 800−1.5
−1
−0.5
0
0.5
1
1.5
symbol index (n)
Re{
z 1(n)}
(d)
Figure 4.13: CHE-DFE evolution through a FMT1 packet for (a) simple recursion method, (b) fixed
threshold, (c) varying threshold and (d) minimization of the L0-norm approximation.
this convergence time by 100 symbol intervals. In Fig. 4.15(c) can be seen the estimate c(n) where it
is possible to distinguished the pre-echo that is also present in Fig. 4.1(a). Finally, the constellation
resulting from the fusion process is shown in Fig. 4.15(d).
Although all the results presented so far, relative to FMT processing, have used 4 receivers, this hasn’t
been justified yet. The gains for a using several receivers can be observed in Figs. 4.2(a)-(f) for both
methods presented in section 3.4. As in Section 4.1 for OFDM, the receivers were chosen from shallowest
to deepest. In all cases there’s a significant improvement up to 4 receivers and increasing the number of
receivers to 6 still adds some gain. The use of 4 receivers is considered to yield good results and since a
setup with more receivers means an increase of cost it was considered that it might not compensate to use
more receivers. Besides, this choice is the same for the OFDM receiver, which ensures that none of the
modulations has the upper hand for the remaining generated results. From 6 receivers up, the evolution
depends on the chosen method. For the simple combination of the received signals in one CHE-DFE, the
MSE is almost stable, usually varying less than 1 dB. When the pre-combiner is considered, the MSE
decreases in most of the situations, sometimes up to 2 dB. This is presumed to be due to the fact that
as the signals from more distant receivers are included, their disparity is excessive for the pre-combiner
method to maintain its gain. Note that the performance should improve when the receivers are chosen
54
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0 0.5 1 1.5 2−50
−40
−30
−20
−10
MS
E (
dB)
||vtx
−vrx
|| (m/s)
threshold=0threshold=0.2varying thresholdmin{L0−norm}fusion
(a)
0 0.5 1 1.5 2−50
−40
−30
−20
−10
MS
E (
dB)
||vtx
−vrx
|| (m/s)
threshold=0threshold=0.2varying thresholdmin{L0−norm}fusion
(b)
0 0.5 1 1.5 2−50
−40
−30
−20
−10
MS
E (
dB)
||vtx
−vrx
|| (m/s)
threshold=0threshold=0.2varying thresholdmin{L0−norm}fusion
(c)
Figure 4.14: MSE evolution with increasing Doppler shifts for the different channel estimation methods
in packets of (a) FMT1 (b) FMT2 and (c) FMT3.
in order to maximize their spacial diversity like it happens with the more simple method.
Comparing the absolute values of the error, and not just its evolution, it is clear that the pre-combiner
method had a poorer performance in all cases, except to the FMT1 packet. A possible explanation is
that for more subcarriers the signal is more sensitive to the Doppler effect and, since the pre-combiner is
applied before Doppler compensation, the pre-combiner might not be capable of handling such Doppler
shifts. Other possibility, directly related, is that for signals with more Doppler shift the intrinsic ambiguity
of the receiver with competing processing blocks is more likely to affect the results, especially in this case
where two RLSλ algorithms are employed.
Section 3.2.1 mentioned a problem of ambiguity between the values of the Doppler compensation and
the coefficients of feedforward filter. This happened because the PLL and the RLSλ algorithms where
driven by the same error. On experimental results the ambiguity problem doesn’t appear in a clear
way, but it did for some simulations as illustrated Fig. 4.2. For the lower velocities the Doppler effect is
apparently correctly compensated. When the speeds increases to 0.8m/s, the PLLs for all the receivers
diverge and the CHE-DFE convergence for the best carrier is shown below in in Fig. 4.2. The overall
MSE for that packet was of -0.65dB.
