A polynomial phase model for

estimation of underwater acoustic

channels using superimposed pilots

Felix Trulsson

Engineering Physics and Electrical Engineering, master's level

2019

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

ABSTRACT

In underwater acoustic communications the time variation in the channel is a huge chal-

lenge. The estimation of the impulse response at the receiver is crucial for the decoding

of the signal to become accurate. One way is to transmit a superimposed pilot sequence

along the unknown message, and by the knowledge of the sequence have the possibility

to continuously track the variation in the channel over time.

This thesis investigates if it is possible by the aid of superimposed pilot sequences to

separate the taps in the channel impulse response and using a parametric method to

describe the taps as polynomial phase signals.

The method used for separation of the taps was a moving least squares estimator.

Thereafter each tap was optimised to a polynomial phase signal (PPS) using a weighted

non-linear least squares estimator. The non-linear parameters of the model was then

determined with the Levenberg-Marquardt method. The performance of the method was

evaluated both for simulated data as well as for data from field tests. The performance

was determined by calculating the mean squared error (MSE) of the model over different

frame lengths, signal to noise ratio (SNR), weights for the superimposed pilots, rapidness

of time variation and impulse response lengths.

The method was not sensitive to the properties of the channel. Even though the model

had high performance, the complexity of the computations generated long compilation

times. Hence, the method needs further work before a real time implementation could

be possible.

iii

PREFACE

I am grateful for the opportunity to perform the work of this thesis at FOI. I would

personally like take the opportunity to give a special thanks to Dr. Magnus Lundberg

Nordenvad for introducing me to the thesis topic as well as the overall interesting field

of underwater communication and to Professor Jaap van de Beek for the interesting

discussion the last few months. Both of which have been my supervisors during the

work.

I would also like to thank Bernt Nilsson for helping me extract field test data and

for the discussions of the content and Professor Johan Carlson for providing the LaTeX

template.

The fantastic illustrations in Fig. (1.3) and on the cover deserves some extra attention,

which all goes to Frida Georgsson.

Last but absolutely not least, a great thanks to my friends and family for all the support

during my time at the university.

Felix Trulsson

v

CONTENTS

Chapter 1 – Introduction 1

1.1 Purpose and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to digital communication . . . . . . . . . . . . . . . . . . . 2

1.3 Underwater acoustic communication channels . . . . . . . . . . . . . . . 4

1.3.1 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.3 Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.4 Time variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.5 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 A review of underwater acoustic communications . . . . . . . . . . . . . 6

1.5 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2 – Theory 11

2.1 Signal representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Superimposed pilots . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Channel representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Channel impulse response . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Polynomial phase signals . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Moving linear least squares estimator . . . . . . . . . . . . . . . . . . . . 17

2.4 Weighted non-linear least squares estimator . . . . . . . . . . . . . . . . 20

2.5 Exhaustive search method . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Levenberg-Marquardt method . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 3 – Method 27

3.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Separation of taps in time variant impulse response . . . . . . . . . . . . 29

3.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 4 – Results 31

4.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Chapter 5 – Discussion 39

5.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 6 – Conclusion 43

viii

CHAPTER 1

Introduction

1.1 Purpose and outline

Underwater wireless communication has played an important role in commercial systems

as well as in the military. It is necessary for national security and defence, the off-shore

oil industry, pollution monitoring, collection of scientific data, disaster detection and

early warnings, and also for the discovery of new resources. As the technology develops

rapidly these fields grows explosively. Hence, the need for stable, high rate underwater

communication is and will be a field that have to evolve [1]. This thesis serves as an

attempt to accurately estimate all taps in the impulse response from the channel of an

underwater acoustic communication channel.

The thesis is composed as follows. Chapter 1 first, apart from the purpose and outline,

gives a review in the field of underwater communication, its applications and the proper-

ties of the communication channel. It then introduces the concept of channel estimation

and shortly describes different different method which have been used in present time.

The last part of this chapter presents the scope of this thesis. It describes in which

way the specific work presented fills a former gap in the field of underwater acoustic

communication.

In Fig. (1.1) a visualisation of a communication system is presented which also con-

cludes the work performed during the thesis. The theory describing this is presented in

chapter 2. The chapter therefore begins by explaining how a signal s can be constructed

before transmitted from the transmitter Tx. Thereafter the convolution between the sig-

nal s and the channel H is thoroughly described in section 2.2. Also how a time variant

channel can be described using polynomial phase signals is presented in section 2.2.2.

The channel H and the disturbance or noise e will distort the signal. Thus, when the

receiver Rx samples the signal r as a distorted version of s an estimation of the channel

H is needed in order to reverse the distortion. The last part of chapter 2 will therefore

describe the theory of how an estimation of H can be performed where this estimation

1

2 Introduction

can be represented as H.

Tx H Rxs r H

e

Figure 1.1: Visualisation of a communication system describing the thesis content.

The testing of the algorithms were in a large extent done by simulation. Therefore,

the simulation of the systems transmitter and the propagation through its channel is

described it chapter 3. Both of which refers back to the theory presented in the be-

ginning of chapter 2. The second half of chapter 3 describes how the time variant taps

of the channel impulse response were separated and how the full impulse response was

estimated. This section connects the theory to the specific problem.

The results generated from performing the methods described in chapter 3 will be

presented in chapter 4. Results both from simulated data as well as from data collected

during field tests are presented.

Chapter 5 evaluates how well the method performs, which are its strengths and which

improvements can be made. The chapter does also include a section describing future

work suited as the next step regarding the method presented.

Chapter 6 will present the conclusion of the methods performance and in which state

the method is when presented.

1.2 Introduction to digital communication

Telecommunication has been evolving since the discovery of electricity in the 1800’s. An

example is the line telegraphy which was perfected by Morse in 1844 [2]. The mathe-

matical theory of communication and its possibilities was published by Shannon in 1948

[3], where after the amount of applications started to arise [2]. The development of com-

munication systems and the capacity restrictions in analogue systems made the need of

digital communications during the later half of the 1900’s.

A digital communication system consists of a transmitting part, a channel and a receiv-

ing part, roughly described. Both the transmitter and the receiver consists of multiple

blocks, making communication through the system possible. A wireless communication

system divided into the essential blocks is shown in Fig. (1.2).

When transmitting digital information over a physical channel the information must be

represented in terms of analogue waveforms. A process called modulation [4]. Modulation

1.2. Introduction to digital communication 3

Source Source Encoder Channel Encoder Modulator

channel

DemodulatorChannel DecoderSource DecoderDestination

Transmitter

Receiver

Figure 1.2: Block diagram of a wireless communication system [2].

techniques when transmitting analogue waveforms can be categorised into three main

modulation methods [2]. These are phase-shift keying (PSK), amplitude-shift keying

(ASK) and frequency-shift keying (FSK). Among these the modulation technique most

often used in communication is PSK. An other method called QAM, which is a hybrid

between PSK and ASK is also widely used in communications [2]. There are multiple

variations within the modulation techniques giving a variety in robustness and complexity

of the system.

From modulation, the symbols are still discrete, the transmitter therefore uses a mod-

ulator translating the discrete symbols into continuous time analogue waveforms to be

transmitted over the physical channel [4].

The physical characteristics of a channel varies widely as the channel may be a wire,

a band of radio or audio frequencies, or a beam of light [3]. The characteristics of

the channel distorts the transmitted wave. Communication over wireline can be well

modulated as a linear time-invariant system [4]. The transfer function can then, by

a feedback loop from the receiver, be assumed to be known at the transmitter. For

wireless communications this is not the case. Due to mobility between the transmitter

and receiver the channel may vary, leading to that an accurate channel feedback will be

unavailable [4].

