nder ndi ng ho he eng na di but ed he eq uency do n el he eng hs her bi en na cen he des l er ended ex ct uppr es he na no hi el he ca det er ni na nd ur ns ut us ue he ca ndo na ns nce ea ur ed ef u
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nder ndi ng ho he eng na di but ed he eq uency do n el he eng hs her bi en na cen he des l er ended ex ct uppr es he na no hi el he ca det er ni na nd ur ns ut us ue he ca ndo na ns nce ea ur ed ef u

he ndo pr ces ho e ur ns ut ha he ey cus he he na hi ea ur na eng ha es hes ni cel he eco ndm en ha ct er za ns ha SS pr ces es ho hi ha pt er pr ces ha eco ndo der er di c hi co es nd he er er ea za n duce he di cus us ng he ca SS pr ces es but he

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nder ndi ng ho he eng na di but ed he eq uency do n el he eng hs her bi en na cen he des l er ended ex ct uppr es he na no hi el he ca det er ni na nd ur ns ut us ue he ca ndo na ns nce ea ur ed ef u




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Presentation on theme: "nder ndi ng ho he eng na di but ed he eq uency do n el he eng hs her bi en na cen he des l er ended ex ct uppr es he na no hi el he ca det er ni na nd ur ns ut us ue he ca ndo na ns nce ea ur ed ef u"— Presentation transcript:


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nder ndi ng ho he eng na di but ed he eq uency do n, el he eng hs her bi en na cen he des fil er ended ex ct uppr es he na no hi el he ca det er ni na nd ur ns ut us ue he ca ndo na ns nce, ea ur ed ef udi na del ed ndo pr ces nce he eci fic udi na n no n) ddi di ur ba nce na bui pa fil er ex ct he udi nd uppr es he di ur ba nce na ul need deci de her pl ce he cut eq uency he fil er her edi ha eng es co nf ng find ppr pr eq uency -do des cr pt SS ndo pr ces ndi dua pl unct ns pi ca do n ha ns ha di na el -b eha ed unct ns eq uency her hei ns nl

defined he ens ener zed unct ns Seco nd, nce he pa cul pl unct det er ned he ut co pr ba bi ex er en ea ur es ct ua ndo ha ea ea ur es he ns ha epr es en he ho cl pl unct ns e. he ndo pr ces ho e. ur ns ut ha he ey cus he he na hi ea ur na eng ha es hes ni cel he eco nd-m en ha ct er za ns ha SS pr ces es ho hi ha pt er pr ces ha eco nd-o der er di c, hi co es nd he er er ea za n. duce he di cus us ng he ca SS pr ces es but he ca er WE WE ed ua ns hi he cr cur en hr ug h) uni es ef er he he na hen SS, he ns neo us er en xx xx 183 nh nd
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184 Chapter 10 er ral here xx is

the FT of the ut co el on unct on xx ). ur her re, when is er dic in co el on, so hat me er ges and ens em ble er ges re eq ual in co el on co put ns, then so epr es en ts the e-a er ge er in ens em ble em er. te hat ince xx xx ), no xx is ys al and en in pler no on uc as xx her ef re ha een re ppr pr te or t, but ha ll to xx to id pr er on of no nal co en ns, and to eep ppa en the act hat his ua is the ur ier ns rm of xx ). The eg al ug es ts hat ble to co ns ider the ex ected n- neo us) er r, um ing the pr cess is er dic, the e-a er ge er in equency band of dth d to en xx ex ine his ho ug ur

her, co ns ider ex ct ing band of equency co nen ts of pa ing hr ugh an deal ba ndpa ss fil er, ho wn in ure IG RE 10.1 deal ba ndpa ss fil ter to ex act band of eq uencies om nput, ). eca use of the re bt ning om ), the ex ected er in the utput can er pr eted as the ex ected er hat has in the el ected pa band. ing the act hat xx 2) see hat his ex ected er can co puted as 0) d xx d 3) nd us xx d 4) nd is indeed the ex ected er of in the pa band. It is her ef re ea na ble to ca ll xx the er tr al (P D) of ). te hat the ns neous er of ), and hence the ex ected ns neous er ], is ys no

