1 Introduction A 3 model The Y-chart model Application description Preliminary specfications... 10

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1 Contents I Introduction 1 1 Introduction A 3 model The Y-chart model II Application 7 2 Application description Preliminary specfications Introduction to sonar The brief history The general principle Sonar theory The speed of sound The Sonar equation Transmission Loss Target Strength Noise Level Transmitted pulse Sampling frequency Pulse type Continuous-wave pulse Coded pulse Received pulse Continuous-wave pulse Linear chirp Application Sonar system Preprocessing Matched Filter Inverse filter Filter comparison Project scope Project delimitation Problem statement

2 CONTENTS III Algorithm 35 7 Introduction 37 8 Matched filter Time domain convolution Frequency domain convolution Implementation choice Detector algorithm Description Normaliser Detector Nonlinearity Detector algorithm Implementation Normalizer Estimation of SNR SST-Page test algorithm Simulation IV Architecture Introduction Hardware mapping Cost function Graphical representation of the Algorithm Scheduling Finite-state machine Finite-state machine with datapath Matched filter 59 Bibliography 63 V Appendix 65.1 Basebanding Worksheet discussing target tracking CONTENTS

3 Part I Introduction

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5 Introduction 1 Something is missing here A 3 model The A 3 model, also known as the A-cube model, is the basic design model used in this project to explore and thereby find a suitable and efficient solution to the problem. The A-cube model is a design model developed and used in a wide range of projects at Aalborg University. The model build upon three general domains: Application, Algorithm and Architecture, thereby yielding the three A s. The three domains is usually illustrated as on figure 1.1. Figure 1.1: The A-cube model. Solid lines refers to the multiple solutions. Dashed lines refers to the iterative processes. The model takes the Application domain as a starting point. The Application domain explores and analyses the different parts of the application. At first the problem is the subject of analysis, thereby deriving at a problem statement. The further analysis will explore the different aspects of the theory behind the problem. This will in most cases give several results, as there can be more than one approach to the solving the problem. One application is chosen and a requirement specification is derived. To meet this requirement specification there probably exist several algorithms in the Algorithm domain which has to be analysed in order to find the most suitable one. The design space of each algorithm has to be explored, as to decide if the algorithm is the most suitable one. This exploration is a iterative process, that bounces back and forth between the Application and Algorithm domains. The process may even lead to changes of the chosen application and thereby the requirement specification. In the end the most suitable algorithm is found and this can be taken to the Architecture domain. As in the previous two domains, the chosen algorithm can be mapped into a whole set of different implementations, which therefore gives rise to an exploration of the Architecture domain. This is as before an iterative process that analyses and compares the different architectures as to obtaining the most efficient solution. The process may as before lead to a change in the Algorithm domain and even all the way back to the Application domain. To end the circle the solution from the Architecture domain has to be tested in relation to the requirement specification. If the test is not completed with success, the design process has to be started all over again, probably with a revised requirement specification. The model is used in this project, as a result of this, the report is structured thereafter. The report is divided into parts where three of these is, Application, Algorithm and Architecture. This gives a structured overview of the 3

6 The Y-chart model design process and makes it easier to follow this. 1.2 The Y-chart model Another model which is specially developed for designing and implementing hardware is Gajski s Y-chart. The Y-chart is a model which works with different abstraction levels that can be used to model the hardware. In each of the abstraction level there are three design representations, or domains of the hardware, The behavior, structure and physical representation. The behavior representation describes what behavior or functionality the application is having at the abstraction level, the structure is showing which kind structure the application is working on, and the physical representations described how the application is physical implemented. In table 1.1 examples of the what could be in each of the domains, for each abstraction levels. Abstraction level High Representation Behavior Structure Physical Executable specifications (c, python) Processor, ASIC Medium (RTL) Flow chart, Instr set, FSMD Adders, Multipliers, Shifters, Busses Low Boolean equations, Differential equations Gate, Transistors, Flip-flops PCB FPGA, Discrete components Digital, analog Table 1.1: Table showing an example what could be in the different representations at different abstraction levels. The y-chart assume that all hardware design can be modeled using three domains, which emphasize different properties of the design. The three domains are the three representations described above. Each domain can be described in different abstraction levels. And for each abstraction levels there are generated a component of the design [Gajski et al., 2009]. On figure 1.2 the y-chart is shown: Figure 1.2: Y-chart model showing three design domains with three abstraction levels, high, medium and low. On figure it is seen that the after having specified the behavior of a design, synthesised it into a structure and 4 Introduction

7 The Y-chart model defined a physical implementation the model moves a abstraction level down. This movement in abstraction level can be performed when the design has been verified on the abstraction level. When moving down an abstraction level it is possible to elaborate further on the components in the design. For each abstraction level it is typically not necessary to go through all domains, for example it is not necessary to define the physical design on the high and medium abstraction levels. On these levels it can be enough to define a functional model defining important metrics such as performance, delay, cost, power etc. and then the design is represented as logic gates and flip-flops it is possible to perform layout and routing for these standard blocks [Gajski et al., 2009]. Introduction 5