Looking carefully at the beginning of phase compensation for the four cases from Figs. 4.2(a)-(d), a
sudden transition can be seen. Until this transition the overall phase distortion was so little that the PLL
couldn’t track it, probably because the RLSλ algorithm still hadn’t converge and the filter coefficients
55
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−10 −5 0 5 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Mag
nitu
de
Time [symbols intervals]
(a)
0 100 200 300 400 500 600 700 800−1.5
−1
−0.5
0
0.5
1
1.5
symbol index (n)
Re{
z 1(n)}
(b)
0 50 100 150 200 250 300 3500
0.01
0.02
0.03
0.04
0.05
0.06
Mag
time
Channel estimate for receiver 2Channel estimate for receiver 6Channel estimate for receiver 10Channel estimate for receiver 14
(c)
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
2 222
2
22 2
2
2 2
2
22
2
22
2
2
2
2222
2 2
22
21
Real Part
Imag
inar
y P
art
(d)
Figure 4.15: Results for the fusion process in a FMT1 packet: (a) Channel estimates, (b) High resolution
channel estimate (c(n)), (c) CHE-DFE evolution and (d) constellation of the symbol estimates.
where influencing the phase of the error to vary almost randomly. Then, because the RLSλ algorithm
starts to converge or because the phase offset becomes so large that the PLL can lock onto it, the phase
starts to be correctly compensated. This generates the abrupt transition at the beginning of each packet.
In the last case, Fig. 4.2(d), the error in the phase compensation is too high at this transition time and the
PLL diverges. It’s important to note for the speed of 0.2m/s and 0.4m/s the final phase compensation was
of about 7 rad and 14 rad. Since the velocity of the transmitter is constant, the Doppler effect increases
proportionally to the velocity, so in third the case the final phase compensation should be about 21 rad,
although it isn’t quite, probably because the ambiguity was already causing problems. Besides the phase
compensation isn’t as linear as it is in the first cases, containing some oscillations. In the fourth case, the
final phase compensation should be about 28 rad, but none of the PLLs manages to do this.
It’s important to point out that the FMT receiver isn’t limited to signals with such low Doppler values,
in fact results with higher speeds have already been presented in Fig. 4.2. So the bad performances due
to the ambiguity on the receiver don’t depend only on the Doppler effect, which further emphasizes the
fact that there’s always a risk of losing one FMT packet due to this.
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0 2 4 6 8 10 12 14 16−20
−18
−16
−14
−12
−10
−8
−6
Number of Receivers
MS
E [d
B]
||v
tx−v
rx||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(a)
0 2 4 6 8 10 12 14 16−20
−15
−10
−5
MS
E [d
B]
Number of Receivers
||v
tx−v
rx||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(b)
0 2 4 6 8 10 12 14 16−18
−16
−14
−12
−10
−8
−6
−4
Number of Receivers
MS
E [d
B]
||v
tx−v
rx||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(c)
0 2 4 6 8 10 12 14 16−22
−20
−18
−16
−14
−12
−10
−8
−6
Number of Receivers
MS
E [d
B]
||v
tx−v
rx||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(d)
0 2 4 6 8 10 12 14 16−15
−14
−13
−12
−11
−10
−9
−8
−7
−6
−5
Number of Receivers
MS
E [d
B]
||vtx
−vrx
||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(e)
0 2 4 6 8 10 12 14 16−16
−14
−12
−10
−8
−6
−4
Number of Receivers
MS
E [d
B]
||vtx
−vrx
||=0 m/s
||vtx
−vrx
||=1 m/s
||vtx
−vrx
||=2 m/s
||vtx
−vrx
||=3 m/s
||vtx
−vrx
||=4 m/s
||vtx
−vrx
||=5 m/s
||vtx
−vrx
||=6 m/s
(f)
Figure 4.16: MSE evolution for an increasing number of receiver (a) combining the signals in a single
CHE-DFE from an FMT1 packet, (b) from an FMT2 packet, (c) from an FMT3 packet and (d) using
the pre-combiner method for an FMT1 packet, (e) for an FMT2 packet and (f) for an FMT3 packet.
