So, given the value x generated by the computer, we solve this equation for y, which now has the

Transcription

1 Calculaio ad odelig of radar perforace Sessio 9 // Cluer siulaio echiques Iroducio I he previous sessio we discussed he odelig of o-gaussia sea cluer ad is effecs o radar perforace. he copoud for of he K disribuio proved o be paricularly valuable; he classic, Gaussia oise, resuls were redeeed by iegraio over a gaa disribuio of local power. Noeheless here are ay siuaios ha are o aeable o direc aalyic aack; his approach ay o be he os revealig ad coveie, eve whe i is possible. So, o coplee he aalyic approach of he previous sessio, we will look a he siulaio of cluer reurs; his provides us wih aoher roue alog which we ca approach radar perforace probles. Our aalyic odels for cluer have bee eclusively saisical; his prejudice is sill evide i he siulaio ehods we look a. I essece we us cofro he proble of geeraig correlaed rado ubers wih prescribed oe ad wo poi saisics (i.e. PDF ad correlaio fucio. I he case where he siulaed process is Gaussia reasoable progress ca be ade; his provides us wih boh a good place o sar ad a valuable siulaio ool. he corolled siulaio of correlaed o-gaussia processes is uch ore difficul. Foruaely we are aily ieresed i siulaig cluer, for which he copoud odel has proved o be very effecive; he Gaussia siulaio echiques allow us o ge a he speckle copoe wihou oo uch difficuly. he pricipal difficuly we he ecouer is i he siulaio of he correlaed variaio i he local power copoe. Sigifica progress ca be ade owards he soluio of his proble, adopig a fairly pragaic, kife ad fork, approach. Oce we ca siulae a gaa process i his way, we are able o geerae K disribued cluer. Whe we have doe his we will have a reasoable, physically oivaed cluer siulaio capabiliy ha should sad us i good sead for he soluio of pracical radar perforace probles. Before we ca do ay of his, however, we cosider he basic buildig block of all hese siulaio ehods, he geeraio of u-correlaed rado ubers wih a kow (usually Gaussia PDF. Geeraig u-correlaed rado ubers wih a prescribed PDF Moder copuers ca readily supply us wih a successio of u-correlaed rado ubers, uiforly disribued bewee ad ; his we ake as our sarig poi, wihou furher eplaaio. (I Maheaica Rado[]does his for us. For he oe we will use eclusively o deoe his uiforly disribued rado variable, laer o we will doubless use for oher higs as well hese P y y by equaig he ca be apped oo a collecio of u-correlaed ubers y wih a PDF cuulaive disribuio of each. hus we have P C, < P ( ' d' C ( y P ( y' y y So, give he value geeraed by he copuer, we solve his equaio for y, which ow has he y. his relaioship is fudaeally ipora; we will reur o i whe we coe o PDF P y siulae correlaed o-gaussia processes. I soe cases he rasforaio fro o y is relaively sraighforward. hus, if y is required o have a epoeial disribuio, we have P y ( y ep( y, y ep y log( y ( y' dy' ep( y y dy' ( (

2 Calculaio ad odelig of radar perforace Sessio 9 // As - ad have he sae disribuio we ca also wrie his rasforaio bewee uiforly ad epoeially disribued rado ubers as y log. he geeraio of u-correlaed Gaussia rado ubers is paricularly ipora i pracical applicaios. A firs sigh we igh hik ha we us solve y ( y' dy' ( y ep erf π ( a each sep i he siulaio; foruaely his ca be avoided by he followig rick. Cosider wo idepede Gaussia rado variables, y,y wih he joi PDF P ( y y ep( y y π If we ow represe hese i polar for he he joi PDF of hese polar variables is y y, ( y r cosθ, y r siθ (5 P P rθ r ( r, θ P ( r P ( θ ; r ( r r ep( r, P ( θ θ θ π (6 hus he polar agle is uiforly disribued, ad so ca be siulaed hrough We also see ha θ π (7 r is epoeially disribued; hus r ca be siulaed hrough r log(. (8 hus, havig geeraed wo idepede uiforly disribued rado ubers,, we produce wo idepede Gaussia rado variables y,y as y ( cos( π, y log( si( π log. (9 We oe ha hese y variables each has a zero ea ad a variace of ; a Gaussia rado variable wih a ea of µ ad a variace of is obaied whe we he for µ y. A Gaussia rado variable wih zero ea ad ui variace is required i ay of our siulaios, we deoe i by g. Idepede rado variables wih oher PDFs ca be geeraed siilarly; cosiderable cuig has bee deployed i he efficie soluio of (. Maheaica (ad oher sofware packages iplee ay of hese as buil i fucios; should you eed o geerae idepede o-gaussia rado variables for ay reaso you should always see if he hard work has bee doe for you already.