nneg e, no ter ho na the pa band, It ws ha t, in ddi on to eing eal and ev en in the PSD is ys no nneg e, xx or ll hi le the PSD xx is the ur ier ns rm of the ut co el on unct on, it an V. nh im and ge C. e, 10
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Section 10.2 Einstein-Wiener-Khinchin Theorem on Expected Time-Averaged Power 185 is us eful to ha na me or the ns rm of the ut co el on unct on; ha ll efer to xx as the ex SD. ct ly pa lel es ul ts pply or the DT ca se, ea ding to the co ncl us on hat xx is the er ect al dens of ]. .2 WI IN EM ON ED ED WER The pr ev ous ect on defined the PSD as the ns rm of the

ut co el on unc- on, and pr ided an er pr et on of his ns m. no dev el op an er na ute to the SD. ns ider ndom ea za on of WSS pr ces s. ha eady en oned the di fficul ies th ing to the FT of di ect so pr ceed ndi ect Let the nal bt ined ndo ing ), so it eq ua ls in the er al but is ut ide his er l. us 5) here define the ow unct on to or and her se. Let deno te the ur ier ns rm of ); no te hat eca use the nal is no nzero nly er the fin te er al ), ts ur ier ns rm is pi ca ly ell defined. no hat the gy tr al (E D) xx of is en xx 6) and hat his ESD is ct ua ly the ur ier ns

rm of ), here ). us ha the FT pa ir d 7) r, di ding th ides hi is id, ince ca ing nal co ns ca les ts ur ier ns rm the me un ), d 8) The ua on the is hat defined or the DT ca se) as the am of the fini e-l eng th nal ). eca use the ur ier ns rm er on is nea r, the ur ier ns rm of the ex ected lue of nal is the ex ected lue of the ur ier ns m. her ef re ex ect ons of th ides in the pr eceding eq ua on. Since )] xx ), co nclude hat xx Λ( 9) here Λ( is ng ul ar pul se of hei at the in and deca ing to at the es ult of ca ing out the co ut on and di ding an V. nh im and ge C.

e, 10
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186 Chapter 10 er ral ing the it as es to at the es ul t 1 xx xx 0) im his is the te Wi in heo rem pr ed ener, and nde- enden ly Khi nc hin, in the ea ly s, but as nly ecen ly eco nized ted ns ein in ). The es ult is to us eca use it under ies ba ic et ho or es ing xx ): th en co pute the er do am or ev er al ea za ons of the ndom pr cess e., in ev er al ndep enden ex er en ), and er ge the es ul s. ncr ea ing the um er of ea za ons er hi the er ing is done ll reduce the no se in the es te, hi le ep ea ing the en re pr cedure or ger ll pr the equency es ut on of the es

te. .1 Sy tem en ficat on Us ng an om ces ses as ut ns ider the pr blem of det er ning or den ng the pul se es nse of ble TI tem om ea ur em en ts of the input and utput ], as ndi ca ted in ure IG RE 10.2 Sy tem th pul se es nse to det er ined. The st rd ppr is to ho se the input to unit pul ], and to ea ure the co es nding utput ], hi defini on is the pul se es nse. It is ten the ca se in pr ct ice, ho ev er, hat do not sh to or re una ble to pi his ple nput. or ns ance, to bt in el ble es te of the pul se es nse in the pr esence of ea ur em en er s, sh to use re ener et ic nput,

one hat ex ci tes the tem re ng here re ener ly ts to the pl tude can use on the input na l, so to get re ener gy ha to ca use the input to act er nger me. co uld then co pute ev ua ing the er se ns rm of ), hi in urn co uld det er ined as the io ). re has to en, ho ev er, to ens ure hat or Ω; in ther ds, the input has to uffici en ly h. In pa cul r, the input ca nnot ust fini te near co bi na on of us ids unl ess the TI tem is uc hat no edg of ts equency es nse at fini te um er of eq uencies er es to det er ine the equency es nse at ll eq uencies hi uld the ca se th um