4.3 Results Relative to Single-Carrier modulation
The receiver for single-carrier modulation is very similar to the FMT receiver, as explained in section
3.5, so most of the results are similar for both cases. One of the main questions was about the difficulty in
parameterizing each of the receivers, this is addressed by Fig. 4.19 where the evolution of the CHE-DFE
output is depicted for one simulated single-carrier packet.
The conditions are the same as in the packet from Fig. 4.13a and in both cases no sparsity was forced
into the channel estimate. The overall MSE in the single-carrier transmission is -20.23 dB, which is
slightly worse than -20.86 dB for the FMT packet of Fig. 4.13a. The feedforward filter for the single-
carrier packet spans 18 symbols and the CHE-DFE converges in about 200 symbol intervals, whereas in
FMT the filter spans 7 symbols and converges in 30 symbol intervals. Note that FMT symbols are longer
than those for the single-carrier packet; convergence times, in seconds, for both packets are similar.
The impact of the training method, the introduction of sparsity in the channel estimates and the use
of more than one receiver are the same for both modulations.
4.4 Comparison Between OFDM and FMT Results
The main reason to search for an alternative modulation to OFDM was that it is very sensitive to
the Doppler effect, as can be seen in Fig. 4.4. These results are comparable to Fig. 4.14 for FMT since 4
receivers where used in both cases and the simulation conditions were the same. The results from OFDM
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0 100 200 300 400 500 600 700−1
0
1
2
3
4
5
6
7
symbol index (n)
θ1(n
) [r
ad]
Receiver #2Receiver #6Receiver #10Receiver #14
(a)
0 100 200 300 400 500 600 700−2
0
2
4
6
8
10
12
14
16
18
symbol index (n)
θ1(n
) [r
ad]
Receiver #2Receiver #6Receiver #10Receiver #14
(b)
0 100 200 300 400 500 600 700−2
0
2
4
6
8
10
12
14
16
18
θ1(n
) [r
ad]
symbol index (n)
Receiver #2Receiver #6Receiver #10Receiver #14
(c)
0 100 200 300 400 500 600 700−10
0
10
20
30
40
50
60
symbol index (n)
θ1(n
) [r
ad]
Receiver #2Receiver #6Receiver #10Receiver #14
(d)
Figure 4.17: Doppler compensation for a simulated FMT packet where the transmitter is approaching
the receiver at a constant speed of (a) 0.2 m/s, (b) 0.4 m/s, (c) 0.6 m/s and (d) 0.8 m/s.
don’t include sparsity in the channel response, but it has already been verified that such techniques
didn’t prove themselves useful in this case. Comparing with the case where no sparsity algorithms were
used in FMT, OFDM achieves better results by about 5 to 10dB for channel tracking methods when
both the transmitter and receiver and stationary. As the Doppler effect starts to distort the signal FMT
modulation behaves much better and outperforms OFDM by up to 15 dB if the results from FMT1 packet
are compared to ones from OFDM2 or OFDM3 packets. For the simple threshold methods for channel
sparsity, the MSE in FMT is already very similar to OFDM even for a situation where the Doppler effect
isn’t present. Using the more powerful methods of the L0-norm minimization and the fusion process,
the FMT surpasses largely OFDM by more than 15 dB, reaching up to 20 dB in some cases. Still,
the introduction of sparsity in the CHE-DFE has a low resistance to the Doppler effect and these gains
quickly decrease and become similar to the case where the raw channel estimate is used, only slightly
better.
The fact that the sparsity methods have much bigger impact on FMT is explained by the duration of
the symbols of each modulation. It was already mentioned that the shortest OFDM symbols last about
58 ms and by Fig. 4.1(a) it is clear that the largest delay is less than 30 ms, this is enough to cause ISI,
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0 100 200 300 400 500 600 700 800−6
−5
−4
−3
−2
−1
0
1
2
3
4
Re{
z 1(n)}
symbol index (n)
Figure 4.18: Evolution of the CHE-DFE for an FMT1 packet where the receivers ambiguity destroyed
the RLSλ convergence.
