.8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd thrmal ois ad is gratd by th radom motio of chargs i rsistiv typs of dvics. O of th arly attmpts to charactriz thrmal ois was prformd by Nyquist, ad o of his thorms cocrig thrmal ois is that th ma-squar voltag apparig across th trmials of a rsistor of ohms at a tmpratur T dgrs Klvi, i a frqucy bad B Hrtz, is giv by v 4 ktb V (-93) rms 3 whr k.38 w-s K is Boltzma s costat. Th ois powr associatd with th rsistor is v 4 ktb w. (-94) rms.8. Thvi Equivalt Circuit of a Noisy sistor Th Thvi quivalt circuit of a oisy rsistor is as show i igur -8. It cosists of a ois sourc with a voltag dfid by Equatio (-93) ad a oislss rsistor with a valu of. igur -8 Thvi Equivalt Circuit of a Noisy sistor If w coct th oisy rsistor,, to a oislss rsistor,, w ca fid th powr dlivrd to by usig th quivalt circuit of igur -8, computig th voltag across ad th usig this voltag to fid th powr dlivrd to. Th rsultig circuit is show i igur -9 ad th voltag across is giv by v v rms Th powr dlivrd to. (-95) is rom ricipls of Commuicatios by Zimr ad Tratr, ourth Editio. M. C. Budg, Jr 34
v v ktb 4 rms If th load is matchd to th sourc rsistac, that is if. (-96), w hav ktb (-97) Which is th familiar form usd i th radar rag quatio. igur -9 Diagram for Computig th owr Dlivrd to a oad.8.3 How to Hadl Multipl Noisy sistors If w hav a twork that cosists of multipl oisy rsistors w fid th Thvi quivalt circuit by usig a modifid vrsio of suprpositio. To s this w cosidr th xampl of igur -. I th figur, th lft schmatic shows two, paralll, oisy rsistors ad th ctr schmatic shows th quivalt circuit basd o igur -8. Th right schmatic is th ovrall Thvi quivalt circuit for th pair of rsistors. To fid v o w first cosidr o voltag sourc at a tim ad short all othr sourcs. Thus, with oly sourc v w would gt v o v ad with oly sourc v w would gt (-98) v o v. (-99) igur - Schmatic Diagrams for th Two-rsistor roblm M. C. Budg, Jr 35
To gt th total Thvi quivalt voltag w must cosidr that v ad v ar oiss. As such, w must add thir squars. With this, w gt If w us v v v v v o o o v 4kT B ad v o 4kT B w fid that. (-) v 4kTB 4kTB. (-) W fid th Thvi quivalt rsistac by th stadard mas of shortig all voltag sourcs ad fidig th quivalt rsistac. Th rsult of this is. (-) This lads to th Thvi quivalt circuit rprstd by th right schmatic of igur -..8.4 Whit Nois Assumptio Although ot statd abov, o of th assumptios w plac o th oisy rsistor is that its ois powr dsity is costat ovr th badwidth of B. That is, N kt w Hz ovr B. (-3) I fact, although ot ralistic, w assum that th ois powr dsity is costat for all frqucis. I othr words, w assum that th ois is whit. This is a good assumptio i practic bcaus radars ar grally dsigd so that th ois spctrum ito a dvic is flat ovr th badwidth of th dvic. This is spcifically do to assur that th whit ois assumptio ca b ivokd..8.5 Effctiv Nois Tmpratur for Activ Dvics or passiv dvics such as rsistiv attuators w ca fid th ois powr dlivrd to a load by a xtsio of th tchiqu usd i th abov xampl. or activ dvics this is ot possibl. or ths dvics th oly way to dtrmi th ois powr dlivrd to a load is through masurmt. I gral, th ois powr dlivrd to th load will dpd upo th iput ois powr to th dvic ad th itrally gratd ois. Th stadard mthod of rprstig this is to writ th ois powr dlivrd to th load as whr out Gi it GkTB GkT B (-4) G is th gai of th dvic, ktb is th iput ois powr (i a badwidth of B ), M. C. Budg, Jr 36
