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Johnson, Sandra Gomulka.
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Antenna array output power minimization using steepest descent adaptive algorithm
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by Sandra Gomulka Johnson.
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[Tampa, Fla.] :
University of South Florida,
2004.
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Thesis (Ph.D.)--University of South Florida, 2004.
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Includes bibliographical references.
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Includes vita.
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Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
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Document formatted into pages; contains 95 pages.
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ABSTRACT: A beamforming antenna array is a set of antennas whose outputs are weighted by complex values and combined to form the array output. The effect of the complex valued weights is to steer lobes and nulls of the array pattern to desired directions. These directions may be unknown and so the antenna weights must be adjusted adaptively until some measure of array performance is improved, indicating proper lobe or null placement. An adaptive algorithm to adjust the complex weights of an antenna array is presented that nulls high power signals while allowing reception of GPS signals as long as the signals arrive from different directions. The GPS signals are spread spectrum modulated and have very low average power, on the order of background thermal noise. Simulations of the adaptive algorithm minimize the output power of the array to within 5 dB of the background noise level.The adaptive algorithm, named the Hilbert-space-based (HSB) gradient method, is based on the steepest descent algorithm and implements an efficient, exact gradient calculation. With M antennas in the array, only M-1 weights are adjustable; one antenna weight is held constant to prevent the algorithm from minimizing the output power trivially by zeroing all weights thus preventing the reception of any signal by the array. It appears that M-1 adjustable antenna weights can null M-1 unwanted signals (jammers). However, in the course of the algorithm development, a few configurations of antennas and jammer arrival directions were found where this is not true. Even when the jammer arrival directions are known ('oracle') certain configurations are mathematically impossible to cancel. The oracle solution has a matrix formulation and under certain conditions an exact solution for antenna weights to annihilate the jammers can be found.This provides an excellent comparison tool to assess the performance of other adaptive algorithms. The HSB gradient adaptive algorithm and the oracle solution are both implemented in Matlab. Outputs of both are plotted for comparison.
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Adviser: Snider, Arthur David.
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signal processing.
Hilbert space gradient.
exact gradient.
GPS reception.
jammer nulling.
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Dissertations, Academic
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x Electrical Engineering
Doctoral.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.561
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An tenna Arra y Output P o w er Minimization Using Steep est Descen t Adaptiv e Algorithm b y Sandra Gom ulk a Johnson A dissertation submitted in partial fulllmen t of the requiremen ts for the degree of Do ctor of Philosoph y Departmen t of Electrical Engineering College of Engineering Univ ersit y of South Florida Ma jor Professor: Arth ur Da vid Snider, Ph.D. P aul Flikk ema, Ph.D. Vija y K. Jain, Ph.D. Nagara jan Ranganathan, Ph.D. Mourad Ismail, Ph.D. Date of Appro v al: No v em b er 16, 2004 Keyw ords: signal pro cessing, jammer n ulling, GPS reception, exact gradien t, Hilb ert space gradien t c r Cop yrigh t 2004, Sandra Gom ulk a Johnson
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Dedication I dedicate this dissertation to m y h usband, William Jo el Dietmar Johnson, and to m y daugh ter, Cora Julia Kim b erly Johnson. It is through his generosit y and motiv ation, her patience, and their lo v e that this w ork has b een accomplished.
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Ac kno wledgmen ts I w ould lik e to ac kno wledge m y advisor, Dr. A. Da vid Snider, for his patience, encouragemen t, and p ersistence during the course of our researc h. I m ust also ac kno wledge Dr. P aul Flikk ema for in tro ducing me to this researc h problem. I thank him as w ell as Chris Sp erandio for their help with L A T E X. The friendships of Carlos and Florence Briceno and the LaP oin te family ha v e b een in v aluable. Also I m ust include m y paren ts, Ric hard and Rosemarie Gom ulk a, for their un w a v ering encouragemen t. Finally all thanks b e to Go d from Whom all go o d things come.
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T able of Con ten ts List of T ables iii List of Figures iv Abstract v Chapter 1 In tro duction 1 1.1 Motiv ations 1 1.2 An tenna Arra y F undamen tals 2 Chapter 2 An tenna Arra y Notation 5 Chapter 3 Existing Algorithms and Assessmen t Metho dology: Comparison with the Oracle Solution 10 3.1 Existing Adaptiv e Algorithms 11 3.1.1 Wiener Solution 11 3.1.2 Least Mean Square Solution 12 3.1.3 P o w er In v ersion Solution 13 3.1.4 Subgradien t Searc h T ec hnique 13 3.2 A New Candidate: Hilb ert-Space-Based (HSB) Gradien t Algorithm 14 3.3 Algorithm Comparison Considerations 15 3.3.1 Oracle Solution 15 3.3.2 Condition Num b er 21 3.4 Finding Confounding Congurations 21 Chapter 4 Hilb ert-Space-Based (HSB) Gradien t Algorithm 28 4.1 P o w er Minimization and W eigh t Calculation 28 4.2 Exact Gradien t Calculation 32 4.3 Noise Sim ulation 35 4.4 Dierences in Algorithm and Oracle Solution 36 Chapter 5 Matlab Implemen tation 38 5.1 setup.m 38 5.2 adapt.m 40 5.3 motion.m 44 Chapter 6 Algorithm and Oracle P erformance 47 6.1 Summary of P erformance 47 6.1.1 No Motion 48 i
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6.1.2 Y a w Motion 48 6.1.3 F ull Motion 49 6.2 Catalog of Output 49 6.2.1 No Motion 49 6.2.2 Y a w Motion 51 6.2.3 Dieren t Jammer Amplitude Lev els 53 6.2.4 F ull Motion 56 6.3 A Note on Out-of-Plane Jammers 58 6.4 Noise Imm unit y 58 Chapter 7 Summary and F uture W ork Directions 61 References 62 App endices 64 App endix A Sym b ol Glossary 65 App endix B Matlab Source Co de 67 App endix C GPS Signal Characteristics 81 App endix D Alternativ e Gradien t F orm ula 84 Ab out the Author End P age ii
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List of T ables T able 1. Constan ts in setup.m 39 T able 2. V ariables in setup.m Initialized b y User. 40 T able 3. T ypical Data for No Motion Cases. 50 T able 4. T ypical Data for Y a w Motion Cases. 51 T able 5. T ypical Data for No Motion Cases with Dieren t Jammer Amplitudes. 53 T able 6. T ypical Data for Y a w Motion Cases with Dieren t Jammer Amplitudes. 53 T able 7. T ypical Data for F ull Motion Cases. 56 T able 8. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=0 dB). 59 T able 9. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=1 dB). 59 T able 10. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=3 dB). 60 iii
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List of Figures Figure 1. A 3 Elemen t Linear Arra y 3 Figure 2. Jammer Propagation Dela y 6 Figure 3. 3 Elemen t An tenna Arra y with 2 Jammers. 8 Figure 4. Confounding Conguration Example. 19 Figure 5. Example of ~ a m and ~ u n 41 Figure 6. Six Jammers, No Motion. 50 Figure 7. Six Jammers, Y a w Motion, Near Confounding Conguration. 52 Figure 8. F our Jammers 50 dB, 30 dB, 40 dB, 20 dB; No Motion. 54 Figure 9. Six Jammers 50, 30, 40, 20, 40, 50 dB; Y a w Motion. 55 Figure 10. Six Jammers 50, 30, 40, 20, 40, 50 dB; F ull Motion. 57 iv
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An tenna Arra y Output P o w er Minimization Using Steep est Descen t Adaptiv e Algorithm Sandra Gom ulk a Johnson ABSTRA CT A b eamforming an tenna arra y is a set of an tennas whose outputs are w eigh ted b y complex v alues and com bined to form the arra y output. The eect of the complex v alued w eigh ts is to steer lob es and n ulls of the arra y pattern to desired directions. These directions ma y b e unkno wn and so the an tenna w eigh ts m ust b e adjusted adaptiv ely un til some measure of arra y p erformance is impro v ed, indicating prop er lob e or n ull placemen t. An adaptiv e algorithm to adjust the complex w eigh ts of an an tenna arra y is presen ted that n ulls high p o w er signals while allo wing reception of GPS signals as long as the signals arriv e from dieren t directions. The GPS signals are spread sp ectrum mo dulated and ha v e v ery lo w a v erage p o w er, on the order of bac kground thermal noise. Sim ulations of the adaptiv e algorithm minimize the output p o w er of the arra y to within 5 dB of the bac kground noise lev el. The adaptiv e algorithm, named the Hilb ert-space-based (HSB) gradien t metho d, is based on the steep est descen t algorithm and implemen ts an ecien t, exact gradien t calculation. With M an tennas in the arra y only M 1 w eigh ts are adjustable; one an tenna w eigh t is held constan t to prev en t the algorithm from minimizing the output p o w er trivially b y zeroing all w eigh ts th us prev en ting the reception of an y signal b y the arra y It app ears that M 1 adjustable an tenna w eigh ts can n ull M 1 un w an ted signals (jammers). Ho w ev er, v
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in the course of the algorithm dev elopmen t, a few congurations of an tennas and jammer arriv al directions w ere found where this is not true. Ev en when the jammer arriv al directions are kno wn (`oracle') certain congurations are mathematically imp ossible to cancel. The oracle solution has a matrix form ulation and under certain conditions an exact solution for an tenna w eigh ts to annihilate the jammers can b e found. This pro vides an excellen t comparison to ol to assess the p erformance of other adaptiv e algorithms. The HSB gradien t adaptiv e algorithm and the oracle solution are b oth implemen ted in Matlab. Outputs of b oth are plotted for comparison. vi
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Chapter 1 In tro duction 1.1 Motiv ations An an tenna is a device to transmit and receiv e electromagnetic radiation. The an tenna creates a pattern of radiation that spatially describ es areas of gain (lob es) or atten uation (n ulls) for the signals it transmits or receiv es. The radiation pattern of a single isotropic an tenna is simpler than that of an arra y of m ultiple an tennas; the radiation pattern of an arra y consists of a w eigh ted sup erp osition of eac h an tenna's radiation pattern. The lo cation of lob es and n ulls in an arra y radiation pattern can b e con trolled with a b eamforming net w ork consisting of a signal pro cessor implemen ting an algorithm that w eigh ts the outputs of eac h an tenna and com bines them to form the arra y output [9 ]. The sub ject of this dissertation is the signal pro cessing algorithm. The RF data receiv ed b y the an tennas is assumed to b e do wncon v erted to a digital baseband stream and the signal pro cessing constituting the prop osed algorithm o ccurs completely in the digital domain. The ob jectiv e of the prop osed Hilb ert-space-based (HSB) gradien t adaptiv e algorithm is to receiv e signals from the Global P ositioning System (GPS) of satellites b y n ulling all jammer signals receiv ed b y the arra y The desired GPS signals ha v e v ery lo w p o w er, on the order of thermal noise, -160 dBw, so if the output p o w er of the arra y is ab o v e this lev el, it is assumed to come from jammer signals. The adaptiv e algorithm reduces the p o w er output of the arra y b y adapting w eigh ts of the an tennas to n ull p o w er. The adaptiv e algorithm is not based on forming lob es in the direction of arriv al of the GPS signals. The dissertation is organized as follo ws. The remainder of Chapter 1 in tro duces what a b eamforming net w ork is and includes an example to illustrate the eect of an tenna w eigh ts on the arra y output. 1
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Chapter 2 in tro duces the v ector notation used to describ e the an tenna arra y geometry and denes v ariables describing the input and output signals of the arra y Chapter 3 includes a surv ey of existing adaptiv e algorithms related to the HSB gradien t algorithm and in tro duces the oracle solution. The oracle solution is a standard for assessing realistic solutions for an tenna arra y w eigh ts based on arra y geometry and kno wn jammer arriv al directions. Chapter 4 describ es the theory of the HSB gradien t algorithm including details of the exact gradien t calculation and noise sim ulation. Chapter 5 describ es the implemen tation of the HSB gradien t algorithm in Matlab. Chapter 6 con tains output of the Matlab sim ulation of the HSB gradien t algorithm, illustrating its adaptiv e p erformance. 1.2 An tenna Arra y F undamen tals This section will briery describ e the radiation pattern of an arra y of an tennas and the eect the an tenna w eigh ts ha v e on this pattern. The radiation pattern of an arra y is the sup erp osition of the individual radiation patterns of eac h an tenna in the arra y The principle of pattern m ultiplication [1 ] states that the radiation pattern of an arra y of iden tical an tennas is determined b y the pro duct of the element factor and the arr ay factor The element factor is the radiation pattern of an individual an tenna in the arra y The arr ay factor dep ends on the geometry of the arra y an tennas as w ell as on the complex w eigh t of eac h an tenna in the arra y By adjusting the complex w eigh ts of the an tennas in the arra y the arra y factor and th us, the o v erall arra y radiation pattern, can b e mo died. Adaptiv e algorithms describ e ho w to adjust the w eigh ts and a b eamforming net w ork implemen ts these w eigh t adjustmen ts. A b eamforming net w ork assigns an adjustable w eigh t to eac h an tenna output signal. Eac h w eigh t is a complex n um b er whic h has a v ariable gain (magnitude) and a v ariable phase. Adjusting the w eigh t of eac h an tenna mo dies the arra y factor and has the eect of steering lob es and n ulls to v arious p ositions. Signals arriving at the arra y in a n ull lo cation will b e sev erely atten uated or ev en eliminated b y the arra y; signals arriving at a 2