Figure 4.19: Evolution of CHE-DFE on a single-carrier packet.
but not a significantly long or sparse channel impulse response. In FMT, the largest symbols duration
is 14.2 ms, which is about half the largest possible delay which is already a significant difference from
OFDM. The case of FMT1 packets, where the symbols only last 3.56ms, is the worst in terms of ISI since
the channel impulse response spans over more than 10 symbols, considering that there’s a slight pre-echo.
Yet, by introducing sparsity this is the case that achieves better results, specially for the cases where the
Doppler effect is present and ICI has to be dealt with.
The other case where it’s important to compare both modulations is in the multireceiver scenario, for
this Figs. 4.1 and 4.2 should be considered. For both cases it’s clear that the most significant gains are
obtained where the number of receivers rises from 1 to 4, continuing to increase up to 8 receivers. From
this point on the MSE is stable for FMT if the multireceiver CHE-DFE is considered, but starts to decrease
in the case where the pre-combiner of to the OFDM final LS estimation. The overall improvements in
the MSE are about 15 dB for OFDM when the channel tracking method is used whereas in FMT this
gain is near 9 dB. For both modulations, these increases in performance are sustained as the Doppler
starts to increase.
Finally, besides the MSE, the achieved bit rate should be analyzed since it is the parameter that
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directly concerns the users. In this work it was assumed that the coding performance is the same for
both modulations so the differences in efficiency come only from the parameterizations and the shapes of
the signaling pulses and from the training methods that are used in the receivers.
The bit rates in all cases are shown in Table 4.1, where it can be seen that OFDM has higher values,
specially for the OFDM3 packets. Still, pilot subcarriers in OFDM and symbols for training in FMT
must be considered. For a situation with no Doppler effect, the best result in OFDM was about -32 dB
for an OFDM3 packet with 12.5% of the subcarrier being used was pilots (Fig. 4.2), this corresponds to a
useful bit rate of 5817.875 bit/s. For FMT, in the same situation, better MSE is achieved with the more
advanced sparsifying methods and training length can be as low as 10 symbols with the developed method
(Fig. 4.14. This means, that for a FMT1 packet the useful bit rate would be 4435.6 bit/s. Considering
that the transmitter is moving towards a stationary receiver at 2 m/s, and referring again to Figs. 4.2 and
4.14, the best result for OFDM is -10 dB using 25% of the subcarriers as pilots in a OFDM1 packet, while
FMT can achieve a better performance with the same training length of 10 symbols. The useful bit rate
for OFDM decreases to 4312.5 bit/s and FMT successfully maintained the values of 4435.6 bit/s. This
points out that, although OFDM may achieve higher bit rates than FMT, if the two methods are to have
the same performance the first has to sacrifice a lot more bit rate. Eventually, for example for significant
Doppler effect, FMT surpasses the useful bit rate of OFDM while still achieving better performance.
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Chapter 5
Conclusion
In this chapter the main contributions and innovations are reviewed in section 5.1 and related to the
goals of this thesis. Potential new developments and some of the unsolved problems are addressed in
section 5.2.
5.1 Achievements
The goal of this work was successfully fulfilled by developing and implemented new techniques for
FMT modulation resulting in a strong alternative to OFDM modulation in underwater communications.
The tackling of problems like channel estimation by contextualizing them with the known characteristics
of this type of communications paid off, yielding significant increases in the overall performances.
For OFDM, channel estimation and tracking were implemented, including sparsity methods resorting
to thresholding. To track the channel a new method was proposed, consisting only in tracking the phase
considering that the amplitude of the response channel estimate is constant through the packet. In section
4.1, it was confirmed it performs very similarly to the case where the whole channel response is updated
from one symbol to the next with reduction in the number of algebraic operations.
The basic receiver for FMT was chosen as a CHE-DFE due to the direct physical meaning of its
components, namely the coefficients of feedforward filter predict ISI from future symbols, whereas the
channel estimates enable the removal of ISI from the past symbols. A problem of bad initial convergence
of the RLSλ algorithm that updates the filter coefficients was successfully identified and corrected by
changing the initialization value of the inverted correlation matrix P .