3 Calculaio ad odelig of radar perforace Sessio 9 // Geeraig correlaed Gaussia rado ubers We recall ha he su of wo Gaussia rado variables iself has Gaussia saisics; subjecig a Gaussia process o ay liear operaio (e.g. filerig, iegraio or differeiaio siilarly yields aoher Gaussia process. he qualiaive effecs of he liear operaio are aifes i he correlaio properies of he oupu process, raher ha i is sigle poi saisics. A siple, ad pracically useful, siulaio of a correlaed Gaussia process is as follows. Cosider he siple recurrece (or feedback βg. ( he g are zero ea, ui variace idepede Gaussia rado ubers; is chose o have agiude less ha o keep higs sable. If we sar higs of wih we ca wrie he soluio o ( as r β g ( r r Apar fro he rasie er, his ca be viewed as he oupu of a filer acig o he successio (or ie series of gs. he ea value of, averaged over realisaios of he gs cosiss of he rasie er while is ea square is r β g r ( r β β β r, s r r ( r s g r g s ( Oce he rasie ers have died dow, we see ha he oupu of he process is a zero ea Gaussia process wih a odified variace. More iporaly, however, correlaio has bee iduced i his oupu, ha was abse i he successio of u-correlaed gs ha serve as is ipu. his correlaio ca be calculaed direcly

4 Calculaio ad odelig of radar perforace Sessio 9 // ; s s r s r s s r r r r r r r g g g g β β β β ( hus we see how he liear feedback process has iduced a correlaio i he oupu Gaussia process, ha we ca corol hrough he paraeer ; β allows us o corol he power of he oupu process. If we chose β (5 we esure ha he oupu has a ui variace, oce rasies have died dow. A siple correlaed Gaussia geeraed i his way ca iself be a very useful siulaio ool. We see fro ( ha his ehod provides a eaple of he use of a filer o iduce correlaio i a Gaussia process; his i ur relaes direcly o he descripio of he oupu process i ers of is power specru. I is ieresig o see how his approach is iiaely relaed o he sochasic differeial equaio descripio of rado processes. We recall he Orsei Uhlebeck process (over daped Browia haroic oscillaor, evolvig subjec o he Lagevi equaio f d d α (6 Here f is whie oise for which ' ' f f αδ (7 We iegrae his up, forally a leas, o give us soehig very siilar o ( ep ep d f α α (8 hus, if we propagae he process over he ie-sep bewee ad ad copare ( wih (8 we ca ake he ideificaios ep ep ep α β α β α f f d d g d f g (9

5 Calculaio ad odelig of radar perforace Sessio 9 // We ca also regard he firs er i (8 as a rasie; a log ies, whe such effecs have seled dow, we have d d ep ( α( f( f( ( (7 ebodies a siple flucuaio-dissipaio resul, which requires he ipu whie oise (flucuaio ad he liear resisace (dissipaio ers o balace ou o esablish a equilibriu ea square value. he power specru of he process ca be derived fro eiher (9 or he uderlyig SDE as α S ( α A siple geeralisaio of he OU process ha is paricularly useful i cluer odelig is is cople versio; he aologue of ( ( ( i ( ( ep( α( ( g ig z ep α z I Q ( his geeraes a cople Gaussia process, wih a characerisic Doppler frequecy, wih a ui ea power ad a epoeially decayig iesiy acf. Realisaios of his process cropped up i he Fourier aalysis sessio; we recall ha hey looked sufficiely lifelike o be isake for saples of real daa. Higher diesioal Gaussia processes Goig for he uli-diesioal case we adop a vecor oaio ad se up he Lagevi equaio as d d f A ( he ari A is cosa; o esure ha he syse is sable all is eigevalues us be egaive (i he real par. Oce agai we iegrae i up forally ep( A ep( A( f d ( he whie oise vecor f has a correlaio ari of he for (which we do kow ye ( f( ( G f δ (5 I he log ie lii, where all he rasies have died dow, we ca cosruc he covariace ari B of, jus as we did i ( above

6 Calculaio ad odelig of radar perforace Sessio 9 // li d d ( A( G A ( ep ep δ B dep ( A Gep( A ( (6 A quick iegraio by pars ell us ha d AB BA d ( ep( A Gep( A G d (7 his is ea; i ells us, oce we have prescribed our equilibriu saisics B ad he relaaio processes A, he way i which we us drive he Lagevi equaios i order o aiai hese. hus give he dissipaio, we ca ideify he flucuaios cosise wih he prescribed covariace. Should he ari G have ay egaive eigevalues, presuably you are askig for a syse ha cao be realised (i would have a egaive power specru. o iplee his as a siulaio we ca discreise he evoluio equaio as H V H ep V VV ( A( ep ( A Gep( A d (8 We ow subsiue our epressio for G, whereupo everyhig sors iself ou very icely VV d ep BHBH ( A( AB B A ep( A ( ep( A Bep( A d d (9 I is reassurig o oe ha his resul ies i wih he followig, raher slapdash, aalysis:

