I

Modelling and Simulation of an Underwater Acoustic Communication Channel

Submitted by: Kalangi Pullarao Prasanth

A Thesis approved on

by the following committee:

Hochschule Bremen University of applied sciences

Bremen, Germany.

Date

Kraus, Dieter, Prof. Dr.-Ing.

Wenke, Gerhard, Prof. Dr.-Ing.

II

ACKNOWLEDGEMENTS

I take this opportunity to thank all those magnanimous persons who stood behind me as

an inspiration and rendered their full service throughout my thesis. I am deeply indebted

to my thesis supervisor, Prof. Dr.-Ing. Dieter Kraus for timely and kind help, guidance,

providing me with valuable suggestions when ever I used to digress away from the aim

of project work and also the most essential materials required for the completion of this

report. He stood as an inspiration through out my project work and explained me even

the minute details very patiently at various stages of the project.

I would like to thank Prof. Dr.-Ing. Gerhard Wenke, for his support and cooperation in

this project. Finally, I want to thank my parents and sisters for providing me mental and

emotional support through my endeavour. I want to thank all my friends who have

distinguished themselves by giving me strength, encouragement, guidance, and support

to persevere throughout this project despite many difficult obstacles.

III

ABSTRACT

Underwater acoustic communication is a rapidly growing field of research and

engineering. The wave propagation in an underwater sound channel mainly gets

affected by channel variations, multipath propagation and Doppler shift which keep lot

of hurdles for achieving high data rates and transmission robustness. Furthermore, the

usable bandwidth of an underwater sound channel is typically a few kHz at large

distances. In order to achieve high data rates it is natural to employ bandwidth efficient

modulation.

Thus we present a reliable simulation environment for underwater acoustic

communication applications (reducing the need of sea trails) that models the sound

channel by incorporating multipath propagation, surface and bottom reflection

coefficients, attenuation, spreading and scattering losses as well as the

transmitter/receiver device employing Quadrature Phase-Shift Keying (QPSK)

modulation techniques. To express the quality of the simulation tool various simulation

results for exemplary scenes are presented.

IV

TABLE OF CONTENTS

1 Introduction...............................................................................................................1

2 Fundamentals of Ocean acoustics .............................................................................4

2.1 Sound Velocity in the Ocean.............................................................................4

2.2 Dependence of c on T, S and z ..........................................................................5

2.3 Typical Vertical Profiles of Sound Velocity.....................................................6

2.3.1 Underwater Sound Channel (USC)...........................................................6

2.3.2 Surface Sound Channel .............................................................................8

2.3.3 Antiwaveguide Propagation......................................................................9

2.3.4 Sound Propagation in Shallow Water .......................................................9

2.4 Propagation Loss of Sound .............................................................................10

2.4.1 Spreading Loss ........................................................................................10

2.4.2 Sound Attenuation in water.....................................................................11

2.4.3 Sound Attenuation in sediment ...............................................................14

2.4.4 Reflection & Transmission Coefficients, R and T ..................................15

2.4.5 Surface and Bottom Scattering................................................................19

2.4.6 Ambient Noise ........................................................................................20

3 Sound Propagation ..................................................................................................22

3.1 The Wave Equation.........................................................................................22

3.2 Helmholtz Equation ........................................................................................26

3.3 Sound Propagation in Homogenous Waveguide ............................................27

3.3.1 Image or Mirror Method .........................................................................28

3.3.2 Grazing angles.........................................................................................30

3.3.3 Travel Times ...........................................................................................32

3.3.4 Transmission loss for each ray................................................................33

4 Modulation ..............................................................................................................35

4.1 Digital Modulation Techniques.......................................................................35

4.1.1 ASK.........................................................................................................35

4.1.2 FSK .........................................................................................................36

4.1.3 PSK .........................................................................................................37

V

4.2 Bit rate and Symbol rate..................................................................................38

4.3 Representation of Signals................................................................................39

4.3.1 Baseband and Bandpass Signals .............................................................39

4.3.2 Baseband vs. Bandpass ...........................................................................39

4.4 Modulation – QPSK........................................................................................41

4.5 Pulse shaping...................................................................................................46

5 System description ..................................................................................................48

5.1 Simulation system ...........................................................................................51

5.2 Transmitter ......................................................................................................52

5.2.1 Training Sequence...................................................................................52

5.2.2 QPSK mapping .......................................................................................53

5.2.3 Pulse shaping...........................................................................................53

5.2.4 Carrier modulation ..................................................................................53

5.3 Channel ...........................................................................................................53

5.4 Receiver ..........................................................................................................54

5.4.1 Bandpass Filtering...................................................................................55

5.4.2 Down conversion and Sampling .............................................................56

5.4.3 Matched Filtering....................................................................................56

5.4.4 Synchronization ......................................................................................57

5.4.5 Sampling .................................................................................................58

5.4.6 Phase Estimation .....................................................................................58

5.4.7 Decision ..................................................................................................59

6 Observations and Results ........................................................................................60

7 Summary and Concluding Remarks........................................................................71

Appendix .........................................................................................................................73

References .....................................................................................................................107

VI

LIST OF FIGURES Fig. 1: Temperature vs. Depth .......................................................................................5

Fig. 2: Sound velocity vs. Depth ...................................................................................5

Fig. 3: Sound velocity vs. Salinity.................................................................................6

Fig. 4: Underwater sound channel of the first kind (co < ch).

(a) profile c(z), (b) ray diagram..........................................................................7

Fig. 5: Underwater sound channel of the second kind (co > ch).

(a) profile c(z), (b) ray diagram..........................................................................7

Fig. 6: Surface sound channel. (a) profile c(z), (b) ray diagram ...................................8

Fig. 7: Formation of a geometrical shadow zone when the

velocity monotonically decreases with depth. ...................................................9

Fig. 8: Sound propagation in a shallow sea. (a) profile c(z), (b) ray diagram ...............9

Fig. 9: Diagram indicating empirical formulae for different frequency domains .......13

Fig. 10: General diagram indicating the three regions of B(OH)3,

MgSO4 and H2O ...............................................................................................13

Fig. 11: Attenuation plot for various salinities & for temperature a) 20°C b) 30°C .....14

Fig. 12: Reflection and Transmission at a fluid-fluid interface.....................................15

Fig. 13: Ambient Noise Level for different domains at vw = 20 kn ...............................21

Fig. 14: Hierarchy of underwater acoustic models........................................................22

Fig. 15: Schematic diagram indicating displacement of a particle

from x to x + dx in water column.....................................................................23

Fig. 16: Homogenous waveguide with source S and receiver R....................................27

Fig. 17: Reflections of a wave from the boundaries of a layer, and the

image sources ...................................................................................................28

Fig. 18: Diagram illustrating dependence of 1R on grazing angle,

frequency and two wind speeds........................................................................31

Fig. 19: Diagram illustrating dependence of 2R on grazing angle

and two bottom types........................................................................................31

Fig. 20: Multipath propagation depicting delays in 2D-view........................................32

Fig. 21: Multipath propagation depicting delays in 3D-view........................................33

Fig. 22: Multipath propagation depicting transmission loss..........................................34

VII

Fig. 23: Baseband information sequence - 0010110010 ...............................................36

Fig. 24: Binary ASK (OOK) carrier ..............................................................................36

Fig. 25: Binary FSK carrier ...........................................................................................36

Fig. 26: Binary PSK carrier (note the 180° phase sifts at big edges) ............................37

Fig. 27: The relation between a Bandpass signal and its Baseband equivalent signal ..40

Fig. 28: Converting a Bandpass signal into its Baseband equivalent signal. ................40

Fig. 29: QSPK state diagram .........................................................................................42

Fig. 30: General block diagram QPSK transmitter........................................................43

Fig. 31: Data sequence transmitted................................................................................44

Fig. 32: Modulated carrier signal for I channel .............................................................44

Fig. 33: Modulated carrier signal for Q channel............................................................45

Fig. 34: QPSK signal for the given data sequence ........................................................45

Fig. 35: Root raised cosine pulse with a roll-off factor, 0.5β = ..................................47

Fig. 36: Underwater Acoustic simulation system..........................................................48

Fig. 37: The baseband equivalent system......................................................................49

Fig. 38: Oversampling the system .................................................................................49

Fig. 39: Moving the anti-alias filter and the sampling device in front of

the matched filter..............................................................................................50

Fig. 40: The equivalent discrete time baseband system. ...............................................51

Fig. 41: The Simulation system considered...................................................................51

Fig. 42: The transmitter .................................................................................................52

Fig. 43: Mapping of bits into QPSK symbols ...............................................................53

Fig. 44: Underwater Acoustic Channel Model ..............................................................54

Fig. 45: The receiver......................................................................................................55

Fig. 46: Output from the matched filter for successive signaling in absence of noise. .57

Fig. 47: Example of cross-correlating the received sequence with the training

sequence in order to find the timing.................................................................58

Fig. 48: Simulation results showing relative travel times for various

receiver locations of a sinc-pulse without including any transmission

loss phenomenon..............................................................................................61

Fig. 49: Simulation results showing relative travel times for various

transmitter and receiver locations of a sinc-pulse including the

transmission loss phenomenon.........................................................................64

VIII

Fig. 50: Simulation results showing relative travel times for two different

vertical depths of transmitter and receiver of a sinc-pulse including

the transmission loss phenomenon...................................................................66

Fig. 51: BER plot direct-path for the above Environmental scenario 1. .......................67

Fig. 52: BER plots multi-path propagation for the above Environmental

scenario 2, case 1 a) linear scale b) log scale...................................................68

Fig. 53: BER plots multi-path propagation for the above Environmental

scenario 2, case 2 a) linear scale b) log scale...................................................69

Fig. 54: Received QPSK states for direct path ..............................................................70

Fig. 55: Received QPSK states for multi path...............................................................70

Fig. 56: Schematic diagram for Simulation...................................................................74

1

1 Introduction

The need for underwater wireless communications exists in applications such as remote

control in off-shore oil industry, pollution monitoring in environmental systems,

collection of scientific data recorded at ocean-bottom stations, speech transmission

between divers, and mapping of the ocean floor for detection of objects, as well as for

the discovery of new resources. Wireless underwater communications can be

established by transmission of acoustic waves.

Underwater communications, which once were exclusively military, are extending into

commercial fields. The possibility to maintain signal transmission, but eliminate

physical connection of tethers, enables gathering of data from submerged instruments

without human intervention, and unobstructed operation of unmanned or autonomous

underwater vehicles (UUVs1 , AUVs2).

Underwater communications in general mainly gets affected due to

• Channel Variations

Channel variations are variations in:

- Temperature

- Salinity of water

- pH of water

- Depth of water column or pressure and

- Surface/bottom roughness.

• Multipath Propagation

The channel can be considered as a wave guide and due to the reflections at surface

and bottom we have the consequence of multipath propagation of the signal.

• Attenuation

Acoustic energy is partly transformed into heat and lost due to sound scattering by inhomogeneities.

1 UUV – Unmanned underwater vehicle 2 AUV – Autonomous underwater vehicle

2

• Doppler Shift

- Due to the movement of the water surface, the ray getting reflected from surface can

be seen as a ray actually getting transmitted from a moving transmitter, and thereby,

having Doppler shift in the received.

- When the receiver and transmitter are moving with respect to each other, the emitted

signal will either be compressed or expanded at the receiver. Thereby, Doppler

effect is observed.

Channel variations and multipath propagation keep a lot of hurdles for the achievement

of high data rates and robust communication links. Moreover, the increasing absorption

towards higher frequencies limits the usable bandwidth typically to only a few kHz at

large distances.

In this report, the channel has been modeled by considering multipath propagation,

surface and bottom reflection coefficients. In order to achieve high data rates it is

natural to employ bandwidth efficient modulation. In our case Quadrature Phase-Shift

Keying (QPSK, which is equivalent to 4-QAM) modulation techniques have been used

for transmitter and receiver.

A random bit generator is employed as the bit source. The transmitter converts the bits

into QPSK symbols and the output from transmitter is fed into “Underwater Acoustic

Channel”. The receiver block takes the output from the channel, estimates timing and

phase offset, and demodulates the received QPSK symbols into information bits.

The QPSK modulation technique is extensively being used in several applications like

CDMA (Code Division Multiple Access) cellular service, wireless local loop, Iridium (a

voice/data satellite system) and DVB-S (Digital Video Broadcasting-Satellite). In our

case the idea of receiver design has been taken from these applications.

In this report, we have considered in depth the channel variations and multipath

propagation as our investigation. Thus we present a reliable simulation environment for

underwater acoustic communication applications (reducing the need of sea trails) that

models the sound channel by incorporating multipath propagation, surface and bottom

reflection coefficients, attenuation, spreading and scattering losses as well as the

transmitter/receiver device employing Quadrature Phase-Shift Keying (QPSK)

modulation techniques.

3

To express the quality of the simulation tool various simulation results for exemplary

scenes are presented. In the following, chapters 2 and 3 describe completely about

underwater acoustic channel, its variations and effects, the multipath propagation

phenomenon, the channel design, etc. Chapters 4 and 5 present a detailed description

about the QPSK modulation techniques used in this thesis, transmitter design, receiver

design and the complete communication part of the system. A part of this thesis is

already published in cf. [11].