The receiver consisting of a demodulator and a channel decoder that processes the

received analogue waveform. A task for the demodulator is to synchronise the received

signal with the transmitted, compensating for phase, frequency and time shifts induced

by the channel. Another is to compensate for intersymbol interference (ISI) which the

dispersion in a channel induces [4]. An optimal demodulator maps each received waveform

to the corresponding bit sequence, making a tentative decision on the transmitted symbol.

The sequence can then be fed directly to the channel decoder which goal is to produce

an estimate of the information sequence put into the channel encoder [4][2].

4 Introduction

1.3 Underwater acoustic communication channels

Due to the electromagnetic- and optical waves poor propagation properties underwater

[5], acoustical waves are most often used for underwater communication. This even

though the method suffers from limited bandwidth, the mediums refractive properties,

Doppler-shift, rapid time variation, extended multipath and severe fading [6]. Thus,

water is referred to as one of the most difficult medium working with, in use today [7].

One of the significant properties is the low propagation speed of approximately 1500

m/s. Since the wave also has multipath propagation, these properties lead to a delay

spread in the order of tens or even hundreds of milliseconds. For an acoustical wave to

propagate in water over long distance a low carrier frequency must be used. Therefore,

even if the bandwidth is in the kHz region, no narrowband approximation can be done.

For a broader understanding the channel properties attenuation, noise, multipath, time

variability and Doppler effect are described below in separate sections.

1.3.1 Attenuation

A signal transmitted in water will experience losses of different forms, one being the loss

due to absorption. The absorption of the signal will affect the propagation range and is

frequency dependent. Apart from the loss due to absorption the signal will suffer from

spreading loss. A loss increasing by the propagation distance. The total loss can be

modelled as a function of the signal frequency f and the distance of the transmission as,

A(l, f) =

(l

lr

)ka(f)l−lr , (1.1)

where lr represents some reference distance and models the loss due to spread and a(f)

the frequency dependent absorption coefficient[7].

1.3.2 Noise

Ambient and site-specific noise are the types of noise present during underwater acoustic

communication [7]. The ambient noise is always present in the background in the quiet

deep-sea. Ambient noise comes from distant shipping, turbulence, breaking waves among

others. Different rules for site-specific noise, which only exists in certain areas. As the

name tells it comes from more specific phenomena of the area such as snapping shrimp in

the south or acoustic noise from ice cracking in polar regions. The site-specific noise can

not be seen as Gaussian as it most often contains significant non-Gaussian components.

Ambient noise can be approximated to a Gaussian distribution although not always

white[7].

1.3. Underwater acoustic communication channels 5

Figure 1.3: The multiple paths of transmitted signals towards the receiver.

1.3.3 Multipath

Two effects that will induce multipath formation of a transmitted signal is reflection and

sound refraction in water. Reflection of the signal is due to reflection at the surface,

at the bottom and also at different kind of objects such as large rocks, islands etc [7].

Refraction in the water is a result of the sound speeds spatial variability. The variability

is a consequence of that the sound speed depends on the temperature, salinity and

pressure. In shallow water these factors depends on environmental circumstances such as

season, depth and location. Breaking waves among others will in many cases also induces

multipath propagation due to scattering [8].

As is shown in Fig. (1.3) these different effects will affect a transmitted signal beam

of rays, each ray might take a slightly different path to the receiver. The receiver then

observes multiple signal arrivals. In accordance to Snell’s law a ray will always bend

towards a region of lower propagation speed. Therefore, even though a ray has a longer

propagation distance it might reach the receiver before one of shorter distance [7].

1.3.4 Time variability

Time variability can be caused by either inherent change in the propagation medium or

because the transmitter/receiver is in motion. Where the first can be caused by both on

6 Introduction

a very long time scale such as monthly changes in temperature, which does not affect

the signal instantaneously, as well as waves breaking which will affect the signal due

to scattering of the signal and Doppler spread. The Doppler spread is induced by the

changing in path length [7].

1.3.5 The Doppler effect

The motion of transmitter or receiver does not only induce time variability, it also con-

tributes to the change in channel response through the Doppler effect. It causes both

frequency shifting and frequency spreading with a magnitude proportional to the ratio

between the sound speed and the velocity the transmitter and receiver have in relation to

each other. This effect is in radio- and electromagnetic communication negligible since

the velocity of the signal is in a much greater order then any transmitter-receiver motion.

In acoustic communication this is not the case. A small motion such as if a transmit-

ter/receiver moves unintentionally, the relation to the very low speed of sound is not even

then negligible. An underwater autonomous vehicle can move in the order of a few m/s

which creates a severe distortion[7].

1.4 A review of underwater acoustic communications

Historical perspective

The upcoming of submarines started the development of underwater acoustic communi-

cation as there was a need for communication. An underwater telephone using analogue

modulation was developed in 1945 [9]. The phone was developed by the U.S Navy and

was called ”Gertrude” [6]. Similar systems have been used even in present time [6].

The technique uses single side band modulation as carrier with analogue filters for

pulse spectral shaping around the human voice band at the transmitter. The filtering,

reproduction and demodulation performance is often of low accuracy. However, as the

human mind is able in some extent to process distorted speech the system does work [6].

In the 1960’s the awareness of signalling and modulation in imperfect channels in-

creased as the development of digital communication did as well. The throughput in

these channels were limited leading to extreme low data rates. From then researchers

have tried to increase the throughput for underwater propagation [6]. Hence, increasing

the performance in digital underwater acoustic communication.

In the 1970’s the attempts to unravel multipath channels in order to increase the

throughput started [10]. Since then researchers over the world have been able to develop

and use more sophisticated processing methods, but in principle the problem still remains

[6].

1.4. A review of underwater acoustic communications 7

Outline of underwater acoustic communication

The earlier sections stated the multiple challenges regarding communication under water.

Due to these complications techniques developed for wired or wireless communications

for terrestrial use does have to be significantly modified for underwater use [6].

Before transmission in water a robust system is crucial. In usage of coherent signalling

PSK and QAM, mentioned in section 1.2, does for the medium provide a robust modu-

lation [11].

To estimate the channel in communication, sequences known to both the transmitter

and the receiver are used to estimate the change in the channel. These sequences are

called pilot sequences. Two competitive symbol based schemes are conventional pilot

sequences and superimposed pilot sequences [12]. The difference between them will be

described thoroughly in the theory. A brief description, however, will be made here.

The approach of conventional pilots is often used, meaning a sequence of know pilot

symbols will be added both before and after a transmitted signal, so called preamble

and post amble. This method restricts the length of a signal possible to decode in the

receiver depending on how rapid the channel is shifting. Another concern is the waste

of bandwidth during the pilot sequence as it bears no information [12]. When using

superimposed pilots the preamble and post amble are still added. However, parallel

with the unknown signal a known pilot sequence is transmitted. This makes it possible

to continuously track change in the signal along the unknown symbol sequence [13].

Therefore, the throughput of the channel can be increased in comparison with other

conventional pilot distributions as the length of the signal between preamble and post

amble can be increased [12].

To estimate the channel for the signal to propagate in, one choice to be made is if a

parametric or non-parametric estimation is to be used. The parametric estimation is

defined as, to compute the estimate of a sample in a received symbol the gain and delay

of different paths are estimated and used to describe the sample. A non-parametric

estimator on the other hand estimates the sample directly from the received symbol and

ignores the propagation path and the structure underlying the symbol [14].

Methods used for estimation

The section highlight major challenges in underwater acoustic communication and how

researchers in present time have developed methods to avoid them.

In channels induced by high Doppler spread PSK modulation together with decision

feedback equalisers (DFE) and spatial diversity have been a computationally complex

but effective combination for communication [15].

Due to discrete arrivals of multipath spread have lead to lower complexity, enhanced

performance and faster channel tracking. In these cases a sparse structure of the equaliser

has been used [16], [17], [18].