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lob e will b e enhanced b y the arra y Th us the an tenna arra y p erforms spatial ltering using the mainlob e, sidelob es, and n ulls of its radiation pattern. These are somewhat analogous to the passband, transition band, and stop band from traditional circuit lter theory The spatial areas in the arra y pattern of gain and atten uation are frequency dep enden t. A deep n ull at 1 ma y not b e as deep, or ev en a n ull, at another frequency 2 The bandwidths of the desired and jammer signals m ust b e tak en in to accoun t when designing the an tenna arra y as w ell as the b eamforming net w ork. Eac h an tenna w eigh t in a narro wband arra y is a complex n um b er; the an tenna w eigh ts con trolled b y the prop osed adaptiv e algorithm are implemen ted as suc h in the sim ulations to follo w. F or wideband arra ys, the an tenna w eigh ts m ust pro vide the abilit y to n ull jammers o v er a range of frequencies. These w eigh ts, b eing frequency dep enden t, can b e implemen ted at eac h an tenna as a linear lter or a digital tapp ed dela y line [10 ]. General notation for a jammer impinging on an an tenna arra y is dev elop ed in Chapter 2. A simple example to illustrate the eect of gain and phase adjustmen ts in a narro wband arra y follo ws. Consider a linear an tenna arra y of 3 an tennas eac h separated b y a distance r as in Figure 1. The signal arriving at the arra y is considered to b e a plane w a v e mo deled #2 #3 #1 r r signal d 2 d 1 signal w a v e fron ts x Figure 1. A 3 Elemen t Linear Arra y as a sin usoid with frequency arriving at an angle with resp ect to the axis of the arra y A particular w a v efron t of the signal will arriv e at an tenna #3 rst and after a dela y of = d 2 =c = r cos =c seconds ( c is the propagation sp eed of the signal) the same w a v efron t of the signal will arriv e at an tenna #2 and nally after an additional dela y of ( d 1 d 2 ) =c = r cos =c = seconds, it arriv es at an tenna #1. The output of the arra y is 3
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w 3 cos( t + q 3 ) + w 2 cos( ( t ) + q 2 ) + w 1 cos ( ( t 2 ) + q 1 ) ; (1) where w m represen ts the gain of the m th an tenna and q m represen ts the phase shift in troduced b y the m th an tenna. If the an tenna phases are adjusted to comp ensate for the propagation dela ys so that the signal app ears to ha v e arriv ed at eac h an tenna at the same time, the signal will app ear to b e b o osted in amplitude at the output of the arra y This is accomplished b y adjusting the phase of an tenna #2 to b e q 2 = and the phase of an tenna #1 to b e q 1 = 2 while k eeping w 1 = w 2 = w 3 = 1. This results in the arra y output of cos ( t ) + cos ( ( t ) + ) + cos ( ( t 2 ) + 2 ) = 3 cos ( t ). The an tenna w eigh ts can also b e adjusted so that the arra y output is eliminated. With the an tenna gains set to w 2 = 1 = 2 and w 1 = 1 = 2 and q 2 and q 1 dened as ab o v e, the output of the arra y is cos ( t ) 1 2 cos ( ( t ) + q 2 ) 1 2 cos( ( t 2 ) + q 1 ) = cos ( t ) 1 2 cos ( t ) 1 2 cos ( t ) = 0 and the signal has b een eliminated at the output of the arra y These examples sho w that the c hoice of an tenna w eigh ts can signican tly aect the arra y output. Normally it is up to the w eigh t adjustmen t algorithm to distinguish friendly signals from jammer signals for prop er lob e and n ull placemen t. The prop osed algorithm do es not distinguish the t yp e of incoming signal although it assumes dieren t p o w er lev els b et w een the desired and un w an ted signals; it simply trac ks the output p o w er of the arra y and adjusts the an tenna w eigh ts un til the output p o w er is minimized. Adjusting an tenna w eigh ts to ac hiev e the desired arra y output when the input to the arra y is not w ell dened is the dilemma all adaptiv e algorithms m ust o v ercome, including the prop osed algorithm to b e describ ed in Chapter 4. 4
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Chapter 2 An tenna Arra y Notation This c hapter will in tro duce the notation used to describ e the conguration of an tennas in an arra y as w ell as the input and output of the arra y The geometry of the an tenna arra y and the incoming signals will b e expressed mathematically using v ector notation. The incoming signals to the arra y are of 3 t yp es: the GPS signals, the jammers, and noise. The GPS signals and noise are considered to ha v e lo w p o w er on the order of bac kground noise lev els, while the jammers are assumed to ha v e a m uc h higher p o w er lev el. The basic strategy of the prop osed HSB gradien t algorithm fo cuses on reducing the arra y output jammer p o w er to a lev el comparable to the output GPS signal p o w er, so that the latter can b e detected with spread-sp ectrum tec hnology An arra y can consist of an y n um b er of an tennas (assumed iden tical) arranged in an y conguration. One an tenna in the arra y is selected as the reference an tenna. The origin of a reference co ordinate system is c hosen to coincide with the lo cation of this reference an tenna. The other an tennas in the arra y are referred to as p eripheral an tennas. The an tennas are lo cated at p ositions ~ a m m = 1 ; 2 ; : : : ; N ant where N ant is the total n um b er of an tennas in the arra y All p eripheral an tennas ha v e adjustable gain w m and adjustable phase q m represen ted as a complex term A m = w m e iq m The w eigh t of the reference an tenna, m = 1, is xed at a constan t v alue A 1 1. The total GPS signal receiv ed b y an tenna # m is denoted ( m ) GP S ( t ). The n um b er of individual GPS signals, eac h from a dieren t GPS satellite, is not tak en in to accoun t nor are their directions of arriv al at the arra y The noise at an tenna # m is N m ( t ). Details of the noise mo del used in the sim ulation are presen ted in Chapter 4. 5
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The jammers, mo deled as plane w a v es, arriv e from directions ~ j n n = 1 ; 2 ; : : : ; N j am where N j am is the total n um b er of jammers impinging on the arra y eac h with frequency n and (complex) amplitude J n The phase of jammer # n at the reference an tenna, n is incorp orated in to J n The ~ j n are unit v ectors. A jammer signal receiv ed b y the reference an tenna as a sin usoid with frequency is (the real part of ) J e i! t = j J n j e i ( t + n ) : As stated previously in section 1.2, it tak es a nite time for a jammer w a v efron t to propagate across the arra y and so the jammer w a v efron t reac hes eac h p eripheral an tenna with a corresp onding phase dela y with resp ect to the phase of the jammer w a v efron t at the reference an tenna. These dierences in propagation distance b et w een the p eripheral an tennas and the reference an tenna are giv en b y ~ a m ~ j n the pro jection of the lo cation v ector of an tenna # m on to the v ector represen ting the arriv al direction of jammer # n Figure 2 sho ws a jammer arriving at a 3 elemen t an tenna arra y The (negativ e) distance x 2 equals ~ a 2 ~ j 1 and the distance x 3 equals ~ a 3 ~ j 1 Eac h of these propagation distance dierences x y #1 #2 #3 ~ a 2 ~ a 3 ~ j 1 jammer #1 x 3 x 2 Figure 2. Jammer Propagation Dela y 6
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has an asso ciated time dela y of = ~ a m ~ j n =c where c is the sp eed of propagation of the jammer signal (tak en to b e the sp eed of ligh t), and a corresp onding phase dierence of = ( ~ a m ~ j n ) c = 2 f ( ~ a m ~ j n ) c = 2 ( ~ a m ~ j n ) (2) where = 2 f and f c = 1 F rom gure 2, the jammer w a v efron t reac hes an tenna #3 x 3 =c seconds b efore it propagates to the reference an tenna and the jammer propagates to an tenna #2 x 2 =c seconds after it has reac hed the reference an tenna. Eac h of the m an tennas in the arra y w eigh ts its input b y A m and the arra y output is the sum of these w eigh ted signals. The output signal at an tenna # m consisting of a GPS signal, noise, and N j am jammers is then A m ( m ) GP S ( t ) + A m N m ( t ) + N j am X n =1 J n e i! n t e i 2 n ~ a m ~ j n A m : (3) The total output of the arra y with N j am jammers arriving at eac h an tenna in the arra y is the sum of the output of eac h an tenna in the arra y N ant X m =1 A m ( m ) GP S ( t ) + N ant X m =1 A m N m ( t ) + N j am X n =1 N ant X m =1 J n e i! n t e i 2 n ~ a m ~ j n A m ; (4) or in matrix form [ (1)GP S ( t ) ( N ant ) GP S ( t )] 2 666666664 A 1 A 2 ... A N ant 3777777775 + [ N 1 ( t ) N N ant ( t )] 2 666666664 A 1 A 2 ... A N ant 3777777775 +[ J 1 e i! 1 t J 2 e i! 2 t J N j am e i! N j am t ] 2666664 e i 2 1 ~ a 1 ~ j 1 e i 2 1 ~ a N ant ~ j 1 ... . ... e i 2 N j am ~ a 1 ~ j N j am e i 2 N j am ~ a N ant ~ j N j am 3777775 2666666664 A 1 A 2 ... A N ant 3777777775 7
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x y #1 #2 #3 r r ~ j 1 ~ j 2 jammer #1 jammer #2 Figure 3. 3 Elemen t An tenna Arra y with 2 Jammers. = GP S ( t ) A + N ( t ) A + J ( t ) C A = x ( t ) A (5) where x is the `generic' input v ector to the arra y ( ( m ) GP S ( t ) + N m ( t ) + J ( t ) C ), and C is a matrix whose ( n; m )th elemen t represen ts the phase dierence of jammer # n arriving at an tenna # m C = 26666666664 e i 2 1 ~ a 1 ~ j 1 e i 2 1 ~ a 2 ~ j 1 e i 2 1 ~ a N ant ~ j 1 e i 2 2 ~ a 1 ~ j 2 e i 2 2 ~ a 2 ~ j 2 e i 2 2 ~ a N ant ~ j 2 ... ... . ... e i 2 N j am ~ a 1 ~ j N j am e i 2 N j am ~ a 2 ~ j N j am e i 2 N j am ~ a N ant ~ j N j am 37777777775 : (6) A simple example will illustrate the v ector notation just in tro duced. Consider a 3 elemen t planar arra y with the reference an tenna lo cated at the origin ( ~ a 1 = 0) and p eripheral an tennas lo cated at ~ a 2 = r ^ i; ~ a 3 = 1 2 r ^ i + p 3 2 r ^ j as in gure 3. The arriv al directions of t w o jammers are ~ j 1 = 1 p 2 ^ i + 1 p 2 ^ j and ~ j 2 = p 3 2 ^ i + 1 2 ^ j 8
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The dot pro ducts of the an tenna v ectors and jammer v ectors are ~ a 1 ~ j 1 = 0 ~ a 2 ~ j 1 = r p 2 ~ a 3 ~ j 1 = ( 1 2 p 2 + p 3 2 p 2 ) r ~ a 1 ~ j 2 = 0 ~ a 2 ~ j 2 = p 3 2 r ~ a 3 ~ j 2 = 0 and the arra y output is [ (1)GP S ( t ) (2)GP S ( t ) (3)GP S ( t )] 2 666664 A 1 A 2 A 3 3777775 + [ N 1 ( t ) N 2 ( t ) N 3 ( t )] 2 666664 A 1 A 2 A 3 3777775 + h J 1 e i! 1 t J 2 e i! 2 t i 264 1 e i 2 r = p 2 e i 2 r (1+ p 3) = 2 p 2 1 e i 2 r p 3 = 2 1 375 2666664 A 1 A 2 A 3 3777775 : Since the arra y output is dominated b y the high p o w er jammers, the task of the prop osed algorithm is to adjust the an tenna w eigh ts, A 2 ; : : : ; A N ant so that n ulls are steered to w ard the jammer arriv al directions, minimizing the jammer p o w er to the lev el of the noise and GPS signals. This lea v es the GPS signals in tact to b e pro cessed b y spread sp ectrum demo dulation tec hniques. Recall the reference an tenna w eigh t is xed at A 1 = 1 so that the algorithm minimizes the output p o w er while a v oiding the trivial solution, A m = 0 ; m = 1 ; 2 ; : : : ; N ant Note that ev en though the jammer arriv al directions app ear explicitly in the form ulation just presen ted, in practice one do es not kno w the jammer arriv al directions or frequencies. A t this p oin t of the exp osition, the inclusion of the jammer arriv al directions is necessary to form ulate the phase dierences b et w een the jammers and the an tennas as the jammers propagate across the arra y 9
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Chapter 3 Existing Algorithms and Assessmen t Metho dology: Comparison with the Oracle Solution Adaptiv e algorithms are used in a wide v ariet y of applications. When used with an an tenna arra y the adaptiv e algorithm mo dies the an tenna w eigh ts based on some cost function. When the cost function indicates impro v ed arra y p erformance, the algorithm is adapting the an tenna w eigh ts in the desired fashion. The c hoice of algorithm to adaptiv ely adjust the an tenna w eigh ts is determined b y suc h factors as the p erformance c haracteristics to b e impro v ed, kno wn information ab out the op erating en vironmen t of the arra y and cost of implemen tation. This c hapter will pro vide a review of existing tec hniques used b y adaptiv e algorithms. The prop osed algorithm is related to and shares similarities with these existing algorithms. Also, this c hapter con tains details of the oracle solution. This is an exact solution for antenna w eigh ts based on minimizing jammer output p o w er with kno wledge of the jammer parameters. It is called the `oracle' b ecause the jammer information is una v ailable in practice. The oracle pro vides a metric up on whic h the p erformance of other adaptiv e algorithms can b e assessed. A direct comparison of an tenna w eigh ts b et w een one algorithm and another is not alw a ys p ossible. After the rst iteration of eac h algorithm, the w eigh ts ev olv e along dieren t tra jectories. This c hapter will in tro duce the scenario that an arra y ma y encoun ter certain congurations whic h cause the algorithm to stall. The oracle solution is a history-free standard assessing ho w an algorithm is doing at a particular conguration, regardless of ho w it got there. 10
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3.1 Existing Adaptiv e Algorithms 3.1.1 Wiener Solution A p opular cost function that is used in adaptiv e arra y algorithms is the mean squared error [10 ]. The error is a measure b et w een some reference or desired arra y output signal, d and the actual arra y output signal, x ( t ) A where A is a v ector of an tenna w eigh ts and x ( t ) is the v ector of an tenna input signals (assumed real for this exp osition). The error signal is e ( t ) = d ( t ) x ( t ) A and the mean squared error is MSE = E [ e 2 ( t )] = d 2 ( t ) 2 E [ d ( t ) x ( t )] A + A T E [ x T ( t ) x ( t )] A: The term E [ d ( t ) x ( t )] = r xd is the cross-correlation v ector of the actual input and the desired signal and the term E [ x T ( t ) x ( t )] = R xx is the auto correlation matrix of the actual input signal. The MSE is a quadratic function of the an tenna w eigh ts and can b e visualized as a surface with (t ypically) a unique minim um. The w eigh ts corresp ond to some p oin t on the MSE surface and those that corresp ond to the minim um of this surface are those that are sough t b y the adaptiv e algorithm. A w a y to approac h the minim um is b y adjusting the w eigh t v alues in the direction of the negativ e gradien t, r on this surface A new = A old r where is a constan t that determines the fraction of the gradien t to b e implemen ted p er w eigh t adjustmen t. The gradien t of the MSE (assuming real elemen ts of A ) is r = @ E [ e 2 ( t )] @ A = 2 r xd + 2 R xx A: 11
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The gradien t has a v alue of zero at the minim um p oin t on the surface. Setting the gradien t to zero and solving for A yields the Wiener solution for the an tenna w eigh ts, A = R 1 xx r xd : This solution, ho w ev er, requires the statistics of the input signal to b e kno wn, i.e., r xd and R xx This is rare in practice so, an alternativ e to the Wiener solution is prop osed herein.3.1.2 Least Mean Square Solution An alternativ e to the Wiener solution in v olv es the instan taneous v alue of the gradien t on the squared error surface [11 ]. The squared error is e 2 ( t ) = ( d ( t ) x ( t ) A ) 2 and its gradien t is r ( e 2 ( t )) = @ e 2 ( t ) @ A = 2 e ( t ) x ( t ) : This is only an appro ximation to the Me an Squared Error gradien t b ecause the instan taneous v alues of the error and input signals, instead of their exact statistics as in the Wiener solution, are used in the gradien t calculation. The w eigh t up date equation b ecomes A new = A old + (2 e ( t ) x ( t )) : This is usually called the least mean square (LMS) algorithm. Despite the name of this algorithm, the LMS do es not seek the least me an squared error but rather the least squared error in the solution for A As stated previously the squared error is a quadratic function of the w eigh ts and can t ypically b e in terpreted as a surface with a unique minim um. The LMS algorithm approac hes this minim um with iterativ e w eigh t adjustmen ts in the negativ e gradien t direction. The w eigh t adjustmen ts con tin ue un til the gradien t is zero or un til 12