4

2 Fundamentals of Ocean acoustics

The ocean is an extremely complicated acoustic medium. The most characteristic

feature of the oceanic medium is its inhomogeneous nature. There are two kinds of

inhomogeneities:

regular and

random

Both strongly influence the sound field in the ocean. The regular variation of the sound

velocity with depth leads to the formation of the “underwater sound channel” and, as a

consequence, to long-range sound propagation. The random inhomogeneities give rise

to “scattering of sound” waves and, therefore, to fluctuations in the sound field.

2.1 Sound Velocity in the Ocean

Variations of the sound velocity c in the ocean are relatively small. As a rule, c lies

between 1450 and 1540 m/s. But even, small changes of c significantly affect the

propagation of sound in the ocean.

Numerous laboratory and field measurements have now shown that the sound speed

increases in a complicated way with increasing temperature, hydrostatic pressure (or

depth), and the amount of dissolved salts in water. A simplified formula for the speed in

m/s was given by Medwin in [3]:

( )( )2 2 1449.2 4.6 0.055 0.00029 1.34 0.01 35 0.016 c T T T T S z= + − + + − − + (2.1)

Here the temperature T is expressed in ° C, salinity S in parts per thousand [ ], depth z

in meters, and sound velocity c in meters per second. Eq. (2.1) is valid for:

0° ≤ T ≤ 35°C

0° ≤ S ≤ 45 ppt

and 0° ≤ z ≤ 1000 m

The Eq. (2.1) is sufficiently accurate for most cases. However, when the propagation

distances have to be derived from time-of-flight measurements, more accurate sound

speed formulae may be required (i.e. ≤ 0.1 m/s). These are provided by accurate

velocitometers.

000

5

2.2 Dependence of c on T, S and z

Fig. 1 shows the typical Temperature profile with surface of the sea at higher

temperature than the temperature at the sea bed. Here we can see, temperature decreases

with depth till some depth value 300z = m and after that getting constant. This

corresponds to a summer profile of a typical sea.

Fig. 1: Temperature vs. Depth Sound velocity varies with temperature, salinity and depth. The impact of temperature

and pressure upon the sound velocity, c is shown in Fig. 2. This can be viewed in three

domains. In the first domain, temperature is the dominating factor upon the velocity of

sound. In the second domain or transition domain, both the temperature and depths are

dominating upon the velocity of sound. In the third domain, sound velocity purely

depends on depths. These three domains can be seen in Fig. 2, first domain is till depths

of 200 m, transition domain is from 200-400 m and the third domain is above 400 m.

Fig. 2: Sound velocity vs. Depth

15.5 16 16.5 17 17.5 18 18.5 19 19.5 20-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

Temperature [°C]

Dep

th [m

]

Temperature Vs Depth curve

1518 1519 1520 1521 1522 1523 1524 1525 1526-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

Sound Velocity [m/s]

Dep

th [m

]

Sound Velocity Vs Depth curve

6

Dependence of c on salinity, S is shown in Fig. 3. Here, with the increase of S, velocity

of sound, c, also increases keeping the shape of the profile unaffected.

Fig. 3: Sound velocity vs. Salinity

2.3 Typical Vertical Profiles of Sound Velocity

The shape of the sound velocity profile

( ) ( ) ( )( ), ,c z c T z S z z= (2.2) is the most important for the propagation of sound in the ocean.

The c(z) profiles:

- are different in the various regions of the ocean and

- vary with time (seasons).

At depths below 1 km variations of T and S are usually weak and the increase of sound

velocity is almost exclusively due to the increasing hydrostatic pressure. As a

consequence sound velocity increases almost linearly with depth.

2.3.1 Underwater Sound Channel (USC)

In the deep water regions typical profiles possess:

- Velocity minimum at a certain depth, zm (Fig. 4a)

- zm defines the axis of underwater sound channel

- above zm sound velocity increases mainly due to temperature and

- below it sound velocity increases due to hydrostatic pressure

1500 1505 1510 1515 1520 1525 1530 1535-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

Sound Velocity [m/s]

Dep

th [m

]

Sound Velocity varying with Salinity

S=20S=25S=30S=35S=40

7

If a sound source is on the access of the USC or near it, some part of the sound energy is

trapped in the USC and propagates within it, not reaching the bottom or surface, and,

therefore, not undergoing scattering or absorption at these boundaries, cf. [6].

Underwater Sound Channel of the first kind, co < ch

co – velocity at the surface, ch – velocity at a depth ‘h’

Fig. 4: Underwater sound channel of the first kind (co < ch). (a) profile c(z), (b) ray diagram

Waveguide propagation can be observed in the interval depths of 0 cz z< < . The

depth 0z = and cz z= are the boundaries of the USC. The channel traps all sound

rays that leave a source located on the USC axis at grazing angles;

maxχχ < with ( ) 1 2max 2 o m mc c cχ ⎡ ⎤= −⎣ ⎦ (2.3)

where cm and co are the sound velocities at the axis and boundaries of the channel,

respectively. Hence, the greater the difference co – cm, the larger is the interval of angles

in which the rays are trapped, i.e. the waveguide is more effective, cf. [6].

Underwater Sound Channel of the second kind, co > ch

co – velocity at the surface, ch – velocity at a depth ‘h’

Fig. 5: Underwater sound channel of the second kind (co > ch). (a) profile c(z), (b) ray diagram

8

Here, the USC extends from the bottom up to the depth zc where the sound velocity

equals ch. Two limiting rays are shown in Fig. 5b for this case. Trapped rays do not

extend above the depth zc. Only the rays reflected from the bottom reach this zone.

The depth of the USC axis in deep ocean is usually 1000-1200 m. In the tropical areas it

can range down to 2000 m. The sound velocity ranges from, cf. [6]:

- 1450 m/s to 1485 m/s in the Pacific Ocean.

- 1450 m/s to 1500 m/s in the Atlantic Ocean.

2.3.2 Surface Sound Channel

This channel is formed when the axis is at the surface. A typical profile for this case is

shown in Fig. 6a. The sound velocity increases down to depth z = h and then begins to

decrease. Rays leaving the source at grazing angles bχχ < propagate with multiple

reflections in the surface sound channel, cf. [6].

Fig. 6: Surface sound channel. (a) profile c(z), (b) ray diagram

In the case of a rough ocean surface, the sound energy is partly scattered into angles

bχχ < at each interaction with the surface, i.e.

- rays leave the sound channel

- sound level decays in the surface sound channel and increases below the

surface sound channel

Surface sound channels frequently occur

- in tropical and moderate zones of the ocean, where T and S are constant

due to mixing in the upper ocean layer. c increases due to hydrostatic

pressure gradient.

- if the temperature on the surface decays due to seasonal changes, i.e.

summer → autumn → winter

9

- in Arctic and Antarctic regions, where a monotonically increasing sound

velocity profile from the surface to the bottom can be observed.

2.3.3 Antiwaveguide Propagation

Antiwaveguide propagation is observed when the sound velocity monotonically

decreases with depth (Fig. 7a). Such sound velocity profiles are often a result of

intensive heating by solar radiation of the upper ocean layer.

Fig. 7: Formation of a geometrical shadow zone when the velocity monotonically decreases with depth.

All rays refract downwards. The ray tangent to the surface is the limiting one. The

shaded area represents the geometrical shadow zone (Fig. 7b). The geometrical shadow

zone is not a region of zero sound intensity, cf. [6].

2.3.4 Sound Propagation in Shallow Water

This type of propagation corresponds to the case where each ray from the source, when

continued long enough is reflected at the bottom. A typical profile is shown in Fig. 8a. It

is observed in shallow seas and the ocean shelf, especially during summer-autumn

period when the upper water layers get well heated, cf. [6].

Fig. 8: Sound propagation in a shallow sea. (a) profile c(z), (b) ray diagram

10

2.4 Propagation Loss of Sound

2.4.1 Spreading Loss

Spreading loss is a measure of signal weakening due to the geometrical spreading of a

wave propagating outward from the source.

Two geometries are of importance in underwater acoustics:

1. Spherical Spreading

In a homogenous and infinitely extended medium, the power generated by a point

source is radiated in all directions on the surface of a sphere. This is called spherical

spreading. Since intensity equals power per area, we obtain at ranges ro and r, cf. [2]

Io = 24 o

a

rPπ

, I = 24 rPa

π (2.4)

With

ro - reference distance (= 1m),

Pa - acoustic power of source,

Io - acoustic intensity of source at distance ro,

I - acoustic intensity of source at distance r.

The loss due to spherical spreading is:

( )2

.osphere

sphere o

I rg rI r

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.5)

In the case of spherical spreading, the intensity decreases with r2. For spherical

spreading, the spreading loss is

( ) 10log 20log .osphere

sphere o

I rG rI r

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.6)

2. Cylindrical Spreading Cylindrical spreading exists when the medium is confined by two reflecting planes. The

distance between the planes is supposed to be 10h λ> . Where, λ denotes the

wavelength of the sound wave. Since intensity equals power per area, we obtain at

ranges ro and r (with r h ), cf. [2].

11

2a

oo

PIhrπ

= , 2

aPIhrπ

= . (2.7)

The loss due to cylindrical spreading is

( ) .ocylinder

o

I rg rI r

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.8)

The intensity decreases linearly with distance r. In logarithmic notation, for cylindrical

spreading, the spreading loss is

( ) 10log 10log .ocylinder

cylinder o

I rG rI r

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

Taking n as the exponent, we can express the spreading loss for geometric spreading in

logarithmic notation

( ) 10log 10log 10 log ,n

o

o o

I r rG r nI r r

⎡ ⎤⎡ ⎤= = =⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.9)

where exponent n = 1 for cylindrical spreading; n = 2 for spherical spreading.

2.4.2 Sound Attenuation in water

The acoustic energy of a sound wave propagating in the ocean is partly:

- absorbed, i.e. the energy is transformed into heat.

- lost due to sound scattering by inhomogeneities.

Remark: It is not possible to distinguish between absorption and scattering effects in

real ocean experiments. Both phenomena contribute to the sound attenuation in sea

water.

On the basis of extensive laboratory and field experiments the following empirical

formulae for attenuation coefficient in sea water have been derived.

a) Thorp formula, valid frequency domain see Fig. 9a

2 2

2 2

0.11 44 [dB/km]1 4100

f ff f

β = ++ +

(2.10)

where, [ ] is frequency f kHz

12

b.) Schulkin and Marsh, valid frequency domain see Fig. 9b

( )2 2

3 42 28.686 10 1 6.54 10 [dB/km]T

T T

SAf f Bf Pf f f

α −⎛ ⎞= ⋅ + − ⋅⎜ ⎟+⎝ ⎠

(2.11)

where

62.34 10A −= ⋅ , 63.38 10B −= ⋅ , [ ] is the salinity in pptS ,

2 is hydrostatic pressure P kg cm⎡ ⎤⎣ ⎦ ,

[ ] is frequency f kHz , and

( ) [ ]6 1520 27321.9 10 TTf kHz− += ⋅ ,

is the relaxation frequency with T the temperature in [°C]. While the temperature

ranges from 0° to 30°C, Tf varies approximately from 59 to 210 kHz.

c.) Francois and Garrison, valid frequency domain see Fig. 9c

2

3 4

2 221 1 1 2 2 2

3 32 2 2 21 2 H O, Pure water

B(OH) , Boric acid MgSO , Magnesium sulphate

[dB/km]A P f f A P f f A P ff f f f

α = + ++ +

(2.12)

The first term in Eq. (2.12) corresponds to:

Boric acid B(OH)3

0.78 51

1

12454273

1

8.686 10

1

2.8 1035

ph

T

Ac

P

Sf

−

−+

= ⋅

=

= ⋅

(2.13)

Magnesium sulphate MgSO4

( )

( )

( )

2

4 9 22 max max

19908273

2

21.44 1 0.025

1 1.37 10 6.2 10

8.17 101 0.0018 35

T

SA Tc

P z z

fS

− −

−+

= ⋅ +

= − ⋅ + ⋅

⋅=

+ −

(2.14)

The sound speed is approximately given by

max1412 3.21 1.19 0.0167c T S z= + + + (2.15)

13

Pure water H2O

4 5 7 2 8 3

3 4 5 7 2 8 3

5 10 23 max max

4.937 10 2.59 10 9.11 10 1.5 10 203.964 10 1.146 10 1.45 10 6.5 10 20

1 3.83 10 4.9 10

T T T forT CA

T T T forT C

P z z

− − − −

− − − −

− −

⎧ ⎫⋅ − ⋅ + ⋅ − ⋅ ≤ °⎪ ⎪= ⎨ ⎬⋅ − ⋅ + ⋅ − ⋅ ≥ °⎪ ⎪⎩ ⎭

= − ⋅ + ⋅

(2.16)

with f in [kHz], T in [°C], S in [ppt]. And where maxz , pH and c denote the depth in

[m], the pH-value and the sound speed in [m/s] respectively.

Fig. 9: Diagram indicating empirical formulae for different frequency domains

A general diagram showing the variation of alpha, α with the three regions of Boric

acid, B(OH)3, Magnesium sulphate, MgSO4 and Pure water, H2O is depicted in Fig. 10.