Statistical properties of the signals have been used in the development of blind equali-

8 Introduction

sation. These do not require training sequences for the equaliser to converge. However,

the method converges slower than conventional equalisers resulting in limitations in long

or continuous data streams [19].

As the DFE due to the inaccurate decisions in the feedback loop the method suffers from

error propagation. Therefore, to ensure low bit error rate (BER) forward error correction

codes have been used by researchers while developing turbo equalisation techniques. The

techniques have iteratively interacted between decoder and equaliser resulting in joint

estimation, equalisation and decoding [20].

Maximum a posteriori probability (MAP) equalisers are equalisers of high computa-

tional complexity often used in turbo code. Previous work have developed a soft input

DFE for each receiver with a linear equaliser both lowering the complexity and still

achieving high performance [21].

To avoid the major problem of large Doppler spread in shallow water researchers have

used channel trackers with an embedded linear decoder to tackle the problem [22], [23].

1.5. Contributions of the thesis 9

1.5 Contributions of the thesisIn this thesis a parametric method was used to estimate a time variant channel impulse

response. The method is based on that the transmitted signal did include a superim-

posed pilot sequence [12]. The proposed model was to represent each tap in the impulse

response by a polynomial phase signal (PPS) [24]. The PPS-model have previously been

used in other areas [24],[25], [26].

The scope was to separate the taps in the impulse response by a moving least squares

method [27]. This step resolves the multipath of the channel, see Fig. (1.4). When

separated, each tap was estimated as a PPS with constant amplitude using the weighted

non-linear least squares method [28]. The estimation describes the time variance of the

channel, see Fig. (1.4). The constant amplitude was used due to the properties of the

channel and the fact that a PSK modulation only depends on the phase and not the

amplitude.

Hence, the following were investigated in the thesis:

• Is it possible to describe an underwater acoustical communication channel using

polynomial phase signals?

• How well does the proposed method perform with different channel properties?

• In what extent is the model affected when extending the message block length?

Multipath Time variationr H1 H2

Figure 1.4: The figure shows the two estimation steps needed for a complete estimation using

the method of the thesis.

CHAPTER 2

Theory

The theory chapter is divided into several sections starting with the basics of signal repre-

sentation. It will then describe the different parts required to perform the work presented

in the thesis. The order the sections are structured makes it possible to understand one

part at a time and at the end have the full picture.

2.1 Signal representation

In communications, passband channels are employed, which implies that the transmitter

and receiver must be able to handle passband signals. However, all information carried

in a real passband signal is possible to represent by a complex baseband signal. The

signal representation in a complex baseband is of profound practical significance since

a markedly lower sampling rate can be used and still receive an accurate discrete time

signal representation [4].

QPSK modulation

Wireless digital communication needs to modulate data bits, taking the values of 0 or

1, into an analogue waveform as mentioned in section 1.2. Modulation is used to create

an alphabet which can represent digital information in the analogue wave [2]. With the

alphabet a bit-to-symbol map can be created, were a symbol either just represents a

0 or a 1 alternative a sequence of bits. As stated in section 1.2 one well used form of

modulation is PSK, a method which holds the modulus of symbols constant and only

lets the phase vary, see Fig. (2.1). The figure describes a constellation called Quadratic

PSK (QPSK), each symbol can then carry the information of two bits [4].

11

12 Theory

Re

Im

0100

10 11

◦◦

◦ ◦

1√2

1√2

−1√2

−1√2

−

−

−

−

Figure 2.1: The QPSK-constellation in the complex plane.

Root-raised cosine

Transmitting and receiving a digital signal, intersymbol interference (ISI) is a factor that

has to be minimised. One way to achieve this is by applying a matched filter in the

transmitter and/or in the receiver. In communications one of the most frequently used

is the raised cosine filter [4]. In practice the filter should be evenly divided between the

transmitter and receiver to minimise the ISI [29]. The result is applying a root-raised

cosine both in the transmitter and the same in the receiver [30].

Regarding the Nyquist criterion it applies to the transmitter, channel and receiver.

Therefore the filter in the transmitter GTX(f) and in the receiver GRX(f) is constructed

such that the product GTX(f) GRX(f) is Nyquist in frequency domain [4]. As the raised

cosine filter GRC given as,

GRC(f) =

T |f | ≤ 1−a

2TT2

[1− sin((|f |)πT

a)]

1−a2T≤ f ≤ 1+a

2T

0 otherwise,

(2.1)

where a is the fractional excess bandwidth is Nyquist [4]. The root raised cosine is

then given as,

GRRC(f) =√GRC(f), (2.2)

is a transmitter and receiver filter widely used in practice of digital communications as

GRRC(f) GRRC(f) = GRC(f) [29].

2.1. Signal representation 13

Upconversion and downconversion

After the signal is modulated and filtered in the complex baseband it must be upconverted

to passband representation. The passband signal which is transmitted over the channel

is described as a real sinusoid with a carrier frequency fc representing of the complex

baseband signal.

At the receiver the passbandsignal is downconverted from the real passband to the com-

plex baseband. As the digital signal processing preferably is performed on the baseband

representation of the signal, downconversion is the first step at the receiver minimising

amount of analogue processing. Extraction of the complex signal components are done

separately. For further explanation the reader is recomended to read [4].

2.1.1 Superimposed pilots

In radio communication, adding a preamble and a post amble is commonly used, where

the preamble and post amble contains data symbols known to both the transmitter and

the receiver, these are called pilot sequences. By this knowledge it is possible for the

receiving side to decode the unknown symbols in between, see Fig. (2.2) for setup. The

transmitted signal s[n] will then be formed as,

s[n] =

{ p[n] 1 ≤ n ≤ Np

d[n] Np + 1 ≤ n ≤ N −Np

p[n] N −Np + 1 ≤ n ≤ N,

(2.3)

where N is the total number of samples, p[n] is a known pilot sequence with length Np

and d[n] is the unknown data of length N − 2Np.

As for the case of underwater acoustic communication the coherence time of the wave is

too short for this method to be efficient. Meaning that, the channel the wave propagates

through change too rapidly during the time it takes for the unknown symbols to reach

the receiver for the preamble and post-amble to decode the received data. A method to

handle this problem is then to use a small fraction of the energy used to transmit the

unknown data to add a sequence of known pilots, see Fig. (2.3). This sequence is called,

as the title reviles, superimposed pilots. Mathematically the transmitted signal s[n] is

described as,

s[n] =

{ p[n] 1 ≤ n ≤ Np −Nk

(1− α)d[n] + αp[n] Np −Nk + 1 ≤ n ≤ N −Np +Nk

p[n] N −Np +Nk + 1 ≤ n ≤ N,

(2.4)

where α lays between 0 and 1 which represents the weight of the superimposed pilot

sequence. If set to 1 all data in the signal is known to the receiver and set to 0 the

method collapses to the method of conventional pilots.

14 Theory

If the energy is to be kept constant in the signal Nk is the number of samples which

the preamble and post-amble is shortened with. Nk can therefore be described as,

Nk =

⌊α2(N − 2Np)

2(1− α2)

⌋[13]. (2.5)

If there is no restrictions for the signal energy, Nk is set to zero.

DataPreamble Post-amble

time

N

NpNp N - 2Np

Figure 2.2: Sequence of transmitted data using preamble and post-amble to decode at the receiver.

Superimposed pilotsPreamble Post-amble

Data

time

N

Np - 2NkNp - 2Nk N - 2Np + 2Nk

Figure 2.3: Sequence of transmitted data including superimposed pilots above the usage of pream-

ble and post-amble to decode at the receiver.