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further w eigh t adjustmen ts no longer reduce the squared error. A t this p oin t, the arra y output due to these w eigh t v alues appro ximates the desired signal with a least squared error.3.1.3 P o w er In v ersion Solution Compton has prop osed an algorithm whose ob jectiv e is to minimize the output p o w er of an an tenna arra y and it is called the p o w er-in v ersion algorithm for adaptiv e arra ys [15 ]. It impro v es the signal to in terference ratio in an adaptiv e arra y and is based on the Ho w ellsApplebaum feedbac k [12 ] to minimize arra y output p o w er. The arriv al directions of the desired signals and in terference signals need not b e kno wn and the b est p erformance o ccurs when the desired signal is near thermal noise lev els. The p o w er-in v ersion algorithm requires the designer to c ho ose a v alue for lo op gain whic h dep ends on three things: 1) the required minim um signal-to-noise ratio (SNR) out of the arra y 2) the dynamic range of the signal lev el that the arra y m ust accommo date, and 3) what signals the arra y is receiving: the desired signal only the desired and in terference signals only or desired, in terference and noise signals. The optimal w eigh ts are calculated with an equation dep enden t on the desired signal p o w er p er elemen t, in terference p o w er p er elemen t, desired SNR p er elemen t, and in terference-to-noise ratio p er elemen t. This algorithm requires a substan tial amoun t of information ab out the arra y's op erating en vironmen t b efore eectiv e an tenna w eigh ts can b e calculated. 3.1.4 Subgradien t Searc h T ec hnique Another an tenna arra y p o w er minimization metho d uses a so-called \subgradien t" tec hnique to searc h a p erformance surface for the optimal an tenna w eigh ts [6]. This metho d assumes an arra y con taining a xed-w eigh t reference an tenna with the remaining an tennas, referred to as p eripheral an tennas, ha ving adjustable w eigh ts. The tec hnique sequen tially sets the p eripheral an tenna w eigh ts to v alues from a kno wn, nite list of settings, based on the lo cations of the an tenna elemen ts, un til the output p o w er of the arra y is minimized. The jammers are view ed as phasors whose magnitudes and phases are manipulated b y the 13
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p eripheral an tenna w eigh ts. The algorithm adjusts the p eripheral an tenna w eigh ts un til the o v erall jammer phasor from the p eripheral an tennas cancels or minimizes the jammer phasor from the reference an tenna whose w eigh t is xed. As an example, consider a planar arra y in the xy -plane with N ant = 7 an tennas, 6 p eripheral an tennas equally spaced around a cen tral reference an tenna. Let the origin of the xy axes coincide with the cen tral an tenna. The subgradien t searc h sets the p eripheral an tenna w eigh ts to the v alues from the set f 1 N ant 1 e i ( m ) ; m = 2 ; 3 ; : : : ; N ant g where m is the angle describing the lo cation of the m th p eripheral an tenna measured coun terclo c kwise from the x axis. The nite list of settings comprise these v alues. The tec hnique monitors the output p o w er of the arra y as eac h an tenna w eigh t tak es on eac h v alue from the nite set. When all com binations ha v e b een tried, the tec hnique c ho oses the optimal w eigh ts as those giving the lo w est output p o w er. The results of this tec hnique w ere compared in [6 ] to results of the LMS algorithm for a 4 elemen t planar arra y receiving GPS and jammer signals. The LMS algorithm outp erformed the simple, computationally inexp ensiv e, subgradien t tec hnique in static and dynamic cases (roll motion) but with the tradeo of more complex hardw are implemen tation. Although the subgradien t tec hnique w as outp erformed b y the LMS algorithm, it still pro vided sucien t n ulling for successful GPS signal detection. 3.2 A New Candidate: Hilb ert-Space-Based (HSB) Gradien t Algorithm The algorithms men tioned ab o v e, LMS, Wiener, and Compton, are p o w erful algorithms when information is kno wn ab out the arra y op erating en vironmen t, and ab out the input and desired signal c haracteristics. The desired signal m ust b e appro ximated in practice and an y error in this appro ximation will propagate to the w eigh t solution. The desired signal for the HSB gradien t algorithm to b e describ ed in Chapter 4 is a constan t (zero); the algorithm adjusts the an tenna w eigh ts un til the output p o w er is minimized. The statistics of the input signal are rarely kno wn for all p oten tial inputs to the antenna arra y and the problem is further comp ounded when the input data is not stationary The HSB gradien t algorithm w orks successfully with nonstationary data and the dynamic 14
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range of the input signals and output SNR need not b e kno wn for successful algorithm p erformance. The HSB gradien t algorithm ecien tly calculates the exact gradien t on the p erformance surface based on a time-a v eraged p o w er output of an arra y instead of appro ximating a gradien t with instan taneous signal statistics. The HSB gradien t algorithm w eigh ts are not limited to a nite set of p ossible v alues as are the w eigh ts from the subgradien t searc h tec hnique. Therefore, the p erformance of the HSB gradien t algorithm is exp ected to exceed that of the subgradien t searc h tec hnique. 3.3 Algorithm Comparison Considerations Giv en the v ariet y of adaptiv e algorithms that are a v ailable, one w ould lik e to kno w whic h algorithm will result in the b est p erformance for a giv en application. Comparisons among the outputs of a group of p oten tial algorithms are a w a y to select the b est p erforming algorithm but this requires that all p oten tial algorithms b e implemen ted, to obtain the outputs to mak e suc h comparisons. Therefore, comparing an y n um b er of dieren t algorithms can b e a v ery time consuming task. All algorithms seek a solution that minimizes some measure of the arra y output without kno wledge of jammer parameters. This section in tro duces the `oracle' solution whic h seeks to zero the arra y output with kno wledge of the jammer parameters. The oracle th us pro vides a standard against whic h other algorithms can b e measured. If there is no oracle solution, then no other algorithm can p ossibly succeed in n ulling the output for the giv en jammer parameters.3.3.1 Oracle Solution In the course of sim ulating the HSB gradien t algorithm describ ed in Chapter 4 with an arra y in motion (arra y motion is sim ulated with roll, pitc h, and y a w), o ccasionally a conguration of the an tennas of the arra y and the jammers w as encoun tered that confounded the con v ergence of the algorithm, and the output p o w er w as not minimized. The arra y app eared to mo v e through suc h regions of div ergence and then return to regions of con v er15
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gence, indicating that the motion of the arra y through certain jammer arriv al congurations caused the div ergence of the algorithm. This op ened a new question as to whether it w as p ossible to form ulate congurations of an tennas and jammers that w ere mathematically unsolv able. If suc h congurations could b e found and c haracterized, the congurations that confounded the HSB gradien t algorithm migh t b e explained. The existence of a solution for an tenna w eigh ts to n ull the jammers, presuming kno wn an tenna lo cations and kno wn jammer directions, called the \oracle solution", w as studied. The congurations of an tennas and jammers for whic h no oracle solution exists w ere named \confounding congurations". The oracle solution for the an tenna w eigh ts will b e describ ed here. Recall the output of the arra y from equation (5) is GP S A + N A + J C A The goal of an y adaptiv e algorithm is to minimize the p o w er in this arra y output. Consider, instead, the task of forcing only the jammer comp onen t of the output to b e zero, i.e., J C A = 0. Since the jammer amplitude v ector, J cannot b e con trolled, the oracle cannot exploit an y feature ab out the amplitudes of the jammers, and th us it m ust solv e C A = 0 (for A ). The jammer amplitudes do not aect the v alues of the oracle an tenna w eigh ts; they pro vide no information for these w eigh ts. The arrival dir e ctions of the jammers ~ j n ho w ev er, directly inruence the solution. They app ear in the elemen ts of C : C A = 26666666664 e i 2 1 ~ a 1 ~ j 1 e i 2 1 ~ a 2 ~ j 1 e i 2 1 ~ a N ant ~ j 1 e i 2 2 ~ a 1 ~ j 2 e i 2 2 ~ a 2 ~ j 2 e i 2 2 ~ a N ant ~ j 2 ... ... . ... e i 2 N j am ~ a 1 ~ j N j am e i 2 N j am ~ a 2 ~ j N j am e i 2 N j am ~ a N ant ~ j N j am 37777777775 2666666664 A 1 A 2 ... A N ant 3777777775 = 2666666664 00 ... 0 3777777775 : (7) 16
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All elemen ts of the rst column of C ha v e terms whic h con tain ~ a 1 ~ j n and since ~ a 1 = 0 for the reference an tenna the rst column is a column of 1's. Also, A 1 1 for the reference an tenna. Equation (7) can th us b e rewritten as 26666666664 1 e i 2 1 ~ a 2 ~ j 1 e i 2 1 ~ a N ant ~ j 1 1 e i 2 2 ~ a 2 ~ j 2 e i 2 2 ~ a N ant ~ j 2 ... ... . ... 1 e i 2 N j am ~ a 2 ~ j N j am e i 2 N j am ~ a N ant ~ j N j am 37777777775 2666666664 1 A 2 ... A N ant 3777777775 = 2666666664 00 ... 0 3777777775 : (8) These equations can b e further rearranged b y taking the rst column of C to the righ t hand side. The C matrix with the rst column remo v ed is no w referred to as C r educed with size N j am ( N ant 1). 26666666664 e i 2 1 ~ a 2 ~ j 1 e i 2 1 ~ a N ant ~ j 1 e i 2 2 ~ a 2 ~ j 2 e i 2 2 ~ a N ant ~ j 2 ... . ... e i 2 N j am ~ a 2 ~ j N j am e i 2 N j am ~ a N ant ~ j N j am 37777777775 2666664 A 2 ... A N ant 3777775 = C r educed 2666664 A 2 ... A N ant 3777775 = 2666666664 1 1 ... 1 3777777775 : (9) A criterion for the existence of an oracle solution can b e stated as: An tenna w eigh ts annihilating the jammer output p o w er exist if and only if the v ector [ 1 1 1] T lies in the column space of C r educed When the n um b er of jammers equals the n um b er of p eripheral an tennas, N j am = N ant 1, C r educed is square. If, further, rank( C r educed ) = N ant 1, the unique exact an tenna w eigh ts 17
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are 2666666664 A 2 A 3 ... A N ant 3777777775 = C 1 r educed 2666666664 11 ... 1 3777777775 : (10) If the n um b er of jammers is less than the n um b er of p eripheral an tennas, N j am < N ant 1, the system of equations (9) is underdetermined. In this case, if the rank of C r educed is full, rank( C r educed ) = N j am the system has an innite n um b er of solutions. F or the case when there are more jammers than p eripheral an tennas, N j am > N ant 1, the arra y is ask ed to form N j am n ulls with only N ant 1 < N j am adjustable an tenna w eigh ts. This results in an o v erdetermined system of equations (9) with the p ossibilit y{or rather, lik eliho o d{of no solution for the an tenna w eigh ts. (Of course a least squares solution to equation (9) of minim um norm [2 ] can alw a ys b e expressed using the pseudoin v erse of C r educed [3 ], written as C r educed as 2666666664 A 2 A 3 ... A N ant 3777777775 = C r educed 2666666664 11 ... 1 3777777775 : ) (11) The follo wing is a simple example of an N ant = 3 an tenna arra y receiving N j am = 2 jammers, ( N j am = N ant 1). With 2 adjustable an tenna w eigh ts, it seems reasonable to exp ect that 2 n ulls can b e formed to atten uate 2 jammers. The example will illustrate that this is not alw a ys true. The an tennas are lo cated at ~ a 1 = 0 ; ~ a 2 = r ^ i; ~ a 3 = 1 2 r ^ i + p 3 2 r ^ j and the jammer arriv al directions are ~ j 1 = p 3 2 ^ i + 1 2 ^ j and ~ j 2 = p 3 2 ^ i 1 2 ^ j see gure 4. The jammer frequencies are 18
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x y #1 #2 #3 r r ~ j 1 ~ j 2 jammer #1 jammer #2 Figure 4. Confounding Conguration Example. the same, 1 = 2 The an tenna w eigh ts will n ull the output, from equation (8), if 264 1 e i 2 ~ a 2 ~ j 1 e i 2 ~ a 3 ~ j 1 1 e i 2 ~ a 2 ~ j 2 e i 2 ~ a 3 ~ j 2 375 2666664 1 A 2 A 3 3777775 = 2666664 000 3777775 : Substituting in the v alues for ~ a m and ~ j n C r educed = 264 e i 2 p 3 2 r e i 2 p 3 2 r e i 2 p 3 2 r e i 2 p 3 2 r 375 : Let e i = e i 2 p 3 2 r and equation (9) b ecomes 264 e i e i e i e i 375 264 A 2 A 3 375 = 264 1 1 375 : 19
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Rewrite the left hand side of this equation as A 2 264 e i e i 375 + A 3 264 e i e i 375 = ( A 2 + A 3 ) 2 64 e i e i 375 : But the righ t hand side of the equation is 264 1 1 375 : h e i e i i T is a scalar m ultiple of [ 1 1] T if and only if e i = e i that is, e i 2 = 1 ; = k where k is an in teger. Th us, these jammers can b e n ulled only if = 2 p 3 2 r = k that is, if the jammer w a v elength equals p 3 times the an tenna separation, r divided b y an in teger, i.e. = p 3 r =k : (12) F or this example, ev en though the n um b er of jammers is equal to the n um b er of p eripheral an tennas in the arra y N j am = N ant 1, making C r educed a square matrix, if the jammer w a v elength do es not satisfy equation (12) the v ector [ 1 1] T do es not lie in the column space of C r educed The jammer p o w er, therefore, cannot b e n ulled for this conguration. W e prop ose that whenev er a prescrib ed conguration of jammers and an tennas causes an y adaptiv e p o w er-minimizing algorithm to div erge, the conguration should b e tested for an oracle solution b efore the algorithm is \blamed". If the conguration has no oracle solution, it should b e remo v ed from the test set in assessing the p erformance of the adaptiv e algorithm, b ecause no algorithm could p ossibly giv e satisfactory p erformance. Whenev er an oracle solution do es exist, on the other hand, it completely n ulls the jammer p o w er. Th us, it pro vides a standard against whic h other algorithms can b e measured. T o summarize, if an adaptiv e algorithm fails to p erform as exp ected, the oracle solution analysis can p erhaps pro vide a p oten tial clue b y exp osing a confounding conguration. The oracle solution, if it exists, also pro vides an ev aluation of the eectiv eness of an adaptiv e algorithm solution. 20
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3.3.2 Condition Num b er When no oracle solution exists, the conguration of jammer arriv al directions and an tenna lo cations is lab eled a confounding conguration. Ho w ev er, there exist congurations of jammers and an tennas for whic h there is an oracle solution but it is n umerically unstable; these suc h cases are called near-confounding congurations. A metric used during researc h of confounding congurations w as the condition n um b er of C r educed whic h tests the n umerical robustness of an oracle solution. The condition n um b er measures sensitivit y to error [16 ] and it is calculated b y taking the ratio of the largest singular v alue of the matrix to the smallest singular v alue. The higher the condition n um b er of a matrix, the closer the matrix is to b eing singular [16 ]; a condition n um b er of innit y indicates a matrix that is singular, one without full rank. A confounding conguration can result from a singular C r educed F or cases with N j am = N ant 1 ( C r educed square), if the condition n um b er of C r educed is large, on the order of 1 10 4 the arrangemen t of jammers and an tennas ma y confound the HSB gradien t algorithm and is lab eled a near-confounding conguration. Therefore, the condition n um b er of C r educed pro vides a useful indication as to whether an y adaptiv e algorithm will b e able to nd optimal w eigh ts to n ull jammers. If the condition n um b er is on the order of 1 10 4 or greater the HSB gradien t algorithm ma y not pro duce viable w eigh ts and the an tenna/jammer conguration pro ducing the high condition n um b er should b e remo v ed from the test set. 3.4 Finding Confounding Congurations The previous section describ ed ho w the oracle can b e used to test whether giv en antenna/jammer congurations are confounding or near-confounding. This section examines some asp ects of the oracle solution, exploring the abilit y to construct confounding an tenna/jammer congurations. This ma y b e useful for jamming an enem y when the conguration of the enem y's arra y is kno wn. 21