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9 Part II Application

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11 Application description 2 As described in the A-cube model in section 1.1, the first step of the design process is to describe and thereafter analyse the application. The following chapter will describe the application and introduce a simple set of specifications, which has to be carried out in the final product. What is introduced in this project, is a fish finding sonar device, which will help the fishermen find fish. Fish finding products does already exist for use on boats or as handheld devices. These work in most cases with a single transmitter and receiver pointed directly at the bottom from the boat, thereby only being able to show in what depth there are fish and not the exact position of them. To get a picture of the horizontal plane where the fishes are, the sensor needs to be moved by the fisherman. This is however not always a desirable behaviour as in the case where the fisherman stands at the side of a lake and not in a boat. A final product should have the following features to overcome the limitations of existing devices: Locating fish in three dimensions By locating fish in a three dimensions it is possible to give an exact location and not only determining a distance to the fish. In three dimensions it is possible to determine both a direction, distance and a depth of the fish. Indication of velocity and direction To make it easy for the fisherman to guess where the fishes are heading, the device should facilitate velocity and direction measurements. Estimation of size The size of the fish should be estimated by the device and thereby indicate the interesting targets for the fisherman. User portability The device will be a simple set up with a single sensor buoy, containing all transducers, connected to a handheld processing unit. This makes it easy for the fisherman to deploy the device and start fishing. Handheld As the device will be made for fishermen it should be possible to move the system, thereby demanding the device not be heavy or unwieldy. Graphical output The output of the device should be a simple graphical representation of the surrounding environment, e.g. a sketch of the shore and where the fish are in the relation to this. This will however not be the features of the prototype developed in this project. The devices normal operation environment is in the shoreline of e.g. lakes, fjords or seas, thereby locating fish in the surrounding water of the device. The principle is sketched on figure

12 Preliminary specfications Sensor buoy Fish Finder Fisher Figure 2.1: General principle of device developed in the project. As can be seen on the figure 2.1, the device will be made as a sensor buoy which will float in the shoreline. The sensor buoy will only contain the needed transducers and the necessary amplifiers. The sensor buoy will be connected to a processing unit which will be placed ashore. The processing unit will contain all the needed hardware and software for signal processing. A graphical display on the processing unit will make information from the sensors available to the fisherman in a meaning full way, e.g. in terms of coordinates or similar representations. 2.1 Preliminary specfications The preliminary specfications is introduced to define some of the standards the prototype developed in this project has to overcome. The specifications of the prototype has to be both realisable and usable, as they will describe some of the physical restrictions. The prototype has to be able to: detect the location of an object in a lake. the object will be defined as a box with the dimensions 50 x 10 x 10 cm, which is the estimated size of a proper sized trout. the maximum distance to the object is defined to be 30 m from the transducers. update the fisherman with the location of the object in real-time. real-time is defined in this project, as a new update every quarter of a second. These preliminary specifications will be used in the further work in developing of a prototype, as the standards at which it has to follow. 10 Application description

13 Introduction to sonar 3 The typical technology for underwater navigation, communication and location of objects is Sonar, which is a contraction of the words Sound Navigation And Ranging. Sonar is a technology, much like radar, but is using the theories of underwater acoustics instead of electromagnetic-waves in air. Sonar uses the generation of pressure waves in water as the fundamental element. The following chapter will elaborate upon the history and general principle of Sonar, thereby giving a general introduction to the subject. 3.1 The brief history The first indication of a sonar-like device was proposed by Leonardo da Vinci in 1490, when he wrote: If you cause your ship to stop, and place the head of a long tube in the water and place the outer extremity to your ear, you will hear ships at a great distance from you. It should however take another 400 years before this principle was used in a more scientific matter. The development of the sonar technology mainly took part in the first half of the 20th century and especially during the first world war (WW1). One discovery that made it possible to use sonar was piezoelectricity, which was discovered by the Curie brothers in Piezoelectricity is when a electric charge is produced when applying a force to a material e.g. crystals or quarts. Piezoelectricity found it s first application in sonar, as a quartz crystal was glued between two steel plates to generate an electric charge as a pressure wave influenced the constellation. By transmitting a pressure wave into water, it was possible to measure the time for a reflected wave to hit the transducer. This measurement was of course only possible as the speed of sound in seawater was determined to 1500 and 1435 meters per second respectively by Colladon (1826) and Sturm (1827). The first patented system came in 1912 just months after RMS Titanic sunk after hitting an iceberg. The patent was on a system to detect and warn of nearby icebergs. The first patented and demonstrated system was established by Fessenden in The system later became known as the Fessenden oscillator. The Fessenden oscillation was able to detect an iceberg at three kilometre range, but it was unable to determine the bearing of the iceberg, due to it s low frequency of 500 Hz. Under WW1 both the British and French worked on refining and developing underwater detection systems to detect the growing German fleet of submarines. The British development was mainly on listening devices, passive sonar, the French was on echolocation devices, active sonar. At this time the name for the technology was not sonar, but ASDIC, an acronym for Anti-Submarine Division. A major breakthrough came from Paul Langevin in the years between 1915 and 1918, where he demonstrated how to transmit a signal and detect its reflection on a submarine. This was however to late to be used in the war. Later at WW2 the sonar, as it was called now, was further developed to detect mines and the like. The technology had also, at this time, found its way to civilian applications as determining water depth and even fishing. Today the sonar technology is used in many maritime applications and is found on almost every bigger boats and ships. This of cause while the sonar equipment and the associated computer systems is relatively cheap. Some of the applications is e.g. fish-finding, sea-floor mapping and handheld devices for diving purposes. In warfare applications, the sonar technology is still used to detect submarines, however countermeasures is invented to make submarines harder to find. Modern torpedoes and even mines is also equipped with sonar to make the hitting accuracy more efficient. 3.2 The general principle This project is mainly focused on the sonar application of finding and tracking objects in water. In general sonar can be divided in to two different techniques: active and passive. The principle of a active sonar is sketched on 11