ed em, e., fini e-o rder em, except hat one uld need to no an upp er ound on the rder of the tem so as to ha uffici en um er of us ids co bined in the nput ). an V. nh im and ge C. e, 10
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Section 10.2 Einstein-Wiener-Khinchin Theorem on Expected Time-Averaged Power 187 The co ns ts ug est us ing ly ut na l. or ns ance, upp se let the input li th or ea ing the lue +1 or th eq ual pr ba bi ndep enden ly of the lues en at ther es. his pr cess is ct and) de-s ense na th ean lue and ut co el on unct on xx ]. The co es nding er ect al dens xx is flat at the lue er

the en re equency nge ]; ev den ly the ex ected er of is di buted ev enly er ll eq uenci es. pr cess th flat er ect rum is ef er red to as ite ss (a erm hat is ted the ugh no on hat hi te co ins ll ble eq uencies in eq ual un ); pr cess hat is not hi te is er med co ns ider hat the FT ok or pi cal ple unct on of er no ul li pr ces s. pi cal ple unct on is not bs ut ely um ble or ua re um ble, and so do es not ll to ei ther of the ca eg ies or hi no hat here re ni cely eha ed s. ex ect hat the FT ex ts in me ener zed-f unct on ense ince the ple unct ons re ounded, and her ef re do not ter

han no ly th or ge ), and his is indeed the ca se, but it is not ple ener ized unct on; not ev en as nice as the pul ses or pul se ins or do ubl ets hat re ar th. When the input is er no ul li pr ces s, the utput ll so WSS ndom pr ces s, and ll in not pl ea ns rm to deal th. ev er, eca ll hat xx 1) so if can es te the cr -co el on of the input and ut put, can det er ine the pul se es nse or his ca se here xx ]) as ]. or re ener al ndom pr cess at the nput, th re ener al xx ], can or ing the ur ier ns rm of ), bt ning 2) xx If the input is not cces ble, and nly ts ut co el on or eq ui en ly ts

SD) is no wn, then can ll det er ine the de of the equency es nse, as ong as can es te the ut co el on or SD) of the ut put. In his ca e, ha xx 3) en ddi nal co ns ts or no edge out the em, one can ten det er ine ot re or ev en ev er hi ng) out om no edge of ts ni tude. .2 ng ci do es one es te nd/ or xx in an ex ple uc as the one e? The us ual pr cedure is to ume or pr e) hat the na ls and re hat er di ci er ts as ha no ted ea ier is the epl cem en of an ex ect on or le ge me when co put ing the ex ected an V. nh im and C. e, 10
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188 Chapter 10 er ral lue of ous unct ons of

ndom bles ci ted th na ndom pr ces s. us WSS pr cess uld ca led -e ic if im 4) The co er gence on the hand ide es equence of ndom bl es, so here re ubt et ies ed in defining it pr eci el but pa ss hese sues in .) Si or pa ir of -c -e ic pr ces es, co uld epl ace the cr -co el on the me er ge of ]. hat er di ci ener ly eq ui res is hat lues en pi cal ple unct er me epr es en of the lues en cr ss the ens em ble. ui el hat his eq ui res is hat the co el on et een ples en at di ffer en mes ls o st eno ugh. or ns ance, uffici en co ndi on or WSS pr cess th fini te ance to ea

n-er dic urns out to hat ts ut co ance unct on xx tends to as tends to hi is the ca se th st of the ex ples deal th in hese no es. re pr eci se neces ry and uffici en t) co ndi on or ea n-er di ci is hat the e-a er ged ut co ance unct on xx ], ei ted ng ul ar ndo w, 0: im xx 5) ar em en ho lds in the CT ca se. re ng en co ndi ons ing ur th en ts ther han ust econd en s) re needed to ens ure hat pr cess is eco nd-o rder er dic; ho ev er, hese co ndi ons re pi ca ly sfied or the pr ces ses co ns der, here the co el ons deca ex nen ly th g. .3 el ng ers and en ng ers here re ous det