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The exact an tenna w eigh ts A to annihilate inciden t jammers, the oracle solution, exist if the v ector [ 1 1 1] T lies in the column space of C r educed C r educed 2666666664 A 2 A 3 ... A N ant 3777777775 = 2666666664 1 1 ... 1 3777777775 : (9) In this section, it is desired to nd jammer arriv al directions that defy an oracle solution, i.e., confounding congurations. One w a y among man y to explore this starts with in tro ducing the v ariable ~ v n = 2 ~ j n (13) for more compact notation and factors the matrix C r educed as C r educed = C 0 r educed ( 0 indicates prime, and do es not indicate transp ose) where C 0 r educed = 26666666666664 1 1 1 e i ( ~ v 2 ~ v 1 ) ~ a 2 e i ( ~ v 2 ~ v 1 ) ~ a 3 e i ( ~ v 2 ~ v 1 ) ~ a N ant e i ( ~ v 3 ~ v 1 ) ~ a 2 e i ( ~ v 3 ~ v 1 ) ~ a 3 e i ( ~ v 3 ~ v 1 ) ~ a N ant ... ... . ... e i ( ~ v N ~ v 1 ) ~ a 2 e i ( ~ v N ~ v 1 ) ~ a 3 e i ( ~ v N ~ v 1 ) ~ a N ant 37777777777775 and = 26666666666664 e i ~ v 1 ~ a 2 0 0 0 0 e i ~ v 1 ~ a 3 0 0 0 0 e i ~ v 1 ~ a 4 0 ... ... ... . ... 0 0 0 e i ~ v 1 ~ a N ant 37777777777775 : 22
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(Note that the v ectors ~ v ha v e units of in v erse meters, (m 1 ) and the v ectors ~ a ha v e unit of meters.) So C r educed 2666666664 A 2 A 3 ... A N ant 3777777775 = C 0 r educed 2666666664 A 2 A 3 ... A N ant 3777777775 = 2666666664 1 1 ... 1 3777777775 ; and no w the criterion for existence of a solution is that [ 1 1 ; 1] T m ust lie in the column space of C 0 r educed Note that if C 0 r educed tak es the form C 0 r educed = 2666666664 1 1 1 ... ... ... r r r ... ... ... 3777777775 ; i.e., has a ro w of iden tical en tries (b esides the rst ro w), with r 6 = 1, then the columns cannot span the v ector [ 1 1 1 1] T So if a conguration of jammers is c hosen suc h that all en tries of, sa y the second ro w of C 0 r educed are equal but dieren t from 1, then there are no solutions. W e explore this p ossibilit y The second ro w of C 0 r educed will ha v e equal en tries if ( ~ v 2 ~ v 1 ) ~ a 2 = ( ~ v 2 ~ v 1 ) ~ a 3 + 3 2 = ( ~ v 2 ~ v 1 ) ~ a N ant + N ant 2 ; (14) where 3 ; ; N ant are in tegers. In order to prev en t r = e i ( ~ v 2 ~ v 1 ) ~ a 2 = 1 w e require ( ~ v 2 ~ v 1 ) ~ a 2 6 = 2 (15) for an y in teger Note that equations (14) and (15) are conditions only on the dier enc e ~ v 2 ~ v 1 if there is one solution, there are innitely man y 23
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No w, consider a sp ecic example of nding confounding jammers for a giv en arra y The arra y from gure 4 has ~ a 1 = 0 ; ~ a 2 = ^ i; and ~ a 3 = 1 2 ^ i + p 3 2 ^ j and w e will confound it with 2 jammers. Let ~ v 2 ~ v 1 = ^ i + ^ j Then ( ~ v 2 ~ v 1 ) ~ a 2 = ; ( ~ v 2 ~ v 1 ) ~ a 3 = ( ^ i + ^ j ) ( 1 2 ^ i + p 3 2 ^ j ) = 2 + p 3 2 : T o satisfy equation (14), set ( ~ v 2 ~ v 1 ) ~ a 3 + 3 2 = 2 + p 3 2 + 3 2 = ( ~ v 2 ~ v 1 ) ~ a 2 = and solv e for = 1 p 3 ( + 4 3 ) : (16) Th us ~ v 2 ~ v 1 = ^ i + 1 p 3 ( + 4 3 ) ^ j : (17) Pic k 3 = 1 and = 1, then from equation (16), = 7 : 8325 and ~ v 2 ~ v 1 = ^ i + 7 : 8325 ^ j : Since the direction and amplitude of ~ v 1 is not determined, pic k j ~ v 1 j = 32 : 987m 1 and c ho ose the direction as ~ v 1 = 16 ^ i + 28 : 85 ^ j Then ~ v 2 is, from equation (17), ~ v 2 = 15 ^ i 21 : 0175 ^ j and its magnitude is j ~ v 2 j = 25 : 82m 1 The w a v elength of jammer #1 is 1 = 2 = j ~ v 1 j = 0 : 19047 m (18) and the w a v elength of jammer #2 is 2 = 2 = j ~ v 2 j = 2 = j ~ v 1 + ^ i + 1 p 3 ( + 4 3 ) ^ j j = 0 : 243 m : (19) 24
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The frequencies corresp onding to the w a v elengths of these 2 jammers that will confound the 3 elemen t equilateral triangle an tenna arra y are f 1 = 1 : 575 GHz and f 2 = 1 : 23 GHz, whic h are close to the w a v elengths of the GPS transmissions. Another example of an arra y that can b e confounded with 2 jammers is the cen tered square arra y The an tenna lo cation v ectors are ~ a 1 = 0 ; ~ a 2 = ^ i; ~ a 3 = ^ j ; ~ a 4 = ^ i; and ~ a 5 = ^ j Let ~ v 2 ~ v 1 = ^ i + ^ j : Notice that ( ~ v 2 ~ v 1 ) ~ a 2 = and ( ~ v 2 ~ v 1 ) ~ a 4 = so equation (14) requires = + 4 2 = 4 : If 4 is an ev en in teger, this con tradicts the requiremen t (equation (15)) that ( ~ v 2 ~ v 1 ) ~ a 2 cannot b e an in teger m ultiple of 2 ; so c ho ose to b e an o dd m ultiple of : = (2 N 1 + 1) ; where N 1 is an y in teger. Equation (14) also requires ( ~ v 2 ~ v 1 ) ~ a 3 = and ( ~ v 2 ~ v 1 ) ~ a 5 = so + 3 2 = + 5 2 = ( 5 3 ) (2 N 2 + 1) where N 2 is an y in teger. The cen tered square arra y can b e confounded b y an y 2 jammers related b y ~ v 2 ~ v 1 = (2 N 1 + 1) ^ i + (2 N 2 + 1) ^ j Arbitrarily c ho osing N 1 = 1 and N 2 = 2 results in = 3 and = 5 Again, the direction and amplitude of ~ v 1 can b e an ything. Cho osing j ~ v 1 j = 32 : 987 m 1 and its direction as ~ v 1 = 20 ^ i + 26 : 23 ^ j results in ~ v 2 = 29 : 42 ^ i + 41 : 94 ^ j and j ~ v 2 j = 51 : 23 m 1 25
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The w a v elengths of the jammers are 1 = 2 = j ~ v 1 j = 0 : 1905 m and 2 = 2 = j ~ v 1 + (2 N 1 + 1) ^ i + (2 N 2 + 1) ^ j j = 0 : 1226 m : The frequencies of these jammers are f 1 = 1 : 575 GHz and f 2 = 2 : 45 GHz. Note that w e ha v e confounded a ve an tenna arra y with only two jammers. This particular metho d of nding jammers to confound an arra y will not w ork if all the an tenna lo cation v ectors are related b y ~ a i ~ a j = ~ a k for some i; j; k An example of suc h an arra y is a hexagonal arra y with ~ a 1 = 0 ; ~ a 2 = ^ i; ~ a 3 = 1 2 ^ i + p 3 2 ^ j ; ~ a 4 = 1 2 ^ i + p 3 2 ^ j ; ~ a 5 = ^ i; ~ a 6 = 1 2 ^ i p 3 2 ^ j ; and ~ a 7 = 1 2 ^ i p 3 2 ^ j F or this arra y ~ a 3 ~ a 4 = ~ a 2 ~ a 4 ~ a 3 = ~ a 5 ~ a 3 ~ a 2 = ~ a 4 etc. Substituting one of these, for example ~ a 2 = ~ a 3 ~ a 4 in to equation (14) and rearranging terms results in ( ~ v 2 ~ v 1 ) ~ a 4 = 3 2 = ( ~ v 2 ~ v 1 ) ( ~ a 4 ~ a 3 ) + 4 2 = = ( ~ v 2 ~ v 1 ) ( ~ a N ant ~ a 3 ) + N ant 2 but this lea v es the term ( ~ v 2 ~ v 1 ) ~ a 4 equal to an in teger m ultiple of 2 This violates the condition of equation (15) defeating the ob jectiv e of nding a confounding conguration. Summarizing, note that the HSB gradien t algorithm as w ell as other algorithms fo cus on minimizing the total p o w er output from the arra y j [ GP S + N + J C ] A j 2 while the oracle fo cuses on zer oing the term C A So if the oracle solution exists and GP S and N are small, it is a reasonable standard for assessing the HSB gradien t algorithm; if the oracle solution do es not exist then the goal of the HSB gradien t algorithm ma y b e unac hiev able. The next c hapter describ es the HSB gradien t adaptiv e algorithm that computes an tenna w eigh ts whic h seek to minimize the output p o w er of an arra y A comparison b et w een the HSB gradien t w eigh ts and the oracle w eigh ts is made b y comparing, at eac h step of the sim ulation, the arra y output p o w ers resulting from eac h set of w eigh ts. The comparison 26
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sho ws that the HSB algorithm exhibits excellen t p erformance in minimizing the output p o w er of an arra y 27
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Chapter 4 Hilb ert-Space-Based (HSB) Gradien t Algorithm This c hapter will describ e the theory of the HSB gradien t algorithm. The ob jectiv e of this algorithm is to minimize the output p o w er of the arra y The signals of in terest receiv ed from the GPS satellite net w ork are spread sp ectrum signals of negligible p o w er, ab out the same as the bac kground noise lev el. Reducing the arra y output p o w er to this lev el will allo w GPS signals to b e extracted. 4.1 P o w er Minimization and W eigh t Calculation A t time t r the output of an tenna # m consisting of GPS signals, noise, and N j am jammers is A m ( m ) GP S ( t r ) + A m N m ( t r ) + N j am X n =1 J n e i! n t r e i 2 n ~ a m ~ j n A m where N m ( t r ) conceptually represen ts a random noise comp onen t in an tenna # m The details of the nature of the random noise in the arra y are discussed in section 4.3. The arra y output is the sum of all an tenna outputs N ant X m =1 A m ( m ) GP S ( t r ) + N ant X m =1 A m N m ( t r ) + N ant X m =1 N j am X n =1 J n e i! n t r e i 2 n ~ a m ~ j n A m whic h can b e expressed in matrix form, from (5), as GP S A + N A + J C A = xA: (20) The p o w er minimization and w eigh t calculations will pro ceed as follo ws. F or some initial c hoice of w eigh ts A the output p o w er of the arra y is sampled ev ery seconds and assessed at times lab eled b y the v ariable t r = r ( r is an in teger), of these p o w er measuremen ts 28
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are a v eraged ev ery seconds, and the gradien t of this a v erage (with resp ect to the w eigh ts) is calculated. The p eripheral an tenna w eigh ts are adjusted in the direction of the negativ e gradien t. This constitutes one algorithm iteration. It is unlik ely that one single adjustmen t in eac h w eigh t v alue results in their optimal v alues so these up dated w eigh ts are used to accum ulate another batc h of output p o w er measuremen ts, and the algorithm is rep eated. If the pro cedure con v erges, the searc h b ecomes more precise, with ner adjustmen ts in the w eigh t v alues to w ard their optimal v alues, as the algorithm progresses. The instan taneous output p o w er of the arra y p ( t r ; A ), is p ( t r ; A ) = ( xA )( xA ) y (21) where y represen ts conjugate transp ose. F or eac h v alue of t r p ( t r ; A ) represen ts one sample of the instan taneous arra y output p o w er. During the k th iteration of the algorithm, samples of instan taneous p o w er, p ( t r ; A ), are a v eraged to obtain the a v erage output p o w er of the arra y P ( k ; A ), i.e., P ( k ; A ) = 1 k X t r =( k 1)+1 p ( t r ; A ) : (22) The a v erage output p o w er is a quadratic function of the an tenna w eigh ts. T o help picture this p o w er function, consider an arra y of 3 an tennas, all with real w eigh ts only in an y conguration (2 p eripheral an tennas with adjustable w eigh ts and 1 reference an tenna with a xed w eigh t). The output p o w er of the arra y can b e plotted along the z -axis as a function of the adjustable an tenna w eigh ts represen ted along the x and y axes. The p o w er surface is shap ed lik e a b o wl with a single, unique minim um. The an tenna w eigh ts that corresp ond to this minim um are those that the algorithm seeks. The strategy is to con v erge to the minim um b y con tin uously adjusting the v alues for the an tenna w eigh ts so as to enforce P ( k + 1 ; A ) < P ( k ; A ). Recall, from Chapter 2, A is a 1 N ant v ector with comp onen ts w m e iq m m = 1 ; 2 ; : : : ; N ant The p o w er P ( k ; A ) can b e expressed explicitly in terms of the magnitudes, w m and phases, 29
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q m of the an tenna w eigh ts b y reincarnating A m = w m e iq m as a new v ariable W where W is a column of an tenna w eigh t parameters [ w 1 w 2 w N ant q 1 q 2 q N ant ] T (recall since A 1 1, w 1 = 1 and q 1 = 0). The ob jectiv e of the algorithm is to c ho ose the c hange in the w eigh ts, W m to reduce P ( k ; W m ): P ( k + 1 ; W m + W m ) P ( k ; W m ) ; m = 1 ; 2 ; : : : ; 2 N ant : Without risk of confusion, w e write P ( W ) for P ( k ; A ). Expanding P ( W m + W m ) with a (matrix) T a ylor series to rst order yields P ( W m + W m ) P ( W m ) + W T @ P @ W P ( W m ) + W T r P P ( W m ) ; (23) where W is the column of w eigh t parameters in tro duced ab o v e and r P represen ts the column of all comp onen ts of the gradien t of the output p o w er with resp ect to the an tenna w eigh t parameters, r P = [ @ P @ w 1 ; @ P @ w 2 ; : : : ; @ P @ w N ant ; @ P @ q 1 ; @ P @ q 2 ; : : : ; @ P @ q N ant ] T F or the steep est descen t algorithm, the w eigh ts from iteration k to iteration k + 1 are adjusted in the direction of the negativ e p o w er gradien t with a scale factor the \step size" W = r P : (24) Substituting equation (24) for W T in equation (23) results in P = P ( W + W ) P ( W ) r P T r P : (25) The desir e d c hange in p o w er is kno wn; it is the dierence b et w een the presen t p o w er, P ( k ; A ) = P ( W ) and the minim um p o w er, whic h is appro ximately zero. P = c hange in p o w er = (0 P ( W )) = r P T r P 30
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and solving for the step size one obtains = P ( W ) r P T r P : (26) The w eigh t adjustmen t equation (24) b ecomes W = r P = P ( W ) r P T r P r P : (27) Equation (27) in terms of the w eigh t magnitudes states w i = @ P @ w i = P ( W ) 0BBBBB@ @ P =@ w i N ant X m =2 h ( @ P @ w m ) 2 + ( @ P @ q m ) 2 i 1CCCCCA ; i = 2 ; 3 ; : : : ; N ant (28) and in terms of the w eigh t phases it states q i = @ P @ q i = P ( W ) 0BBBBB@ @ P =@ q i N ant X m =2 h ( @ P @ w m ) 2 + ( @ P @ q m ) 2 i 1CCCCCA ; i = 2 ; 3 ; : : : ; N ant : (29) These w eigh t c hange calculations o ccur once p er algorithm iteration, eac h time with up dated v alues for and r P Our inno v ation for calculating the gradien t comp onen ts, @ P @ w m and @ P @ q m will b e describ ed in the next section. If the arra y is in motion, the optimal w eigh t searc h p erformed b y the algorithm b ecomes nonstationary b ecause the p o w er surface c hanges at eac h iteration. The p o w er surface is a function of the phase matrix C whic h is a function of the c hanging jammer arriv al directions, ~ j n F or an arra y in motion, the p o w er is time a v eraged and eac h w eigh t is up dated p er algorithm iteration as it is for a stationary arra y 31
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4.2 Exact Gradien t Calculation This section will describ e the computation of the gradien t of the a v erage arra y output p o w er. What distinguishes the HSB gradien t algorithm from all other gradien t based adaptiv e algorithms is that it calculates the exact gradien t comp onen ts, not appr oximations of the gradien t comp onen ts. F urthermore, the algorithm calculates the gradien t comp onen ts sim ultaneously and ecien tly The p o w er gradien t, r P is computed using the Hilb ert space inner pro ducts of signals whic h are readily a v ailable at the arra y output b y the follo wing metho d [5 ]. F rom equations (21) and (22), P ( k ; A ) = 1 k X t r =( k 1)+1 p ( t r ; A ) = 1 k X t r =( k 1)+1 ( xA )( xA ) y : Rewrite this summation using inner pro duct bra/k et notation < a; b > = 1 T k T X t =( k 1) T +1 a ( t ) b y ( t ) and the p o w er is represen ted as P ( k ; A ) = < xA; xA > : (30) Since the arra y output results from the N ant an tenna outputs, xA = N ant X m =1 A m x m ( k ) ; (31) the arra y output p o w er b ecomes P ( k ; A ) = < N ant X g =1 A g x g ( k ) ; N ant X g =1 A g x g ( k ) > = N ant X g =1 N ant X h =1 A g A h < x g ( k ) ; x h ( k ) > : (32) 32