2.2. Channel representation 15

2.2 Channel representation

2.2.1 Channel impulse response

A signal s[n] propagating through a channel will be distorted by the channel impulse

response due to the physical properties described in section 1.3. If a channel with Additive

White Gaussian Noise (AWGN) is considered the signal also becomes affected by the noise

in the channel. Therefore, the signal from a such a channel can be defined as an distorted

AWGN-signal and the received signal can be expressed as,

r[n] =K−1∑k=0

hk[n− k]s[n] + e[n]. (2.6)

Where K is the length of the channel response, the notation k denotes a response from

the channel at a specific delay and e[n] is the AWGN.

As Eq. (2.6) describes definition of a convolution between the signal and the channel

it can be written in matrix form as,

r = Hs + e, (2.7)

where H is the N ×N convolution matrix for the channel specified as,

H =

h0[1] 0 0 . . . . . . 0

h1[2] h0[2] 0 . . . . . . 0

h2[3] h1[3] h0[3] . . . . . . 0...

...... . . . . . .

...

hK [K + 1] hK−1[K + 1] . . . h0[K + 1] . . . 0...

......

... . . ....

0 . . . . . . hK [N ] . . . h0[N ].

(2.8)

The diagonals in the matrix H is then the time variant taps of the impulse response of

the channel. Each tap in an impulse response represents a path taken by the transmitted

signal.

2.2.2 Polynomial phase signals

The model proposed in section 1.5 was to estimate the time variance of each path in

underwater acoustic channels as time variant in phase but constant in amplitude. This

can then be described as a complex exponential function. The continuous phase vari-

ation is represented as a function in the exponent, and according to the Weierstrass

approximation theorem, any continuous function can be approximated as a polynomial

of some order on a closed interval [31]. Therefore, any complex phase varying signal can

be expressed as,

16 Theory

Re

Im

0100

10 11

◦◦

◦ ◦

1√2

1√2

−1√2

−1√2

−

−

−

−

◦

◦

◦

Figure 2.4: Showed in the figure is the effect on a symbol due to a time variant channel.

x[n] = γe(jθ[n]). (2.9)

θ[n] is a polynomial of any given order and can be written accordingly,

θ[n] =M−1∑m=0

θm(n)m, (2.10)

where θm is the parameter to the variable of order m for m = 0, 1, 2, ....,M − 1 describ-

ing the phase variation [24].

As described in section 1.3, the rays of a transmitted signal will due to multiple effects

reach the receiver at different times. Hence, the receiver will sample a sum of multiple

signals.Therefore, if h[n] from Eq. (2.6) is described as a polynomial phase signal. A

received signal could then be represented as,

r[n] =K−1∑k=0

γke

(j∑M−1

m=0 θm,k(n−k)m)s[n] + e[n] (2.11)

If describing the channel in an underwater acoustic communication system as Eq. (2.11)

the transmitted symbols will have a change in amplitude at the receiver and depending

on the time of arrival different phase changes. This is visually described in Fig. (2.4).

2.3. Moving linear least squares estimator 17

2.3 Moving linear least squares estimator

To resolve the multipath in the channel, described in section 1.3, the taps of the impulse

response have to be separated. From Fig. (1.4) in section 1.5 this is the first step of the

estimation according to the proposed method of the thesis. Hence, this section describes

one approach to resolve the multipath.

Def 1: Linear least squares estimation

For a linear system Ax=b the least square estimator can be used in order

to determine the vector x. The least square solution will then have an error

e = b - Ax. Hence, the square error of the residuals are given as

||b− Ax||2. (2.12)

This is, in mean square sense the best solution for a linear system. The

error is orthogonal to the solution which gives,

AHe = 0 =⇒ AHAx = AHb. (2.13)

The solution to the problem then becomes,

x = (AHA)−1AHb [32]. (2.14)

According to the definition the method described is only applicable in a linear case,

whereas in a real system that is often not the case [28]. However, considering a non-linear

function f(t) which are slowly varying. The function can then be estimated as a set of

linear functions as is described in the definition of moving least squares estimation [27].

tp

f(t)

t

Figure 2.5: An illustration of how a non-linear function can be divided into P pieces of linear

functions. The rectangular encloses one of the P regions where the function can be seen as

linear.

18 Theory

Def 2: Moving least squares estimation

Given a non-linear function f(t), an estimation can be written as a sum of

linear functions as,

f(t) =P∑p=1

fp(tp), (2.15)

where each function is only active an interval tp, where tp is defined for p =

1, ..., P. For visual representation Fig. (2.5) shows an interval for where a

function fp(tp) is active. Each of the functions fp(tp) can then be estimated

using the least square approximation [27].

Following the definition of the linear least squares method each sequential

function fp(tp) can be solved as

fp(tp) = (AHp Ap)

−1AHp bp, (2.16)

where bp is a N × 1 data vector, A a N ×K system matrix and fp(t) the

K × 1 estimated function.

If combining Eq. (2.4) and Eq. (2.11) a system with multiple path propagation can

be described. The channel impulse responses then distorts the signal as a non-linear

complex exponential function. Then the moving linear least squares method could be

used to estimate all channel components h(k)[n] in Eq. (2.11). The partial function fpthen represents the sum of h(k)[n] taps at a time instance p. If extracting a small fraction

of the signal r[n] from Eq. (2.11), it can be considered constant. All K components

h(k)[n] can then be extracted by constructing a convolution matrix P of an M sized

window of the pilot sequence p[n]. P then becomes an M × K matrix for each time

instance p, where K is the number of components in the channel impulse response and

M the window over time where r[n] can be considered constant. For the system to have

a solution the condition M > K must be fulfilled. From Eq. (2.4) and Eq. (2.16)

estimating the channel response at time instance p then can be described as,

h1,p = (PHp Pp)

−1PHp rp (2.17)

where (.)H is the Hermitian transpose, h1,p is the estimation of all taps at time instance

p and rp is the M -sized window of the received signal around time instance p.

During the preamble and post amble of a signal the full signal is known to the receiver

compared to the block in between where only the fraction containing the superimposed

pilot sequence is known. Therefore, the estimation H1[n] of H[n] is divided into three

sections. One active only during the preamble, one active along the superimposed pilot

sequence and one during the post amble. Eq. (2.17) is rewritten to,

2.3. Moving linear least squares estimator 19

h1,p = (PHp Pp)

−1PHp pp + (PH

p Pp)−1PH

p ep 1 ≤ p ≤ Np

h1,p = (PHp Pp)

−1PHp αpp

+ (PHp Pp)

−1PHp (1− α)dp Np + 1 ≤ p ≤ N −Np

+ (PHp Pp)

−1PHp ep

h1,p = (PHp Pp)

−1PHp pp + (PH

p Pp)−1PH

p ep N −Np + 1 ≤ p ≤ N.

(2.18)

The vectors ep, pp and dp are M -sized windows of e[n], p[n] and d[n] respectively

around time instance p. For the intervals 1 ≤ p ≤ Np and N − Np + 1 ≤ p ≤ N the

signal is fully known therefore Pp is constructed as the convolution matrix of an M -sized

window in p[n]. However for the interval Np+1 ≤ p ≤ N−Np only a fraction if the signal

is known, dependent on the weight α. The convolution matrix Pp is then constructed

from an M -sized window of αp[n].

according to Eq. (2.4).

The noise (PHp Pp)

−1PHp ep is considered Gaussian distributed and the unknown symbols

(PHp Pp)

−1PHp dp together with the Gaussian noise is seen as noise of an other magnitude

(and maybe other distribution). Eq. 2.18 is therefore rewritten to,

h1,p = (PHp Pp)

−1PHp pp + (PH

p Pp)−1PH

p e(1)p 1 ≤ p ≤ Np

h1,p = (PHp Pp)

−1PHp αpp + (PH

p Pp)−1PH

p e(2)p Np + 1 ≤ p ≤ N −Np

h1,p = (PHp Pp)

−1PHp pp + (PH

p Pp)−1PH

p e(1)p N −Np + 1 ≤ p ≤ N

(2.19)

where e(1)p and e

(2)p are the two different noise components ep and ep + dp. The full

estimation H1[n] is then described as,

H1[n] =P∑p=0

h1,p =P∑p=0

(PHp Pp)

−1PHp rp (2.20)

where the columns in H1[n] are the K time variant taps of the impulse response. H1[n]

can therefore be written as,

H1[n] = [h(1)1 [n]T , h

(2)1 [n]T , . . . , h

(K)1 [n]T ]. (2.21)

If a small weight α is used the estimation over the superimposed pilot sequence will

however be of bad signal-to-noise ratio (SNR) as the signal is not fully known.