102 103 104 105 106 10710-6

10-4

10-2

100

102

104

106Attenuation Coefficient in Water from Francois/Garrison method

Frequency [Hz]

Atte

nuat

ion

[dB

/km

]

Pure WaterMagnesiumsulphateBoric AcidAttenuation= Sum of All

Fig. 10: General diagram indicating the three regions of B(OH)3, MgSO4 and H2O

a)

b)

c)

100 Hz 3 KHz0 10 KHz 0.5 MHz 1 MHz f

14

From Fig. 10, it can be observed that for the Boric acid region, Attenuation is

proportional to 2f . And for the regions Magnesium sulphate and pure water also

Attenuation is proportional to 2f . In the transition domains it is proportional to f .

Attenuation increases with increasing salinity and temperature, Fig. 11. Attenuation

increases with increasing frequency.

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

frequency in KHz

Atte

nuat

ion

in [d

B/k

m]

Salinity=15pptSalinity=20pptSalinity=25pptSalinity=30pptSalinity=35pptpH=8Water Temperature=20°C

0 20 40 60 80 100 120 140 160 180 200

0

10

20

30

40

50

60

70

80

90

100

frequency in KHz

Atte

nuat

ion

in [d

B/k

m]

Salinity=15pptSalinity=20pptSalinity=25pptSalinity=30pptSalinity=35pptpH=8Water Temperature=30°C

Fig. 11: Attenuation plot for various salinities & for temperature a) 20°C b) 30°C

2.4.3 Sound Attenuation in sediment

The sound attenuation in sediment mainly varies with the bottom type. Bottom type, in

short represented by bt , defines the sediment material of the ocean. The following table

provides the values of bt for each sediment type.

Sediment type

value of bt

very coarse sand 0

coarse sand 1

medium sand 2

fine sand 3

very fine sand 4

very coarse silt 5

coarse silt 6

medium silt 7

15

fine silt 8

very fine silt 9

clay 10

Table 1 The following empirical formula is provided to find the sound attenuation in the

sediment depending on the bt .

1 1 8.686 1kHz m

n

SfKα ⎛ ⎞ ⎡ ⎤= ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(2.17)

Where,

Sα - attenuation of the sediment The following table provides the values for K and n for four sediment types.

Sediment type

K n

very fine silt 0.17 0.96 fine sand 0.45 1.02 medium sand 0.48 0.98 coarse sand 0.53 0.96

Table 2

2.4.4 Reflection & Transmission Coefficients, R and T

Reflectivity is the ratio of the amplitudes of a reflected plane wave to a plane wave

incident on an interface separating two media. It is an important measure of the impact

of the bottom on sound propagation. The reflection coefficient for a simple case is

derived here.

Fig. 12: Reflection and Transmission at a fluid-fluid interface.

1 1

Medium 1 ,cρ

2 2

Medium 2 ,cρ

Incident Reflected

Transmitted

1θ

2θ

1θ

z

x

16

Fig. 12 shows reflection at an interface separating two homogeneous fluid media with

density iρ and sound speed , 1, 2.ic i = The pertinent incident angles with the horizontal

in the xz -plane are denoted by iθ . Assuming the incident wave to have unit amplitude

and denoting the amplitudes of the reflected and transmitted waves by R and T,

respectively, we can write the acoustic pressures as, Ref. [1]

( )1 1 1 11

exp sin cos , ip ik x z k cωθ θ⎡ ⎤= + ≡⎣ ⎦ , (2.18)

( )1 1 1exp sin cos , rp R ik x zθ θ⎡ ⎤= −⎣ ⎦ (2.19)

( )2 2 2 22

exp sin cos , ,tp T ik x z k cωθ θ⎡ ⎤= + ≡⎣ ⎦ (2.20)

where the common time factor ( )exp i tω− is omitted.

The unknown quantities R, T and 2θ are determined from the boundary conditions

requiring continuity of pressure and vertical particle velocity across the interface at

0z = . With the total pressure in medium 1 given by 1 i rp p p= + and the pressure in

medium 2 by 2 tp p= , the boundary conditions can be mathematically stated as

1 21 2

1 2

1 1, .p pp pi z i zωρ ωρ

∂ ∂= =

∂ ∂ (2.21)

It is easily seen that the requirement of continuity of pressure at 0z = leads to ( )2 2 1 11 exp cos cos ,R T i k k xθ θ⎡ ⎤+ = −⎣ ⎦ (2.22) and

( )1 21 2

1 2

1 cos cos .k kR Tθ θρ ρ

− = (2.23)

Since the left side is independent of x , this yields to Snell’s law of refraction, 1 1 2 2cos cos ,k kθ θ= (2.24)

1 2 2 1

2 1 1 2

sin .sin

k c c nk c c

θ ωθ ω

= = = = (2.25)

This law simply states the invariability of the horizontal component of the wave vector

across the interface. From Eq’s. (2.22) and (2.24), we can now write, 1 .R T+ = (2.26)

With 2

1

m ρρ

= , and together with the equation obtained from the second boundary

condition Eq.(2.25), it follows

17

( ) 1 21 cos cosm R nTθ θ− = . (2.27) Rearranging Eq. (2.27) by substituting Eq.(2.26), lead to the following expressions for

the reflection coefficient R and the transmission coefficient T

2 2

1 11 22 2

1 2 1 1

cos sincos coscos cos cos sin

m nm nRm n m n

θ θθ θθ θ θ θ

− −−= =

+ + − (2.28)

1 12 2

1 2 1 1

2 cos 2 coscos cos cos sin

m mTm n m n

θ θθ θ θ θ

= =+ + −

(2.29)

Computation of Reflection & Transmission Coefficients, R and T with absorption behavior of the media considered

In the above reflection and transmission characteristics without consideration of the

absorption behaviour of the media were deduced. In contrast to sound propagation at the

water-air-boundary layer experimental investigations at the water sediment-boundary-

layer showed that the theory agrees sufficiently exact only with consideration of the

absorption in the sediment with the results of measurement. Due to this reason the

results of section 2.4.3 are now extended for the case of a boundary layer between a

absorption-free medium1 (water) and an absorption-afflicted medium2 (sediment) .

With the introduction of absorption we have a complex wave number,

2 2, 2,R Ik k ik= + (2.30) with

2,2

Rkcω

= and 2, 2Ik α= . Where 2c and 2α denote the velocity of sound and attenuation

of the medium 2 (sediment) respectively. The attenuation of medium 1 (water) is

negligible and it can be neglected.

Eq. (2.25) can be written as,

2, 2,1 2

2 1 1 1

sin .sin

R IR I

k kk i n in nk k k

θθ

= = + = + = (2.31)

18

The wave number 1k and 1θ are always real and therefore, their product 1 1sink θ is also

real. The complex wave number 2k and the product 2 2sink θ are complex.

From all the above we can write,

1 12

2

sinsin kkθθ = (2.32)

which implies,

2 2

2 21 12, 2,2 2 2 1 12

2

sincos 1 sin R Ikk k k n k ik

kθθ θ= − = − = + (2.33)

and 1 1 1sink k θ= . (2.34) Rearranging the above, we can write the pressure transmitted as,

( ){ } ( ){ }2, 1 2,exp 0, exp ,T T

t l Rp T i k r i k k r tω⎡ ⎤= −⎢ ⎥⎣ ⎦ (2.35)

where

( ) ( )1 2, 2, 2,, sin ,cosT T

RP p P Pk k k k θ θ= = (2.36)

( ) ( )2, 2, 2,0, sin ,cosT T

IA A A Ak k k θ θ= = (2.37)

{ }22 2 2 2 2

1 2, 1 1 1sin Re sinRPk k k k nθ θ⎡ ⎤= + = + −⎢ ⎥⎣ ⎦ (2.38)

and { }2 2

2, 1 1Im sin .IAk k k n θ= = − (2.39)

The angles of refraction for the phase and amplitude fronts are,

( ) { }1 1

2, 2 22,1

sinarg arctan arctanRe sin

PPR

kkk n

θθθ

⎛ ⎞⎛ ⎞ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟

−⎝ ⎠ ⎜ ⎟⎝ ⎠

(2.40)

and ( )2, arg 0A Akθ = = (2.41) with the use of

2P

P P

kcω π

λ= = (2.42)

19

the phase velocity of the wave in the sediment can be written as,

{ }1

22 2 2

1 1

.sin Re sin

PP

cck

n

ω

θ θ= =

⎡ ⎤+ −⎢ ⎥⎣ ⎦

(2.43)

2.4.5 Surface and Bottom Scattering

Scattering is a mechanism for loss, interference and fluctuation. A rough sea surface or

seafloor causes attenuation of the mean acoustic field propagating in the ocean

waveguide. The attenuation increases with increasing frequency. The field scattered

away from the specular direction, and, in particular, the backscattered field (called

reverberation) acts as interference for active sonar systems. Because the ocean surface

moves, it will also generate acoustic fluctuations. Bottom roughness can also generate

fluctuations when the source or receiver is moving. The importance of boundary

roughness depends on the sound-speed profiles which determine the degree of

interaction of sound with the rough boundaries.

Often the effect of scattering from a rough surface is thought of simply an additional

loss to the specularly reflected (coherent) component resulting from the scattering of

energy away from the specular direction. If the ocean bottom or surface can be modeled

as randomly rough surface, and if the roughness is small with respect to the acoustic

wavelength, the reflection loss can be considered to be modified in a simple fashion by

the scattering process. A formula often used to describe reflectivity from a rough

boundary is

( ) ( ) 20.5R R eθ θ − Γ= (2.44)

where ( )R θ is the new reflection coefficient, reduced because of scattering at the

randomly rough interface. Γ is the Rayleigh roughness parameter defined as

2 sink θΓ = (2.45)

where 2k π λ= is the acoustic wave number and σ is the rms roughness. As said in

section 2.4.4, the attenuation from water can be neglected and therefore, the reflection

coefficient for smooth ocean surface can be taken as –1. Therefore, the rough sea-

surface reflection coefficient for the coherent field is

( ) 20.5 .R eθ − Γ= − (2.46)

20

The roughness of the ocean surface is due to wind induced waves. It can be calculated

from the spectral density of ocean displacements. It is often modeled by the Neumann-

Pierson wave spectrum. The rms roughness or rms wave height of a fully developed

wind wavefield is then approximately

5 50.324 10 wvσ −≅ ⋅ ⋅ (2.47)

where, wv denotes the wind speed, [m/s].

For ocean bottom, σ is related to the particle size (particle refers to the material of the

sediment, further see section 2.4.3, Table 1) by,

21000

bt

σ−

= m (2.48)

Where bt - represents the bottom type (refer, section 2.4.3, Table 1).

2.4.6 Ambient Noise

An important acoustic characteristic of the ocean is its underwater ambient noise. It

contains a great bulk of information concerning the state of the ocean surface, the

atmosphere over the ocean, tectonic processes in the earth’s crust under the ocean, the

behaviour of marine animals and so on cf.[1], [6].

From, Fig. 13, different dominating levels of ambient noise and total noise level can be

observed and the individual formulae for all these are as stated below:

For shipping noise (traffic) 10-300 Hz

( ) ( ) [ ]8

4 43 1010 log in kHz

1 10trafficNL f ff

⎛ ⎞⋅= ⋅ ⎜ ⎟⎜ ⎟+ ⋅⎝ ⎠ (2.49)

Turbulence noise

( ) ( ) [ ]30 30 log in kHzturbNL f f f= − ⋅ (2.50)

Self noise of the vessel

( ) [ ], in dBvessel sNL f v (2.51)

where f and sv denote the frequency and vessel speed respectively.

21

Biological noise (fishes, scrimps etc.)

( ) [ ], in dBBioNL f S (2.52) where f and S denote the frequency and seasonal dependence.

Sea state noise

( ) [ ], in dBss wNL f v (2.53) The sea state noise can be determined as function of wind speed vw in [kn] and

frequency f in [kHz] by

( ) 53

2

, 40 10 log1

wss w

vNL f vf

⎛ ⎞= + ⋅ ⎜ ⎟

+⎝ ⎠ (2.54)

Thermal noise ( )ThermNL f in [dB]

The thermal noise is due to molecular agitation (Brownian Motion). It can be

expressed as function of frequency f in [kHz] by

( )15 20 logThermNL f= − + ⋅ (2.55) Thus the total noise level can be determined by:

( ) (

)0.1 0.1 0.1

0.1 0.1 0.1

, , , 10 log 10 10 10 10 10 10

traffic turbo vessel

bio ss therm

NL NL NLs w

NL NL NL

NL f v S v = ⋅ + + +

+ + (2.56)

101 102 103 104 105 1060

10

20

30

40

50

60

70

80

90

100

Frequency [Hz]

NL

[dB

]

Ambient Noise Curve Formulation for windspeed=20 kn

NL TrafficNL TurbulanceNL Sea StateNL ThermalNL

Fig. 13: Ambient Noise Level for different domains at vw = 20 kn

22

3 Sound Propagation

Sound propagation in the ocean is mathematically described by the wave equation,

whose parameters and boundary conditions are descriptive of the ocean environment.

As schematically shown in the Fig.14, there are essentially five types of models

(computer solutions to the wave equation) to describe sound propagation in the sea, cf.

[1]:

FFP - fast field program

NM - normal mode

PE - parabolic equation

FD - direct finite-difference

FE - finite-element

Fig. 14: Hierarchy of underwater acoustic models.

3.1 The Wave Equation

The wave equation in an ideal fluid can be derived from hydrodynamics and the

adiabatic relation between pressure and density. The equation for conservation of mass,

Euler’s equation (Newton’s 2nd Law), and the adiabatic equation of state, are as stated

below, Ref. [1], [3], [6]:

Fig. 15 is used to describe the motion of a particle in a water column from where in the

entire derivation of the wave equation has been done.