GkT B is th ois powr gratd by th dvic (i a badwidth of B ) ad T is th quivalt ois tmpratur of th dvic. or rsistors, th quivalt ois tmpratur is a actual tmpratur. or activ dvics th quivalt ois tmpratur is ot a actual tmpratur. It is th tmpratur that would b cssary for a rsistor to produc th sam ois powr as th activ dvic. Both G ad T ca b masurd. I th abov quatio, ad i all calculatios of ois to follow, w vr spcifically stat th valu of th badwidth. W simply carry as it alog as a rquird paramtr..8.6 Nois igur A altrat to usig ffctiv ois tmpratur to charactriz th ois proprtis of dvics is to us ois figur. Th ois figur,, of a dvic is dfid as ois powr out of th actual dvic. (-5) ois powr out of a idal dvic I this dfiitio it is assumd that th ois powr ito th dvic is giv by kt B (-6) i whr T 9 K. To comput th ois out of th idal dvic w assum that th dvic dos ot grat its ow ois. Thus outidal Gi ktbg. (-7) rom (-4), th actual ois powr out of th dvic, wh th iput ois powr is, is i kt BG kt BG. (-8) outactual With this w ca rlat to T as kt BG kt BG T outactual. (-9) outidal ktbg T Altratly, w ca solv for T i trms of as T T. (-) A importat poit from Equatio (-9) is that th miimum ois figur of a. Aothr importat poit i th abov is that ois figur is always dvic is basd o a assumptio that th ois powr ito th dvic drivs from a rsistiv ois sourc at th stadard tmpratur of 9 ºK. I workig radar problms som popl prfr ois figur ad othrs prfr ffctiv ois tmpratur. Most of th ois spcificatios of dvics ad radars ar M. C. Budg, Jr 37
providd i trms of ois figur. Howvr, as w will s shortly, ffctiv ois tmpratur, ad total ois tmpratur, ar oft usful wh charactrizig th combid ffcts of xtral ois sourcs ad rcivr ois. or most dvics, ois figur is dtrmid by masurmt. Th xcptio to this is attuators. or attuators, th ois figur is th attuatio. Thus, if a attuator has a attuatio of (a umbr gratr tha o) th ois figur is. (-) Th ratioal bhid this is that if th attuator is matchd to th sourc ad th load impdac, which ar assumd th sam, th ois powr out of th attuator is qual to th ois powr iput to th attuator. Thr is a furthr, ustatd, assumptio that th ois tmpratur of th rsistiv lmts that mak-up th attuator ar at th sam tmpratur as th ois sourc drivig th attuator. With th abov w ca driv th ois figur of a attuator as follows. If th attuator is cosidrd idal, i.. th rsistiv lmts that mak-up th attuator do ot grat ois, th ois powr out of th attuator is. (-) outattidal iatt Howvr, for a actual attuator w hav outattactual. (-3) iatt By th dfiitio of Equatio (-5) th ois figur of th attuator is outattactual iatt. (-4) outattidal iatt.8.7 Nois igur of Cascadd Dvics Sic a typical radar has svral dvics that cotribut to th ovrall ois figur of th radar w d a mthod of computig th ois figur of a cascad of compots. To this d, w cosidr th block diagram of igur -. I this figur, th circl to th lft is a ois sourc, which i a radar would b th ata or othr radar compots. or th purpos of computig ois figur, it is assumd that th ois sourc has a ffctiv ois tmpratur of T (cosistt with how ois figur is dfid s Sctio.8.6). Th blocks followig th ois sourc rprst various radar compots such as amplifirs, mixrs, attuators, tc. Ths blocks ar rprstd by thir gai, G, ad ois figur,. or purposs of computig ois figur, all of th dvics ar assumd to hav th sam badwidth of B. M. C. Budg, Jr 38
igur - Block Diagram for Computig Systm Nois igur To driv th quatio for th ovrall ois figur of th N dvics w will cosidr first dvic, th dvics ad, th dvics,, ad 3, ad so forth. This will allow us to dvlop a pattr that w ca xtd to N dvics. Sic w hav th ois figur of ach dvic w ca comput th ffctiv ois tmpratur of ach dvic via Equatio (-). Thus, th ffctiv ois tmpratur of dvic k is k T T. (-5) k or Dvic, th iput ois powr is kt B. (-6) i Th ois powr out of a idal Dvic is G kt BG. (-7) outi i Th actual ois powr out of Dvic is G kt BG kt BG k T T BG. (-8) out i it rom Equatio (-5) th systm ois figur from th sourc through Dvic is out out i ktbg T k T T BG T (-9) whr th last quality was a rsult of Equatio (-9). or Dvic, th iput ois powr is i out k T T BG. (-) Th ois powr out of a idal cascad of Dvics ad is G G kt BG G. (-) out i i Th actual ois powr out of Dvic is G k T T BG G kt BG out i it T k T T BG G G Th systm ois figur from th sourc through Dvic is. (-) M. C. Budg, Jr 39