7 Calculaio ad odelig of radar perforace Sessio 9 // H B H B V V B H H B B H H H H V V H H V V H V, ; ( he ari oaio we have used here does raher coceal soe of he copleiy of he algebra ivolved. he uder-daped Browia haroic oscillaor, which fored he subsace of oe of he eercises for Sessio 6, is a relaively siple eaple. We ca ideify he ari A as ζ A ( while he equilibriu covariace ari of he posiio ad velociy variables is fored fro he heral averages k k B. ( Subsiuig hese resuls io (7 we fid he correlaio ari of he whie oise drivig ers required o esablish hese equilibriu values is give by kζ G ( Now we cosider a real life proble, he siulaio of a Gaussia process wih four characerisic relaaio ies, used o odel he aircraf oios, i a lile ore deail. I his applicaio he power specru of he aircraf s displacee fro is avigaed pah is prescribed as k k S ( We have already see ha Fourier rasforaio of he Lagevi equaio leads us rapidly o he power specru of he process. his leads us o cosider he followig se of coupled differeial equaios: f d d d d d d d d (5

8 Calculaio ad odelig of radar perforace Sessio 9 // Epressed oly i ers of his gives us f d d d d d d d d (6 Fourier rasforaio he gives us f i i ~ ~ (7 his ca be wrie i he for f i i i i ~ ~ (8 whe we ake he ideificaios (9 I follows ha he specru of he process is S ff S ( hus he process has a specru of he hoped for for. (, ca be ideified as he plae s velociy ad acceleraio. o propagae his hig o he copuer we draw o he resuls we have jus derived; we kow A ad G ad so us cosruc H ad he variace of V. A ca be wrie i he for M Ω M Ω M A ( (which sees here o have bee plucked ou of hi air; see A.C. Aike Deerias ad Marices, Oliver ad Boyd, 96, pp5-6, for a brief discussio of backgroud aerial. his allows us o wrie he ari wih which o for he rasie par of he propagaig soluio as ep ep M Ω M A ( Ispecio of he coupled differeial equaios (6 ells us ha

9 Calculaio ad odelig of radar perforace Sessio 9 // G ; Gij δi,δ j, ( (Aleraively oe ca ideify he ari A associaed wih he coupled differeial equaios (6 ad evaluae B, he equilibriu covariace ari of,,,, fro he power specru (. If oe he epresses,,, i ers of,,, ad fors he ari AB BA oe fids ha all elees vaish ecep for,. While i s ice o deosrae his cosisecy, i would be overwheligly edious ad ie cosuig o carry hrough such a aalysis if oe did o have Maheaica o ake he srai. You igh like o work hrough his se of calculaios, as uch as a eercise i he use of Maheaica as ayhig else. We ow calculae he variace of he rado icree i : V V τ M ep τ ( A Gep( A ep d; τ ( Ω M G( M ep( Ω d M ( Now ( M ( M i( M j M G (5 i, j Direc evaluaio of he ari iverse ells us ha (usig Maheaica Pluggig his i gives us M k (6 ( k l l k V V ( Ψ ij ep( i( M i( M jep( j MΨM ( ( ( i j τ ( ( M i M j τ d ep (7 (You igh like o check ha he log ie liis of hese resuls ie up wih he covariace ari obaied direcly fro he power specru (; oce agai Maheaica akes he calculaio feasible. his covariace ari is syeric ad posiive defiie ad so ca be facorised io he produc of a ari ad is raspose, he so-called Cholesky decoposiio. So, oce we have evaluaed VV we ca perfor his decoposiio uerically: i j V V PP (8

10 Calculaio ad odelig of radar perforace Sessio 9 // he geerae a realisaio of V as V P (9 g where gis a vecor of zero ea, ui variace, u-correlaed Gaussia rado ubers. his allows us o propagae he process as i (8. Fourier syhesis of rado processes. We have already see how he correlaed sequece of Gaussia rado ubers geeraed by feedback ca be viewed as he oupu of a siple (oe-pole, ifiie respose filer. We will ow look a how his approach igh be eeded. Cosider he applicaio of a filer o a whie oise process f, φ o give he sigal ( ' f( ' d' h We ca evaluae he correlaio fucio of his process as φ (5 φ φ( τ d dh( h( τ f f dh( h( τ (5 Aleraively we ca Fourier rasfor he oupu of he filer o give us ~ φ ad obai is power specru as S h f φφ ~ ~ (5 ~ (5 h S ff hus we ca relae he auocorrelaio fucio ad power specru of he oupu process o he filer fucio h. his relaioship is o uique, as he phase of he filer fucio is o deeried. hus, for eaple we igh ideify he posiive roo of he power specru wih he F of he filer fucio ~ h S φφ ; (5 Oe ca he recosruc he process φ by Fourier iversio. I pracice his is doe discreely, usig a FF ype algorih, which i effec is evaluaig a Fourier series (raher ha a rasfor. hus we cosider a represeaio of he Gaussia process of he for a π π φ a cos b si (55 over a period of legh. he quaiies a, b are idepede zero ea Gaussia rado variables wih a variace give by he power specru of he process o be siulaed, i.e.