FD/FE

Range Independent

Wave Equation

Normal Mode

Coupled FFP

Coupled NM

Adiabatic NM

Ray PE

Range dependent

Fast Field Program

23

Fig. 15: Schematic diagram indicating displacement of a particle from x to x + dx in water column.

In deriving all the following equations we have used the terms gP and gρ . These two

terms are defined as follows:

g op p p= + (3.1) where

gp - Total pressure

op - Static pressure p - change in pressure g oρ ρ ρ= + (3.2) where

gρ - Total density

oρ - Static density ρ - change in density Equation for conservation of mass

( ) ( ) ( ) ( )Resultant mass stream density variation

Mass variation

gg gx dx Av x dx x Av x Adx

tρ

ρ ρ∂

+ + − = −∂

(3.3)

( )Av x dx+

( )Av x - Inward Volume stream at position x

x

pg(x)

v(x)

x

x + dx

pg(x + dx)

v(x + dx) A

- Outward Mass Stream at a displacement of dx ( ) ( )g x dx Av x dxρ + +

- Outward Volume stream at a displacement of dx

24

( ) ( )g x Av xρ

( ) Mass increament+

( ) Mass decreament− Euler’s Equation (Newton’s 2nd Law) From Fig. 15, the Newton’s 2nd law F m a= ⋅ can be written as:

( ) ( )Total Force,

g g gV

F am

dvp x A p x dx A Adxdt

ρ⋅ − + ⋅ = (3.4)

With

v vdv dt dxt x∂ ∂

= +∂ ∂

(3.5)

dv v v dx dv v vvdt t x dt dt t x

∂ ∂ ∂ ∂= + = = +∂ ∂ ∂ ∂

(3.6)

and

( ) ( ) ( ) ( ), ,, , g gg g gp x dx t p x tp x t p x dx t p

dx dx x

⎡ ⎤− + −− + ∂⎣ ⎦= = −∂

(3.7)

Eq. (3.4) can be re-written as

gp p v vvx x t x

ρ∂ ∂ ∂ ∂⎛ ⎞− = − = +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(3.8)

and with

( ) ( ) ( ) ( ) ( ), , , , gg g vx dx t v x dx t x t v x t

dx xρρ ρ ∂+ + −

=∂

(3.9)

Eq. (3.3), can be written as:

( )g gv

x t tρ ρ ρ∂ ∂ ∂

= − = −∂ ∂ ∂

(3.10)

which is known as the equation of continuity.

- Inward Mass stream at position x

25

Adiabatic equation of state

( )2

22

1 ,2

g gg o

g gs s

p pp p ρ ρ

ρ ρ⎡ ⎤ ⎡ ⎤∂ ∂

= + + + ⋅⋅⋅⎢ ⎥ ⎢ ⎥∂ ∂⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(3.11)

and for convenience we define the quantity

2

s

pcρ

⎡ ⎤∂= ⎢ ⎥∂⎣ ⎦

, (3.12)

where c will turn out to be the speed of sound in an ideal fluid. In the above equations,

ρ is the density, v is the particle velocity, p is the pressure, and the subscript s

denotes that the thermodynamics partial derivatives are taken at constant entropy.

For op p and oρ ρ , Eq. (3.11) becomes:

2p c ρ= (3.13)

Considering that the time scale of oceanographic changes is much longer than the time

scale of acoustic propagation, we will assume that the material properties oρ and 2c are

independent of time. Then, taking the partial derivatives of Euler’s equation, Eq. (3.8)

with respect to x and for continuity equation, Eq. (3.10) with respect to t gives

2

2 g gp v vv

x x t x xρ ρ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞− = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(3.14)

and

2 2

2 2 2

1gg

v pvx t x t t c t

ρ ρρ∂⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ = − = −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

. (3.15)

For lower speeds, in Eq. (3.14) we can ignore, gvv

x xρ∂ ∂⎛ ⎞

⎜ ⎟∂ ∂⎝ ⎠ and for oρ ρ , the term

g vx t

ρ∂⎛ ⎞∂⎜ ⎟∂ ∂⎝ ⎠

in Eq. (3.15) can be ignored. Now, Eq’s. (3.14) and (3.15) can be written as:

2

2 gp v

x x tρ∂ ∂ ∂⎛ ⎞− = ⎜ ⎟∂ ∂ ∂⎝ ⎠

(3.16)

2

2 2

1g

v px t c t

ρ∂ ∂ ∂⎛ ⎞ = −⎜ ⎟∂ ∂ ∂⎝ ⎠. (3.17)

Combining Eq’s. (3.16) and (3.17), we get the one dimensional linear wave equation

2 2

2 2 2

1p px c t

∂ ∂=

∂ ∂. (3.18)

26

Extending it to three dimensional equation we get,

2

2 2

1 ppc t

∂Δ =

∂ (3.19)

where

2 2 2

2 2 2x y z∂ ∂ ∂

Δ = + +∂ ∂ ∂

denotes the Laplacian operator.

3.2 Helmholtz Equation

For ( ) ( ), , , expp P x y z t j tω= , we obtain

2 0P k PΔ + = . (3.20)

In spherical coordinates the Laplacian can be expressed by 2 2 2P P R R P RΔ = ∂ ∂ + ∂ ∂ ,

if taken into account that P only depends upon R. Spherical wave solution of the

Helmholtz Equation is given by,

( )exp4

A jkRP

Rπ−

= (3.21)

with

( ) ( ) ( )2 2 2o o oR x x y y z z= − + − + − (3.22)

where , ,o o ox y z are the coordinates of an omni directional point source (pulsating sphere

of small radius). Another simple and important solution is given by plane wave,

( )exp x y zP A j k x k y k z⎡ ⎤= − + +⎣ ⎦ (3.23)

where , and x y zk k k denote the wave numbers that satisfy,

2 2 2x y zk k k= + +Tk k (3.24)

with ( ), ,T

x y zk k k=k the wave number vector.

27

3.3 Sound Propagation in Homogenous Waveguide

A homogenous water column within infinitely extended perfectly reflecting boundaries,

as shown in Fig. 16. is considered in the sequel.

Fig. 16: Homogenous waveguide with source S and receiver R The field produced by a point source at ( )0, Sz in the absence of boundaries is given by,

( ) ( ),4

jkReP r z AR

ωπ

−

= . (3.25)

where,

( ) ( )2 2s sR z z r r= − + −

( ) Amplitude or Source strengthA ω =

Next we need to add a solution to the homogenous Helmholtz equation to satisfy the

boundary conditions of vanishing pressure at the surface and bottom of the waveguide.

The method which we use for this is image or mirror method, and is explained in the

following section. Here, the ocean surface and bottom are considered as two mirrors.

The rays which hit the surface and bottom are then starting exactly at the images of the

actual sources of origin. With this logic the whole image or mirror method is developed

and thereby it is easy to provide mathematics for multipath propagation.

•R

0Sr

D

S

r

1480 c m s=

31000 kg mρ =

sediment

air

2. Boundary

1. Boundary

z

Sz •

r

z

28

3.3.1 Image or Mirror Method

The image method superimposes the free-field solution with the fields produced by the

image sources. In the waveguide case, sound will be multiply reflected between the two

boundaries, requiring an infinite number of image sources to be included see for details

[1,6].

Fig. 17: Reflections of a wave from the boundaries of a layer, and the image sources

Fig. 17 shows a schematic representation of the contributions from the physical source

at depth Sz and the first three image sources, leading to the first 4 terms in the

expression for the total field,

( ) ( ) ( )

( ) ( ) ( )

01 02

03 04

1 0201 02

2 03 1 04 2 0403 04

ˆ, , ,

ˆ ˆ ˆ , , ,

jkL jkL

jkL jkL

e eP r z A RL L

e eR R RL L

ω ω φ ω

φ ω φ ω φ ω

− −

− −

⎧≅ +⎨

⎩⎫

+ + ⎬⎭

(3.26)

( ) ( )ˆ , , ,2i i iR R Rπφ ω φ ω θ ω⎛ ⎞= − =⎜ ⎟

⎝ ⎠, where i = 1,2.

•

•

Z

r0

2R

1R

D

Air

Bottom

Water

Image Surface

Image Bottom

Sz−

2 SD z+

2 SD z−

Sz

03L

01L

02L

04L

z

r

03φ 04φ

02φ

29

( )

( )

( )

( )

2201

2202

2203

2204

,

,

2 ,

2 .

s

s

s

s

L r z z

L r z z

L r D z z

L r D z z

= + −

= + +

= + − −

= + + −

The remaining terms are obtained by successive imaging of these sources to yield the

ray expansion for the total field,

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

1

2

3

1 1 2 10 1

11 2 2 2

2

11 3 2 3

3

ˆ ˆ, , , ,

ˆ ˆ , ,

ˆ ˆ , ,

m

m

m

jkLm m

m mm m

jkLm m

m mm

jkLm m

m mm

eP r z A R RL

eR RL

eR RL

ω ω φ ω φ ω

φ ω φ ω

φ ω φ ω

−∞

=

−+

−+

⎧= ⎨

⎩

+

+

∑

( ) ( )4

1 11 4 2 4

4

ˆ ˆ , ,mjkL

m mm m

m

eR RL

φ ω φ ω−

+ + ⎫+ ⎬

⎭

(3.27)

where, A - Amplitude of the signal,

1R̂ - Surface reflection coefficient,

2R̂ - Bottom reflection coefficient, k - Complex wave number,

1 2 3 4, , ,m m m mL L L L - Length’s of all rays.

with

( )

( )

( )( )

( )( )

221

222

223

224

2

2

2 1

2 1

m S

m S

m S

m S

L r Dm z z

L r Dm z z

L r D m z z

L r D m z z

= + − +

= + + +

= + + − −

= + + + −

(3.28)

and D being the vertical depth of the duct.

30

3.3.2 Grazing angles

The angle with which each ray grazes the boundaries is usually termed as grazing angle.

This is quite important because of its influence on both the bottom and surface

reflection coefficients. With simple mathematics, the grazing angle for the four paths or

for all the rays can be computed, and is given by

( )11

2tan s

m

Dm z zr

φ − ⎛ ⎞− += ⎜ ⎟

⎝ ⎠, (3.29)

( )12

2tan s

m

Dm z zr

φ − ⎛ ⎞+ += ⎜ ⎟

⎝ ⎠, (3.30)

( )( )1

3

2 1tan s

m

D m z zr

φ −⎛ ⎞+ − −

= ⎜ ⎟⎜ ⎟⎝ ⎠

, (3.31)

( )( )1

4

2 1tan s

m

D m z zr

φ −⎛ ⎞+ + −

= ⎜ ⎟⎜ ⎟⎝ ⎠

, (3.32)

where

1 2 3 4, , ,m m m mφ φ φ φ - grazing angles of all the rays,

D - depth of the duct or channel,

0 to m = ∞ ,

sz - depth of the source in meters,

z - depth of the receiver in meters,

r - distance of the receiver in meters.

Further, from Eq. (3.27), we can write down the influence of

• Wind speed, frequency and grazing angle on Surface reflection coefficient, 1R̂ ,

• Bottom type and grazing angle on Bottom reflection coefficient, 2R̂ .

The functional dependence of wind speed wv , frequency f and grazing angle mφ on

Surface reflection coefficient is illustrated in Fig. 18.

31

0 10 20 30 40 50 60 70 80 90-800

-700

-600

-500

-400

-300

-200

-100

0

grazing angle [deg], f=25kHz

scat

terin

g lo

ss

a) wv =5 kn

0 10 20 30 40 50 60 70 80 90-15000

-10000

-5000

0

grazing angle [deg], f=25kHz

scat

terin

g lo

ss

b) wv = 15 kn.

Fig. 18: Diagram illustrating dependence of 1R on grazing angle, frequency and two wind speeds From Fig. 18, it can be observed that with an increase of grazing angle the scattering

loss also increases. In the same way with the increase of wind speed, there is an increase

in scattering loss.

Similarly we can also observe the dependence of Bottom reflection coefficient, 2R on grazing angle mφ and bottom type bt . This is illustrated in Fig. 19.

0 10 20 30 40 50 60 70 80 90-7

-6

-5

-4

-3

-2

-1

0

grazing angle [deg]

refle

ctio

n lo

ss [d

B]

a) bt = coarse sand

0 10 20 30 40 50 60 70 80 90-35

-30

-25

-20

-15

-10

-5

0

grazing angle [deg]

refle

ctio

n lo

ss [d

B]

b) bt = very fine sand

Fig. 19: Diagram illustrating dependence of 2R on grazing angle and two bottom types

32

3.3.3 Travel Times

The travel time of each ray is the time taken for it to reach the receiver. From the above

discussion it is vivid that, all the other paths need more time compared to the direct

path. Travel times for all the rays can be easily computed provided we know the

length’s or distances of all rays and the velocity of each ray. From Eq.(3.28), we know

the distances of all rays and the velocity of each ray is the speed of sound, c. Thereby,

we can write as

1 1

2 2

3 3

4 4

,

,

,

.

m m

m m

m m

m m

T L c

T L c

T L c

T L c

=

=

=

=

(3.33)

where,

c - Sound velocity in meters per second.

1 2 3 4, , and m m m mL L L L - Length’s of each ray in meters.

1 2 3 4, , and m m m mT T T T - Travel times of all rays in seconds.