A new method for the training of the CHE-DFE was developed yielding very promising results in
terms of the final convergence of the RLSλ algorithm and with coherent and significant reduction of
the necessary training length. The performance of the CHE-DFE was further improved by applying
the same sparsity methods used in OFDM for the channel estimates, plus a more advanced method of
L0-norm minimization was also considered. Moreover, the channel estimates from different subcarriers
where combined to improve each of them in a process of fusion that had previously been proposed, but
was reformulated to include sparsity in the final estimates.
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Multireceiver scenarios were successfully considered in the reception of both modulations with sig-
nificant performance improvements. In FMT, the CHE-DFE was redesigned to enable the input of the
signals from all receivers in the same process of decision. An alternative method of pre-combining the
signals before the CHE-DFE was implemented, but with poor results.
A single-carrier receiver was successfully adapted from FMT modulation that allowed for comparing
both in terms of their performances and parameterizations. It was confirmed that the windows length for
single-carrier have to be longer than in FMT to achieve good results, so the finding the optimal window
size is harder for single-carrier.
Both the modulations were compared for simulated and real data and it was observed that the FMT
outperforms OFDM whenever significant Doppler is present. For situations where the transmitter and
receiver are static, FMT still outperforms OFDM by using the processes of L0-norm minimization and
fusion to compute the channel estimates. When the simpler thresholding methods for exploiting channel
sparsity are considered, OFDM and FMT perform similarly and OFDM only surpassed FMT when a
simple channel estimate, without sparsification, was used in the CHE-DFE.
The downsides of FMT is that for most of the considered cases it achieves less bit rate than OFDM and
the receiver complexity significantly increases for some of the considered methods. The parametrization
of the FMT receiver can be harder than its OFDM counterpart since there’s not an automatic method
or absolute rules for choosing the best window for the feedforward filter and the channel estimate.
5.2 Future Work
The implementations and developed methods open several possibilities for further improvements,
especially for the FMT receiver.
In the case of OFDM it should be interesting to explore the possibility to track the channel with fewer
subcarriers and find simpler ways of tracking the channel estimate. In particular, if a phase estimate
could be computed, instead of a complete channel response, this would mean a reduction to half the
coefficients that are estimated in the current tracking method. The sparsity inducing methods should be
tested in channels where the echoes are more distant to see what kind of improvements they can bring to
OFDM. In such a situation it would also be important to develop other methods or adapt the ones used
in FMT. The fact that for many receivers the multireceiver gain starts to decrease suggests that better
combining methods could be developed. Finally, the lack of robustness to Doppler should be addressed
since it is the most limiting aspect of this modulation.
For FMT, the developed training method should be improved to asserting that the impressive gains
are attained by a majority of cases, instead of the current small fraction of packets. The relationship
with the convergence of the RLSλ and the PLL algorithms might provide answers to this.
Further investigations may/should seek a better understanding of how the threshold should vary in
each transmitted packet and finding less complex methods for introducing sparsity than the approximate
L0-norm minimization approach without sacrificing performance. Such methods could easily be incorpo-
rated in the fusion method and that adaptation might provide higher gains. The methods implemented
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in this work were found to have a low robustness to the Doppler effect, so new methods should take that
into consideration.
New methods to improve the gain from multiple receivers can also be sought, but more importantly
would be to investigate the best placement of the receivers, namely, the distance between them. This is
a point of common interest with the OFDM and single-carrier modulations.
The phase ambiguity problem in the FMT receiver was proven to be very limiting in terms of how
much Doppler effect can be supported. Developing more powerful phase synchronization methods might
thus lead to improvements in the overall convergence speed and residual MSE of FMT receivers. It’s
believed that the remaining components of the FMT receiver, namely, the methods for obtaining sparse
channel estimates, the fusion process and the adaptation of the CHE-DFE to a multireceiver scenario,
are all capable of supporting higher Doppler rates if the ambiguity problem was solved.
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