11 Calculaio ad odelig of radar perforace Sessio 9 // a π π b S (56 If ow we wish o realise a sequece of N values of he rado variable φ we ca obai his by akig he real par of he oupu of he FF fored as φ N ( S ep i ( g ig ππ N π N I Q. (57 he Fourier syhesis ehod has wo advaages over he real ie siulaio ehod, i ha i ca odel correlaio fucios ha cao be epressed as a su of epoeially decayig ad oscillaig ers, ad ca be eeded o he siulaio of rado fields as well as of ie series. Is pricipal disadvaage is ha i produces saples i baches, raher ha oe a a ie. Approiae ehods for he geeraio of correlaed gaa disribued rado ubers. A rado variable y is said o be gaa disribued if he pdf of is values y akes he for P y γ ν b ν y ep( by y Γ ( ν y < ; (58 b ad ν are referred o as he scale ad shape paraeers respecively. Much as i he oe diesioal Gaussia case, he correlaio fucio y R y y γ (59 ca be defied. he for of his correlaio fucio ad he requiree ha he argial y ad y have he for (58 are o sufficie o specify heir joi pdf uiquely. disribuios of Furherore, a liear rasforaio of a gaa disribued process eed o geerae a process ha iself has gaa sigle poi saisics; his is a cosequece of he gaa disribuio s propery of ifiie divisibiliy, which corass wih he sabiliy of he Gaussia disribuio. As a resul he copuer geeraio of a gaa process wih a specified correlaio fucio is less sraighforward ha he correspodig Gaussia siulaio. Oe approach o his proble was developed i he SAR group by Oliver ad co-workers. I his a array of ucorrelaed gaa variaes y j γ, is geeraed, he weighed o iduce he required correlaio y N Γ, i ik,k j w yγ. (6 Uforuaely he higher order oes of y Γ, i, which ca be characerised i ers of hose of y γ, j ad he weighs w ik, are o cosise wih a gaa disribuio. Furherore, fudaeal resricios are placed o he correlaio properies ha ca be odeled i his way; i paricular i is

12 Calculaio ad odelig of radar perforace Sessio 9 // o able o geerae a correlaio fucio ha akes values less ha he oally decorrelaed value y Noeheless his ehod is widely used ad has bee discussed eesively i he lieraure. A aleraive approach o he siulaio of a correlaed gaa process akes a correlaed Gaussia process of zero ea ad ui variace as is sarig poi. his is he apped oo a gaa process y by he eoryless o-liear rasfor (MNL geeraed by he soluio of he equaio π ν ep ' d' ' ep( ' ' ( ν y y dy. (6 Γ y Used i cojucio wih sadard ehods for geeraig Gaussia ie series ad rado fields wih prescribed correlaio properies, his ehod ca geerae correlaed ie series ad rado fields ha ehibi gaa sigle poi saisics. For a log ie i was hough ha his MNL pus forh a gaa variae whose correlaio properies cao be relaed i a siple ad iverible way o hose of he ipu Gaussia process. Cosequely a epirical elee has eered io he odelig of he correlaio i he oupu gaa process. For eaple, i has bee foud ha a Gaussia process wih a epoeially decayig acf is rasfored by he MNL io a gaa process whose acf also displays a seeigly epoeial decay over several decades. he followig forula has bee devised o relae he observed characerisic decay ies of he Gaussia(τ G ad gaa (τ γ processes ad he ν paraeer of he gaa disribuio. τ G 5.. (6 τ 7. ν γ Epirical sudies of he correlaio properies of gaa processes derived fro ipu Gaussia processes wih Gaussia ad power-law correlaio fucios have revealed he sae appare ivariace i fucioal for of he corrrelaio fucio uder he MNL ad rules of hub aalogous o (6 have bee devised ha ecapsulae he resuls of hese uerical sudies. he geeraio of a correlaed o-gaussia process by MNL he MNL ehod discussed i he previous secio ca be adaped o he geeral o-gaussia case. he required MNL or poi o-liear rasforaio is defied by equaig he cuulaive disribuio of a zero ea ui variace Gaussia process, evaluaed a he value ake by his process, wih he cuulaive disribuio of he required process, hus deeriig he laer s value. So, if he pdf of he values of is P dis (, we se ( ' ' ep ( ' ' Pdis d d erfc π (6 where, i he secod equaliy, we have ideified he copleeary error fucio. he copleeary quaile fucio Q dis ( ς of he required disribuio is ow defied by