ect on and es on pr bl ems hat re el ely ea sy ul te, e, and na yze when me ndom pr cess hat is ed in the pr blem or ns ance, the set of ea ur em en ts is hi te, e., has flat ect al dens When the pr cess is co red ther han hi te, the ea ier es ul ts om the hi te ca se can ll ten ed in me ppr pr te f: (a) the co red pr cess is the es ult of pa ing hi te pr cess hr ugh me TI ng or ng fil er, hi ha es the hi te pr cess at the input to one hat has the ect al ha ct er ics of the en co red pr cess at the ut put; or (b) the co red pr cess is ns ble to hi te pr cess pa ing it hr ugh an TI

ng fil er, hi fla tens out the ect al ha ct er ics of the co red pr cess pr es en ted at the input to ho se of the hi te no se bt ined at the ut put. an V. nh im and ge C. e, 10
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Section 10.2 Einstein-Wiener-Khinchin Theorem on Expected Time-Averaged Power 189 us, del ing or ha ping fil ter is one hat co er ts hi te pr cess to me co red pr ces s, hi le hi ening fil ter co er ts co red pr cess to hi te pr ces s. An es ult hat ws om hi nk ing in er ms of del ing fil ers is the ing ted and us ified ther nf ly here re ca eful ea en is ey ond our

co e): ey ct: eal unct on xx is the ut co el on unct on of ea -v lued WSS ndom pr cess if and nly if ts ns rm xx is ea l, ev en and no n- neg e. The ns rm in his ca se is the PSD of the pr ces s. The ty of hese co ndi ons on the ns rm of the ca ndi da te ut co el on unct on ws om pr er ies ha eady es bl shed or ut co el on unct ons and SD s. gue hat hese co ndi ons re so ffic upp se xx has hese pr p- er es, and ume or pl ci hat it has no pul pa t. Then it has eal and ev en ua re t, hi deno te xx co ns ruct bly no nca us l) ng fil er ho se equency es nse eq ua ls his ua re t; the uni

-s ple ep nse of his fil ter is ound er e-t ns ing xx ). If then pply to the input of his fil ter zer -m ean) uni -v ance hi te no se pr ces s, e. ., er no ul li pr cess hat has eq ual pr ba bi ies of ing +1 and at ea DT ns ndep enden ly of ev ery ther ns t, then the utput ll WSS pr cess th PSD en xx ), and hence th the ecified ut co el on unct on. If the ns rm xx had an pul se at the in, co uld ca pt ure his dding an ppr pr te co ns det er ined the pul se eng th) to the utput of del ing fil er, co ns ucted as us ing nly the no n-i pul pa rt of the ns m. or pa ir of pul

ses at eq uencies in the ns m, co uld ly add erm of the rm co Θ), here is det er ni ic and det er ined the pul se eng th) and is ndep enden of ll ther ther bl es, and uni rm in ]. Si ar em en ts can ade in the CT ca se. us te el the ic ed in des ning hi ening fil ter or pa cul ar ex ple; the ic or del ing fil ter is ar ct ua er se) to hi s. ns ider the ing di cr et e-t me tem ho wn in ure IG RE 10.3 di cr et e-t me hi ening fil er. Supp se hat is pr cess th ut co el on unct on xx and PSD xx ), e., xx xx uld to hi te no se utput th ance an V. nh im and C. e, 10
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190 Chapter 10 er ral no hat xx 6) r, xx 7) his then el ls us hat the ua red ni tude of the equency es nse of the TI tem ust to bt in hi te no se utput th ance If ha xx ble as nal unct on of or can del it hat ), then can bt in ppr pr te ct za on of XAM LE .1 en ng fil ter Supp se hat xx co Ω) 8) Then, to hi ten ), eq ui re ble TI fil ter or hi (1 9) (1 or eq ui en (1 0) (1 The fil ter is co ns ined to ble in rder to pr duce WSS ut put. One ho ice of hat es ul ts in ca us al fil ter is 1) th eg on of co er gence OC) en his tem unct on co uld ul pl ied the

tem unct on of ss em, e., tem unct on ing 1, and ll pr duce the me hi ening ct on, eca use 1. .3 NG OF ED OM ES WSS ndom pr cess is er med if ts PSD is ba ndl ted, e., is zero or eq uencies ut ide me fini te band. or det er ni ic na ls hat re ba ndl ted, can ple at or the ui st te and eco er the nal ex ct ex ine here hether can do the me th ba ndl ted ndom pr ces es. In the di cus on of pl ing and DT pr ces ing of CT na ls in our pr or co ur es, the der ons and di cus on ely hea ly on pi ct ur ing the effect in the equency an V. nh im and ge C. 10
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Section 10.3