20 Theory

2.4 Weighted non-linear least squares estimator

The time variation of the channel, described in section 1.3, does separately affect each

tap of the impulse response. Therefore an estimation over time for each taps is required

in order to resolve the time variation of the full channel. The following section propose

an estimation method performed on each tap that was separated in section 2.3 which

will describe its variation over time. This procedure is illustrated in Fig. (2.6).

H1[n]

WNLS1

WNLS2

WNLSK

H2[n]

h(1)1 [n]

h(2)1 [n]

h(K)1 [n]

h(1)2 [n]

h(2)2 [n]

h(K)2 [n]

Figure 2.6: The figure shows how the taps separated in the moving least squares estimator are

separately estimated using the weighted non-linear least squares (WNLS). These together then

describes the full impulse response of the channel.

Given a system model h(k)2 [n] and data h

(k)1 [n] from one tap. The model best describing

the data is the one minimising the error between them. Therefore, an error function J

can be set up as

J =N−1∑n=0

(h(k)1 [n]− h(k)2 [n])2, (2.22)

were h(k)2 [n] is a non-linear parametric model. To solve the problem one approach, if

possible, is to separate the linear parameters from the non-linear parameters and therefore

simplify the complexity of the problem [28]. The model can then be described as

h(k)2 [n] = H(θ)φ, (2.23)

where H(θ) describes the non-linear part of the model and φ the linear part. If the

2.4. Weighted non-linear least squares estimator 21

model is to be described by Eq. (2.9) of order two the parts in Eq. (2.23) can be written

as,

θ =[θ1, θ2]T , (2.24)

H(θ) =[c01, c1e

(j(θ1+θ2)

), . . . , cN−1e

(j(θ1(N−1)+θ2(N−1)2)

)]T , (2.25)

φ =Ae

(jθ0

), (2.26)

where in this case the linear parameters are constant. c that is element wise multiplied

to the non-linear part of the model is given accordingly,

c =[c0, c1, . . . , cN−1]T , where

ci =1 for i = 0, . . . , Np and i = N −Np + 1, . . . , N

ci =α for i = Np + 1, . . . , N −Np.

(2.27)

This follows the theory in section 2.1.1 as of how large part of the data consists of

superimposed pilots.

In Eq. (2.22) no assumption of a statistic model or weight between samples have been

made. If the reliability of one sample is greater than another, it would also be of higher

importance. Therefore, specific weights for each sample can be added to Eq. (2.22).

Giving the weighted model

J =N−1∑n=0

w[n](h(k)1 [n]− h(k)2 [n])2, (2.28)

where w[n] is the weight for the model at time instance n. The time variant channel

tap h(k)1 [n] represented as a vector can be written as h

(k)1 [n] = h. Eq. (2.28) can then be

written in matrix form as

J(θ,φ) = (h−H(θ)φ)HW(h−H(θ)φ), (2.29)

where the linear and non-linear parts are separated. W is the weighting matrix for the

model. Setting the derivative of the error function J(θ,φ) equal to zero, then φ can be

estimated accordingly

φ = (HHWH)−1HHWh. (2.30)

Hence, making the parameters of φ only dependent on θ. The error function J is then

reduced to a function of θ. The minimisation problem then becomes,

22 Theory

J(θ) = hH(W−WH(HHWH)−1HHW)h (2.31)

Knowing the nonlinear parameters in θ the linear or constant parameters in φ are

estimated by eliminating the non-linear part from the data in h(k)1 [n]. These parameters

are then given according to the following equations [28].

γ =1

N

∣∣∣N−1∑n=0

h(k)1 [n]e

(−j(θ1n+θ2n2)

)∣∣∣ (2.32)

θ0 = ∠N−1∑n=0

h(k)1 [n]e

(−j(θ1n+θ2n2)

), (2.33)

Maximum likelihood estimator

If the data in h(k)1 [n] is Gaussian distributed then Eq. (2.31) is the estimation of the

mean. Hence, giving the probability density function (pdf) [28],

G(θ,W) =1√

π||W||exp

(− J(θ)

). (2.34)

Maximising Eq. (2.34) gives the maximum likelihood estimation of the pdf. The

maximum of Eq. (2.34) is found when minimising J(θ). Hence, given that h(k)1 [n] is

Gaussian distributed and W being the covariance matrix of h(k)1 [n], minimising Eq. (2.31)

also gives the maximum likelihood estimation [33].

2.5 Exhaustive search method

When solving the minimisation problem of Eq. (2.31) the direction of arrival estima-

tions are not always optimised simultaneously. This can be solved by using an uniform

exhaustive search. If a K-dimension parametric search is considered as,

[θ1, . . . θK ] = arg minθ1,...θK

{J(θ)}. (2.35)

The search range is set uniformly around between [−a, a], where a is a positive real

value, for each search parameter. The spacing between grid points are set to a step size

∆, spanning the grid according to Fig. (2.5). At each grid point the function J(θ) is

evaluated after which the point minimising J(θ) is selected [34]. The method therefore

does find the global optima.

Due to the fact that the method does evaluate all discrete points in the area of interest

it will also, using a fine discretisation, become a computational expensive method for

finding an optima.

2.6. Levenberg-Marquardt method 23

θ2

θ1

∆1

∆2

Figure 2.7: The figure visualises the search grid for a two dimensional exhaustive search.

2.6 Levenberg-Marquardt method

If evaluating all points in the criterion function is not an option an optimisation algo-

rithm handling the problem in a more effective manner is needed, where the Levenberg-

Marquardt algorithm could be used in such a case.

The Levenberg-Marquardt algorithm is a nonlinear optimisation algorithm used for

constrained nonlinear problems [35]. As described in section 2.4 the function to be

minimised is the weighted non-linear least square,

minθ

J(θ) = ||J(θ)||2 =N−1∑i=0

(Ji(θ))2, (2.36)

where J(θ) is given according to Eq. (2.31). A problem suited to solve using the

Levenberg-Marquardt method.

The Gauss-Newton method follow the same optimisation process as the Newton method

with line search. The difference being that the Gauss-Newton method uses the convenient

and often efficient approximation [35],

∇2J = ∇JH∇J, (2.37)

of the Hessian. ∇J being the Jacobian of the function J and (.)H the Hermitian

transpose.

24 Theory

Def 3: The Gauss-Newton method

The function to be minimised in the Gauss-Newton method is,

J(θ) =1

2

m∑i=1

(ri(θ))2, (2.38)

where ri are the residuals. In difference to the Newton method which

uses the true Hessian the Gauss-Newton method uses the approximation

∇JH∇J, giving the system,

(∇JHk ∇Jk)pk = −∇JHk rk, (2.39)

to be solved for the step length pk, then minimising the function J(θ) by

iterating,

θk+1 = θk + pk (2.40)

The approximation can also be used to obtain the Levenberg-Marquardt method, but

the method replaces the line search with a trust region strategy. A trust region strategy

avoids the weakness of the Gauss-Newton method, namely, the behaviour of the algorithm

as the Jacobian ∇J becomes rank-deficient or nearly so [35].