13 Calculaio ad odelig of radar perforace Sessio 9 // Qdis Pdis ( d ς ; (6 ( ς usig his we ca wrie he MNL ha akes he ipu Gaussia rado values io he correspodig values of he required o-gaussia rado variable as Q dis erfc. (65 he cosrucio of his appig is i geeral a o-rivial uderakig. he sofware package Maheaica supplies buil i quaile fucios for a wide variey of o-gaussia processes; hese are, however raher slow i eecuio. A rapidly evaluable represeaio of (65 is a prerequisie if he MNL approach is o be pracically feasible. We e evaluae he correlaio fucio of he process. his ca be epressed i he for where ( G G(,, G. (66 d d ( ( P R ( RG RG PG,, R ep π R G he wo diesioal iegraio iplici i (66 ca be avoided by recallig he epasio discussed i sessio 5 (,, G P R G ep ( ( H( H( π where he H are Herie polyoials defied by H z d ( z ( z! R G (67 (68 ep ep. (69 dz (68 provides us wih a epasio of he propagaor of he Orsei-Uhlebeck process i ers of eigefucios of is associaed Fokker-Plack equaio; he Herie polyoials effecively ecode he facorisaio properies of he correlaio fucios of he Gaussia process. his epasio of he joi pdf as a power series i he ipu correlaio fucio leads o a represeaio of he oupu correlaio fucio ha is copuaioally useful ad highlighs he ee o which he laer is odified relaive o he forer. Usig (68 we fid ha R G d ( H Q ( π ep dis erfc. (7!

14 Calculaio ad odelig of radar perforace Sessio 9 // Oce we have evaluaed he iegrals dep ( H ( Q dis erfc (7 we have a power series represeaio of he appig bewee he correlaio fucios of he ipu Gaussia ad oupu o-gaussia processes. his series is rapidly coverge, or aalyically suable, i os cases of ieres. I paricular we oe ha, whe he series (7 coais oly wo o-egligible ers, i reduces o R (7 ad he correlaio fucios of he ipu ad oupu processes are esseially ideical. I geeral he higher order ers i (5 characerise he disorio i he correlaio fucio iduced by he MNL ha we sough o capure i he epirically derived rules of hub used i approiae siulaios. So far, we have esablished a readily evaluable ad iverible appig bewee he correlaio fucios of he ipu Gaussia ad oupu o-gaussia processes relaed by he o-liear rasforaio (65. Usig his we ca ow ailor he correlaio properies of he ipu Gaussia process, hrough he ehods described i he iroducio, o corol he correlaio fucio of he oupu o-gaussia process. G Correlaed epoeial ad Weibull processes We ow discuss he correlaio properies of MNL geeraed o-gaussia processes ha provide useful odels of cluer, eiher as a represeaio of he evelope of he process or of is local power, wihi he copoud odel. As a firs eaple we cosider heν gaa (i.e. egaive epoeial disribued variable y wih he pdf ( y P y I his case he required MNL is ideified as he soluio of ep (7 y ( ( ep y' dy' ep ' d π (7 or, ore copacly, ( y ( ep erfc. (75 hus we see ha he appig ( ( y log erfc (76

15 Calculaio ad odelig of radar perforace Sessio 9 // applied o a Gaussia process will geerae a epoeially disribued process. Our geeral resul (7 reduces i his case o y y R G d ( H ( ( ep log erfc ; (77 π! Noe ha, o leig RG go o zero ad akig accou of he iegral ( ep log erfc d π log (78 we recover he oally decorrelaed resul y. he uerical iegraios over he Herie polyoials i (77 prese o real probles ad reveal ha he series is very rapidly coverge. hus, oce we have calculaed ad sored he firs few of hese iegrals, (7 provides us wih a racable ad sigle valued fucioal relaioship bewee he correlaio fucios of he ipu Gaussia ad oupu egaive epoeially disribued processes. his he les us ap he required correlaio fucio of he oupu process poi by poi oo ha of he correspodig ipu Gaussia process. Assuig ha he laer ca be cosruced by he Fourier syhesis, we are ow able o geerae he required correlaed, epoeially disribued process by MNL. Figure shows he appig fro RG o Rep. I is reassurig o oe ha, by seig R o G uiy, we recover he fully correlaed resul y. By allowig RG o ake egaive values he MNL ca geerae a epoeially disribued process whose correlaio fucio akes values ha, while hey are ecessarily greaer ha zero, are less ha he fully de-correlaed lii y. I is precisely his ype of correlaio ha Oliver s filerig ehod cao reproduce. Figure shows he correlaio fucios of he ipu ad oupu processes, he forer has bee give a siple epoeially daped oscillaory for. Deailed copariso of he wo curves shows ha he udershoo below he fully de-correlaed value is less arked i he oupu process ha i is i he ipu. Mos ieresigly, Figure ad he log log plo i Figure show ha, o a very good approiaio, a posiive correlaio i he ipu process is reproduced uchaged i he correlaio i he oupu epoeially disribued process. A sligh deviaio fro his siple behaviour is oiceable asr G approaches, ad is ephasised by he liear coiuaio of he sall R G behaviour io his regio. his observaio accous, i he ν case, for he ivariace uder he Gaussia-o-gaa MNL of he correlaio fucios sudied i earlier work.