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No w the m th comp onen t of the gradien t with resp ect to the w eigh t magnitude w m is (the time dep endence of x is dropp ed for clarit y) (recall A g = w g e iq g ) @ P @ w m = @ ( P N ant g =1 P N ant h =1 A g A h < x g ; x h > ) @ w m = X g X h < x g ; x h > A g @ w h e iq h @ w m + X g X h < x g ; x h > @ w g e iq g @ w m A h : On the righ t hand side of this equation, the partial deriv ativ e with resp ect to w m is non-zero only when h = m for the rst term and only when g = m for the second term: @ P @ w m = X g X h < x g ; x h > A g hm e iq m + X g X h < x g ; x h > g m e iq m A h = X g < x g ; x m > A g e iq m + X h < x m ; x h > e iq m A h : Rearranging terms and iden tifying the summation of the an tenna outputs as the arra y output signal, xA from equation (31), yields @ P @ w m = X g < A g x g ; e iq m x m > + X h < e iq m x m ; A h x h > = < xA; A m x m > =w m + < A m x m ; xA > =w m : The terms in the last line are complex conjugates of eac h other. A quan tit y added to its conjugate lea v es only t wice the real part @ P @ w m = 2 < < A m x m ; xA > w m : (33) The c omp onents of the gr adient with r esp e ct to the weight magnitudes ar e simply the Hilb ert sp ac e inner pr o ducts of the r e al p arts of the appr opriately sc ale d (by 1 w m ) individual output signals of e ach antenna A m x m ( k ) with the arr ay signal output, x ( k ) A That is, they ar e c orr elations (at zer o delay). 33
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Similarly the m th comp onen t of the gradien t with resp ect to the phase q m of the w eigh t is @ P @ q m = @ ( P N ant g P N ant h A g A h < x g ; x h > ) @ q m (34) = X g X h < x g ; x h > A g @ w h e iq h @ q m + X g X h < x g ; x h > @ w g e iq g @ q m A h : (35) On the righ t hand side of this equation, the partial deriv ativ e with resp ect to q m is non-zero only when h = m for the rst term and only when g = m for the second term: @ P @ q m = X g X h < x g ; x h > A g hm w h ( i ) e iq h + X g X h < x g ; x h > g m w g ( i ) e iq g A h = i X g < x g ; x m > A g A m + i X h < x m ; x h > A m A h : Rearranging terms and iden tifying the summation of the an tenna outputs as the arra y output signal, xA from equation (31), yields @ P @ q m = i X g < A g x g ; A m x m > + i X h < A m x m ; A h x h > = i < xA; A m x m > + i < A m x m ; xA > = 2 = < A m x m ; xA > : (36) The c omp onents of the gr adient with r esp e ct to the weight phases ar e simply the Hilb ert sp ac e inner pr o ducts of the imaginary p arts of the sc ale d individual output signals of e ach antenna A m x m ( k ) with the arr ay signal output, x ( k ) A With all comp onen ts of the gradien t a v ailable, they are then used to determine the an tenna w eigh ts to minimize the output p o w er from the arra y as in equations (28) and (29). T o reiterate, the comp onen ts of the gradien t are found using the equations @ P @ w m = 2 < < A m x m ; xA > w m 34
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and @ P @ q m = 2 = < A m x m ; xA > : These comp onen ts, again, are not appr oximations of the gradien t, they are the exact comp onen ts of the gradien t. The estimates of the gradien t used b y other algorithms only appro ximate the gradien t. Algorithms that use nite dierences to appro ximate the gradien t lose accuracy particularly near the minim um of the quadratic surface, and tak e m ultiple time steps to ev aluate, since eac h term in the nite dierence requires a separate p o w er a v erage. The sim ultaneous, ecien t, exact calculation of all comp onen ts of the gradien t b y the metho d prop osed herein leads to faster, more robust jammer cancellation p erformance. Although w e do not exploit it in this study it is of in terest to note that these inner pro ducts can b e ev aluated using p o w er a v erages alone, for judiciously c hosen signal com binations @ P @ w m = 1 w m f < A m x m + xA; A m x m + xA > < A m x m ; A m x m > < xA; xA > g and @ P @ q m = i f < A m x m + ixA; A m x m + ixA > < A m x m ; A m x m > < xA; xA > g : 4.3 Noise Sim ulation Up to no w this c hapter has fo cused on the implemen tation of the algorithm; this section will describ e details ab out noise in the sim ulation of the algorithm. Noise is added as random `input' to the arra y; it is sub ject to scaling b y the an tenna w eigh ts as are the jammer and GPS input signals. The ob jectiv e of the HSB gradien t algorithm is to minimize the output p o w er of the arra y During eac h iteration, noise is sim ulated as random samples and input to eac h antenna. These samples are added to the jammer and GPS input signals and these comp osite signals are scaled b y the an tenna w eigh ts and com bined to form the arra y output. 35
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Referring to equation (20), the sample at time t r of the arra y output signal is mo deled as GP S A + N A + J C A = xA: The noise term in this equation is N A = N ant X m =1 N m ( t r ) A m where at eac h time step N m ( t r ) is mo deled as an indep enden t sample of a random v ariable with a Gaussian distribution of zero mean and v ariance 2 The p o w er in this noise at the output of the arra y m ust equal the (sp ecied) bac kground noise lev el, It is kno wn that the sum of N ant random v ariables, eac h with the same mean, and v ariance, 2 has mean equal to N ant and v ariance equal to N ant 2 [8 ]. Setting the standard deviation of eac h sample of noise at eac h an tenna equal to p = N ant xes the p o w er lev el of the noise at the output of the arra y to matc h that of the bac kground noise. 4.4 Dierences in Algorithm and Oracle Solution A direct comparison of the an tenna w eigh ts of the HSB gradien t algorithm with another algorithm is not p ossible b ecause the an tenna w eigh ts approac h their optimal v alues via dieren t paths, dep ending on the details of the adaptiv e algorithm. Ho w ev er, as discussed in Chapter 3, the results of an y iteration of the an tenna w eigh ts of the HSB gradien t algorithm can b e compared with the oracle w eigh ts for the corresp onding an tenna p osition. Some dierence b et w een the algorithm and oracle w eigh ts is exp ected b ecause the oracle pro vides the exact w eigh ts annihilating only the output p o w er comp onen t due to jammers, j J C A j 2 while the algorithm w eigh ts minimize the output p o w er of the arra y due to all inputs, j GP S A + N A + J C A j 2 The HSB gradien t algorithm and the oracle also dier in their dep endence on the jammer amplitudes. The HSB gradien t algorithm uses the jammer amplitude information in nding the solution for the an tenna w eigh ts. The jammer amplitudes and arriv al directions aect 36
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the output signal of the arra y the output p o w er of the arra y the gradien t, and the w eigh t up date equation. Ev ery an tenna in the arra y is aected b y ev ery jammer arriving at the arra y A w eak jammer cannot remo v e or reduce the eect that a particular an tenna w eigh t has in the arra y nor can a strong jammer cause a particular an tenna w eigh t to dominate the algorithm solution. All jammers aect all an tenna w eigh ts. What determines the algorithm's solution for eectiv e p o w er minimizing an tenna w eigh ts is ho w close the searc h of the p o w er surface in the negativ e gradien t direction approac hes the minim um. The oracle solution do es not tak e in to accoun t an y information ab out the amplitudes of the jammers. The oracle solv es C A = 0 for the optimal w eigh ts to annihilate the arra y output. The jammer amplitudes, J do not aect the v alues of the oracle an tenna w eigh ts. The arriv al directions of the jammers, ho w ev er, are critical to the solution. They are used in the elemen ts of C 37
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Chapter 5 Matlab Implemen tation This c hapter will describ e the implemen tation of the HSB gradien t algorithm in Matlab. Since the theory of the algorithm has already b een presen ted in the previous c hapter, this c hapter is included only for readers who are in terested in co ding details. The results of the sim ulations are rep orted in Chapter 6. The input to the arra y is mo deled as high p o w er jammers and lo w p o w er noise. The GPS signals are not mo deled explicitly although they w ould app ear similar to the noise. The algorithm do es not p erform an y demo dulation of the GPS signals. The HSB gradien t algorithm is implemen ted in Matlab with 3 les: setup.m adapt.m and motion.m 5.1 setup.m The le setup.m denes and initializes constan ts necessary for the adaptiv e algorithm implemen tation. See T able 1 for a list of these constan ts. Setup.m also denes v ariables that are initialized b y the user. T able 2 lists these v ariables along with t ypical v alues they migh t b e assigned. The p eripheral an tennas in the arra y can b e arranged in an y conguration around a cen tral an tenna. Here the an tennas are arranged in a plane with the origin of the reference co ordinate system coinciding with the cen tral an tenna. The p eripheral an tenna lo cations are measured coun terclo c kwise from the x -axis at angles r m ; m = 2 ; 3 ; : : : ; N ant Eac h jammer arriv es at the arra y with an azim uth angle measured coun terclo c kwise from the x -axis and with an elev ation angle measured do wn from the z -axis. 38
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T able 1. Constan ts in setup.m V ariable Name V alue Description deg to rad = 180 degrees to radians con v ersion constan t jmath p 1 imaginary n um b er denition A 2 Nant matrix, all ro w 1 en tries = 1, all ro w 2 en tries = 0 initial an tenna w eigh t magnitudes and phases a 1 Nant complex w eigh t v alues Ann 2 Nant initial an tenna w eigh t magnitudes and phases for no noise case ann 1 Nant complex w eigh t v alues for no noise case speed of light 3x10 8 m/s sp eed of ligh t constan t P int 20x10 6 sec p o w er in tegral duration LOfreq 1.575x10 9 Hz lo cal oscillator frequency omegaLO 2 LOfreq rad/sec lo cal oscillator frequency jam freq LOfreq frequency of jammers omega jam 2 jam freq frequency of jammers motion vector [ roll pitch yaw ] v ector of v ariables indicating arra y motion az el rr spherical co ordinates of jammer arriv al angles phi jam az azim uth jammer arriv al angle, coun terclo c kwise from x -axis theta jam ( = 2)el elev ation jammer arriv al angle, do wn from z -axis gamma 1 ( Nant -1) v ector of angles describing an tenna locations, coun terclo c kwise from x -axis The sampling rate of the jammer and noise signals is set to 20 MHz ( = 50 ns). Ev ery iteration of the algorithm pro cesses = 400 samples or 20 s of data. These 400 samples of instan taneous arra y output are a v eraged to form one sample of the a v erage arra y output p o w er p er iteration. The v ariable P int represen ts the duration of one iteration. The phase matrix, C is calculated in setup.m The matrix is made up of the exp onen tial phase dierences b et w een the jammer w a v efron ts impinging on eac h p eripheral an tenna with resp ect to the cen ter an tenna in the arra y C has size Njams Nant (from Chapter 3, N j am = Njams and N ant = Nant ) and the ( n; m )th elemen t has the form using the jammer arriv al directions of and and an tenna lo cations of r C nm = e i 2 ~ a m ~ j n = e i 2 cos( r m n ) sin n 39
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T able 2. V ariables in setup.m Initialized b y User. V ariable Name T ypical V alue Description Nant 7 n um b er of an tennas in arra y array radius 0.1 radius of arra y in meters a min 0.1 minim um an tenna w eigh t magnitude a max 10 maxim um an tenna w eigh t magnitude step control 1 fraction of gradien t correction to b e implemen ted noise floor 1.25 (1 = 0 dB) target output p o w er v alue Njams 6 n um b er of jammers inciden t on arra y theta jam [90,90,90,90,90,9 0,90,90,90] v ector of jammer elev ation angles, reduced to 1 Njams phi jam [30,90,150,210,27 0, 330,15,75,135] v ector of jammer azim uth angles, reduced to 1 Njams J [100,100,100,100 ,100,100,100,100, 10 0] v ector of jammer amplitudes, reduced to 1 Njams sim time 0.5 sec length of sim ulation, eac h iteration of algorithm is P int long roll 0 roll rate in deg/sec, + y on to + z pitch 0 pitc h rate in deg/sec, + z on to x yaw 360 y a w rate in deg/sec, + y on to + x beta 0 initial oset of arra y in degrees, -roll where, in general, for a planar arra y ~ a m = cos r m ^ i + sin r m ^ j ; ~ j n = cos n sin n ^ i + sin n sin n ^ j + cos n ^ k and ~ a m ~ j n = cos ( r m n ) sin n : (37) 5.2 adapt.m The le adapt.m p erforms the p o w er, gradien t, and w eigh t adjustmen t calculations. The algorithm adapts the an tenna w eigh ts for the duration sim time a length of time en tered b y the user. Since eac h iteration of the algorithm co v ers P int =20 s, the n um b er of w eigh t adjustmen t iterations p erformed is sim time / P int A t the start of an iteration, 40
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x y z ~ a 1 ~ a 2 ~ a 3 ~ j 1 jammer r Figure 5. Example of ~ a m and ~ u n motion vector is c hec k ed for a nonzero v alue. If it is nonzero, the arra y has roll, pitc h, and/or y a w motion and the le motion.m is called. This le up dates the phase matrix C due to the arra y motion. Details of motion.m are describ ed in the next section. With the phase matrix comp onen ts calculated either from setup.m or from motion.m the oracle w eigh ts are computed as in equation (10) 2666666664 A 2 A 3 ... A N ant 3777777775 or = C 1 r educed 2666666664 11 ... 1 3777777775 or (11) 2666666664 A 2 A 3 ... A N ant 3777777775 or = C r educed 2666666664 11 ... 1 3777777775 41
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where the subscript or indicates or acle w eigh ts. With the oracle w eigh ts kno wn, the output p o w er corresp onding to the oracle w eigh ts is calculated using equation (21), p or acl e ( t r ; A or ) = ( xA or )( xA or ) y : The condition n um b er of C r educed is calculated p er iteration using the cond function in Matlab. T o calculate the gradien t of the p o w er surface, the arra y output signal and arra y output p o w er m ust b e calculated rst. As stated in section 4.3, the arra y output signal ma y con tain noise as w ell as jammers. The sim ulated output signal of the arra y is N ( t r ) A + J ( t r ) C ( t r ) A = x ( t r ) A where the initial v alues of an tenna w eigh ts, A declared in setup.m are used for the rst iteration. During eac h iteration, 400 samples of arra y output are sim ulated, t r = ; 2 ; : : : ; 400 F or eac h v alue of t r the noise, N ( t r ), is a 1 N ant v ector of white noise samples sim ulated with the randn Matlab function. The Matlab randn function generates zero mean, unit v ariance white noise samples. The v ariance of the noise m ust b e mo died as describ ed in section 4.3 to sim ulate a certain bac kground noise lev el in the output signal of the arra y This is accomplished b y m ultiplying eac h noise sample at eac h an tenna b y the scalar q noise floor Nant Com bining all Nant an tenna outputs results in a noise signal at the output of the arra y with v ariance equal to noise floor The 400 noise samples are added to as man y jammer comp onen ts to form the input to eac h an tenna. These signals are scaled b y the an tenna w eigh ts then summed together to form the arra y output. Note that the phase matrix C and the an tenna w eigh t v ector A are eac h mo died only once p er iteration. A snapshot of the arra y and jammer arriv al directions is tak en at the start of ev ery algorithm iteration and held constan t for the duration of one iteration. Recall that the ob jectiv e of the algorithm is to minimize the output p ower of the arra y to within 5 dB of the bac kground noise lev el. Eac h of the 400 samples of instan taneous output p o w er 42