14 The general principle figure 3.1. Reflected wave Transmitter/ Receiver Object Original wave Distance Figure 3.1: Principle of a active sonar distance measurement [Wiora, 2005]. Active sonar works in the way that a transmitter emits a sound wave and this wave reflects on objects in the water. The reflection can then be detected by the receiver and it is thereby possible to calculate direction and distance to the object. In passive sonar there is no transmitter, and the sonar is detecting the noise that objects emits. This could be sounds of movement, engine noise or something else that generates pressure waves. It will for instance not be possible to detect the sea-floor, rocks etc.. This is why most sonar devices is using active sonar to locate objects in water. 12 Introduction to sonar

15 Sonar theory 4 In order to make a signal model for the environment, it is necessary to have some basic knowledge of how sonar work. To be able to locate fish, the application will make use of the active sonar principle. Most of the sonar literature today focus solely on the sonar principles in the oceans. These principles are also applicable in a lake, but there are some conditions one need to be aware of. In this chapter the sonar theory needed for the application will be described. 4.1 The speed of sound In water sound travels as longitudinal waves. These waves are also called compression waves. A key parameter for sound is pressure. When the sound is moving it pushes or pulls water molecules apart, and this movement is felt locally as a pressure change. The peak pressure of the sound wave is the amplitude. The sound wave is described by its speed c and frequency f, related through equation 4.1 where λ is the wavelength c = λ f (4.1) In water the speed of sound can be seen as function of ambient temperature (T), ambient pressure (p), and the salinity (S) of the water [Hodges, 2010]. So c can be written as: c = F(T, p,s) (4.2) An example of the T parameter can be that a temperature increase by 1 C result in an increase of c with 4.5 m/s. Similar will an increase in both p and S also increase the sound speed, and a reduction will decrease the sound speed. But it is the temperature which has the biggest effect on c [Hodges, 2010]. An empirical version of equation 4.2 is shown in equation 4.3 c(t,d,s) = A 1 + A 2 T + A 3 T 2 + A 4 T 3 + A 5 (S 35) + A 6 D + A 7 D 2 + A 8 T (S 35) + A 9 T D 3 (4.3) Where the A constants are A 1 = A 4 = A 7 = A 2 = A 5 = A 8 = A 3 = A 6 = A 9 = And T is the temperature in C S is salinity in parts per thousand ( ) D is the depth in meters In equation 4.3 the depth has replaced pressure from equation 4.2. For T = 10 C, D = 0 m and S = 35, equation 4.3 gives c = 1490 m/s. 13

16 The Sonar equation Equation 4.3 is an example of an empirical equation to calculate the sound speed in water, other empirical equations to calculate the speed of sound in water based on T, D and S has also been proposed [Hodges, 2010]. Another important feature of sound is that the wave spead spherical. Because it spreads spherical the acoustic intensity, I, are inverse proportional with the range, R, squared. The relationship is shown in equation 4.4 Where I = W 4πR 2 (4.4) I is the acoustic intensity, meaning the amount of power per area unit. W is the power R is the range On figure 4.1 the spherical radiation is shown. Figure 4.1: Spehrical spread of sound. Taken from [Ainslie, 2010, p. 33] skal lige have kigget på at lave et 3d billed BK 4.2 The Sonar equation In the following section the sonar equation will be described. First a general introduction to the sonar equation will be given, and afterward the different parameters in the sonar equation will be described in details. When a sound pulse are transmitted from a transmitter into the water, there are several factors in the environment which will reduce the strength of the transmitted pulse, before it is reflected of a target. The reflected signal are also reduced by these factor before it is received by the hydrophone. The sonar equation is a way evaluate the sonar system against these environmental factors. Monostatic? John Wayne For an active monostatic sonar the sonar equation can be written as [Hodges, 2010]: Where E L = S L 2 T L + T S [db] (4.5) E L is the received echo level. 14 Sonar theory