16 Calculaio ad odelig of radar perforace Sessio 9 // R G Figure : he appig bewee he ipu ad oupu correlaio fucios uder he MNL ( RG 6 8 (i Rep 6 8 (ii Figure : Ipu (i ad oupu (ii correlaio fucios displayig oscillaory behaviour - log R G 6 8 log ( Rep Figure : A log-log plo deosraig he effecive ivariace of posiive correlaio fucios uder he MNL (8 he Weibull process akes values ξ whose pdf is α α PW ξ αbξ ep bξ ; ξ ; ξ < (79

17 Calculaio ad odelig of radar perforace Sessio 9 // ad has bee widely used as a odel for cluer. We see ha he aalysis we have jus oulied ca be applied o he correlaio properies of a Weibull process geeraed by soe MNL. he required MNL is defied by he equaio (we ca se b equal o uiy, wihou ay real loss of geeraliy or ( ξ α ep erfc (8 ( α ξ log erfc. (8 he correlaio i he ipu Gaussia process is apped over io he correlaio i he MNL geeraed Weibull process by ξ ξ R ( G α d ( H ( ( ( π ep log erfc (8! ha ca be evaluaed by di of soe sraighforward uerical iegraios. Oce his has bee doe, ad he appig bewee ξ ξ( ad RG has bee ivered, Weibull disribued oise wih a prescribed correlaio fucio ca be geeraed. Figure shows a se of log-log plos aalogous o hose i Figure, ad reveals he ee o which he correlaio fucio is preserved i his Weibull case. As he Weibull process becoes progressively ore o-gaussia we see ha deviaios fro a siple liear relaio bewee he ipu ad oupu correlaio fucios becoe icreasigly ore arked for values of R G close o uiy ( ( log z z z z z α α α α 5 Figure : A log-log plo deosraig he effec of icreasigly o-gaussia saisics o he appig of he correlaio fucio uder he Gaussia-Weibull MNL I is useful o copare he siulaio of epoeial ad Weibull processes jus discussed wih ehods based o a cople Gaussia process. hese are sipler o iplee, bu afford us cosiderably less freedo i he odellig of he correlaio properies of he process.

18 Calculaio ad odelig of radar perforace Sessio 9 // A cople Gaussia process cosiss of real ad iagiary pars,y ha are heselves Gaussia processes wih zero eas ad equal (ui variaces; a ay oe ie or place,y, are ucorrelaed. he power z y (8 of he process has a pdf ha akes he paricularly siple for where ( z P z ep, (85 y, while is correlaio fucio follows direcly fro he Gaussia facorisaio propery ad he correlaios bewee,y: ( z z y y y ( k (86 hus we see ha z z ecessarily akes values bewee hose of z ad z ; a udershoo i he correlaio fucio below he value of he square of he ea, such as ha show i Figure, cao be odeled hrough (8. he epoeial for ake by (85 faciliaes he rasforaio of z io a process wih a Weibull pdf (79; he required MNL is siply ξ z α. (87 o relae he correlaio fucio of he oupu process o ha of he ipu cople Gaussia process we recall he joi pdf of he z process akes he for (, P z z ep ep ( ( z z k ( z z ( k ep ( k ( z z L ( z L ( z k ( ( ( k I k z z he epasio is quie aalogous o (68: he L are Laguerre polyoials defied as a special case (α of α L z ( z α ( z ( z (88 ep d ep (89 α z! dz hese ecode he facorisaio properies of he uderlyig Gaussia process, i uch he sae way ha he Herie polyoials do i (68.

19 Calculaio ad odelig of radar perforace Sessio 9 // Argues aalogous o hose leadig o (8 show ha he correlaio fucio of he Weibull variable (87 ca be epressed as α dz dz ( z z αp( z, z ξ ξ k dzz α ep( z L ( z (9 he iegral over he Laguerre polyoial ca be deduced fro (89 o be ( α z L z α dzz α ep Γ ; (9! o brigig hese resuls ogeher we fid ha ( F k ξ ξ Γ α α, α;; (9 where we have ideified he hypergeoeric fucio. he resul (9 reduces o he epeced ucorrelaed lii whe we se k ; he fully correlaed lii is recovered whe we se k ad use Gauss forula for he hypergeoeric fucio of ui argue Γ F ( α, α; ; Γ ( α ( α. (9 Much as i he sipler epoeially disribued case we see ha he correlaio fucio of a Weibull process geeraed fro he power of a cople Gaussia process cao ake values less ha ha of he square of is ea; his cosrai is reoved whe we perfor he appropriae MNL direcly o a correlaed oe diesioal Gaussia process. We also oe ha his ehod has bee adaped o he geeraio of cohere Weibull cluer, whose power has he disribuio (79 ad whose real ad iagiary pars are hose of. y α iy y (9 hus, i his case, he power specru of he ipu Gaussia process corols he phase of he oupu process, eacly as i does ha of he ipu process, ad he correlaio fucio of he power of he oupu process. We should also sress ha he process geeraed hrough (9 is o a saisfacory odel for cohere sea cluer as i does o icorporae he large separaio i ie ad legh scales characerisic of he decays i he correlaios i he local speckle ad uderlyig odulaio processes prese i he cluer. We will discuss he odelig of cohere cluer i ore deail shorly.