Travel time of each ray in seconds

Num

ber o

f ray

s

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

1

2

3

4

5

6

7

80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 20: Multipath propagation depicting delays in 2D-view.

33

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.2

0.4

0.6

0.8

1

1.2

1.4

Travel time of each ray in seconds

Number of rays

Am

plitu

de

Fig. 21: Multipath propagation depicting delays in 3D-view.

Fig. 20 and Fig. 21 show the delay of rays in 2 dimensional and 3 dimensional views. It

can be clearly observed the delay of sinc pulse from ray 1 till ray 8. Here, sinc pulse is

just taken as an example to show the concept of delay.

3.3.4 Transmission loss for each ray

The transmission loss or sometimes referred as propagation loss is nothing but the sum

of all the losses, a ray gets effected. So, transmission loss can be written as,

0404 1 2

04

1 ˆ ˆltl e R Rl

α= (3.34)

In the above equation the transmission loss is written only for ray 4, as an example. In

terms of Eq. (3.27), spreading loss is due to the terms 1 2 3 4

1 1 1 1, , ,m m m ml l l l

(as discussed in

sec. 2.4.1). The attenuation or absorption is from the imaginary part of the complex

wave number k , as discussed in sec. 2.4.3, refer to Eq. (2.30). The Total Reflection loss

can be seen in this way. It is a sum of,

• reflection loss, loss which is caused when a ray travels from medium1 to medium2

due to the refraction of the ray, due to the reflection of the ray.

• scattering loss, loss which is caused by the roughness of the boundary. That is rays

get scattered in an un orderly fashion.

34

In Eq. (3.27), it is caused due to the terms 1 1 1 11 2 1 2 1 2 1 2, , ,m m m m m m m mR R R R R R R R+ + + + . The

following figure illustrates the transmission loss phenomenon.

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.005

0.01

0.015

0.02

Travel time of each ray in secondsNumber of rays

Am

plitu

de

Fig. 22: Multipath propagation depicting transmission loss.

As an example, a sinc pulse is taken to present the transmission loss phenomenon. In

Fig.22, one can observe clearly the degradation process in amplitude from ray 1 till

ray 8.

35

4 Modulation

Ordinarily, the transmission of a message signal (be it in analog or digital form) over a

band-pass communication channel (e.g., telephone line, satellite channel) requires a

shift of frequencies contained in the signal into other frequency ranges suitable for

transmission, and a corresponding shift back to the original frequency range after

reception. A shift of the range of frequencies in a signal is accomplished by using

modulation, defined as the process by which some characteristic of a carrier is varied

in accordance with a message signal. The message signal is referred to as the

modulating signal, and the result of the modulation process is referred to as the

modulated signal. At the receiving end of the communication system, we usually

require the message signal to be recovered. This is accomplished by using the process

known as demodulation, which is the inverse of the modulation process [7].

4.1 Digital Modulation Techniques

With a binary modulation technique, the modulation process corresponds to switching

or keying the amplitude, frequency, or phase of the carrier between either of two

possible values corresponding to binary symbols 0 and 1. This results in three basic

signaling techniques, namely, amplitude shift-keying (ASK), frequency shift-keying

(FSK) and phase shift-keying (PSK), as described herein [7]:

4.1.1 ASK

In ASK the amplitude of the signal is changed in accordance to the information and all

else is kept fixed. 1 is transmitted by a signal of particular amplitude. To transmit 0, we

change the amplitude keeping the frequency constant. On-off keying (OOK) is a special

form of ASK, where one of the amplitudes is zero as shown below;

36

Fig. 23: Baseband information sequence - 0010110010

( ) ( ) ( )sin 2ASK t s t ftπ= (4.1)

Fig. 24: Binary ASK (OOK) carrier

4.1.2 FSK

In FSK, we change the frequency in response to the information, one particular

frequency for a 1 and another frequency for a 0 as shown below for the same bit

sequence as above. In the example below, frequency 1f for a 1 is higher than 2f used

for the 0.

( )( )( )

1

2

sin 2 for 1

sin 2 for 0

f tFSK t

f t

π

π

⎧⎪= ⎨⎪⎩

(4.2)

Fig. 25: Binary FSK carrier

37

4.1.3 PSK

In PSK, we change the phase of the sinusoidal carrier to indicate the information. Phase

in this context is the starting angle at which the sinusoid starts. Depending on the start

of the binary sequence, to transmit 0 or 1, we shift the phase of sinusoid by 180°. Phase

shift represents the change in the state of information in this case.

If ( )b t represents the binary sequence then we can write

( ) ( ) ( )sin 2PSK t ft b tπ= ⋅ . (4.3)

Fig. 26: Binary PSK carrier (note the 180° phase sifts at big edges) Remarks ASK

• Pulse shaping can be employed to remove spectral spreading.

• One binary digit is represented by presence of carrier, at constant amplitude. Other

binary digit by absence of carrier.

• ASK is susceptible to sudden gain changes and demonstrates poor performance.

FSK

• Bandwidth occupancy of FSK is dependant on the spacing of the two symbols. A

frequency spacing of 0.5 times the symbol period is typically used.

• FSK can be expanded to a M-ary scheme, employing multiple frequencies as different

states.

PSK

• PSK can be expanded to a M-ary scheme, employing multiple phases as different

states.

38

• Filtering can be employed to avoid spectral spreading.

4.2 Bit rate and Symbol rate

Symbols, Bits and Bauds A symbol is quite apart from a bit in concept although both can be represented by a

sinusoidal or wave functions. Where bit is the unit of information and symbol is the unit

of transmitted energy. The definition of a symbol is always little bit ambiguous. In

general or according to communication aspects, it can be broadly defined as: “A symbol

is a set of bits”, not just one bit. The size of the set depends upon the modulation

scheme which you are using. For example, in BPSK, symbol has only a single bit and in

QPSK, 2 bits constitute a symbol. In our case considered, the number of bits in a symbol

can be written as, 2log M . Where M represents the number of phase shifts.

To understand and compare different modulation format efficiencies, it is important to

first understand the difference between bit rate and symbol rate. The signal bandwidth

for the communications channel needed depends on the symbol rate, not on the bit rate,

cf. [9].

bit rateSymbol ratethe number of bits transmitted with each symbol

=

Bit rate is the frequency of a system bit stream. Take, for example, an 8 bit sampler,

sampling at 10 kHz. The bit rate, the basic bit stream rate, would be eight bits multiplied

by 10K samples per second or 80 Kbits per second. (For the moment we will ignore the

extra bits required for synchronization, error correction, etc.).

The symbol rate is the bit rate divided by the number of bits that can be transmitted with

each symbol. If one bit is transmitted per symbol, as with BPSK, then the symbol rate

would be the same as the bit rate of 80 Kbits per second. If two bits are transmitted per

symbol, as in QPSK, then the symbol rate would be half of the bit rate or 40 Kbits per

second. A Baud rate is same as the Symbol rate. If more bits can be sent with each

symbol, then the same amount of data can be sent in a narrower spectrum. This is why

modulation formats that are more complex and use a higher number of states can send

the same information over a narrower piece of the RF spectrum.

39

4.3 Representation of Signals

4.3.1 Baseband and Bandpass Signals

In many communication systems the baseband signal that conveys the message to be

transmitted is up-converted (i.e., translated in frequency) in order to better suit the

characteristics of the channel. An example of such a system is a QPSK system. The

QPSK modulation process can be viewed as a two step procedure. First, a baseband

signal, consisting of a series of complex valued pulses, is formed, cf. section 4.5. This

signal is then up-converted to the desired carrier frequency. The result is a bandpass

signal which can be transmitted over a physical channel [10].

4.3.2 Baseband vs. Bandpass

In general any amplitude or phase modulation technique can be described by the

relation

( ) ( ) ( )cos 2 cx t A t f t tπ φ⎡ ⎤= +⎣ ⎦ (4.4)

where cf is the carrier frequency. The bandwidths of the phase function ( )tφ and the

amplitude function ( )A t are in general much lower than the carrier frequency. Hence,

the rate-of-change in these signals is typically much lower than cf . Consequently, ( )x t

is a bandpass signal with its spectrum concentrated around the carrier frequency cf .

Using trigonometric relations, it is possible to write the same function as

( ) ( ) ( ) ( ) ( )cos 2 sin 2I c Q cx t x t f t x t f tπ π= − (4.5)

where

( ) ( ) ( )( )cosIx t A t tφ= (4.6)

( ) ( ) ( )( )sinQx t A t tφ= (4.7)

represent the quadrature components. Here, ( )Ix t and ( )Qx t is the in-phase (I) and the

quadrature component (Q) respectively. Eq. (4.5) can be rewritten using complex

numbers as

( ) ( ){ }2Re cj f tbbx t x t e π= (4.8)

where

( ) ( ) ( )bb I Qx t x t jx t= + (4.9) is the baseband equivalent signal.

40

It is always instructive to study the baseband and bandpass equivalent signals in the

frequency domain. As illustrated in Fig. 27, the baseband equivalent signal is obtained

from the bandpass signal by removing the image of the signal on the negative frequency

axis, scaling the remaining spectrum by a factor of two and moving the result to the

baseband by shifting the spectrum cf Hz to the left. This frequency domain result, as

well as baseband and bandpass signals in general, is thoroughly discussed in chapter 4

of [4].

Fig. 27: The relation between a Bandpass signal and its Baseband equivalent signal

in the frequency domain.

A baseband equivalent signal can be obtained by using a down-converter. Such a device

might be implemented in several ways. One example of an implementation is shown in

Fig. 28. The signal is first multiplied by 22 cj f te π− in order to shift the spectrum and scale

it by a factor of two. The low-pass filter removes the negative frequency image. It is

assumed that the low-pass filter is ideal and has a sufficiently large bandwidth so as not

to alter the shape of the positive frequency image. A down-converter is found in most

bandpass communication receivers.

Fig. 28: Converting a Bandpass signal into its Baseband equivalent signal. Communication signals are usually represented using just the complex signal ( )bbx t in

Eq.(4.9), which is called as baseband representation as opposed to the bandpass

representation ( )x t , which is a real valued signal. The baseband representation is much

easier to work than the bandpass representation, as will be illustrated below.

( )x t

⊗ LP ( )Ix t

⊗ LP ( )Qx t

( )2sin 2 cf tπ

( )2cos 2 cf tπ

41

Advantages of the baseband representation

The use of a baseband representation simplifies communications system simulation and

analysis in a number of ways. A simulation or analysis of a baseband system is not tied

to any particular carrier frequency and can be reused if the carrier frequency is changed.

Certain very useful operations become extremely simple when using the complex

representation. For example, a frequency shift by shiftf is done by multiplying the signal

by 2 shiftj f te π . A phase shift of shiftϕ is done by multiplying the signal with shiftje ϕ .

4.4 Modulation – QPSK

In binary data transmission, we send only one of two possible signals during each bit

interval bT . On the other hand, in an M-ary data transmission system we send any one

of M possible signals, during each signaling interval T. For almost all applications, the

number of possible signals 2nM = , where n is an integer, and the signaling interval

bT nT= . It is apparent that a binary data transmission system is a special case of an

M-ary data transmission system. Each of the M signals is called a symbol. The rate at

which these symbols are transmitted through a communication channel is expressed in

units of bauds (as explained in the above section). For M-ary data transmission, it

equals to 2log M bits per second.

Quadrature phase shift keying (QPSK) is an example of M-ary data transmission with

4M = . In QPSK, one of four possible signaling elements is transmitted during each

signaling interval, with each signal uniquely related to a dibit (pairs of bits are termed as

dibits).

For example, we may represent the four possible dibits 00, 10, 11, and 01 in Gray-

encoded form (further on Gray encoding cf. [4], p.170), by transmitting a sinusoidal

carrier with one of four possible values, as follows:

42

( )

cos 2 , dibit 0043cos 2 , dibit 104

cos 2 , dibit 0143cos 2 , dibit 114

c c

c c

c c

c c

A f t

A f ts t

A f t

A f t

ππ

ππ

ππ

ππ

⎧ ⎛ ⎞+⎜ ⎟⎪ ⎝ ⎠⎪⎪ ⎛ ⎞+⎪ ⎜ ⎟

⎝ ⎠⎪= ⎨⎛ ⎞⎪ −⎜ ⎟⎪ ⎝ ⎠

⎪⎛ ⎞⎪ −⎜ ⎟⎪ ⎝ ⎠⎩

(4.10)

where 0 t T≤ ≤ ; we refer to T as the symbol duration. Fig. 29 depicts the signal state

diagram of Eq.(4.10).

Fig. 29: QSPK state diagram Clearly QPSK represents a special form of phase modulation. This is done by

expressing ( )s t succinctly as

( ) ( )cos 2c cs t A f t tπ φ⎡ ⎤= +⎣ ⎦ (4.11)

where the phase ( )tφ assumes a constant value for each dibit of the incoming data

stream. A further insight into the representation of QPSK can be developed by

expanding the cosine term in Eq.(4.11) and rewriting the expression for ( )s t as,

( ) ( ) ( ) ( ) ( )cos cos 2 sin sin 2c c c cs t A t f t A t f tφ π φ π⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦ (4.12) According to this representation, the QPSK wave ( )s t has an in-phase component equal

to ( )coscA tφ⎡ ⎤⎣ ⎦ and a quadrature component equal to ( )sincA tφ⎡ ⎤⎣ ⎦ .