k T T T G BG G T T out. (-3) out i ktbg G T G T Or, usig Equatio (-9). (-4) G It is itrstig to ot that th ois figur of th scod dvic is rducd by th gai of th first dvic. W will xami this agai i a xampl to b prstd shortly. or ow w procd to dtrmi th systm ois figur from th sourc through th third dvic. Th ois powr out of a idal cascad of Dvics, ad 3 is G G G kt BG G G. (-5) out 3i 3 i 3 Th actual ois powr out of Dvic 3 is G G (-6) out 3 i3 it3 3 out it3 or, substitutig for out from Equatio (-8), T out3 k T T BGG G3 kt3bg3 G. (-7) T T 3 k T T BGG G3 G GG Th systm ois figur from th sourc through Dvic 3 is 3 out3 k T T T G T G G BG G G kt BG G G 3 3 out3i 3 T T T T G T G G T 3 Or, agai usig Equatio (-9). (-8) 3 3. (-9) G GG Hr w ot that th ois figur of Dvic 3 is rducd by th product of th gais of th prcdig two dvics. With som thought w ca xtd Equatio (-6) to writ th systm ois figur from th sourc through Dvic N as 3 4 N N. (-3) G GG GG G3 GG G3 GN M. C. Budg, Jr 4
It will b lft as a xrcis to show that th ffctiv ois tmpratur of th N dvics is T T T T 3 N N T. (-3) G GG GG GN I th abov w foud th systm ois figur btw th iput to Dvic through th output of Dvic N. If w watd th ois figur btw th iput of ay othr dvic, say Dvic k, to th output of som othr succdig dvic, say Dvic m, w would assum that th sourc of igur - was coctd to th iput of Dvic k ad w would iclud trms lik thos of Equatio (-3) that would carry to th output of Dvic m. Thus, for xampl, th ois figur from th iput of Dvic to th output of Dvic 4 would b. (-3) 3 4 4 G GG3.8.8 A Itrstig Exampl W ow wat to cosidr a xampl that illustrats why radar dsigrs ormally lik to iclud a amplifir as th first lmt i a rcivr. I this xampl w cosidr th two optios of igur -. I th first optio w hav ad amplifir followd by a attuator ad i th scod optio w rvrs th ordr of th two compots. Th gais ad ois figurs of th two dvics ar th sam i both cofiguratios. or Optio, th ois figur from th iput of th first dvic to th output of th scod dvic is igur - Two Cofiguratios Optios 4 5 w/w or 7 db. (-33) o G M. C. Budg, Jr 4
or th scod optio th ois figur from th iput of th first dvic to th output of th scod dvic is 4 4 w/w or 6 db! (-34). o G This rprsts a dramatic diffrc i ois figur of th combid dvics. This diffrc is du to th aformtiod proprty that th ois cotributd to th systm by a idividual dvic is a fuctio of th ois figur of that dvic ad th gais of all dvics that prcd th dvic. I gral, if th prcdig dvics hav a t gai, th ois cotributd by a dvic will b rducd rlativ to its idividual ois figur. If th prcdig dvics hav a t loss, th ois cotributd by th dvic will b icrasd rlativ to its idividual ois figur. I Optio of th abov xampl, th ois figur of th two dvics was clos to th ois figur of simply th amplifir. Howvr, for th scod optio th ois figur was th combid ois figurs of th two dvics. This is why radar dsigrs lik to iclud a amplifir arly i th rcivr chai: it sstially sts th ois figur of th rcivr..8.9 Output Nois owr Wh th Sourc Tmpratur is ot T I th abov, w cosidrd a sourc tmpratur of T. W ow wat to xami how to comput th ois powr out of a dvic wh th sourc tmpratur is somthig othr tha T. rom Equatio (-4) w hav out Gi it GkTB GkT B (-35) whr i ktb ad T is th ois tmpratur of th sourc. If w wr to rwrit Equatio (-35) usig ois figur w would hav out i it G GkTB GkT B. (-36) If w us a cascad of N dvics, G is th combid gai of th N dvics, T is th ffctiv ois tmpratur of th N dvics (s Equatio (-3)) ad is th ois figur of th N dvics (s Equatio (-3)). M. C. Budg, Jr 4