17 The Sonar equation S L is the source level, which is the transmitted pulse strength T L is the transmission loss, which consists of damping from the fact that the sound wave propagates spherically, and some absorption in the water. T S is the target strength, it is the echo level from a target which is hit by the sound wave. In the environment there also are a level of noise which the echo level has to compete with [Hovem, 2007]. The received signal can the be expressed as a signal to noise ratio (SNR). In [Hodges, 2010] the SNR equation is given as: Where SNR = S L 2 T L AG N L (4.6) A G is the array gain of the receiving hydrophone array N L is the total noise level of environment In the sonar literature it is more common to write the SNR in db and thus equation 4.6 is written as: SNR = S L 2 T L + T S N L + A G [db] (4.7) In order for the sonar system to be able to detect or track a target the SNR is required have a certain size which is called the detection threshold (D T ). The detection threshold is defines as the SNR required for a 50 % probability of detection [Hodges, 2010]. The detection threshold is use to define a factor called the signal excess (S E ), which is the amount the SNR is higher than the detection threshold. The signal excess is given as: S E = SNR D T [db] And the signal excess is a parameter used to measure the performance of a sonar system. This concludes the general introduction to the sonar equation and in the following there will be elaborated further on the parameters transmission loss, target strength and noise level Transmission Loss The transmissions loss is defined in [Hodges, 2010] as a loss of intensity between a point of interest and a reference point, normally at one meter from the transducer array. The transmissions loss consist of two factors, a factor from geometrically spreading, and some absorption in the water. The geometrically factor comes from the fact the sound spreads spherical symmetrically in an uniform medium. The intensity of the energy on the surface area, can because of the assumption of energy conservation. The relationship between the intensity at a given range R compared to a reference point R 0 is the given as: The transmission loss is the given as [Hodges, 2010, p. 91] with R 0 = 1m I R 4π R 2 = I R0 4π R 2 0 (4.8) ( IR T L = 10log = 10log I R0 ) ( 4π R 2 ) 0 4π R 2 = 20log(R) [db] (4.9) Sonar theory 15

18 The Sonar equation The absorption loss of the sound wave depends on the frequency. Low frequency sounds travels longer up to thousands of kilometers while high frequency sounds are attenuated faster. In equation 4.10 the expression for the attenuation constant α water is shown. α water = α viscosity + α chemical [db/km] (4.10) skal det i ordlisten eller forklares her? BK Draw figure here? John Wayne Its seen that the attenuation depends on the viscosity of the water and the chemical relaxation * For frequencies above 1 MHz most of the attenuation is due to the waters viscosity. At lower frequencies up to 300 khz chemical relaxing are important [Ainslie, 2010]. The attenuations coefficients are gives as: [ ( T α viscosity = exp 27 + D )] f 2 [db/km] (4.11) α chemical = A B f 2 f 2 + f 2 B f 2 + A Mg f 2 + fmg 2 [db/km] (4.12) Where f is calculated in khz, and f B and f Mg are the relaxation frequencies of boric acid and magnesium sulfate. The relaxation frequencies depends on S and T, and are given as: The coefficients A B and A Mg are gives as: ( ) S 1/2 f b = (0.78 khz) e T /26 35 f Mg = (42 khz) e T /17 A B = f B K ( A Mg = T )( ) S f Mg e D/6000 where K are a variable based on the waters ph values and given as K = 10 ph w 7.85 The equations in this section are empirical equations and may therefore deviate from other empirical equations dealing with the same problems. The equations has been shown to agree with a more complex set of empirical equations describing the same set of problems [Ainslie, 2010]. A inspections of the equations used to calculate α water shows that the attenuation also is a function of the temperature, depth and salinity like in the case of the sound The total transmission loss can then be written as: Target Strength T L = 20log(R) + α 10 3 R (4.13) The target strength or acoustic cross-section area of a object is defined as the ratio between the intensity of the incident sound and the reflected sound measured at unit distance[balk, 2001]. For use in the sonar equation it is calculated as: 10log ( Ir I i ) (4.14) *chemical relaxation: McGraw-Hill Science and Technology Dictionary: (chemistry) The readjustment of a chemical system to a new equilibrium after the equilibrium of a chemical reaction is disturbed by a sudden change, particularly in an external parameter such as pressure or temperature. from: 16 Sonar theory

19 The Sonar equation where I r is the reflected acoustic intensity and I i is the incident acoustic intensity [Hodges, 2010,p. 168]. The single thing contributing most to the acoustic size of a fish is the swim bladder. The reason to this is that it contains air and therefore have a large difference in acoustic impedance compared to water. Experiments have shown [Hodges, 2010,p. 178] that the target strength for a fish can be estimated from the length as, T S( f ish) = 20log(L) 29.6 db (4.15) This equation holds only if the wave length of the transmitted signal is small compared to the size of the bladder. If the frequency of the transmitted pulse is not high enough compared to the size of the bladder, the target strength will be smaller than calculated. Because most fishes is not spherical the orientation of the fish compared to the incident wave also influence the target strength. Equation (4.15) are derived under the assumption that the swim bladder has a elliptic shape, and the length of the swim bladder are 1/12 of the fish lengths [Hodges, 2010]. According [Balk, 2001] to the radius of the swim bladder should have a radius 10 times larger than the fish s acoustic size. The acoustic size are defined as [Balk, 2001]: a = λ 2 π The frequency which is high enough so that equation 4.15 can now be calculated as: (4.16) ( ) λ 10 = L 2 2π 12 λ = L π 60 because λ = c f f = 60 c L π [Hz] (4.17) If the application should be able to find trouts with a length of 0.5 m, f can be calculated to be: f = pi 60 [khz] (4.18) so if the frequency of the application is chosen 60 khz it s assured that equation 4.15 holds Noise Level The noise level consist of the ambient noise and reverberation, where the application are placed. The ambient noise is introduced by marine life, shipping, ocean turbulence, rain, thermal and seismic noise. But in modern high power sonar the dominant factor in the noise level, are the reverberation [Hodges, 2010, p. 143], therefore there will only focused on the reverberation when describing the noise level. Reverberation is the received echoes, from when an active sonar transmit at pulse, which does not come from the target. The reverberation sources are the surface, the bottom and the volume of the water. The reverberation from the surface or the bottom, arises when the sound wave is reflected of one of these surface. Depending on the material the surface consist of not all of the energy are reflected out with the reflected wave, the remaining energy are either absorbed or scattered by the surface. On figure 4.2 the concept are shown. Sonar theory 17