Def 4: The trust region approach

In help of the a quadratic model the trust region method generates a step

for its objective function. A region is defined around the current iterate

for which the objective function is adequately represented within. The

step provides the approximate minimisation of the objective function in the

region. If the step is not acceptable the size of the region is reduced and a

minimiser of the new area is approximated. The direction of the step is in

general changed as the region size is.

From the trust region, for each iteration, the subproblem to be solved is,

mindk

1

2||∇Jkdk + rk||2, subject to ||dk|| ≤ ∆k, (2.41)

dk being the step length, rk the residual and ∆k the trust region radius. If the step

length for the Gauss-Newton method lies strictly within the trust region, meaning dk <

∆k. The Levenberg-Marquard can be collapsed to the Gauss-Newton. If not, there is a

λ > 0 such that ||dk|| = ∆k. Then, from Eq. (2.41), the problem to be solved becomes,

(∇J(θk)H∇J(θk) + λkI)dk = −∇J(θk)

Hrk. (2.42)

The parameter to determine in this case is the step length for the residuals to be

2.6. Levenberg-Marquardt method 25

minimised. Therefore, dk is extracted from the equation and then becomes,

dk = −(∇J(θk)

H∇J(θk) + λkI

)−1(∇J(θ)HJ(θ)

)(2.43)

If J(θ + dk) < J(θ) then λk+1 = λk and the parameters of θ is updated as,

θk+1 = θk + dk. (2.44)

If J(θ + dk) < J(θ) then λk+1 = λk10

before reevaluating Eq. (2.43). This procedure is

iterated until a suitable step length is found [35].

CHAPTER 3

Method

The work that generated the thesis was performed both on simulated data as well as data

collected from field tests performed by FOI. Therefore, methods used to perform cases

are described in this chapter. First the parts only performed in the simulated case is

described in the section Simulation setup, including the transmission and the simulated

channel with AWGN. Where after the section Experimental setup gives a description of

the work performed on the field test data, leading up to the data used for the method.

As the channel estimation and the parameter estimation was performed both on the

simulated data and the real data, both is described in the corresponding sections. The

parts described in this chapter were all implemented in separate functions using MATLAB

2019a.

3.1 Simulation setup

3.1.1 Transmission

In line with the theory in section 2.1 a N − 2Np-sample random signal was modulated.

In the same way was a N -sample pilot sequence generated. It made it possible for the

signal and pilot sequence to be sent alongside one another according to the theory of

superimposed pilots in section 2.1.1. In the transmission process a preamble was placed

in the beginning of the signal and a post amble in the end. In between the unknown

signal was added along side the superimposed pilot sequence which proportion of the

signal energy was varied from one transmission to another. In practice it meant varying

weight of α from Eq. (2.4). As the fraction of the signal energy used by the superimposed

varied as did the length of the preamble and the post amble. It varied in accordance to

Eq. (2.5).

27

28 Method

3.1.2 Channel

The channel block simulated how the channel affected the transmitted signal, as described

in section 2.2.1. In the simulation each impulse response of the channel was constructed

as a polynomial phase signal with constant amplitude according to section 2.2.2 and Eq.

(2.9). The parameters of the polynomial phased signal for every impulse response was

randomly generated inside a constrained range. The channel should vary more than π

over a message frame in order for the method to be evaluated, but it was restricted by

the assumption that the channel varied slowly. Otherwise the theory in section 2.3 does

not apply.

A convolution matrix for the channel response was constructed according to Eq. (2.8).

By multiplying the convolution matrix with the signal the channel distorted the signal

s[n] according to Eq. (2.7). As an AWGN was assumed complex valued Gaussian noise

was added to the distorted signal. The output from the channel was therefore a sum of

distorted noisy signals. As the channel response was generated as a polynomial phase

signal, the output followed Eq. (2.11).

3.2 Experimental setup

This section describes the transmission of messages during field tests earlier performed

by FOI. It also describes the procedure in the receiver done before the channel estimation

was performed. This was done using programs previously written by employees at FOI.

Before the data from the field test was transmitted all information was saved in order

to determine if messages was received.

When the field tests were performed the message was filtered in transmission using

a root raised cosine filter and up-converted to passband frequency. The receiver then

sampled the data with a minimum sampling rate as that of the transmitter. The received

message was first downconverted to the baseband. As the transmitted signal was filtered

through a root raised cosine filter, the received signal was filtered through the same filter

at the receiver according to the theory in section 2.1. The downconverted message was

correlated with the preamble for time synchronisation of the message frame. As the

message was transmitted in a rate higher than one sample per symbol the message was

resampled. Thereafter one sample corresponded to one symbol. Therefore, from that

point the data was processed in symbols instead of samples. By correlating the preamble

with the received symbols the length of the channel response could be determined as

the preamble had maximum correlation at the taps, both causal and non-causal. The

strongest correlation was set as the first causal tap of the impulse response. The received

data with corresponding parameters were saved in a file for post processing.

3.3. Separation of taps in time variant impulse response 29

3.3 Separation of taps in time variant impulse response

According to the assumption described in section 1.5, the channel varied slowly in time

and it was assumed that the coherence time in a small region around a single time

instance was long enough for the signal to be considered constant. Therefore, a least

squares approximation could be performed for a small interval around a time instance

n. The performance was divided into three sections, the first during the preamble, the

second during the superimposed sequence and the third during the post amble. The

method followed the theory in section 2.3 in general and particularly Eq. (2.19). As

the length of the channel response in this step was known, so was the dimension K. As

one estimation was performed for each time instance the only unknown dimension was

therefore the length of the window M , which was set to a value larger then two times

K. This created an over determined system and each value h(k)1 [n] was calculated as a

mean of at least two received samples. The first section, during the preamble, an M ×Kconvolution matrix was constructed using the MATLAB function convmtx() for each

time instance in the preamble.

Solving Eq. (2.19), the response for each channel tap was separated and estimated at

the specific time instance n, according to Fig. (2.5). Therefore, the output from each

time instance was a K × 1 vector containing the value of the K channel taps at time n.

The second section, during the superimposed sequence, only a small weight α of the

received signal was known. The convolution matrix P therefore was weighted with α.

When weighted, the procedure followed exactly the one of the first section. The second

section produced a noisy estimation as the method had multiple noise components in

accordance with Eq. (2.19).

As for the third section, during the post amble, exactly as in the first section the full

signal was known. Hence, the procedure was the same for the first and the third section.

When separated, a parametric solution could be performed for one tap at a time.

3.4 Parameter estimation

The output from section 3.3 was a matrix of noisy estimations of the time-varying impulse

responses. These estimations were to be estimated to a parametric model. The model

used was that of polynomial phase signals with constant amplitude according to section

2.2.2 and Eq. (2.9). The estimation method used to match the model to the data was the

weighted non-linear least squares estimator described in section 2.4. The optimisation

method used to determine the parameters was the Levenberg-Marquardt method from

section 2.6.

The error between the model and h(k)1 [n] was to be minimised. Therefore, an error

function was constructed according to Eq. (2.28). The amplitude and phase was sep-

arated from the non-linear part of the model as they were considered constant. They

were then estimated according to Eq. (2.30) and placed into the error function resulting

30 Method

in Eq. (2.31). The minimisation problem was then only dependent of the non-linear

parameters in θ. These were estimated using the Levenberg-Marquardt method accord-

ing to section 2.6. The Jacobian of the minimisation problem was numerically estimated

using symmetric difference quotient for each of the non-linear parameters. A good initial

guess was needed for the optimisation algorithm to find the global optima. Therefore,

Eq. (2.31) was evaluated in multiple points, making an exhaustive search with a sparse

grid. The minimum value from the grid search was set as the initial guess for the op-

timisation algorithm. Then by using the MATLAB built in function lsqnonlin() the

Levenberg-Marquardt method was used to find the optima of the error function. Know-

ing the non-linear parameters, the amplitude and phase were estimated using Eq. (2.32)

and Eq. (2.33) respectively.