Sampling of Bandlimited Random Processes 191 do in, e., ing the ur ier ns rm of the co uo us -t me nal hr ugh the /D pl ng) and /C eco ns uct on) pr ces s. hi le the um en ts can er na ely ca ied out di ect ly in the me do in, or det er ni ic fini e- ener gy na ls the equency do in dev el pm en eems re co ncept ua ly cl ea r. As ou ex ect, es ul ts ar to the det er ni ic ca se ho ld or the e- co ns uct on of ba ndl ted ndom pr ces ses om pl es. ev er, ince hese ha ic na ls do not ess ur ier ns ms except in the ener ized ense, ca ry out the dev el pm en or ndom pr ces ses di ect ly in the

me do in. An es en ly pa lel um en co uld ha een used in the me do in or de- er ni ic na ls ex ning the al ener gy in the eco ns uct on er or ther han the ex ected ns neous er in the eco ns uct on er r, hi is hat cus on el ). The ba ic pl ing and ba ndl ted eco ns uct on pr cess ho uld ar our pr or udies in na ls and em s, and is depi cted in ure .4 el w. In his fig ure ha ex pl ci ly used ld upp er -ca se ls or the na ls to under co re hat they re ndom pr ces es. /D nT /C nc nT here inc IG RE 10.4 /D and /C or ndom pr ces es. or the det er ni ic ca se, no hat if is ba ndl ted to ess han

then th the /C eco ns uct on defined as nc nT 2) it ws hat ). In the ca se of ndom pr ces es, hat ho el is ha t, under the co ndi on hat ), the er ect al dens of ), is ba ndl ted to ess hat the ean ua re lue of the er or et een and is zer o; e., if 3) an V. nh im and C. e, 10
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192 Chapter 10 er ral then 4) hi s, in effect, ys hat here is zero er in the er r. An er na pr of to the one el is ut ined in blem 13 at the end of his ha pt er .) dev el op the es ul t, ex pand the er or and use the defini ons of the /D or pl ng) and /C or deal ba ndl ted er on) er ons

in ure .4 to bt in 5) fir st co ns ider the st er m, nT nT nc nT nT nc 6) 7) here, in the st ex pr es on, ha ed the et ry of nc( to ha nge the gn of ts um en om the ex pr es on hat precedes t. ua on can ev ua ted us ing ev s el on in di cr ete me, hi tes hat 8) pply ev s el on, no te hat nT can ew ed as the es ult of the /D or pl ing pr cess depi cted in ure in hi the input is co ns dered to unct on of the ble /D nT IG RE 10.5 /D ppl ied to ). The FT in the ble of is ), and ince his is ba ndl ted to the FT of ts pled er on nT is 9) an V. nh im and ge C. e, 10
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Section

10.3 Sampling of Bandlimited Random Processes 193 in the er al Si the FT of inc( nT is ns eq uen under the co ndi on hat is ba ndl ted to d 0) 0) ex t, ex pand the ddle erm in eq ua on ): nT mT nc nT nc mT nT mT nc mT nc mT 1) th the ubs ut on can ex pr ess 31 as inc( mT nc mT 2) ing the den nc( nc( nc( 3) hi in co mes om ev s heo rem see blem 12 at the end of his ha pt er ), ha nc 0) 4) ince nc( if and zero her se. Subs ut ing 31 and 34 to bt in the es ult hat 0, as des red. an V. nh im and ge C. e, 10
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194 Chapter 10 er ral an V. nh im and ge C. e, 10
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