20 Calculaio ad odelig of radar perforace Sessio 9 // Geeraio of correlaed gaa processes by MNL he gaa process plays a ceral role i he odelig of sea cluer, represeig he odulaio of he scaerig due o he sall scale srucure of he ocea surface by is ore slowly varyig large scale srucure, which is, i effec, resolved by he radar. Cosequely uch of he iiial pracical applicaio of he aalysis we have developed so far is epeced o be o he geeraio of correlaed gaa disribued ie series ad rado fields. he Maheaica add-o package Saisics`Coiuous Disribuios` coais fucios ha geerae quailes for a wide variey of o-gaussia disribuios; i he ai, however, hese are raher oo slow for use i pracical siulaios. Noeheless we ca use hese fucios o evaluae he iegrals i (5 for he gaa disribuio wih a selecio of shape paraeers. he resuls we obaied reproduced he hoped-for fully correlaed ad wholly ucorrelaed liiig values of he correlaio fucios ad apped ou he relaioship bewee he ipu ad oupu correlaio fucios i deail. Soe ypical resuls are show i Figure 5. he correspodig log-log plos, aalogous o ha show i Figure, are qualiaively very siilar o hose obaied i he Weibull case. he appare ivariace uder MNL of he correlaio fucios sudied i earlier work ca be udersood i ers of hese resuls. Deviaios fro a liear relaio bewee he ipu ad oupu correlaio fucios are ore proouced i he eree o- Gaussia regie; i appears ha i is hese deparures, aifes i he behaviour of he correlaio fucio close o he fully correlaed lii, ha are capured by he rules of hub suggesed by earlier uerical work. We ca ow replace hese epirically derived rules wih uabiguous calculaios, ha eed oly be ade oce for each value of he shape paraeer ν ad sored for subseque use. I coras o hese rules of hub, he full calculaios are also applicable o ipu correlaio fucios ha ake egaive values ad correspod o values of he oupu correlaio fucio ha are less ha value of he square of he ea of he oupu process. I is ieresig o oe ha successively larger ubers of ers are required i (7 as ν becoes saller ad he gaa pdf ehibis progressively ore sigular behaviour a he origi. Whe he shape paraeer ν ook values sigificaly greaer ha uiy we foud ha he series (7 was icreasigly doiaed by is firs wo ers, wih he cosequece ha he fucioal for of he correlaio fucio was barely affeced a all by he MNL. hus whe ν he firs wo ers i (7 accou for 9% of he oupu correlaio fucio whe R G ad 96% wher G ; whe ν 5. oly abou a % coribuio coes fro he hird ad higher order ers i boh he eree correlaed ad ai-correlaed liis. his observaio suggess ha he MNL ay ed o he for y ν ν (95 which esablishes a process whose ea ad ea square values correspod wih hose of he required gaa disribuio ad which akes he correlaio fucio over uchaged. Direc evaluaio shows, for eaple, ha i he case whereν. he liear relaioship (95 capures he behaviour of he MNL for sall values of bu becoes progressively less saisfacory wih icreasig >..

21 ( ν ν Calculaio ad odelig of radar perforace Sessio 9 //.5 (iv (iii (ii (i RG Figure 5: Correlaio fucios of MNL geeraed gaa processes displayed as fucios of he correlaio fucio of he ipu Gaussia process: (i ν, (ii ν, (iii ν 7,(iv ν We ow give a couple of eaples of ie series ad rado fields wih gaa sigle poi saisics ad corolled correlaio fucios. Figures 6 ad 7 below show a eaple of each. Figure 6 is a ie series wih ν., a ea apliude of, ad a correlaio fucio give by ( cos( ep 8. (96 ν Figure 6: A apliude ie series of a Gaa process wih ν. ad a correlaio fucio give by equaio (68 Figure 7 i a rado field of ee 8 i boh ad y, wih ν5 ad a correlaio fucio give by (, (, y y πy ep cos 8. (97 ν

22 Calculaio ad odelig of radar perforace Sessio 9 // Boh eaples are chose o have he propery of he correlaio fucio fallig below he ea value squared. his shows i he periodic flucuaios i he siulaios ad is o accessible wih oher ehods of siulaio. Figure 7: A Gaa disribued rado field wih ν5 ad a correlaio fucio give i equaio (69 Fially we discuss briefly how cohere sea cluer ca be siulaed; we fid ha he copoud represeaio allows us o reproduce he salie feaures of is behaviour uch beer ha ca oher ehods described i he lieraure. Figure 8 shows he variaio wih rage of he Doppler specru of high-resoluio cohere sea cluer; is odulaio by he parially resolved large-scale srucure of he sea is clearly evide. he basis of he cohere odel is a cople Gaussia process whose correlaio properies are relaed o is power (Doppler specru by he Weier-Khichie heore. As we eioed earlier i has bee suggesed ha o-gaussia cohere sea cluer igh be odeled by subjecig such a cople Gaussia process iy o a o-liear rasforaio ha iposes he required sigle poi saisics. I is possible o ailor he correlaio properies of he ipu cople Gaussia o ach hose required of he oupu Weibull process reasoably well, provided he acf does o ake values less ha he fully de-correlaed lii.. However, boh he ipu ad oupu processes i his schee are saioary; he Weibull cluer does o ehibi he breahig odulaio see i he resuls i Figure 8.