The representation of Eq. (4.12) provides the basis for the general block diagram of the

QPSK transmitter shown in Fig. 30. It consists of a serial-to-parallel converter, a pair of

product modulators, a supply of the two carrier waves (inphase, quadrature) and a

00

01

10

11

I

Q

43

summer. The function of the serial-to-parallel converter is to represent each successive

pair of bits of the incoming binary data stream ( )m t as two separate bits, with one bit

applied to the in-phase channel of the transmitter and the other bit applied to the

quadrature channel.

Fig. 30: General block diagram QPSK transmitter

It is apparent that the signaling interval T in a QPSK system is twice as long as the bit

duration bT of the input binary data stream ( )m t . That is, for a given bit rate 1/ bT , a

QPSK system requires half of the transmission bandwidth of the corresponding binary

PSK system.

Assuming the coding arrangement of Eq. (4.10), for the following data sequence, signal

wave forms are shown for modulated carrier for I channel, Q channel and the final

QPSK carrier.

Example to illustrate QPSK modulation

Assumed data sequence = [0 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1] to be transmitted.

( )cos 2c cA f tπ

Serial- to-parallel converter

Oscillator

( )sin 2c cA f tπ

∑ QPSK signal

2π

I data, odd bits

Q data, even bits

⊗

⊗

Binary data stream ( )m t

44

0 0.5 1 1.5 2 2.5 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5data sequence

time in seconds

mag

nitu

de

Fig. 31: Data sequence transmitted

0 0.5 1 1.5 2 2.5 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5modulated carrier Signal for I channel

time in seconds

mag

nitu

de

Fig. 32: Modulated carrier signal for I channel

45

0 0.5 1 1.5 2 2.5 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5modulated carrier Signal for Q channel

time in seconds

mag

nitu

de

Fig. 33: Modulated carrier signal for Q channel

Here the mapping of the bits is done according to the state diagram of QPSK, Fig. 29.

0 0.5 1 1.5 2 2.5 3

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5QPSK Signal

time in seconds

mag

nitu

de

Fig. 34: QPSK signal for the given data sequence

46

4.5 Pulse shaping

The resulting QPSK symbols are passed through a pulse shaping filter. The rectangular

pulses are not practical to send and require a lot of bandwidth. So, in their lieu we send

shaped pulses that convey the same information but use smaller bandwidths and have

other good properties such as intersymbol interference rejection. One of the most

common pulse shaping used with QPSK is “root raised cosine”, in short RRC. This

pulse shaping has a so called roll-off parameter which controls the shape and the

bandwidth of the signal.

Some common pulse shaping methods are

• Root Raised cosine (used with QPSK)

• Half-sinusoid (used with MSK)

• Gaussian (used with GMSK)

The root raised pulse shape is given by,

( )( ) ( ) ( ) 1

2 2 2

cos 1 sin 1 44

1 16T

t T t T t Tt

T t T

β π β π βψ β

π β

−⎡ ⎤ ⎡ ⎤+ + −⎣ ⎦ ⎣ ⎦=⎡ ⎤−⎣ ⎦

(4.13)

where T, is the symbol time and β is the roll-off factor. The roll-off factor β usually

lies between 0 and 1 and defines the excess bandwidth100 %β . Using a smaller β

results in a more compact power density spectrum, but the link performance becomes

more sensitive to errors in the symbol timing. A typical root raised pulse with a roll-off

factor of 0.5β = is shown in Fig. 35.

47

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Root raised cosine pulse

delay t/T

Am

plitu

de

Fig. 35: Root raised cosine pulse with a roll-off factor, 0.5β =

48

5 System description

In sec. 4.3, we have dealt about bandpass and baseband descriptions. As we have seen

there are many advantages when we use baseband representation over bandpass in terms

of simulations. So, here we have used baseband system representation. A detailed

description about the simulation of a continuous-time baseband system is provided here.

Fig. 36: Underwater Acoustic simulation system.

The communication system considered is shown in Fig. 36. This is a typical set up

which can represent any kind of system using quadrature amplitude modulation (QAM).

This QPSK system is used in our investigations. A brief overview of the system now

follows. At the transmitting side, the sequence of symbols ( )d n is converted to a

continuous-time baseband signal ( )bbs t by a pulse amplitude modulator (PAM). Note

that ( )d n takes the values from discrete set of complex-valued symbols. Up-conversion

is performed by multiplying with 2 cj f te π , resulting in a bandpass signal ( )s t , being

transmitted over the channel (refer chapters 2 and 3). In order to remove the carrier, the

received signal ( )r t is processed by a down-converter which outputs the corresponding

baseband equivalent signal ( )bbr t . The down-converter is followed by a low pass and

then by a matched filter. The detector gives the estimates of the transmitted symbols. As

we already said above, baseband representation is useful in order to be able to simulate

the system using, for example, Matlab, where only time discrete signals can be

represented. Fig. 37 represents the baseband equivalent system.

nT

2 cj f te π−

( )d̂ n

2 cj f te π

( )d n ( )r t det⊗ BP

( )bbr tPAM ⊗( )bbs t ( )s t

Noise

Under- water acoustic channel

( )p t−( )p t

49

Fig. 37: The baseband equivalent system

UAC – Underwater Acoustic channel. A simulation is often based on oversampled system, i.e. the sampling rate is higher than

the symbol rate. In general, a higher sampling rate will more accurately reflect the

original system. However, this comes at the cost of a longer simulation time since more

samples need to be processed. It is common to use an oversampling rate that is a

multiple of the symbol rate. The number of samples per symbols, here denoted by Q is

then an integer.

In order to arrive at the desired discrete time system, we will take the continuous-time

baseband equivalent system, introduce an ideal anti-alias filter at the output of the

matched filter and then oversample its output. This is depicted in Fig. 38, where also

down-sampling a factor Q is assumed to be chosen so large that the bandwidth of the

matched filter is smaller than the bandwidth 2Q T of the anti-alias filter. Consequently,

the anti-alias filter does not change the signal output from the matched filter. The signal

at the input of the detector is same as for the continuous-time system. Thus, this

oversampled system is equivalent to the original system.

Fig. 38: Oversampling the system Below, a series of equivalent systems are presented where the anti-alias filter and the

sampling device step by step, is moved to the left until a completely discrete system

remains. As illustrated in Fig. 39, the first step is to switch the order of the matched

filter and the anti-alias filter and also use a discrete time matched filter ( )p n− . With a

nT Q

( )d̂ ndet ( )bbr t( )p t−PAM ( )d n ( )p t

( )bbs t

Noise

UAC

nT Q

( )d̂ n

2QT2

QT

−f

1

detQ( )bbr t

( )p t−PAM ( )d n ( )p t( )bbs t

Noise

UAC

50

slight change of notation, ( )x n denotes the discrete-time signal, ( )x nT Q . The

matched filter has a bandwidth smaller than the Nyquist frequency 2Q T so this

reordering operation does not affect the signal. This can be further understood with a

simple mathematical motivation by considering the sampled output of the continuous-

time convolution.

( ) ( ) ( )y t h t x t= ∗

and approximating the integral with a summation to yield

( ) ( ) ( )s sy nT h x nT dτ τ τ∞

−∞

= −∫

( ) ( )( )s s sk

T h kT x n k T∞

=−∞

≈ −∑

( ) ( )sT h n x n= ∗

where,

sTTQ

=

The relation holds exactly if both ( )h t and ( )x t has a bandwidth less than the Nyquist

frequency 1 2 sT . As a result, sampled continuous-time convolutions can be computed

using discrete-time processing and if the output is scaled by the sample period sT .

Fig. 39: Moving the anti-alias filter and the sampling device in front of the matched filter. As a next step it should be obvious that the anti-alias filter and the sampler can be

moved in front of the summation without changing the signal at the detector. The anti-

alias filter after PAM can be removed (since the bandwidth ( )p t is smaller) which

means that we can write the sampled transmitted signal as

( )d̂ ndetQ

2QT2

QT

−f

1 nT Q

⊗

T Q

( )bbr tPAM ( )d n ( )p t( )bbs t

Noise

UAC ( )p n−

51

( ) ( ) ( ) .bbk

s n d k p n kQ T Q∞

=−∞

⎡ ⎤= −⎣ ⎦∑

Clearly, this summation can be implemented by up-sampling ( )d n by a factor of Q ,

followed by filtering the resultant sequence with ( )p n .

Fig. 40: The equivalent discrete time baseband system. It is finally possible to summarize the above development in the equivalent discrete-

time-system shown in Fig. 40.

5.1 Simulation system

The simulation system is illustrated in Fig. 41. It consists of a bit source, transmitter,

channel, receiver and a bit sink. The bit source generates the random binary sequence

that is to be transmitted by the transmitter. Typically a random bit source is employed in

simulations and this is the case in our simulation as well. The transmitter converts the

bits into QPSK symbols, applies pulse shaping and up-conversion is done to the desired

carrier frequency.

Fig. 41: The Simulation system considered The output from the transmitter is fed through the underwater acoustic channel. The

receiver block takes the output from the channel, estimates phase and timing offset, and

demodulates the received QPSK symbols into information bits which are fed to the bit

sink. Here, the bit sink counts the number of errors that occurred to gather the statistics

used for investigating the performance of the system.

Bit source Transmitter Underwater acoustic channel

Receiver Bit sink

( )d̂ ndet

Noise

Q( )bbr n

Q( )d n ( )bbs n( )p n ( )p n−

UAC

52

5.2 Transmitter

The transmitter which is used in this report is illustrated in Fig. 42. It consists of blocks

for training sequence generation, QPSK mapping, pulse shaping put together known as

QPSK modulation and carrier modulation block.

Fig. 42: The transmitter

5.2.1 Training Sequence

The training sequence generator generates a known data sequence which is transmitted

prior to any data transmission. Its purpose is to provide the receiver with a known

sequence, which can be used for phase estimation and synchronization. The training

sequence is multiplexed with the data sequence before QPSK modulation as shown in

Fig. 42. In this report, multiplexing is done such that the whole training sequence is

transmitted before the data sequence, but any other scheme can also be used. Keeping

the training sequence in the middle of the data, i.e. half the data bits followed by

training sequence, followed by other half of the data bits, is another common scheme.

Training sequence carry no information and it is therefore to be seen as “useless”

overhead. A shorter training sequence is preferred from a overhead point of view, while

a longer one usually results in better performance of the synchronization and phase

estimation algorithms in the receiver. The length of the training sequence in general is

not fixed any where; it depends on the receiver design and modulation scheme which is

used. Later in this report performance results are shown for shorter and longer train

sequences.

Traning sequence

QPSK mapping

Pulse shaping

Carrier modulation

QPSK modulation

Train

Data transmitted

53

5.2.2 QPSK mapping

The bits are mapped onto corresponding QPSK symbols using Gray coding, as shown in

Fig. 43. Each QPSK symbol is represented by I Qd jd+ , corresponding to real-valued I

and Q channels, respectively. This is covered completely in sec. 4.4.

Fig. 43: Mapping of bits into QPSK symbols

5.2.3 Pulse shaping

The resulting QPSK symbols are passed through pulse shaping filter. Often a

rectangular pulse shape is used in simulations, although a root raised cosine pulse is

common choice in a real system. Here, we have used a root raised cosine pulse shape.

The complete description of the RRC pulse shaper is given in sec. 4.5.

5.2.4 Carrier modulation

Subsequent to pulse shaping is carrier modulation, taking the complex valued pulse

shaped QPSK symbols in the baseband, shifting them in frequency finishes the process

of carrier modulation. Which carrier frequency that one has to choose depends upon the

channel. Underwater acoustic channel is a low frequency channel and here, the carrier

that is chosen is in the range of 20-30 KHz. But, the carrier frequency is almost always

subsequently higher than the baseband frequency determined by symbol rate.

5.3 Channel

The complete description of the channel can be understood from chapters. 1, 2 and 3.

Nevertheless, a brief summary of it is again provided here. The main problems of this

channel are its multipath propagation, thereby a cause of interference. And next are

1,0 0,0

1,1 0,1

Id

Qd

54

channel variations, variations in physical parameters of the ocean such as temperature,

pH, salinity, pressure or depth of water. All these are extensively discussed in the above

chapters mentioned. This report considers almost all the parameters into consideration

while modeling the channel. The simulation block diagram of the channel can be found

in Appendix of the report. Fig. 44 represents the Underwater acoustic channel model

used in this simulation. ( )1h t , represents the direct path or first ray with zero delay

(relative) and ( )Nh t represents the Nth ray with a delay of Nτ with respect to the direct

path.

Fig. 44: Underwater Acoustic Channel Model

5.4 Receiver

The receiver design in any communication system is usually complicated compared to

transmitter and channel design. But, here as the channel is considered extensively the

complexity of the receiver design appears to be a bit reduced compared to channel

design. Fig. 45 depicts the receiver block diagram used in this report. The following

sections explain the functionality of each and every block used here.

Rx signal

⊕ ⊕Tx signal

Noise

( )1h t

( )nh t

( )Nh t

( )h t

55

5.4.1 Bandpass Filtering

The first block in the receiver is a bandpass filter with center frequency equal to the

carrier frequency cf and a bandwidth matching the bandwidth of the transmitted signal.