20 The Sonar equation Incomming wave Reflected wave Scattered energy θi θr Surface Figure 4.2: Sound scattering from a surface The backscattering strength in the given as [Hodges, 2010, p. 144]: Er det muligt at sige noget mere om hvordan frekvens og reverberation fra alger eks. hænger sammen? MJ Where A is the area of the impact of the wave S(θ I,θ R ) is the angular distribution σ bs (θ I ) = A S(θ I,θ R ) cos(θ I ) (4.19) The volume scattering is a little different, here the sound scattering primary comes from marine life, but air bubbles near the surface can for frequencies above 10 khz also be a source for sound scattering. The volume scattering can therefore be very variable because it depends on amount of fish, algae etc, and the wind speed which makes the surface bubbles. The volume scattering can expressed similar as the surface scattering [Hodges, 2010, p. 144]. σ v (θ I ) = V S(θ I,θ R ) cos(θ I ) (4.20) When calculating the reverberation level it is also necessary to take the beam pattern of the transmitter and receiver into account. For a surface or bottom layer the reverberation level can be calculated as: [Hodges, 2010, p. 154] Where B t B r is the beam pattern for the transmitter is the beam pattern for the reciver I 0 is the transmitted source level [da] is the area element ( I0 R L = 10log r 4 S s ) B t (θ,φ)b r (θ,φ)da (4.21) If its assumed that the transmit and receive pattern are the same, then the integral becomes independent of the vertical angle φ and da becomes a planar scatter area, equation 4.21 reduces to. ( ) ( I0 R L = 10log r 4 S cτ 2π ) s 2 r + 10log B t (θ,φ)b r (θ,φ)dθ 0 (4.22) where τ is the pulse time On figure 4.3 a figure of the scatter area is shown. 18 Sonar theory

21 Transmitted pulse ) ~ { Figure 4.3: Scattering area for surface or bottom reverberation. The square is the transducer Hodges [2010] For a omnidirectional transducer the integral part of equation 4.22 would be 2π and the reverberation level can then be expressed as: ( cτ ) R L = S L 40log(r) + 10log S s 2 rθ [db] (4.23) Like fore the volume scattering the reverberation level for the volume scattering can be calculated in a similar way to the surface scattering. If the transmitter and receiver beam pattern are the same, the reverberation level is given as [Hodges, 2010, p. 156] ( ) ( I0 R LV = 10log r 4 S cτ 2π s 2 r2 + 10log and for an omnidirectional transducer it can be calculated as: where Ψ is the beam solid angle 0 π/2 π/2 ) B t (θ,φ)b r (θ,φ)cos(φ)dθ (4.24) ( cτ ) R Lv = S L 40log(r) + 10log S v 2 r2 Ψ [db] (4.25) The total noise level can the be calculated as the sum of the surface, bottom and volume reverberation. i (hodges p 156) er integraledelen i RL udregnet for nogle simple arrays BK 4.3 Transmitted pulse The following section will elaborate on different kinds of waveforms for the transmitted pulse. A general description about some of of the known problems in transmitting such pulses will also be touch upon. The transmitted signal in active sonar can be of many different types, but there are basic types: continuous-wave (CW), coded pulses (CP), pseudo random pulses (PRN), and explosive or impulsive pulses [Hodges, 2010,p. 330]. Pulses generated by detonating explosives is there not elaborated furthere on in this project. The choice of pulse depends on the information wanted about the target. By using CW it is possible to detect the Doppler shift in the signal reflected from the target and thereby estimate it s velocity. The drawback in using CW pulses with good Doppler resolution is that the pulse need to be long, leading to a relative poor range resolution. Coded pulses are frequency modulated pulses and facilitate good range resolution but are bad to estimate the Doppler shift. Codec pulses can also exploite pulse compression to make the SNR better when the noise on the signal is addictive and gaussian as will be explained lateron. PRN is done by transmitting a known broadband signal, thereby including both low range resolution and good Doppler detection capabilities. The use of PRN is however very difficult to implement in practice, while transducers will not work well and the algorithms will become very complex [Hodges, Sonar theory 19