The same procedure was performed for each one of the taps in the impulse response.

CHAPTER 4

Results

This chapter contains the collected results both from simulations and from experimental

data. The results from the methods described in chapter 3 are presented. The method

was in large extent tested and evaluated using simulated channels before testing on

experimental data. Therefore, the chapter starts by presenting the results from the

simulation and the methods performance as different parameters of the system varied.

4.1 Simulation results

Simulating the system made it possible to evaluate the performance of the proposed model

in multiple aspects. The aspects studied was variance in the length of the message frame,

the weights of the superimposed pilot sequence, in the length of the channel response

and the rate of phase change over time.

Evaluation of the method

For accuracy validation of the method used in the thesis, it was evaluated by comparison

to results known to be exact. The moving least squares estimator were compared to

estimate the pilot sequence accurately. Another aspect for the moving least squares

estimator was if it estimated the channel well enough for the weighted non-linear least

squares estimator to be able to make an accurate estimation.

Fig. (4.1) describes how the moving least squares estimator makes an estimation of

one tap in the impulse response. It also visualises how well the weighted non-linear least

squares estimator could track the channel impulse response in the noise from the middle

of the moving least squares estimation. Both being compared to the true time variety of

the tap.

The Levenberg-Marquard method was compared to a grid search of the weighted least

square function. This to evaluate the accuracy of the method as an exhaustive search

by definition finds the global optima, as described in section 2.5. The result is shown in

31

32 Results

0 500 1000 1500 2000 2500

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

True channel

MLS

WNLS

Figure 4.1: The figure shows a comparison of the change in the channel after estimation with

the moving least squares and the weighted least squares to the actual channel impulse response.

-60 -40 -20 0 20 40 6010

-2

100

102

104

106

Levenberg-Marquard

Exhaustive Search

Figure 4.2: The figure shows the MSE dependent on SNR for the Levenberg-Marquard method

and an exhaustive search. The SNR is defined in dB and the MSE is shown in log10-scale.

Fig. (4.2).

When generating the results the following data was used: frame length N = 2500

symbols, channel length K = 11 taps, length of preamble and post amble Np = 250

symbols, the weight of the superimposed sequence was α = 0.4 and the channel varied a

maximum of 4π radians over one frame.

A visualisation of the area created by the 2 parametric function J(θ) from Eq. (2.31)

is shown in Fig. (4.3). The global optima in this case is placed at coordinates (θ1, θ2) =

(−0.001835, 4.367 ∗ 10−7) , even though there exists multiple local optima.

4.1. Simulation results 33

-5500

4

1000

1500

10-7

2 0

2000

10-3

2500

0-2 5

-4

Figure 4.3: The image shows the surface of the parametric search area for θ.

Superimposed pilot weights

For evaluation of performance using different weights for the superimposed pilots, α was

varied. The values assigned were [1, 0.4, 0.25, 0.1], where the value 1 corresponded to the

full signal energy, Hence, in that case the whole signal was known. The result from the

simulation is presented in Fig. (4.4).

-30 -20 -10 0 10 20 3010

-5

100

105

= 1

= 0.4

= 0.25

= 0.1

Figure 4.4: The figure shows the MSE depending on the SNR when the weight of the superim-

posed pilots varied. The SNR is defined in dB and the MSE is shown in log10-scale.

When generating the results the following data was used: frame length N = 2500

symbols, channel length K = 11 taps, length of preamble and post amble Np = 250

34 Results

symbols and the channel varied a maximum of 4π radians over one frame.

Frame length

The result when the length of the message frame is presented in Fig. (4.5). The figure

visualises the performance of the algorithm while the length of the signal frame was

N = [900, 2050, 3200, 4350, 5500] symbols.

-30 -20 -10 0 10 20 3010

-2

100

102

104

106

N = 900

N = 2050

N = 3200

N = 4350

N = 5500

Figure 4.5: The figure shows the MSE depending on the SNR when varying the frequency rate

in the channel. The SNR is defined in dB and the MSE is shown in log10-scale.

When generating the results the following data was used: the weights of the superim-

posed sequence α = 0.4, channel length K = 11 taps, length of preamble and post amble

Np = 250 symbols and the channel varied a maximum of 4π radians over one frame.

Time variance

The performance when changing how rapid the phase varied in the channel over time is

presented in Fig. (4.6). The maximum phase change over one frame was [π, 2π, 3π, 4π, 5π]

radians.

When generating the results the following data was used: frame length N = 2500

symbols, channel length K = 11 taps, length of preamble and post amble Np = 250

symbols and the weights of the superimposed sequence α = 0.4.

Channel response length

The channel response length, as dependent on the channel properties, was a variable for

evaluation. The number of taps were set to the values [5, 8, 11, 14, 17, 20]. The perfor-

mance for different SNR is presented in Fig. (4.7).

When generating the results the following data was used: frame length N = 2500

symbols, the channel varied a maximum of 4π radians over one frame, length of preamble

4.2. Experimental results 35

-30 -20 -10 0 10 20 3010

-2

100

102

104

106

2

3

4

5

Figure 4.6: The figure shows the MSE depending on the SNR when varying the rate of the time

variation in the channel. The SNR is defined in dB and the MSE is shown in log10-scale.

-30 -20 -10 0 10 20 3010

-2

100

102

104

106

K = 5

K = 8

K = 11

K = 14

K = 17

K = 20

Figure 4.7: The figure shows the MSE depending on the SNR when varying the length of the

channel. The SNR is defined in dB and the MSE is shown in log10-scale.

and post amble Np = 250 symbols and the weights of the superimposed sequence α = 0.4.

4.2 Experimental resultsThe results presented in this section is performed on data received from FOI. The data

was generated and collected during former field tests. The pre-processing of the data,

following the method in section 3.2, was performed using programs written at FOI in

MATLAB.

Frame lengths

Fig. (4.8) presents the result from estimating the channel during a field test earlier

performed by FOI. The weight used for the superimposed pilot sequence was α = 0.2.

The properties of the channel from the test in The frame lengths tested in Fig. (4.8) was

N = [2546, 4341, 6136, 7931, 9726] with preamble and post amble of length Np = 1023.

36 Results

0 2000 4000 6000 8000

0.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

Figure 4.8: The figure presents the MSE of the estimation depending on the length of the

message frame using α = 0.2.

The length of the channel impulse response in this case was K = 37.

During the field tests which results are presented in Fig. (4.9) and Fig. (4.10), data

was transmitted using the weights α = 0.4 and α = 0.25 for the superimposed pilot

sequences. The transmissions were performed during the same field test, therefore also

under similar conditions.

When performing the channel estimation presented in Fig. (4.9) was, the block lengths

of the message was N = [1010, 2805, 4600, 6395, 8190] symbols, the channel response

length K = 21 taps, the preamble length Np = 255 symbols.

The parameters when performing the channel estimation presented in Fig. (4.10) was,

the block lengths of the message was N = [1010, 2805, 4600, 6395, 8190] symbols, the

channel response length K = 21 taps, the preamble length Np = 255 symbols.

4.2. Experimental results 37

0 2000 4000 6000 8000

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Figure 4.9: The figure presents the MSE of the estimation depending on the length of the

message frame using α = 0.4.

0 2000 4000 6000 8000

0.01

0.015

0.02

0.025

Figure 4.10: The figure presents the MSE of the estimation depending on the length of the

message frame using α = 0.25.

CHAPTER 5

Discussion

This chapter will evaluate the work throughout the thesis. This includes the assumptions

from section 1.5, how well these held up and the performance of the proposed method.

It will discuss if the model could be useful, which parts needed improvements and the

appropriate next steps if the work does proceed.