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p er iteration is computed as ( xA ) ( xA ) : These 400 samples of instan taneous p o w er are a v eraged to get the a v erage output p o w er of the arra y P av This a v erage p o w er represen ts the p o w er in tegrated o v er a P int = 20 s in terv al, one iteration of the algorithm. Finally the gradien t comp onen ts of the output p o w er are computed as in equations (33) and (36), @ P @ w m = 2 < < A m x m ; xA > w m and @ P @ q m = 2 = < A m x m ; xA > : The term A m x m represen ts the a v erage signal of 400 samples of output from the m th an tenna, A m x m = N A m + J C A m where N is the a v erage noise signal for the curren t lo op iteration. The term xA is the a v erage output signal of the arra y Both A m x m and xA are scalars, they are m ultiplied together and scaled to form 2 < A m x m ; xA > The gradien t comp onen ts are the real (scaled b y the recipro cal of eac h w eigh t magnitude) and imaginary parts of this scaled pro duct. In the Matlab sim ulation, the gradien t comp onen ts o ccup y a 2 ( Nant -1) matrix 264 @ P @ w 2 @ P @ w 3 @ P @ w N ant @ P @ q 2 @ P @ q 3 @ P @ q N ant 375 : The v ector of an tenna w eigh ts is adjusted using the steep est descen t algorithm as follo ws. F rom equation (27), the amoun t of w eigh t c hange is calculated as W = r P = (target p o w er P av ) r P T r P r P where target p o w er is tak en to b e the constan t noise floor W represen ts the w eigh ts as a 2 ( N ant 1) v ector 264 w 2 w 3 w N ant q 2 q 3 q N ant 375 : 43
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Finally the w eigh t v ector is up dated as W W + W : This completes one iteration of the algorithm. The pro cess rep eats un til the n um b er of iterations determined b y the user en tered v ariable sim time has elapsed. The algorithm is designed to reduce the arra y output p o w er to within 5 dB of the target lev el, noise floor or less. If the output p o w er ev er exceeds this lev el, the a v erage n um b er of iterations it tak es for the algorithm to return the output p o w er to within 5 dB of the noise floor lev el is rep orted on the output graphs of adapt.m The le adapt.m generates 3 output plots: the arra y output p o w er v ersus time, the oracle p o w er v ersus time, and the condition n um b er of C r educed v ersus iteration n um b er. 5.3 motion.m If the arra y platform mo v es with roll, pitc h, or y a w, the le motion.m is called from eac h iteration of adapt.m to up date the phase matrix C As the arra y rotates, the arriv al angles of the jammers c hange from their original v alues. The v elo cit y v ector of a jammer with resp ect to the arra y is ~ v = ~ ~ R where ~ is the angular v elo cit y of the arra y ~ = roll ^ i + pitch ^ j + yaw ^ k and ~ R is the v ector describing the lo cation of a jammer direction of arriv al ( represen ts azim uth arriv al direction, represen ts elev ation arriv al direction) expressed in Cartesian co ordinates with resp ect to the cen ter of the arra y ~ R = sin cos ^ i + sin sin ^ j + cos ^ k : 44
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The comp onen ts of ~ v are then ~ ~ R = ( pitch cos yaw sin sin ) ^ i + ( yaw sin cos roll cos ) ^ j + ( roll sin sin pitch sin cos ) ^ k but the v elo cit y of the arra y is also the c hange in time of the arriv al directions of the jammers ~ v = d ~ R =dt = sin sin ^ i + cos cos ^ i + sin cos ^ j + sin cos ^ j sin ^ k : Ev aluating the ^ k comp onen ts on b oth sides of d ~ R =dt = ~ ~ R leads to the expression for the c hange in the elev ation angle due to roll, pitc h, and y a w motion = roll sin + pitch cos : (38) Ev aluating the ^ j comp onen ts on b oth sides of d ~ R =dt = ~ R leads to the expression for the c hange in the azim uth angle = yaw roll cot cos pitch cot sin : (39) Equations (38) and (39) are the c hange in jammer arriv al angles with resp ect to time. The le motion.m is called once p er iteration of adapt.m and eac h iteration is P int or 20 seconds in duration. The amoun ts of c hange in the jammer arriv al directions during an iteration are then P int and P int The v alues of and are up dated b y the amoun ts + P int and + P int : 45
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These new v alues for and are used to up date the comp onen ts of the phase matrix C for use in the p o w er, gradien t, and w eigh t calculations in the next iteration of adapt.m as the arra y rotates. 46
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Chapter 6 Algorithm and Oracle P erformance The HSB gradien t algorithm and oracle p erformance will b e presen ted for v arious input scenarios in this c hapter. The user can congure man y inputs when running the algorithm suc h as the n um b er of jammers impinging on the arra y the jammer arriv al angles, the jammer frequencies, the n um b er of p eripheral an tennas, the arra y conguration, the input noise lev el, arra y motion of roll, pitc h, or y a w, and an y initial arra y oset in the roll direction. The gures included in this c hapter are represen tativ e of the p erformance of the HSB gradien t algorithm for a planar circular arra y V arious input scenarios are included to sho w the robustness of the HSB gradien t algorithm. 6.1 Summary of P erformance This section will pro vide a brief summary of the p erformance of the algorithm and the oracle. The p erformance will b e rep orted for 3 categories: no motion, y a w motion, and full (roll, pitc h, and y a w) motion. The input to the arra y is mo deled as high p o w er jammers and lo w p o w er noise. The GPS signals are not mo deled explicitly although they w ould app ear similar to the noise. The jammer amplitudes are represen ted as v oltages and the noise amplitude is represen ted in w atts. During eac h algorithm iteration, the arra y output p o w er is sampled ev ery = 50 ns and = 400 of these samples are a v eraged to represen t the a v erage arra y output p o w er. The a v eraging in terv al, one algorithm iteration, is 20 s. The gradien t of this a v erage output p o w er is calculated and the an tenna w eigh ts are up dated p er iteration of the HSB gradien t algorithm. 47
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6.1.1 No Motion The algorithm reduces the arra y output p o w er to within 5 dB of the 1 dB noisero or more than 99.8% of the time. The algorithm tak es, on a v erage, 10 iterations or 200 s to reduce the initial arra y output p o w er to within 5 dB of the noisero or. The few er the n um b er of jammers, the faster the initial con v ergence. The p erformance do es not deteriorate with an increasing n um b er of jammers, as long as N j am < N ant nor do es p erformance suer for v arying jammer amplitudes, up to 50 dB. The algorithm main tains the output p o w er b elo w the noisero or after the initial con v ergence. The oracle reduces the arra y output p o w er to within 5 dB of the 1 dB noisero or 99.9999998% of time when the jammer arriv al directions do not result in a confounding conguration or near-confounding conguration (as determined b y the condition n um b er of the phase matrix C r educed section 3.3.2). 6.1.2 Y a w Motion The p erformance of the algorithm, sim ulated with y a w motion (500 /second or less), is only sligh tly w eak er than the results when there is no motion. The algorithm reduces the arra y output p o w er to within 5 dB of the 1 dB noisero or more than 99.3% of the time and initial con v ergence tak es, on a v erage, 10 iterations or 200 s. The p erformance do es not deteriorate with an increasing n um b er of jammers or increasing y a w rate, as long as N j am < N ant and v arying jammer amplitudes, up to 50 dB, do not ha v e an adv erse eect on p erformance. When the output p o w er is greater than 5 dB ab o v e the noisero or, the n um b er of algorithm iterations to reduce the p o w er b elo w this lev el increases with the n um b er of jammers but is unaected b y the y a w rate and on a v erage is less than 3 iterations or 60 s. The oracle reduces the arra y output p o w er to within 5 dB of the 1 dB noisero or 99.9999998% of time when the jammer arriv al directions do not result in a confounding conguration or near-confounding conguration. 48
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6.1.3 F ull Motion The p erformance of the algorithm and oracle with full roll, pitc h, and y a w motion mirror the p erformance of the algorithm and oracle with only y a w motion. The maxim um motion rates sim ulated w ere 180 /second for roll, 180 /second for pitc h, and 500 /second for y a w. 6.2 Catalog of Output The gures in this section sho w the output of the Matlab le adapt.m whic h sim ulates the HSB gradien t algorithm. Eac h gure con tains three plots: the arra y output p o w er due to the HSB gradien t algorithm w eigh ts, the arra y output p o w er due to the oracle w eigh ts, and the condition n um b er of the phase matrix C r educed The arra y used in these sim ulations consisted of 7 an tennas in a planar conguration of 1 cen tral, xed-w eigh t an tenna and 6 p eripheral an tennas equally spaced around the cen tral an tenna at 0 ; 60 ; 120 ; 180 ; 240 ; 300 6.2.1 No Motion F or the follo wing plots no arra y motion is sim ulated. The n um b er of jammers inciden t on the arra y is indicated in the plot title. The jammers arriv e in the plane of the arra y with elev ation angle = 90 and azim uth angles = 120 ; 330 ; 25 ; 204 ; 0 and 108 The jammer amplitudes are sim ulated with constan t amplitudes of 100 V (or 40 dB). The noise added to the input of the arra y is 1.25 (or 1 dB) and the algorithm attempts to reduce the output p o w er lev el to this lev el. The n um b er of iterations sim ulated is 25000, a duration of 0.5 seconds. As the plots sho w, b oth the oracle and algorithm successfully reduce the output p o w er of the arra y to the noisero or. 49
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T able 3. T ypical Data for No Motion Cases. noisero or=1 dB No Motion, 25000 Iterations (.5 sec) Maxim um J 1 = = J N j am Output P o w er > # iterations to return Condition = 40 dB Allo w ed Lev el b elo w allo w ed lev el Num b er # jammers algorithm oracle algorithm oracle 2 0.104% 1.6e-07% 3 0 1.27 4 0.032% 1.6e-07% 9 0 3.54 6 0.036% 1.6e-07% 10 0 14.02 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.036002% of time power exceeds allowed level Average number of iterations to return below allowed level is 100 rollrate;0 yawrate; 0 pitchrate (deg/sec) timepower integral (dB)Hilbert Space Based Gradient 6-jammer minimization with 7 antennas algorithm output power 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 1.6001e-007% of time power exceeds allowed level Average number of iterations to return below allowed level is 0power integral (dB)time oracle power output 0 0.5 1 1.5 2 x 10 4 0 5 10 15 maximum condition number is 14.0185 iterationcondition number condition number Student Version of MATLAB Figure 6. Six Jammers, No Motion. 50
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6.2.2 Y a w Motion The input conditions for the plots in this section are the same as in the previous section except no w the arra y is sim ulated with a y a w rate of 360 = second. The n um b er of jammers inciden t on the arra y is indicated in the plot title. The jammers arriv e in the plane of the arra y = 90 and initially with = 120 ; 330 ; 25 ; 204 ; 0 and 108 The jammer amplitudes are all equal to 100 V (40 dB). The noise added to the input of the arra y is 1 dB and the algorithm attempts to reduce the output p o w er to this lev el. As the plots sho w, b oth the oracle and algorithm successfully reduce the output p o w er of the arra y to the noisero or. Note the o ccurrence of three near-confounding congurations in gure 7. Although the oracle sho ws dicult y annihilating the jammers at these near-confounding congurations, the HSB gradien t algorithm is able to minimize the jammers at these same congurations. Also, when the output p o w er exceeds 5 dB ab o v e the noisero or, the HSB algorithm requires less w eigh t up dates to restore the output p o w er to within 5 dB of the noisero or than the oracle, see T able 4. T able 4. T ypical Data for Y a w Motion Cases. noisero or=1 dB Y a w Motion (360 /sec), 25000 Iterations (.5 sec) Maxim um J 1 = = J N j am Output P o w er > # iterations to return Condition = 40 dB Allo w ed Lev el b elo w allo w ed lev el Num b er # jammers algorithm oracle algorithm oracle 2 0.7% 1.6e-07% 2 0 3.13 4 0.504% 1.6e-07% 3 0 4.84 6 0.564% .924% 2 34 85590.32 51
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.56402% of time power exceeds allowed level Average number of iterations to return below allowed level is 20 rollrate;360 yawrate; 0 pitchrate (deg/sec) timepower integral (dB)Hilbert Space Based Gradient 6jammer minimization with 7 antennas algorithm output power 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.92404% of time power exceeds allowed level Average number of iterations to return below allowed level is 34power integral (dB)time oracle power output 0 0.5 1 1.5 2 x 10 4 0 2 4 6 8 x 10 4 maximum condition number is 85590.3243 iterationcondition number condition number Student Version of MATLAB Figure 7. Six Jammers, Y a w Motion, Near Confounding Conguration. 52
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6.2.3 Dieren t Jammer Amplitude Lev els The input conditions for the plots in this section are similar to those of the previous section except that no w the jammer amplitudes are at dieren t lev els. As noted (section 4.4), this do es not aect the oracle solution. The jammers arriv e in the plane of the arra y as b efore, = 90 and = 120 ; 330 ; 25 ; 204 ; 0 and 108 but no w the jammer amplitude v oltages are 316 ; 31 : 6 ; 100 ; 10 ; 10 0 ; 3 16 (50 dB, 30 dB, 40 dB, 20 dB, 40 dB, 50 dB). The arra y is sim ulated with and without motion; this is indicated in the plot caption. T able 5. T ypical Data for No Motion Cases with Dieren t Jammer Amplitudes. noisero or=1 dB No Motion, 25000 Iterations (.5 sec) Maxim um dieren t jammer Output P o w er > # iterations to return Condition amplitudes Allo w ed Lev el b elo w allo w ed lev el Num b er # jammers algorithm oracle algorithm oracle 2 (50, 30 dB) 0.244% 1.6e-07% 2 0 1.27 4 (50, 30, 40, 20 dB) 0.084% 1.6e-07% 3 0 3.54 6 (50, 30, 40, 20, 40, 50 dB) 0.128% 1.6e-07% 3 0 14.02 T able 6. T ypical Data for Y a w Motion Cases with Dieren t Jammer Amplitudes. noisero or=1 dB Y a w Motion (360 /sec), 25000 Iterations (.5 sec) Maxim um dieren t jammer Output P o w er > # iterations to return Condition amplitudes Allo w ed Lev el b elo w allo w ed lev el Num b er # jammers algorithm oracle algorithm oracle 2 (50, 30 dB) 1.956% 1.6e-07% 2 0 3.13 4 (50, 30, 40, 20 dB) 0.988% 1.6e-07% 2 0 4.84 6 (50, 30, 40, 20, 40, 50 dB) 3.088% .924% 3 34 85590 53