22 Transmitted pulse 2010]. The following section is focused on CW and CP Sampling frequency The sampling frequency also influence the range resolution in the way that the time spacing between each sample corresponds to a spacing in distance. The transmitted signal gets modulated by the objects it hits as illustrated on figure carrier envelope Figure 4.4: Example of how a object modulate the carrier. nyt envelope signal - enheder/tekst på akserne MD As seen on the figure it will be enough to sample the envelope of the signal in order to detect the echo from an object leading to a needed sample rate that can be lower than two times the signal bandwidth. To accommodate the Nyquist sampling theorem for the example with a range resolution of 0.5 m leading to a pulse length of 667 µs the sample rate in distance should be at least = 0.25 m leading to a minimum sampling frequency of: f min = 1 T 12 = 1 ( ) = 3 khz (4.26) The Doppler shift can not be determined from the signal envelope so to find it, the sampling frequency need to be high enough to sample the whole signal and take the maximum possible positive Doppler shift into account Pulse type Whether the transmitted pulse is CW or CP, the pulse will be generated by a window convolved width a carrier to convert the final finite pulse signal. The window of choice can be many, but a simple rectangular window is chosen ret så det bliver rigtigt MJ for the further analysis. This will in frequency domain look as a sinc-function which is illustrated on figure Sonar theory

23 Transmitted pulse Magnitude /T -3/T -2/T -1/T f 0 1/T 2/T 3/T 4/T Frequency Figure 4.5: The frequency domain representation of a rectangular window function. The window function on figure 4.5 is centered around the carrier frequency f 0 and has side lobes which repeats as a function of the pulse period, T.It is important to notice that the 3 db bandwidth of such a function is [Wilkinson, 2010]: B 0.89 (4.27) T This window function is convolved by the frequency domain representation of the carrier signal, which will not be same for CW and CP. The difference will be clear as the following two sections will elaborate on the two waveforms Continuous-wave pulse The CW signal is represented by a sinusoidal signal for a curtain time period. This is illustrated on figure Amplitude Time [s] Figure 4.6: Example of the continuous-wave pulse with a frequency of 10 Hz stating at zero and stopping at one. The CW pulse can be written in time-domain as: { As cos(2π f v(t) = 0 t) if 0 t < T 0 Otherwise where A s is the amplitude of the transmitted pulse and in the case of figure 4.6 it is 1. f 0 is the frequency of the wave and T is the length of the pulse. The pulse has a value of A s for as long as t is less than T. This corresponds to a windows function times a sinusoidal with constant frequency, which will be a single tab in frequency domain convolved with the frequency response of a rectangular window, the sinc function on figure 4.5 will simply be Sonar theory 21

24 Transmitted pulse moved to be symmetric around the f 0 frequency. The CW pulse can be written in frequency-domain as [Wilkinson, 2010]: V ( f ) = T sin(πt ( f f 0)) (4.28) πt ( f f 0 ) As the frequency response of the CW pulse is just the response of a rectangular window, it will have a bandwidth of: B 0.89 (4.29) T Coded pulse The coded pulse, is here introduced as a linear chirp, which is a pulse that makes a sweep in a range of frequencies, as can be seen on figure Amplitude Time [s] Figure 4.7: A chirp pulse with f 0 = 10 and k = 30. The pulse has length one and starts from zero. The reason for doing the frequency sweep, is that the bandwidth could be controlled by changing the range of the sweep, thereby holding the pulse period constant. The chirp pulse is defined as: ( t ) ( v(t) = rect cos 2π[ f 0 t + 1 ) T 2 Kt2 ] (4.30) where the factor K is the chirp rate, and defined as: K = f T An important factor to introduce, is the time-bandwidth product or dispersion factor, which is defined as: (4.31) f 0 er ikke centerfrekvensen efter hvordan pulsen er defineret i eq 4.30 MJ D = f T = KT 2 (4.32) For a dispersion factor greater than 50, the passband of the Fourier spectrum of the pulse will start to look rectangular [Wehner, 1987], thereby approximating a bandwidth equal to f. If the dispersion factor is very low, the passband will become more flat, thereby introducing a reduction bandwidth [Wehner, 1987]. The passband will be symmetrical around the carrier frequency f 0. The sinc function on figure 4.5 will be convolved with the Fourier spectrum of the linear chirp pulse, which will consist of the sweep frequencies, why the resulting bandwidth will be wider than at the CW pulse. The chirp rate is controlling the range resolution indirectly. This gives the chirp pulse an advantage, as the range resolution can become better by still using the same pulse period. The Fourier transform of the chirp function: V ( f ) = F {v(t)} (4.33) is a very complicated function [Wehner, 1987, p. 123] which will not be shown here. 22 Sonar theory