Performance

In previous chapter it was shown the moving least squares estimator was able to separate

the taps in the impulse response and weighted non-linear least squares estimator did

find a solution comparing to the exact solution, as shown in Fig. (4.1). It validated the

combination of the moving least squares method and the weighted non-linear least squares

did have the ability to separate the taps and estimate each tap. Together describing the

full impulse response.

The weighted non-linear least squares estimator also found the optimal solution for

different SNR comparing to the exhaustive search when evaluating Fig. (4.2). The

exhaustive search was guarantied, using a fine grid, to find the global optima since it

evaluates all points in the region of interest. Hence, the weighted non-linear least squares

estimator was validated to perform an accurate estimation of the defined problem.

The frame length of the unknown symbol was varied in both the case of the simulated

data and the field test data. From these, Fig. (4.5), (4.8), (4.9), (4.10) determines

the method does perform well for different frame lengths. Fig. (4.5) also validates the

method for different SNR. In Fig. (4.9) and (4.10) there is not much change in the MSE,

to tell from the magnitude of the MSE this could because the channel is stable for all

the lengths evaluated. The difference to be of notice is between the different weights

of the superimposed pilots. There can be seen that the weight does have an impact

towards generating an even better estimation. Fig. (4.8), which had a wider spread in

the impulse response was also harder to resolve using the method. This leading to a

larger MSE compared to both the simulations and the other field test results. However,

39

40 Discussion

the pattern in Fig. (4.8) did show expected results as the channel should be harder to

estimate with a longer frame as the time between fully known symbols did increase. The

accurate estimation even when using long message frames was of great interest since if

the gap between training sequences can be increased so can the data rate.

The weight of the superimposed pilot sequence are of greater importance in lower

SNR, shown in Fig. (4.4). In high SNR a lower α would be preferred as the SNR for the

unknown symbols then becomes higher. Therefore, for channels of low SNR the weight

should be increased as it is better to be able to perform an estimation with low SNR for

the decoding then to not be able to perform one at all.

The length of the impulse response does affect the result, seen in Fig. (4.7). This

since the longer the length of the impulse response the closer the window M used in the

moving least squares estimator comes to the coherence length of the channel. If a longer

window then the coherence is needed there is, at the time of the writing, no method to

resolve such a channel [36].

In the same way as a longer channel response creates a channel more difficult to decode

so does a more rapid time variation. Due to the same reason. The result shown in Fig.

(4.6) shows that the assumption of being able to estimate the impulse response in a

time variant channel holds for a maximum phase change of 5π radians over one frame.

This variation is to large for conventional pilots to unwrap and the change can still be

considered slow. Hence, the coherence time is long enough for any estimator at all to

work.

Together Fig. (4.2)-(4.10) validates the proposed method and its possibilities. This

leads to the possibility to increase the frame lengths during transmission even in a highly

varying channel, leading to higher bit rate though the channel.

Even though the method performed well for the different SNR tested this factor seemed

do represent the larger impact to the performance. A large SNR is of course highly valued

and not always possible, this is however using the proposed method the largest factor

of performance. From evaluating Fig. (4.3), low SNR could result in a local optima

becoming the global one if unfortunate, resulting in an optimised around the wrong

parametric values.

Non-linear parametric search

As the problem is of the structure where it is highly likely to be ill conditioned for

the Gauss-Newton method to be suitable. Hence, the Levenberg-Marquardt method was

used. However, a multiple parametric non-linear search is computational complex leading

to long run-times. The usage of Levenberg-Marquardt method does result in an accurate

estimation as shown in Fig. (4.2) but there might be the need for a trade off to cut down

the run time. Another approach is to estimate the largest trend first and subtracting it,

leading to multiple one dimensional searches. These as well as the discrete polynomial

transform are likely to be less accurate but will be faster then the current parameter

estimation method [25].

5.1. Future work 41

Superimposed weights

It will be harder to perform the estimation if smaller weights are used for the superim-

posed pilots. Noisier and therefore both larger and more local optima will be induced,

see Fig. (4.3), which makes it more likely to find a local optima then the global one.

On the other hand, if it is possible to perform an accurate estimation with small weights

the SNR of the transmission will be much better. This since a larger part of the signal

energy is used for the unknown symbols instead of a known sequence.

Computational expense

As shown in Fig. (4.3) multiple optima were present. Therefore, the initial guess was of

greatest importance. However, the evaluation of the criterion function J(θ) was compu-

tationally expensive and grid search needed to evaluate this function multiple times.

As Eq. (2.31) performs multiple operations in the order of N2 each evaluation is of

high cost. As the industry wants to increase the throughput in communication, the

length of message frames shall increase resulting in heavier computations. Therefore, the

number of evaluations needs to be kept to a minimum for the method to be efficient.

In an exhaustive search the evaluation is performed for each grid point. Therefore,

searching for an optimal initial guess is computational expensive. As the taps of the

response have travelled in the same channel, there might be a chance they are correlated

and/or dependent in some extent. If so is the case the search for the initial guess of the

strongest tap in the impulse response could be used for the rest, decreasing the number

of evaluations of Eq. (2.31).

As mentioned in previous work [16] impulse responses in underwater acoustic com-

munication, even though they are long, can often be treated as sparse. Because of the

computational expense, if considering the impulse response as sparse it would result in

fewer parametric searches and therefore lower the computational expenses.

5.1 Future work

The thesis and the work it included have shown it to be possible to represent the impulse

response of a underwater acoustic channel using polynomial phase signals. However,

further studies are still needed investigating if the method could be used for real time

implementation. The following describes potential improvements to the method.

Analytical determination

The exhaustive search will give an exact solution if the grid spacing is infinitely small.

Since in practice this is not possible to perform this will not define the analytically best

solution. A method used to derive the analytical definition of the lower bound of the

MSE in a system is the Cramer-Rao lower bound (CRLB) [26], [37]. By deriving the

42 Discussion

CRLB, if it exists, it would define how well the model could perform. Therefore, this

needs to be derived to determine how close to the lower bound the present performance

is or how much it can be improved.

Algorithm improvements

Investigation of the dependencies between the taps in the impulse response should be

performed. As the taps does propagate in the same channel arriving close in time, some

dependence between then might be found. If there is any dependence the initial value

grid search for each tap could be weighted between the taps and the number of grid points

could then be lowered. This could lower the computational expense of finding a initial

value. The impulse response is also likely to be of a sparse structure [16]. One plausible

way to make the model more adaptive to the specific channel is to only consider the

taps that actually does carry information which would lower the number of parameter

estimations needed.

5.2 EthicsThe purpose of the thesis was to develop a method towards stable underwater commu-

nications. The method itself could be applied in many different kinds of communication

systems. As an example it could be used in underwater autonomous vehicles for mapping

the underwater environment. However, even though the work was performed for scien-

tific purposes the method could be implemented in equipment used to do harm as well.

Using the work presented should consider the ethics and the possible negative impacts

the methods could have to the world.

CHAPTER 6

Conclusion

This thesis has examined the possibility to estimate the impulse response for an underwa-

ter acoustic communication channel using polynomial phase signals and estimating the

parameters of these signals. The approach was to separate the time variant taps of the

impulse response using a moving least squares estimator. Each tap was then estimated

to a polynomial phased signal with constant amplitude using the weighted non-linear

least squares estimator. The Levenberg-Marquardt method was used for the weighted

non-linear least squares estimates.

The thesis shows it to be possible to describe the change in at least some underwater

acoustic channels over time using polynomial phase signals with constant amplitude. The

method is not markedly sensitive to the frame length of the message, the length of the

channel impulse response or a rapid time variance even in low SNR. Overall it could

be concluded the method did perform a accurate estimation of an underwater acoustic

communication channel. However, due to the complexity of multi-parameter optimisation

it is of need of further work before it is ready for real time implementation.

43

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