The purpose of the bandpass filter is to remove out-of-band noise. Choosing bandwidth

of the bandpass filter should be taken some care. If the bandwidth is chosen too large,

more noise enters than necessary will pass on to the subsequent stages. On the other

hand, if it is too narrow, the desired signal is distorted.

Fig. 45: The receiver

Decision

BP

Phase estimation

Down conversion

Front end

Matched filter

Phase estimation

Sync.

Phase correction

Training

56

5.4.2 Down conversion and Sampling

The down conversion block down-converts the received bandpass signal resulting into

complex valued baseband signal. In down-conversion operation the input signal is

multiplied with the local oscillator signal. Here a local oscillator is not used and the

same carrier frequency is assumed in both receiver and transmitter. So, there won’t be

any affect in terms of carrier frequency. One aspect of the local oscillator signal in the

down-conversion block is how to set the initial phase. In Fig. 45, a connection from the

optional phase estimation block to the down-conversion block is shown with dashed

lines and phase estimate obtained from the phase estimator is used for the initial phase

of the local oscillator. Another approach, which is common in practice, is not to lock the

phase of the local oscillator, but instead to do a phase compensation of the baseband

signal. This is done in our case after the matched filtering, where phase compensation is

simply a rotation of the signal constellation. The latter approach is shown with solid

lines in Fig. 45.

5.4.3 Matched Filtering

The matched filtering block contains a filter matched to the transmitted pulse shape. The

matched filter operation can be done on a discrete time signal or a continuous time

signal. The two possibilities are equivalent, but from the implementation point of view,

operating on the discrete time signal is to prefer.

In case of a rectangular pulse-shape, the matched filter is an integrate-and-dump filter.

In Fig. 46, the output signal from the matched filter (either I or Q channel) is shown for

the case of rectangular pulse shapes. The black dots represent the sampled signal in the

receiver, assuming four samples per symbol. The optimal sampling instants are

illustrated with small arrows. In the figure, the sampling of the matched filter happens

to be at one of the samples of the discrete signal, but this is typically not the case. If the

matched filter is to be sampled between two solid dots, interpolation can be used to find

the value between two samples, or, simpler but with loss in performance the closest

sample can be chosen.

57

Fig. 46: Output from the matched filter for successive signaling in absence of noise.

The solid line is the resulting output signal from the matched filters and the dotted lines

are the contributions from the first two bits (the remaining bits each has a similar

contribution, but this is not shown). The small arrows illustrate the preferred sampling

instants.

5.4.4 Synchronization

The synchronization algorithm is crucial for the operation of the system. Its task is to

find the best sampling time for the sampling device. Ideally, the matched filter should

be implemented such that the signal to noise ratio for the decision variable is

maximized. For a rectangular pulse shape, the best sampling time sampt is at the peak of

the triangles coming out from the matched filter, illustrated with small arrows in

Fig. 46.

The synchronization algorithm used in this report is based on the complex training

sequence. During the training sequence, it is known to the receiver what the transmitter

is transmitting. Hence, one possible way of recovering the symbol timing is to cross-

correlate the complex valued samples after the matched filter with locally generated

time-shifted replica of the training sequence. By trying different time-shifts in steps of

T Q , where Q is the number of samples per symbol, the symbol timing can be found

with a resolution of T Q . Keeping it mathematical terms, if ( ){ } 1

0

L

nc n

−

= is the locally

generated symbol-spaced replica of the QPSK training sequence of length L and ( )r n

denotes the output from the matched filter, the timing can be found as

sampt sampt T+ samp 6t T+

1+ 1+ 1+ 1+1− 1− 1−

t

58

( ) ( )1

samp samp0

arg max .L

kt r kQ t c k

− ∗

=

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑ (5.1)

190 200 210 220 230 2400

20

40

60

80

100

120

140

Fig. 47: Example of cross-correlating the received sequence with the

training sequence in order to find the timing. In this example, the delay was estimated to 211 samples (corresponding to the

maximum) and, hence, the matched filter should be sampled at 211, 211 + Q , …. in

order to recover the QPSK symbols. The correlation properties of the training sequence

are important as they affect the estimation accuracy. Ideally the autocorrelation function

for the training sequence should be equal to a delta pulse, i.e. zero correlation

everywhere except at lag zero. Therefore, a training sequence should be carefully

designed.

5.4.5 Sampling

The output from the matched filter is down-sampled with a sampling rate of 1 Q , i.e.

every thQ symbol in the output sequence is kept. The position for the sample (illustrated

by arrows in Fig. 46 controlled by the synchronization device previously described.

5.4.6 Phase Estimation

The phase estimator estimates the phase of the transmitted signal, which is necessary to

know in order to demodulate the signal. Phase estimation especially in the low SNR

region, is a hard problem and several different techniques are available. The phase

59

estimation algorithm used in this report is as follows. Using a complex baseband

representation, the sub samples of the matched filter output is a sequence of the form

( ) ( ) ( ) ( )1 1 2..., , , , ,...,n n n nj j j je e e eϕ ϕ ϕ ϕ− + +Φ + Φ + Φ + Φ + (5.2) where { }4, 3 4n π πΦ ∈ ± ± is the information bearing phase of the thn symbol and ϕ

is the unknown phase offset caused by the channel. If nΦ is known, which is the case

during the training sequence, the receiver can easily remove the influence from the

information in each received symbol by element-wise multiplication with complex

conjugate of a QPSK modulated training sequence replica, generated by the receiver.

The value of ϕ can then easily obtained by averaging over the sequence. In other

words, if ( ){ } 1

0

L

nr n

−

= denotes the L received QPSK symbols (i.e. received signal after

down sampling) during the training sequence and ( ){ } 1

0

L

nc n

−

= is the local replica of the

complex training sequence, an estimate of the unknown phase offset can be obtained as

( ) ( )( )1

0

1ˆ argL

kr k c k

Lϕ

−∗

=

= ∑ . (5.3)

The longer the training sequence, the better the phase estimate as the influence from

noise decreases. A longer training sequence, on the other hand, reduces the amount of

payload that can be transmitted during a given time.

5.4.7 Decision

The decision device is a threshold device comparing the I and Q channels, respectively,

with threshold zero. If the decision variable is larger than zero, a logical “0” is decided

and if its less than zero a logical “1” is decided.

60

6 Observations and Results

This chapter presents you with some exemplary simulation results along with some

interesting observations. First we look into the Underwater Acoustic Channel then we

cover the Communication part of the system.

As discussed in the above chapters 2 and 3, the major impact in an underwater acoustic

channel would be its multipath propagation. Always our desired goal is to achieve high

data rates at a decent geometry of the transmitter and receiver (low BER is implied).

Here, the term geometry means the physical positioning of a transmitter and receiver in

an underwater acoustic channel of depth D and infinite length. At shorter distances the

multipath reaches the receiver at a much longer time compared to the direct path. This

statement may appear some what contrasting to what we think. But, it certainly makes

sense when we look into it in a deeper view. Here, we are not speaking about the time

taken for each ray to reach the receiver. But, instead we are referring to the “Relative

time” of all the other rays comparing to direct path.

Fig. 48 presents the simulation results for a particular environmental scenario varying

the receiver location. This figure explains the impact of distances, (indirectly its grazing

angles which play a major role) on time delays of multipath propagation for the

following environmental scenario. Here, the wind speed and bottom type are not

included as we are representing only the time delay concept without any transmission

loss phenomenon included.

Environmental Scenario (for Fig. 48)

Source location ( ) ( ), 0, 20S Sr z = m

Receiver locations ( ) ( )1 1, x,20R Rr z = m, x 10,100,500,1000=

Sound velocity c = 1500 m/s Water depth 40D = m Salinity 35S = ppt Water temperature T = 14 °C pH value pH = 8

61

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

Number of ray

Relative travel time [s]

Am

plitu

de

a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

Number of ray

Relative travel time [s]

Am

plitu

de

b)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

Number of ray

Relative travel time [s]

Am

plitu

de

c)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21

2

3

4

5

6

7

8

0

0.2

0.4

0.6

0.8

1

Number of ray

Relative travel time [s]

Am

plitu

de

d)

Fig. 48: Simulation results showing relative travel times for various receiver locations of a sinc-pulse

without including any transmission loss phenomenon.

The relative times of all the 8 rays comparing to direct and the grazing angles for each

case are provided in the following.

a) ( ) ( )1 1, 10,20R Rr z =

T = [0 20.8207 20.8207 47.0817 47.0817 73.6106 73.6106 100.2081]

Angles = [0 75.9638 75.9638 82.8750 82.8750 85.2364 85.2364 86.4237]

b) ( ) ( )2 2, 100,20R Rr z =

T = [0 5.1355 5.1355 18.7083 18.7083 37.4700 37.4700 59.1197]

Angles = [0 21.8014 21.8014 38.6598 38.6598 50.1944 50.1944 57.9946]

62

c) ( ) ( )3 3, 500,20R Rr z =

T = [0 1.0650 1.0650 4.2397 4.2397 9.4656 9.4656 16.6508]

Angles = [0 4.5739 4.5739 9.0903 9.0903 13.4957 13.4957 17.7447]

d) ( ) ( )4 4, 1000,20R Rr z =

T = [0 0.5331 0.5331 2.1299 2.1299 4.7828 4.7828 8.4794]

Angles = [0 2.2906 2.2906 4.5739 4.5739 6.8428 6.8428 9.0903]

There is a huge difference in relative travel times for very shorter distances of 10 m,

case (a), compared to a desirable range of 1000 m, case (d). This can be understood

when we observe the corresponding grazing angles for each case. In case (a), the

grazing angles are very high due to shorter distances [Refer. Eq’s. (3.29) - (3.32)]

where, as in case (d) you observe very low grazing angles. Another observation is the

same, relative travel times and grazing angles for rays hitting surface or bottom,

surface-bottom-surface or bottom-surface-bottom, etc. This is due to the location of

both transmitter and receiver at exactly half of channels depth.

Here, we did not show any impact of Reflection loss and Spreading loss, only the time

delay concept has been focused. From now on, we refer mainly to grazing angles to

explain the behaviour of the system as the distance or lengths of each ray are included

when you calculate the grazing angle. The following simulation results are exclusively

presented to show the impact of transmission loss (including time delays) on multipath

propagation at various vertical depths of transmitter and receiver along with various

horizontal distances. Changing the vertical depths means, placing the receiver not

exactly at the half of the water channel depths but instead placing it, either nearer to the

bottom or nearer to the surface to look what exactly is happening for the bottom and

surface reflection coefficients.

When the separation between them is 10-200 m you don’t have that much impact of

multi-path propagation and thereby, receiver design complexity is much reduced. But,

in practical applications the distance between the transmitter and receiver is generally

desired beyond 500 m. So, all our simulation results are presented considering a 1000 m

separation between the transmitter and receiver. But, nevertheless, we do present some

simulation results when the receiver is at smaller distances from the transmitter.

Another criterion which is considered in the following simulation results is the change

63

of transmitter and receiver vertical depths. These are presented only to show the impact

of the distances between transmitter and receiver. For example, when we think of the

water channel depths to be 40 m and if the receiver is located at depths of 35 m, it

signifies that there would be certainly a more amount reflection of the signal from

Surface compared to Bottom. We always mean relative time delays with respect to the

direct path when we speak about time delays.

Environmental Scenario (for Fig. 49)

Source location ( ) ( )1

, 0, yS Sr z = m, y 10,35=

Receiver locations ( ) ( )1 1, x, yR Rr z = m, x 200,1000=

Sound velocity c = 1500 m/s

Water depth 40D = m

Salinity 35S = ppt

Water temperature T = 14 °C

pH value pH = 8

Wind speed 8wv =

Bottom type bt = coarse silt

Fig. 49 presents, simulation results including the transmission loss phenomenon for two

transmitters and receivers at different locations. First we look into Fig. 49a and 49b.

This for the case where the receiver is placed at a shorter distance of 200 m and vertical

depths of transmitter and receiver are swapped between 10 and 35 m. The complete

environmental scenario that has been chosen is given above. As said above, here, we

observe only the direct path and the multi-paths are completely suppressed. This is due

to the lower reflection coefficients at higher grazing angles and thereby, the

transmission loss of each ray becomes quite negligible. The transmission loss considers

the number of reflections (in turn reflection coefficients) when a ray hits the boundaries

along with the spreading loss (1 L ). So, the direct path will never have any reflection

loss. Apart from the direct path we observe the 3rd ray in both the cases (a) and (b), but

not the 2nd ray. This is due to zero reflection of the 2nd ray when it hits the surface.

1R – Surface reflection coefficient

2R – Bottom reflection coefficient

64

a) ( ) ( ) ( ) ( )1 1 1 1, 0,10 and , 200,35S S R Rr z r z= =

Angles = [7.1250 12.6804 9.9262 15.3763 27.6995 32.0054 29.8989 34.0193]

R1 = [0.2766 0.0179 0.0835 0.0028 0.0000 0.0000 0.0000 0.0000]

R2 = [0.2002 0.0591 0.1084 0.0342 0.0427 0.0481 0.0457 0.0501]

b) ( ) ( ) ( ) ( )2 2 2 2, 0,35 and , 200,10S S R Rr z r z= =

Angles = [-7.1250 12.6804 9.9262 27.6995 15.3763 32.0054 29.8989 42.7688] R1 = [0.2766 0.0179 0.0835 0.0000 0.0028 0.0000 0.0000 0.0000]

R2 = [0.9772 0.0591 0.1084 0.0427 0.0342 0.0481 0.0457 0.0559]

0

0.05

0.1

0.15

0.2

0

2

4

6

80

1

2

3

4

5

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

a) ( ) ( ) ( ) ( )

1 1 1 1, 0,10 and , 200,35S S R Rr z r z= =

0

0.05

0.1

0.15

0.2

0

2

4

6

80

1

2

3

4

5

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

b) ( ) ( ) ( ) ( )

2 2 2 2, 0,35 and , 200,10S S R Rr z r z= =

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.2

0.4

0.6

0.8

1

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

c) ( ) ( ) ( ) ( )

3 3 3 3, 0,10 and , 1000,35S S R Rr z r z= =

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.2

0.4

0.6

0.8

1

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

d) ( ) ( ) ( ) ( )

4 4 4 4, 0,35 and , 1000,10S S R Rr z r z= =

Fig. 49: Simulation results showing relative travel times for various transmitter and receiver locations of a sinc-pulse including the transmission loss phenomenon.