25 Received pulse 4.4 Received pulse The following section will introduce the signal model, which will describe how the signal is affected when traveling through water Continuous-wave pulse As described consist the returning signal of echoes of the transmitted signal coming back attenuated by β and with a time delay of t r that depends on the distance to the object that generated the echo. With the assumption made earlier is the returning signal for the continuous-wave on this form: { βas cos(2π f r(t) = 0 t) + σ(t)v(t) if t r t < t r + T σ(t)v(t) Otherwise (4.34) Where β is the attenuation factor of the specific echo and A s is the amplitude of the transmitted signal. f 0 is the frequency of the transmitted pulse. σ(t) is a unit variance Gaussian process and v(t) is at time varying reverberation and noise power factor. When one have knowledge of the waveform for the transmitted and received pulse and the only noise present is addictive Gaussian noise the optimum method to maximize the SNR of the receive signal is by matched filtering it as described in On figure 4.8 is three echoes from a continuous-wave pulse shown before and after matched filtering. Amplitude Transmitted and received pulses Received echoes Transmitted pulse Skriv noget her... John Wayne Amplitude Time [s] Received pulses after matched filter Received echoes Time [s] Figure 4.8: Top subplot is a continuous-wave pulse with a duration of one second and a frequency of 5 Hz starting at zero. Three echoes of the transmitted pulse is present at time 3, 5 and 8 and they have an amplitude of 0.7, 0.3 and 0.1. On the bottom subplot are the same echoes after matched filtering. If the received signal consists of multiple echoes as if the case on figure 4.8 then we need to introduce the term range resolution. The range resolution of the transmitted signal determines the minimum distance the received echoes must be from each other in order to make it possible to resolve the them from. This means that the range resolution is the minimum distance two objects can be apart and still be resolved. This is sketched on figure 4.9 Sonar theory 23

26 Received pulse where it is apparent that if object 1 is to close to object 2 with respect to the pulse length, the two returning echoes will overlap each other and appear as a single echo. T 1 2 L1 L2 1 2 Received echo Time Figure 4.9: Two objects, 1 and 2, are placed in reference to the transducer, T, as shown on the top image. The returning echoes from two objects are shown on the bottom image. The range resolution is give as: rr = c 2B (4.35) and in terms of pulse time for the rectangular pulse which has a bandwidth of B = 0.89 T : rr = c T = ct 1.78 (4.36) The concept with echoes separated less than the range resolution is show on figure 4.8 where the echoes from a pulse with a length of one second is separated by 0.8 second. From the range resolution equation it is given that for a rectangular pulse of length one, the objects generating the echoes need to be separated by at least: T min = T 1.78 = 1 = 0.56 s (4.37) 1.78 in units of the pulse propagation speed. Out of this it is clear that the returning echoes should be separated by 2 T min in order to be resolved a to objects. In the case on figure 4.8 with the 0.8 s separated of the returning echoes it is also seen that it is not possible to distinct the two echoes from each other. 24 Sonar theory

27 Received pulse Amplitude Overlapping pulses Received echoes Transmitted pulse Amplitude Time [s] Overlapping pulses after matched filter Received echoes Time [s] Figure 4.10: Top subplot is a continuous-wave pulse with a duration of one second and a frequency of 5 Hz starting at zero and some echoes. The echoes has the same amplitude as the echoes on figure 4.8 but the two first now overlap each other because they are pleased at 3 and 3.8. The third echo is at 8. The bottom subplot is the same echoes after matched filtering and it is obvious that the two overlapping pulses cannot be resolved as independent echoes. From equation 4.35 it obvious that the best range resolution is obtained by having the widest bandwidth possible, which in theory can be obtained by transmitting a Dirac delta impulse. A reason for not just transmitting such a very short pulse, is that the signal-to-noise ratio will decrease with a shorter pulse from that the SNR is given as: SNR = β2 A 2 s T σ (4.38) where σ is the variance of the addictive noise. The pulse length should therefore be based on requirements for the minimum range resolution and SNR requirements. However for the coded pulse it is possible to improve the SNR without affecting the range resolution by the use of pulse compression as will be covered in the following under linear chirp Linear chirp The following will describe the received signal from the linear chirp pulse and how pulse compression can be used to improve the SNR while maintaining the same range resolution. As for the continuous-wave pulse the returning signal contains echoes of the transmitted signal: { βas cos ( 2π [ f r(t) = 0 t Kt2]) + σ(t)v(t) if t r t < t r + T σ(t)v(t) Otherwise (4.39) A signal like that described in equation 4.39 is visualized on figure First is seen the transmitted pulse plus echoes and second is the same pulses show plus addictive Gaussian noise. From the figure it is not possible to see Sonar theory 25

28 Received pulse where the echoes are located because of the noise but from the fact that there is knowledge of that was transmitted and the transmitted pulse was a linear chirp the signal to noise ratio can be improved via matched filtering. Amplitude Transmitted and received pulses Received echoes Transmitted pulse Amplitude Time [s] Pulses + unit variance Gaussian noise Received echoes Time [s] Figure 4.11: The top subplot "Transmitted and received pulses" show a chirp pulse with a period of one second swept from 5 Hz to 30 Hz starting at time zero. On the same figure is three receive echoes of this chirp at time 3, 5 and 8 second with amplitudes 0.7, 0.3 and 0.1. On the bottom subplot is these echoes embedded in unit variance Gaussian noise. When the linear chirp is matched filter the pulse is also compressed in time meaning that the energy in the pulse gets concentrated. As the noise is Gaussian there is no correlation between that and the transmitted pulse so the matched filter will not compress the noise. On figure 4.12 is the echoes from figure 4.11 showed after the matched filtering and it is clear that the echoes are compressed in time after the filtering and it is now easy to find the two first echoes embedded in the noise. It is still hard to tell where the third pulse that has a magnitude of 0.1. For this pulse the compression ratio is: D = f T = (30 5) 1 = 25 (4.40) Where f comes from that the pulse is swept from 5 to 30 Hz. As seen is the pulse compression ratio 25 which means that after the SNR is 25 times the SNR before the filtering enabling the detection of the hidden signals. In table is the SNR for the three pulses before and after the filtering listed. Echo position [s] Amplitude SNR [db] SNR after filter [db] Table 4.1: SNR before and after matched filter for three example echoes. The SNR s before filtering are calculated in the following way by the use of equation 4.38: 26 Sonar theory