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.084004% of time power exceeds allowed level Average number of iterations to return below allowed level is 30 rollrate;0 yawrate; 0 pitchrate (deg/sec) timepower integral (dB)Hilbert Space Based Gradient 4jammer minimization with 7 antennas algorithm output power 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 1.6001e007% of time power exceeds allowed level Average number of iterations to return below allowed level is 0power integral (dB)time oracle power output 0 0.5 1 1.5 2 x 10 4 0 2 4 6 maximum condition number is 3.5391 iterationcondition number condition number Student Version of MATLAB Figure 8. F our Jammers 50 dB, 30 dB, 40 dB, 20 dB; No Motion. 54
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 3.0881% of time power exceeds allowed level Average number of iterations to return below allowed level is 30 rollrate;360 yawrate; 0 pitchrate (deg/sec) timepower integral (dB)Hilbert Space Based Gradient 6jammer minimization with 7 antennas algorithm output power 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.92404% of time power exceeds allowed level Average number of iterations to return below allowed level is 34power integral (dB)time oracle power output 0 0.5 1 1.5 2 x 10 4 0 2 4 6 8 x 10 4 maximum condition number is 85590.3243 iterationcondition number condition number Student Version of MATLAB Figure 9. Six Jammers 50, 30, 40, 20, 40, 50 dB; Y a w Motion. 55
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6.2.4 F ull Motion The arra y is sim ulated with roll=20 = second, pitc h=50 = second, and y a w=180 = second. The jammers arriv e with dieren t amplitudes, indicated in the plot caption and the arriv al directions as in previous sections, = 90 and = 120 ; 330 ; 25 ; 204 ; 0 and 108 The noisero or the algorithm attempts to reac h is 1 dB. T able 7. T ypical Data for F ull Motion Cases. noisero or=1 dB F ull Motion, 25000 Iterations (.5 sec) Maxim um dieren t jammer Output P o w er > # iterations to return Condition amplitudes Allo w ed Lev el b elo w allo w ed lev el Num b er # jammers algorithm oracle algorithm oracle 2 (50, 30 dB) 0.784% 1.6e-07% 2 0 3.063 4 (50, 30, 40, 20 dB) 0.516% 1.6e-07% 2 0 4.831 6 (50, 30, 40, 20, 40, 50 dB) 1.664% 0.792% 2 41 62644 56
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 0.51602% of time power exceeds allowed level Average number of iterations to return below allowed level is 220 rollrate;180 yawrate; 50 pitchrate (deg/sec) timepower integral (dB)Hilbert Space Based Gradient 4jammer minimization with 7 antennas algorithm output power 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 20 40 60 1.6001e007% of time power exceeds allowed level Average number of iterations to return below allowed level is 0power integral (dB)time oracle power output 0 0.5 1 1.5 2 x 10 4 0 2 4 6 maximum condition number is 4.8309 iterationcondition number condition number Student Version of MATLAB Figure 10. Six Jammers 50, 30, 40, 20, 40, 50 dB; F ull Motion. 57
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6.3 A Note on Out-of-Plane Jammers F or a planar arra y an out-of-plane jammer is mathematically equiv alen t to an in-plane jammer at a lo w er frequency A jammer at w a v elength 1 arriving at the arra y from a direction ( ; ) generates an arra y output J C A = J 1 e i! 1 t [ e i 2 1 ~ a 1 ~ j 1 e i 2 1 ~ a N ant ~ j 1 ] 2666666664 1 w 2 e iq 2 ... w N ant e iq N ant 3777777775 : Recall, from equation (37), that the arra y output from this jammer impinging on the m th an tenna is J 1 e i 2 1 r cos( r m ) sin w m e iq m : W e can dene a new, longer w a v elength 0 = 1 = sin so that the jammer signal at the arra y output is equiv alen t to J 1 e i 2 0 r cos ( r m ) w m e iq m in the plane of the arra y ( = 90 ). 6.4 Noise Imm unit y The ob jectiv e of this section is to in v estigate ho w far ab o v e the noisero or an y additional input noise can b e b efore the algorithm fails. Recall the desired GPS signals are on the order of or b elo w the noisero or but are spread sp ectrum mo dulated. As long as the arra y radiation pattern n ulls do not coincide with the arriv al directions of the GPS signals, the latter will pass through the arra y and b e demo dulated accurately If the input noise and jammer amplitudes are on the order of the noisero or or are b elo w the noisero or, the algorithm has nothing to do; the output is already at the noisero or. The algorithm adaptiv ely reduces the output p o w er of the arra y only when the input to the arra y assumed to b e primarily 58
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from undesired jammer signals, is ab o v e the noisero or. V arious cases w ere run to test b oth the HSB gradien t algorithm and the oracle for noise imm unit y The arra y used in these sim ulations consisted of 7 an tennas in a planar conguration of 1 cen tral, xed-w eigh t an tenna and 6 p eripheral an tennas at r = 0 ; 60 ; 120 ; 180 ; 240 ; 300 The jammers arriv e in the plane of the arra y = 90 and = 120 ; 330 ; 25 ; 204 ; 0 and 108 with dieren t amplitudes of 316 (50 dB),31.6 (30 dB), 100 (40 dB),10 (20 dB),100 (40 dB), 316 (50 dB). The arra y is in motion with y a w= 360 /second. Three categories of cases w ere run in whic h the algorithm attempted to reduce the arra y output p o w er to within 5 dB of noisero ors of 0 dB, 1 dB, and 3dB. The noise added as input to the arra y ab o v e the noisero or is indicated in the tables. The results sho w that the algorithm is quite sensitiv e to noise added as input ab o v e the noisero or. F ailure is said to o ccur when the algorithm cannot reduce the output p o w er b elo w the noisero or. An in teresting feature is that the algorithm has a failure rate that decreases with the n um b er of inciden t jammers. With more n ulls in the arra y radiation pattern, more of the input noise can b e atten uated. T able 8. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=0 dB). noisero or=0 dB Njam 2 4 6 added noise algorithm oracle algorithm oracle algorithm oracle 5 dB 36.113% 1.6e-07% 17.373% 1.6e-07% 23.345% 100% 6 dB 51.85% 1.6e-07% 31.873% 1.6e-07% 37.213% 100% 8 dB 80.563% 96.172% 59.586% 96.172% 62.342% 100% indicates near-confounding conguration T able 9. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=1 dB). noisero or=1 dB Njam 2 4 6 added noise algorithm oracle algorithm oracle algorithm oracle 5 dB 20.709% 1.6e-07% 9.692% 1.6e-07% 14.633% 100% 6 dB 35.153% 1.6e-07% 17.613% 1.6e-07% 22.677% 100% 8 dB 71.207% 8.976% 46.722% 8.976% 48.794% 100% indicates near-confounding conguration 59
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T able 10. Algorithm F ailure Rate with Increasing Noise Lev els (noisero or=3 dB). noisero or=3 dB Njam 2 4 6 added noise algorithm oracle algorithm oracle algorithm oracle 5 dB 5.284% 1.6e-7% 2.276% 1.6e-7% 4.548% 99.672% 6 dB 10.64% 1.6e-7% 4.828% 1.6e-7% 7.368% 99.708% 8 dB 34.381% 1.6e-7% 16.621% 1.6e-7% 20.705% 99.98% 10 dB 66.139% 7.12% 45.062% 7.12% 48.41% 100% indicates near-confounding conguration 60
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Chapter 7 Summary and F uture W ork Directions The main con tributions of this w ork are as follo ws. The exact comp onen ts of the gradien t are calculated using inner pro ducts of signals readily a v ailable at the individual an tenna outputs and at the arra y output. The computation is fast and all of the gradien t comp onen ts are calculated p er iteration. The condition n um b er is used to indicate the lik eliho o d of the existence of the optimal an tenna w eigh ts. A high condition n um b er indicates a confounding conguration; one without stable oracle w eigh ts. The abilit y to construct confounding congurations is presen ted. Researc h funding is b eing pursued to study other metho ds to construct confounding congurations as w ell as the geometric in terpretations, if an y of them. The oracle solution calculates the exact an tenna w eigh ts to n ull the jammer comp onen t of the arra y output. The w eigh ts dep end only up on the presen t conguration of an tennas and jammers and so pro vide a c hec k of the eectiv eness of w eigh ts at an y conguration from an y adaptiv e algorithm. The HSB gradien t algorithm, implemen ted in Matlab, successfully minimizes the p o w er output of the arra y Areas for future w ork concern the in v estigation of an adaptiv e accum ulation time for the calculation of the a v erage arra y output p o w er. As the arra y motion increases, the accum ulation of instan taneous arra y output p o w er measuremen ts m ust decrease to capture the dynamic information of the non-stationary arra y en vironmen t. 61
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References [1] Rob ert E. Collin. A ntennas and R adiowave Pr op agation section 3.6, pages 107{121. McGra w-Hill Series in Electrical Engineering. McGra w-Hill, Inc., New Y ork, New Y ork, 1985. [2] Bisw a Nath Datta. Numeric al Line ar A lgebr a and Applic ations c hapter 7, pages 316{ 324. Bro oks/Cole Publishing Compan y P acic Gro v e, California, 1996. [3] Bisw a Nath Datta. Numeric al Line ar A lgebr a and Applic ations section 7.9, pages 348{351. Bro oks/Cole Publishing Compan y P acic Gro v e, California, 1996. [4] P aul G. Flikk ema. Spread sp ectrum tec hniques for wireless comm unications. IEEE Signal Pr o c essing Magazine 14(3):26{28, Ma y 1997. [5] An ton Gecan, P aul G. Flikk ema, and Arth ur Da vid Snider. Jammer cancellation with adaptiv e arra ys for gps signals. IEEE Southe astc on Confer enc e Pr o c e e dings pages 320{323, 1996. [6] An ton S. Gecan and Mic hael D. Zolto wski. P o w er minimization tec hniques for gps n ull steering an tennas. pr eprint 1995. [7] Jr. J. J. Spilk er. Gps signal structure and p erformance c haracteristics. Navigation 25(2):121{146, July 1978. [8] Harold J. Larson. Intr o duction to Pr ob ability The ory and Statistic al Infer enc e section 5.4, pages 178{179. Wiley Series in Probabilit y and Mathematical Statistics. John Wiley and Sons, Inc., New Y ork, New Y ork, 1969. [9] John Litv a and Titus Kw ok-Y eung Lo. Digital Be amforming in Wir eless Communic ations section 2.4, pages 28{34. The Artec h House Mobile Comm unications Series. Artec h House, Inc., Norw o o d, Massac h usetts, 1996. [10] Rob ert A. Monzingo and Thomas W. Miller. Intr o duction to A daptive A rr ays section 3.3, page 89. A Wiley-In terscience Publication. John Wiley and Sons, New Y ork, 1980. [11] Rob ert A. Monzingo and Thomas W. Miller. Intr o duction to A daptive A rr ays section 4.2, pages 162{178. A Wiley-In terscience Publication. John Wiley and Sons, New Y ork, 1980. [12] Rob ert A. Monzingo and Thomas W. Miller. Intr o duction to A daptive A rr ays c hapter 5, pages 217{292. A Wiley-In terscience Publication. John Wiley and Sons, New Y ork, 1980. 62
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[13] Roger L. P eterson, Ro dger E. Ziemer, and Da vid E. Borth. Intr o duction to Spr e ad Sp e ctrum Communic ations section 2.3, 3.3, pages 52{60, 113{115. Pren tice-Hall, Inc., Upp er Saddle Riv er, New Jersey 1995. [14] John G. Proakis. Digital Communic ations section 13.2, pages 698{699,707{709 McGra w-Hill Series in Electrical and Computer Engineering. Comm unications and Signal Pro cessing. McGra w-Hill, Inc., New Y ork, New Y ork, third edition, 1995. [15] Jr. R. T. Compton. The p o w er-in v ersion adaptiv e arra y: Concept and p erformance. IEEE T r ansactions on A er osp ac e and Ele ctr onic Systems 15(6):803{814, No v em b er 1979. [16] Gilb ert Strang. Intr o duction to Line ar A lgebr a section 3.3, 9.2, pages 125{136,386{388 W ellesley-Cam bridge Press, W ellesley Massac h usetts, 1993. 63
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App endices 64
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App endix A : Sym b ol Glossary Sym b ol Meaning i p 1 sampling in terv al, 50 ns r sampling index n um b er of samples to a v erage t r time c sp eed of ligh t, 3 10 8 meters/second frequency in radians/second f frequency in Hertz w a v elength in meters r distance in meters b et w een reference an tenna and a p eripheral an tenna m indicates one an tenna in an arra y N ant total n um b er of an tennas in an arra y ~ a m p osition v ector of an tenna # m w magnitude of an tenna w eigh t q phase of an tenna w eigh t A m complex w eigh t of an tenna # m w m e iq m GP S v ariable represen ting GPS signal N ( t ) random noise input to an tennas n jammer index N j am total n um b er of jammers impinging on arra y ~ j n v ector indicating direction of arriv al of jammer # n ~ v n unit v ector indicating direction of arriv al of jammer # n scaled b y 2 n J n amplitude of jammer # n 65
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App endix A (Con tin ued) Sym b ol Meaning n azim uth angle of jammer # n n elev ation angle of jammer # n ~ x ( t ) generic input to arra y C N j am N ant matrix of exp onen tial phase dierences b et w een jammers and an tennas ^ i direction unit v ector ^ j direction unit v ector p erp endicular to ^ i d ( t ) desired arra y output signal e ( t ) error b et w een desired and actual signals r xd cross-correlation v ector of signals x and d R xx auto correlation matrix of signal x time index k iteration index p ( ; A ) instan taneous arra y output p o w er at time as a function of an tenna w eigh ts A P ( k ; A ) a v erage arra y output p o w er at iteration k as a function of an tenna w eigh ts A T n um b er of instan taneous p o w er samples a v eraged during eac h iteration W column of an tenna w eigh ts [ w 1 w 2 w N ant q 1 q 2 q N ant ] T gradien t step size OR mean of random v ariable 2 v ariance of random v ariable < real part = imaginary part 66
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App endix B : Matlab Source Co de setup.m % This script is a tool for initializing the adapt simulation. % Not all parameters listed herein are used in any given version % of the simulation % September 24, 2004 deg_to_rad = pi/180; jmath = sqrt(-1); more off; format short; Nant = 7; % number of antennas in array array_radius = .1; %radius of array (meters) A=zeros(Nant,2); %Set up weights A(:,1)=ones(Nant,1);A(:,2)=zeros(Nant,1);a=zeros(Nant,1);Ann=zeros(Nant,2); %weights for noise free simulation Ann(:,1)=ones(Nant,1);Ann(:,2)=zeros(Nant,1);ann=zeros(Nant,1);a_min = 0.1; a_max = 10; % limits on weight mags step_control = 1; %fraction of gradient correction to be implemented step_control_nn = 1; noise_floor = 1.25;%target value for power; 67
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App endix B (Con tin ued) %background noise level ( 1 = 0 dB) speed_of_light = 3.0e+08; %speed of light, m/s P_int = 20e-6; %integration time for power computation LOfreq = 1.575e+09; %local oscillator frequency omegaLO = 2*pi*LOfreq; disp('How many jammers? Enter 1,2,3,...,9 to use these preset values. '); Njams=input('Otherwise, enter -1 [minus one]. '); theta_jam =[90,90,90,90,90,90,90,75 ,100 ] deg_to_rad; theta_jam(Njams+1:9)=[];phi_jam = 359*rand(1,9)*deg_to_rad; %phi_jam = [120,330,25,204,0,108,35 1,13 6,22 5]* deg_ to_r ad; phi_jam(Njams+1:9)=[];phi_init=phi_jam;del_freq = [0 0 0 0 0 0 0 0 0]; del_freq(Njams+1:9)=[];jam_freq = LOfreq + del_freq; omega_jam = jam_freq*2*pi; psijam = [0 0 0 0 0 0 0 0 0]' deg_to_rad; psijam(Njams+1:9)=[];%J=[100,100, 100,100,100, 100,100,100,100]; J=[316,31.6, 100,10,100, 316,100,100,100]; J(Njams+1:9)=[];Ajamdot = [0 0 0 0 0 0 0 0 0]; Ajamdot(Njams+1:9)=[];disp('Enter simulation time in seconds; recommend 15e-3; note that') 68