Coming to Fig. 49c and 49d, we certainly see the impact of multipath growing to greater

extent as the separation between transmitter and receiver is more, i.e. 1000 m. In Fig.

49c, the 4th ray hits the surface 2 times and then bottom one time, i.e. S-B-S and the 5th

ray hits the surface one time and 2 times the bottom, i.e. B-S-B. From the following

results, it is observed that the 4th ray grazes at an angle of 3.1481° and 5th ray with

65

5.9941°. This leads to lower reflection coefficients for 5th ray compared to 4th ray.

Similarly, in Fig. 49d, the 5th ray hits the surface one time and 2 times the bottom, i.e.

B-S-B and the 4th ray hits the surface 2 times and then bottom one time, i.e. S-B-S. From

the results provided in d), it is observed that the 5th ray grazes at an angle of 3.1481° and

4th ray with 5.9941°. This leads to lower reflection coefficients for 4th ray compared to

5th ray.

Here, we see another interesting observation, i.e. in Fig. 49c, the 4th ray has larger

amplitude compared to 5th ray and exactly the opposite is seen in Fig. 49d. This is due to

the swapping of the vertical placements of transmitter and receiver from 10 to 35 m.

The simulation results for grazing angles and reflection coefficients:

c) ( ) ( ) ( ) ( )3 3 3 3, 0,10 and , 1000,35S S R Rr z r z= =

Angles = [1.4321 2.5766 2.0045 3.1481 5.9941 7.1250 6.5602 7.6884]

R1 = [0.9492 0.8447 0.9028 0.7773 0.4021 0.2766 0.3361 0.2242]

R2 = [0.7191 0.5533 0.6306 0.4858 0.2568 0.2002 0.2266 0.1769]

d) ( ) ( ) ( ) ( )4 4 4 4, 0,35 and , 1000,10S S R Rr z r z= =

Angles = [-1.4321 2.5766 2.0045 5.9941 3.1481 7.1250 6.5602 10.4812]

R1 = [0.9492 0.8447 0.9028 0.4021 0.7773 0.2766 0.3361 0.0630]

R2 = [0.9772 0.5533 0.6306 0.2568 0.4858 0.2002 0.2266 0.0960] Fig. 50 represents another simulation result to just show the impact of multi-path at a

little bit lower wind speed of 6 knots and with a bottom type value of 4.

Environmental Scenario (for Fig. 50)

Source location ( ) ( )1

, 0, xS Sr z = m, x 10,35=

Receiver locations ( ) ( )1 1, 1000, xR Rr z = m

Sound velocity c = 1500 m/s

Water depth 40D = m

Salinity 35S = ppt

Water temperature T = 14 °C

pH value pH = 8

Wind speed 6wv =

Bottom type bt = very fine sand

66

Fig. 50: Simulation results showing relative travel times for two different vertical depths of transmitter

and receiver of a sinc-pulse including the transmission loss phenomenon.

By now it is understood, multi-path dominates when the separation between the

transmitter and receiver increases and also varies with the vertical positions of the

transmitter and receiver. Added to this when you have a lower wind speed and a soft

bottom, it becomes more worse. The difference can be clearly observed from Fig. 49c,d

and Fig. 50. Here also the same behavior of amplitude difference is observed for 4th,

5th, 6th, 7th, etc as the geometry is different for both cases.

Finally, you have the case of a constructive interference and destructive interference of

the multipath. When the multi-path gets added to the direct path in accordance with its

phase then we have a constructive interference otherwise a destructive one. So,

sometimes even if the multi-path is not dominative, you may still have a poor BER.

As we have discussed till now the multipath propagation in underwater acoustic channel

and all the channel effects, now we move to communications part of the system. Always

in communications the desired goal is to achieve maximum signal to noise ratio. In

underwater acoustic channel the noise is in two forms, one is the ambient noise

discussed in chapter 2 and the other is the multipath itself. We can also say, here the

signal itself acts as a noise as the multipath is nothing but (delayed versions of direct

path) generated from our signal only. So, here when ever we refer to SNR we imply that

it the ratio between the signal strengths of the direct path and multipath. The following

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.2

0.4

0.6

0.8

1

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

a) ( ) ( ) ( ) ( )1 1 1 1, 0,35 and , 1000,10S S R Rr z r z= =

0

0.05

0.1

0.15

0.2

0

2

4

6

80

0.2

0.4

0.6

0.8

1

x 10-3

Relative travel time [s]Number of ray

Am

plitu

de

b) ( ) ( ) ( ) ( )1 1 1 1, 0,10 and , 1000,35S S R Rr z r z= =

67

are some simulation results which show the Bit Error Ratio for only direct path and

multi-path for 2 different wind speeds and bottom types.

DIRECT PATH

1. Environmental Scenario (for Fig. 51)

Source location ( ) ( ), 0,10S Sr z = m

Receiver locations ( ) ( ), 1000,35R Rr z = m

Sound velocity c = 1500 m/s

Water depth 40D = m

Salinity 35S = ppt

Water temperature T = 14 °C

pH value pH = 8

Wind speed 6wv = knots

Bottom type bt = coarse silt

Fig. 51 represents the BER plot only for direct path. As one can imagine, when we

transmit only the direct path, there will not be any noise present only you have

attenuation of the signal strength. So, the BER of direct path is 0.

1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Eb/No Values

Bit

Erro

r Rat

e

Fig. 51: BER plot direct-path for the above Environmental scenario 1.

68

In the following two cases of multi-path propagation have been considered. One is at a

mud bottom type and lower wind speeds and the other is a bit higher wind speed and

sand bottom type. In case 1, the BER is much higher compared to case 2 as expected.

This is due to much reflections at lower wind speeds and softer bottom types.

MULTI-PATH PROPAGATION

Case 1

1. Environmental Scenario (for Fig. 52)

Source location ( ) ( ), 0,10S Sr z = m

Receiver locations ( ) ( ), 1000,35R Rr z = m

Sound velocity c = 1500 m/s

Water depth 40D = m

Salinity 35S = ppt

Water temperature T = 14 °C

pH value pH = 8

Wind speed 6wv = knots

Bottom type bt = coarse silt

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Eb/No Values

Bit

Erro

r Rat

e

1 1.5 2 2.5 3 3.5 410

-3

10-2

10-1

100

Eb/No Values

Bit

Erro

r Rat

e

Fig. 52: BER plots multi-path propagation for the above Environmental scenario 2, case 1 a) linear scale

b) log scale.

69

Case 2 2. Environmental Scenario (for Fig. 53)

Source location ( ) ( ), 0,10S Sr z = m

Receiver locations ( ) ( ), 1000,35R Rr z = m

Sound velocity c = 1500 m/s

Water depth 40D = m

Salinity 35S = ppt

Water temperature T = 14 °C

pH value pH = 8

Wind speed 8wv = knots

Bottom type bt = very fine sand

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Eb/No Values

Bit

Erro

r Rat

e

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

-4

10-3

10-2

10-1

Fig. 53: BER plots multi-path propagation for the above Environmental scenario 2, case 2 a) linear scal

b) log scale. From Fig’s. 52 and 53 it can observed that when you have higher wind speeds and

rough bottom types, the strengths of all the rays constituting multi-path propagation is

getting minimized. In these type of situations the communication aspect would become

easy compared to the acoustic channel. But, in practical applications, lower wind speeds

are present and thereby, making the communications design more hard. So, we should

always keep the range of wind speeds between 0-20 knots for our desired underwater

acoustic applications and the communication system should be designed robust even at

lower wind speeds.

70

Constellation Diagrams From the following constellation diagrams, you can see an error free propagation for

direct path, Fig. 54 and errors for multi-path, Fig. 55

Fig. 54: Received QPSK states for direct path

Fig. 55: Received QPSK states for multi path

71

7 Summary and Concluding Remarks

Some underwater acoustic applications like simple status reports or transfer time-

position co-ordinates may require a bit rate of 100 1bits s−⋅ . But in several other

applications like seafloor mapping and in some military applications bit rates of several 1kbits s−⋅ are required due to the transfer of high size images. As an initial step to

explore systems for communication that have the potential of transferring data at rates

of multiple 1kbits s−⋅ over distances of several kilometres underwater, we have

developed this simulation tool.

This simulation tool is designed for communication using Quadrature Phase Shift

Keying (QPSK) modulation techniques in an Underwater Acoustic Channel (UAC). It

mainly consists of a transmitter, UAC and a receiver. It provides a thorough insight into

various problems that are encountered by underwater sound channel and also explains

the degradation of bit error rate (BER) due to channel variations and presence of

multipath propagation.

All the oceanographic acoustic fundamentals have been considered in depth while

modelling the UAC. QPSK modulation techniques have been employed for the

transmitter and receiver. This tool works with a very low BER for the direct path even

at higher bit rates and is also robust for all channel variations. In short we can

summarize the following about what this simulation model provides: • a thorough insight into the complexity of an underwater acoustic channel.

• the ability to design and analyse time invariant equalizers with sensitivity to

equalizer mismatch.

• gives the flexibility to change the carrier frequency.

This tool shows the practical poor BER for multi path propagation and it produces

satisfactory results in the bandwidths ranging 1-2 Kbps. The robustness of the system

for multipath propagation drastically decreases when the channel variations are getting

worse. The simulation tool developed here was for fixed transmitter and receiver

locations. As explained in this report, the presence of multipath causes an intersymbol

72

interference (ISI) that destroys the message, due to different travel times for different

rays. Depending on the particular sound underwater channel in question, the ISI can

involve, in tens or even hundreds of symbols. A solution for this problem might be to

employ an adaptive equalizer in the simulation tool (here, adaptive is used as we refer to

a moving transmitter and receiver). An equalizer can be viewed as an inverse filter to

the channel. But, nevertheless, in practical situations even the employment of an

equalizer would not solve the problem of transferring high bit rates. This can pose us to

think of employing modulation techniques like Orthogonal Frequency Division

Multiplexing (OFDM). So, our future outlook for the extension of this simulation tool

would be:

• Incorporation of moving transmitter and receiver.

• Model validation with measurements.

• Investigation of adaptive single input multiple output (SIMO) equalization.

• Application of orthogonal frequency division multiplex (OFDM) communication.

73

Appendix

The following Schematic diagram for simulation gives complete idea how the main

program is structured down into functions and then sub-functions. Later to the

schematic diagram, the complete Matlab code is provided according to each function as

stated in the diagram.

74

Fig. 56: Schematic diagram for Simulation

75

Simulation code – Main.m

76

Main.m (contd.)

77

Main.m (contd.)

78

tansmitter.m

79

tansmitter.m (contd.)

80

tansmitter.m (contd.)

81

root_raised_cosine.m

82

training_sequence.m

83

qpsk.m

84

random_data.m

85

underwater_acoustic_channel.m

86

underwater_acoustic_channel.m (contd.)

87

underwater_acoustic_channel.m (contd.)

88

channel.m

89

channel.m (contd.)

90

attenuation.m

91

attenuation.m (contd.)

92

loss.m

93

loss.m (contd.)

94

ambient_noise.m

95

SRC.m

96

SRC.m (contd.)

97

BRC.m

98

BRC.m (contd.)

99

BRC.m (contd.)

100

receiver.m

101

receiver.m (contd.)

102

receiver.m (contd.)

103

phase_estimation.m

104

detect.m

105

detect.m (contd.)

106

detect.m (contd.)

107

References

[1] F.B. Jensen, W.A. Kuperman, M.B. Porter and H. Schmidt, Computational Ocean

Acoustics (Springer- Verlag, New York, Inc., 2000).

[2] H.G. Urban, Handbook of Underwater Acoustic Engineering (STN Atlas Elektronik

GmbH, Bremen, 2002).

[3] H. Medwin and C.S. Clay, Fundamentals of Acoustical Oceanography (Academic Press,

San Diego, 1998).

[4] John. G. Proakis, Digital Communications, fourth edition (McGraw-Hill, NY, 2001).

[5] Johnny R. Johnson, Introduction to Digital Signal Processing (Prentice-Hall of India Pvt.

Ltd, New Delhi, 1996).

[6] L.M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of Ocean Acoustics (Springer-

Verlag, second edition).

[7] Simon Haykin, An introduction to Analog & Digital Communications (John Wiley &

Sons, Singapore, 1994).

[8] www.complextoreal.com

[9] http://literature.agilent.com

[10] http://www.kth.se

[11] 51-st Open Seminar on Acoustic Program, Gdansk 2004.