23 Calculaio ad odelig of radar perforace Sessio 9 // Figure 8: A rage-doppler-iesiy plo of radar sea cluer ake wih a GHz radar a verical polarisaio fro a cliffop sie o he souh coas of Eglad a a rage of k. he verical ais is rage, sapled a.5 iervals over 75. he horizoal ais is Doppler frequecy, fro o 5 Hz. he copoud represeaio of he o-gaussia cluer process allows us o reedy his defec quie sraighforwardly. Raher ha subjecig he cople Gaussia process iself o a o-liear rasforaio, we uliply i by a correlaed gaa process, geeraed by he ehod we have jus discussed. his produc will auoaically have K disribued evelope saisics; he slowly varyig gaa process iposes he o-saioary breahig odulaio while he relaively high frequecy srucure of he cluer, revealed i he Doppler specru, ca be odelled hrough he specru assiged o he cople Gaussia process. I fac, by allowig his laer specru o deped o he curre value of he gaa variae ad he values of he shape paraeerα ad ea cluer power o vary wih rage, direcio ad eviroeal codiios, a very realisic cohere cluer siulaio ca be developed he differece bewee he wo ehods is illusraed qualiaively i figure 9, which shows he Doppler specra of siulaed cluer as a fucio of rage. Resuls obaied cohere K disribued cluer, siulaed usig he copoud ehod, are show o he lef; hose for cohere Weibull is siulaed usig he direc o-liear rasforaio o he righ. he poi apliude saisics have bee chose o be ideical (by seig α ad β ad he average power specra are approiaely he sae. he o-saioary behaviour i he iage o he lef ad he saioary behaviour i ha o he righ are clearly evide.

24 Calculaio ad odelig of radar perforace Sessio 9 // Figure 9: Cohere o-gaussia cluer specral ie series. he lef iage was geeraed usig he copoud ehod ad he righ usig he direc o-liear rasforaio of a cople Gaussia. he horizoal aes are frequecy ad he verical aes are rage.

25 Calculaio ad odelig of radar perforace Sessio 9 // Eercises A sigifica par of ay siulaio is he geeraio, preseaio ad discussio of uerical oupu. Soe of he followig eercises will require you o do his; use your favorie sofware o pu forh graphical or abular resuls ha ca for he basis of subseque discussio. Oher eercises have you fill i aalyical deails ha have bee glossed over i he sessio.. Siple Browia oio has provided us wih a physical oivaio for quie a lo of our discussio of he siulaio of correlaed Gaussia processes. Siulae he followig ad prese he graphically: a he pah of a diffusig Browia paricle b he displacee fro he origi of a over-daped Browia haroic oscillaor. c he cople Orsei Uhlebeck process odel for cohere cluer Look a he power specra of he processes geeraed i (b ad (c; wha ca you say abou he resuls you obai?. Usig a ie doai echique siilar o ha described i he sessio, siulae a correlaed Gaussia rado variable wih he power specru S ( ( ( Wha is he for ake by he correlaio fucio of his process? Verify he aalogue of ( arisig i his proble. Usig Maheaica o ease he algebraic load, look a he AB BA G flucuaio-dissipaio resul i his case, uch as was oulied for he four relaaio ie proble i he oes. Siulae correlaed Gaussia rado processes wih a variey of power specra (choose your ow usig he Fourier syhesis ehod. Eaie he oupu i each case; ca you discer ay qualiaive relaioship bewee he for of he power specru ad he behaviour of he process as a fucio of ie? he hard-liied sigal derived fro a correlaed Gaussia ipu akes he value whe his is posiive ad whe i is egaive. Ca you relae he correlaio fucio of he hard-liied sigal o ha of he uderlyig Gaussia process? You igh wa o do his i wo ways: direcly fro he joi pdf of he correlaed Gaussia rado variables ad via he epasio i Herie polyoials. Wih ay luck you should ge he sae aswer i each case. he paper by ough ad Ward J. Phys. D,, 75, 999, discusses his ad oher aspecs of he geeraio of correlaed o-gaussia oise by MNL i soe deail. (he resul derived i his eercise is due o Nobel Laureae J.H. va Vleck ad grealy faciliaed he aalysis of correlaio fucios, which were uch easier o easure for hard-liied sigals i he old days. he Malver Correlaor also eploied his rick. 5 Iplee a MNL for geeraig gaa disribued oise fro a Gaussia oise ipu. Assig a correlaio fucio o he gaa process ad evaluae ha of he correspodig ipu Gaussia process. Fourier rasfor his o obai is power specru. (Do you ever fid ha his is egaive?!! Geerae a bi of correlaed gaa oise. Use his o geerae he iesiy ad I ad Q copoes of cohere cluer. ry poppig a sigal io he cluer ad see wha i looks like.

26 Calculaio ad odelig of radar perforace Sessio 9 //