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App endix B (Con tin ued) disp('simulations take ~10-20 seconds of computer time per millisec') disp('of simulation time.') sim_time=input('Simulati on time in sec.; '); disp('Enter vehicle angular velocity, deg/sec.'); roll=input('roll rate: '); %+y onto +z pitch=input('pitch rate: '); % +z onto -x yaw=input('yaw rate: '); %+y onto +x motion_vector=[roll, pitch, yaw]; beta=input('Set initial rotation angle (deg): ') %-roll disp('Data Summary') disp('Number of antennae') disp(Nant)disp('Number of jammers; initial theta''s; initial phi''s') disp(Njams)disp(theta_jam/deg_to_ra d) disp(phi_jam/deg_to_rad)disp('relative frequencies; phases; amplitudes; amplitude rates') disp(del_freq)disp(psijam'/deg_to_rad)disp(J)disp(Ajamdot)disp('roll, pitch, yaw rates; initial rotation') disp(motion_vector)motion_vector=motion_vec tor* deg_ to_ rad; roll=roll*deg_to_rad;... pitch=pitch*deg_to_rad; yaw=yaw*deg_to_rad; disp(beta)beta=beta*deg_to_rad; 69
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App endix B (Con tin ued) [az,el,rr]=cart2sph((sin (the ta_j am) .*co s(ph i_j am)) ... (sin(theta_jam).*sin(ph i_ja m)*c os( beta )+.. cos(theta_jam)*sin(beta )),. .. (cos(theta_jam)*cos(bet a) ... sin(theta_jam).*sin(phi _jam )*si n(b eta) )); phi_jam= az; theta_jam = pi/2 el; disp('Rotated jammer theta"s, phi"s'); disp(theta_jam/deg_to_ra d) disp(phi_jam/deg_to_rad)% Set up initial array manifold vector for each source %(relative phases of each jammer at each antenna). zeta = array_radius*omega_jam.*s in( thet a_ja m)/ spee d_of _li ght; gamma = linspace(0, 2*pi*(Nant-2)/(Nant-1), (Nant-1)); for m=2:Nant, for n=1:Njams, C(n,1) = 1; C(n,m)=exp(jmath*zeta(n) *co s(ph i_ja m(n )-ga mma( m-1 ))); end endadapt.m%adapt.m%% This script computes the average array output power and % its exact gradient. The antenna weights are adapted 70
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App endix B (Con tin ued) % and the oracle weights are computed. % Noise is simulated as input to each antenna with % zero mean and variance equal to the noise_floor. % Plots of the array output power are generated % corresponding to the algorithm weights and the % oracle weights. The condition number of the % phase matrix is also plotted. Niterations = floor(sim_time/P_int); time = zeros(Niterations,1); seed = input('Enter a random number seed. ') randn('seed', seed); % Set dynamic variables flag. moving=norm(motion_vecto r); modulator = Ajamdot'*Ajamdot; %initialize variables/counters/flag s cntr=0;m=1;n=0;count = 0; sp=1;kj=1;oj=1;acc=1;occ=1;below=0;under=0;convg=0; 71
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App endix B (Con tin ued) its=0;prev_sample=0;ops=0; %oracle previous sample allowed_level=10^((5+10* log1 0(no ise _flo or)) /10 ); %Compute power plus noise for i = 1:Niterations, if moving>0; motion; %if array in motion, call motion.m end;if modulator>0; modulate; %if modulated jammers, call modulate.m end;a = A(:,1).*exp(jmath*A(:,2) ); weight_prev=abs(a);sig_only=J*C*a;Cred=C(:,2:Nant);a_opt = Cred\(-C(:,1)); %oracle solution a_opt = [1; a_opt]; %add central antenna weight sig_opt = J*C*a_opt; for integral = 1:400, %create noise samples nois_sample=sqrt(noise_fl oor /Nan t)*r and n(1, Nant ); nois_vec(integral,:)=nois _sa mple ; nois_only=nois_sample*a;Vn=sum(nois_sample);pn(integral)=Vn*conj(Vn);V = sig_only+nois_only; nois_opt=nois_sample*a_op t; 72
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App endix B (Con tin ued) V_opt = sig_opt+nois_opt; pwr(integral)=V*conj(V);pwr_opt(integral)=V_opt*c onj (V_o pt); endnois_ran=sum(nois_vec)/s qrt( 400 ); %average noise %average power signals powerno(i)=sum(pn); %noise only power power_hsb(i)=sum(pwr)/40 0; power_orac(i)=sum(pwr_op t)/4 00; cn(i) = cond(Cred); %condition number of Cred %count iterations to reconverge below allowed level if (power_hsb(i) >= allowed_level) kj=kj+1; elseif (prev_sample >= allowed_level) if (power_hsb(i) < allowed_level) nosta(acc)= kj; %number of iterations to adapt acc=acc+1;kj=1;below=1; end endif (below == 0) %power never went below allowed level nosta = 0; endprev_sample=power_hsb(i) ; %count iterations for oracle to reconverge below allowed level if (power_orac(i) >= allowed_level) 73
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App endix B (Con tin ued) oj=oj+1; elseif (ops >= allowed_level) if (power_orac(i) < allowed_level) rto(occ)=oj;occ=occ+1;oj=1;under=1; end endif (under==0) %oracle power never went below allowed level rto=0; endops=power_orac(i); %ops is previous oracle sample rel_w_err(i)=norm(a_opt a)/norm(a_opt); time(i) = P_int*i; sample(i)=i;%Compute the gradient components for k=2:Nant, a_grad = zeros(Nant,1); a_grad(k) = A(k,1)*exp(jmath*A(k,2)); f=J*C*a_grad + nois_ran*a_grad; % first term in
g=J*C*a + nois_ran*a; G=2*conj(f)*g;grad(k,1) = real(G)/A(k,1); grad(k,2) = imag(G); grad1(k,i)=grad(k,1);grad2(k,i)=grad(k,2); 74
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App endix B (Con tin ued) end%Update the weight vector delta_wt=step_control*gr ad.. *(noise_floor-power_hsb(i ))/n orm (gra d,'f ro' )^2; magwt(i)=norm(delta_wt,' fro' ); A=A+delta_wt;count = count + 1; for k=2:Nant, % Repair unacceptable weights if (A(k,1)<0) A(k,1)=-A(k,1);A(k,2)=A(k,2)+pi; endif (A(k,1)a_max) A(k,1)=a_max; end endwAmp(:,i) = A(:,1); wPh(:,i) = A(:,2); wOpt(:,i)=a_opt; endif (kj > 1) nosta(acc)=kj; endif (oj > 1) 75
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App endix B (Con tin ued) rto(occ)=oj; endfor rows = 1:size(wOpt,1), %change matrix of complex optimal weights to for cols = 1:size(wOpt,2), %matrix of magnitudes and matrix of phases mwO(rows,cols) = sqrt(real(wOpt(rows,cols) )^2. .. +imag(wOpt(rows,cols))^2 ); pwO(rows,cols) = atan(imag(wOpt(rows,cols) )... /real(wOpt(rows,cols))); end end%Output power in dB power_orac_dB = 10*log10(power_orac); power_hsb_dB = 10*log10(power_hsb); %Failure rate (power > allowed_level) b=find(power_hsb_dB >= 10*log10(allowed_level) ); err=100*size(b)/size(pow er_h sb_d B); orac_over=find(power_ora c_dB >= 10*log10(allowed_level)); err_orac=100*size(orac_o ver) /siz e(p ower _ora c_d B); %Initial adaption iteration count for abc = 1:length(power_hsb_dB), if convg == 0 its = its+1; endif (power_hsb_dB(abc) <= 10*log10(allowed_level)) convg = 1; %algorithm has reduced power below 76
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App endix B (Con tin ued) end % allowed level endits=its-1;%Average re-convergence counts if (acc > 1) str=sum(nosta)/(acc-1); else str=0; %power never went above allowed level endif (occ > 1) osc=sum(rto)/(occ-1); %oracle sample count else osc=0; %oracle never went above allowed level end%Plot output figure;subplot(3,1,1) plot(time, power_hsb_dB) legend('algorithm output power') axis([0 P_int*Niterations -10 60]); rep=[num2str(err) '% of time power exceeds allowed level ']; text(.001,55,rep);rul=['Average number of iterations to return below allowed level is '... num2str(round(str))]; text(.001,47,rul);rots=[num2str(roll/deg_t o_ra d) rollrate;' num2str(yaw/deg_to_rad) ... yawrate; num2str(pitch/deg_to_rad ) '... 77
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App endix B (Con tin ued) pitchrate (deg/sec)']; text(.001,39,rots);xlabel('time')ylabel('power integral (dB)'); title(['Hilbert Space Based Gradient ',num2str(Njams),... '-jammer minimization with ',num2str(Nant), antennas']); subplot(3,1,2) plot(time,power_orac_dB)axis([0 P_int*Niterations -10 60]); rprt=[num2str(err_orac), '% of time power exceeds allowed level ']; text(.001,55,rprt)orl=['Average number of iterations to return below allowed level is '... num2str(round(osc))]; text(.001,47,orl);legend('oracle power output') ylabel('power integral (dB)'); xlabel('time'); subplot(3,1,3) ul=round(max(cn));plot(cn);axis([0 Niterations 0 ul+2]); legend('condition number'); cng=['maximum condition number is ',num2str(max(cn))]; text(.001,ul/2,cng)xlabel('iteration');ylabel('condition number'); 78
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App endix B (Con tin ued) motion.m% motion.m % This script is a tool for updating % the orientation matrix d when % the array is rotating. % August 5, 2003 theta_dot = -roll*sin(phi_jam) ... + pitch*cos(phi_jam); phi_dot = cot(theta_jam).*(-pitch*s in(p hi_j am) ... roll*cos(phi_jam)) + yaw; theta_jam = theta_jam + theta_dot*P_int; phi_jam = phi_jam + phi_dot*P_int; % These can be replaced by % nonlinear functions if desired. % Set up array manifold vector for % each source (relative phases of % each jammer at each antenna). zeta = array_radius*omega_jam.*s in( thet a_ja m)/ spee d_of _li ght; for mm=2:Nant, for nn=1:Njams, 79
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App endix B (Con tin ued) C(nn,1)=1;C(nn,mm)=exp(jmath*zeta( nn) *cos (phi _ja m(nn )-ga mma (mm1))) ; end end 80
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App endix C : GPS Signal Characteristics The Hilb ert space based gradien t algorithm is designed to adjust the w eigh ts of antennas in an arra y to detect signals from the Global P ositioning System (GPS) Satellites while sim ultaneously n ulling jammer signals. The GPS signals originate from one of 24 satellites. This net w ork of satellites pro vides extremely accurate lo cation and time data for na vigational purp oses. The signals transmitted from the satellites ha v e v ery lo w p o w er due to the limited fuel source on b oard eac h satellite. The mo dulation sc heme for the satellite signals is called Direct Sequence Spread Sp ectrum mo dulation. Spread sp ectrum mo dulates a narro wband signal in to a wideband signal that app ears to b e random in nature while conserving the p o w er in the signal. With the same amoun t of a v erage p o w er no w o v er a m uc h larger bandwidth, the instan taneous p o w er of the signal is reduced. This is accomplished b y mo dulating the data with a sequence of bits called c hips that tak e on only t w o v alues +1 or -1. The sequence of c hips app ears as random data but, in fact, is not. The sequence is called a pseudorandom or PN sequence b ecause it can b e generated from an equation, a primitiv e p olynomial in GF(2). The sequence can b e generated b y implemen ting the primitiv e p olynomial with a Linear F eedbac k Shift Register. F or demo dulation, the PN sequence that mo dulates the data m ust b e repro duced at the receiv er. Unin tended receiv ers who do not kno w the PN sequence will not b e able to demo dulate the data. The receiv er w orks b y correlating the receiv ed signal with the lo cally generated PN sequence. The sequence has a high correlation with itself only at zero shift. When a high correlation is detected, the receiv er in tegrates the signal o v er a data bit in terv al to reco v er the v alue of the data bit. This despreads the signal from wideband bac k to narro wband. The a v erage p o w er in the signal remains the same but the instan taneous p o w er is increased due to the smaller bandwidth, th us impro ving correct bit detection [4 ], [13 ]. Eac h c hip of the PN sequence is of xed length, T c seconds, but this is m uc h shorter than the length of a data bit, T b seconds. Mo dulating the data bits b y this PN sequence increases the bandwidth or spreads the sp ectrum of the data signal from B = 1 =T b Hz to 81
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App endix C (Con tin ued) W = 1 =T c Hz. The ratio W =B is called the pro cessing gain and is ho w m uc h the bandwidth has increased due to spreading [14 ]. The nonspread GPS data rate is 50 bps BPSK, T b = 1 = 50 seconds. The transmitted, spread GPS signal consists of signals in quadrature. The I and Q comp onen ts are spread sp ectrum mo dulated eac h with a distinct PN sequence. F or the I comp onen t, the data is mo dulo-2 added to a PN sequence called the P (Precision) co de and for the Q comp onen t, the data is mo dulo-2 added to another PN sequence called the C/A (Clear/Acquisition) co de [7 ]. The c hip length for the P co de is T c = 1 = 10 : 23x 10 6 seconds giving a pro cessing gain of W B P = T b T c = 204600 : Therefore, the sp ectral heigh t of the I comp onen t is 204600 times lo w er than it w ould b e if it w ere not spread. The c hip length for the C/A co de is T c = 1 = 1 : 023x 10 6 seconds giving a pro cessing gain of W B C = A = T b T c = 20460 : and so the sp ectral heigh t of the Q comp onen t is 20460 times lo w er than it w ould b e if it w ere not spread. The ratio of signal energy p er data bit, E b = P av T b to jammer p o w er sp ectral densit y J 0 = J av T c is E b J 0 = P av T b J av T c = T b =T c J av =P av (40) The ratio E b =J 0 can b e in terpreted as the SNR, while T b =T c is the pro cessing gain, and J av =P av is the jamming margin [14 ]. If the pro cessing gain and desired SNR lev el are kno wn, the jamming margin from equation (40) represen ts the largest v alue a jammer can tak e with reference to the signal of in terest for the system to still meet the desired SNR lev el. 82
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App endix C (Con tin ued) As an example, a desired SNR lev el of 10.5 dB p er bit or 10 6 error probabilit y for a BPSK signal is c hosen. The pro cessing gain for the P co de yields a jamming margin of 10 log 10 J av P av = 10 log 10 T b T c S N R = 10 log 10 204600 10 : 5 = 42 : 61 dB F or the C/A co de, the jamming margin is 10 log 10 J av P av = 10 log 10 T b T c S N R = 10 log 10 20460 10 : 5 = 32 : 61 dB Giv en that the GPS signals receiv ed b y a 0 dBIC an tenna are on the order of -160 dBw [7 ] and the jamming margin of 32.61 dB, the jammer signal cannot b e stronger than 160 + 32 : 61 = 127 : 4 dBw for this error probabilit y to b e ac hiev ed. This illustrates the adv an tage of using spread sp ectrum mo dulation in that the un w an ted signal ma y actually b e higher in p o w er than the desired signal while still pro viding a lo w probabilit y of bit error. This demonstrates that the goal of the algorithm to reduce the output p o w er of the arra y to within 5 dB of the bac kground noise lev el will allo w a high probabilit y of detection and demo dulation of the GPS signals. The signal pro cessing required to obtain accurate na vigational measuremen ts from GPS signals is not relev an t to this do cumen t. The in terested reader will nd a description of this in [7 ]. 83
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App endix D : Alternativ e Gradien t F orm ula Dr. V. K. Jain has suggested the alternativ e deriv ation of the gradien t form ulas: Since P ( k ) = < xA; xA > = A H x H xA = A H A is a real-v alued function of A, r A P ( k ) = 2 A = 2( F + iG ) A where is Hermitian symmetric: F T = F ; G T = G Also, if w e let A = a + ib then (since P ( k ) is a real-v alued function of A ) r A = r a + i r b : Th us, r A P = 2( F + iG )( a + ib ) = 2[( F a Gb ) + i ( F b + Ga )] and r a P = 2( F a Gb ) r b P = 2( F b + Ga ) : The gradien t of the p o w er with resp ect to the magnitude and phase of the an tenna w eigh ts in terms of the gradien t with resp ect to the real and imaginary parts of the an tenna 84
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App endix D (Con tin ued) w eigh ts can b e found using the c hain rule: @ P @ w = @ P @ a @ a @ w + @ P @ b @ b @ w and @ P @ q = @ P @ a @ a @ q + @ P @ b @ b @ q where @ a @ w = e iq = A w @ b @ w = ie iq = i A w @ a @ q = iw e iq = iA and @ b @ q = w e iq = A 85
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Ab out the Author Sandra Gom ulk a Johnson w as b orn in Johnsto wn, P ennsylv ania. She attended the Univ ersit y of New Mexico in Albuquerque and obtained her BSEE in 1990. She attended graduate sc ho ol at the Univ ersit y of Southern California in Los Angeles obtaining her MSEE in 1992. She w as emplo y ed as a signal pro cessing engineer at the Los Alamos National Lab oratory Los Alamos, New Mexico and at Ra ytheon, St. P etersburg, Florida b efore obtaining her PhD in Electrical Engineering at the Univ ersit y of South Florida in T ampa.