29 Received pulse ( ) ( ) A2 T A 1 SNR = 10 log 10 = 20 log 10 = 20 log 10 (A) (4.41) σ 1 where A is the respective amplitudes and after the filtering as: SNR = 20 log 10 (A D) = 20 log 10 (A 25) (4.42) where D is the compression ratio. Squared magnitude Squared magnitude Received signal without noise after matched filter Received signal after matched filter Time [s] Received signal + noise after matched filter Received signal+noise after matched filter Time [s] Figure 4.12: The echoes from figure 4.11 matched filtered. The top subplot is the signals without noise and the bottom one i with the signals embedded in unit Gaussian noise. The compression ratio of 25 also means that the range resolution is improved with a factor 25 compared to that of the continuous-wave pulse. This fact is illustrated on figure 4.13 where the two first pulses are pleased at 3 and 3.1 but as seen is it no problem to resolve them after the matched filtering. Sonar theory 27

30 Received pulse Amplitude Transmitted and overlapping received echoes Received echoes Transmitted pulse Squared magnitude Time [s] Received overlapping echoes after matched filter Received signal after matched filter Time [s] Figure 4.13: Illustration of how it is possible to resolve the individual echoes even though the are overlapping before the matched filtering. The pulse are pleased at 3, 3.1 and 8 second and swept from 5 to 30 Hz over one second. The conclusion from this is that a chirp pulse combined with matched filter can improve the SNR and range resolution compared to a continuous-wave pulse of the same length Application The linear chirp pulse is chosen as the transmitting pulse, while the controllable range resolution is an advantage which is wanted. The range resolution is chosen to be 7.5cm which will introduce a frequency sweep of: f = c 2 δr = 1500 = 10kHz (4.43) 2 δ0.075 The pulse period can then be calculated as the dispersion factor is chosen to be 50: The chirp rate therefore becomes: T = D f = 50 = 0.005s (4.44) K = f T = = (4.45) The design of the chirp pulse is hereby done, and it can now be used in the further analysis. 28 Sonar theory

31 Sonar system 5 At this point it is not apparent what a sonar system exist of and what blocks is present to make the whole system work. The following chapter will elaborate upon this and make an analysis on some of the necessary blocks. The chapter will use figure 5.1 in the process in describing the system. Figure 5.1: A general block diagram of the sonar system. Figure 5.1 shows how a relatively simple sonar system is put together. The sonar system is build up as a loop which will transmit a signal, receive the reflection and upon the received signal make a signal-to-noise ratio, which then can be used to change the power of the transmitting signal. In this way it will be possible to only use the necessary amount of power to transmit the signal, which thereby reduces the overall power consumption. The description of the transmitting signal-path is as follows: The pulse is generated on the basis of the SNR estimate from the detector in the receiver signal-path. The pulse is thereafter beamformed to exploit the transducer array. This is done by duplicating the pulse N times, and delaying some of these in order to generate a beam from the transducers which is directed in a specific way. The SNR can therefor be better, as the directivity will make is possible to avoid reflective surfaces, as the water surface or the sea bottom. The signals is then multiplexed (MUX) in order to shift between transmitting- and receiving-mode. The last block before the transducers is the analog front-end, which modulates the pulse to a higher carrier frequency in order to make it possible to transmit the pulse in a much longer distance. This can be done as the possible transmitting distance can be improved by transmitting a high frequency signal. The description of the receiving signal path is as follows: As the reflections is received by the transducers, these are moved back to baseband, so the sampling frequency can be lower. This is done by the analog front-end as this demodulates the signal. The MUX is now in the receiver position which leads the signal to the beamformer. The beamformer can again improve the SNR, by exploiting directivity. The output from the beamformer is now a single signal, which now has to be preprocessed to make it possible to detect any objects. The preprocessing is basically a filter and a normaliser, that improves the SNR. The detector estimates the noise level so any reflections from objects can be detected. If any objects is detected the tracker calculates direction and velocity of the object, so the user can follow the object. Some of the above touched techniques will in the following be elaborated upon. 5.1 Preprocessing Filtering and deconvolution is introduced to suppress the additive and convolution noise which is affecting to the transmitted signal. Figure 5.2 illustrates how the transmitted signal is affected by the convolution noise, which is the linear time-invariant (LTI) system, g[n], and the additive noise, v[n]. 29