The Theory of Small Reflections

Transcription

1 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions /9 Th Thory of Smll Rflctions Rcll tht w nlyzd qurtr-wv trnsformr usg th multil rflction viw ot. V ( z) = + β ( z + ) V ( z) = = R + β ( z + ) R = λ 4 W found tht th solution could thus writtn s n fit summtion of trms (th rogtion sris): = n n = whr ch trm hd scific hysicl trrttion, trms of rflctions, trnsmissions, nd rogtions. For xml, th third trm ws th: 3 R R

2 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions /9 ( ) 3 = β Now lt s considr th mgnitud of this th: = = Rcll tht = for rorly dsignd qurtr-wv trnsformr : R = = R + nd so: 3 = = 3 For th cs whr vlus R nd r numriclly clos i.., whn: R R + w fd tht th mgnitud of th rflction coffict will vry smll: R =. R + As rsult, th vlu 3 will vry, vry, vry smll.

3 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 3/9 Morovr, w know (sc th connctor is losslss) tht: nd so: = + = + = W cn thus conclud tht th mgnitud of th 3 is likwis vry, vry, vry smll: = This is clssic cs whr w cn roximt th rogtion sris usg only th forwrd ths!! Rcll thr r two forwrd ths: R R ( + ) β ( ) = + = =

4 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 4/9 Thrfor IF nd R r vry clos vlu, w fd tht w cn roximt th rflctd wv usg only th dirct ths of th fit sris: Thrfor: ( + ) β ( ) = + V ( z ) = + β ( z + ) β + β( z + ) ( ) + Now, if w likwis ly th roximtion tht., w conclud for this qurtr wv trnsformr (t th dsign frquncy): + Thrfor: V ( z ) = ( ) β ( ) = + + β ( z + ) β + β( z + ) ( ) +

5 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 5/9 This roximtion, whr w:. us only th dirct ths to clcult th rogtion sris,. roximt th trnsmission cofficts s on (i.., = ). is known s th Thory of Smll Rflctions, nd llows us to us th rogtion sris s n nlysis tool (w don t hv to considr n fit numr of trms!). Considr g th qurtr-wv mtchg ntwork SFG. Not thr is on rnch ( = S of th connctor), tht is not cludd ithr dirct th. = =

6 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 6/9 With rsct to th thory of smll rflctions (whr only dirct ths r considrd), this rnch cn rmovd from th SFG without ffct. = = Morovr, th thory of smll rflctions imlmnts th roximtion =, so tht th SFG coms:. β =. Rducg this SFG y comg th. rnch nd th β rnch vi th sris rul, w gt th followg roximt SFG: β = =+ = β β Th roximt SFG whn lyg th thory of smll rflctions!

7 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 7/9 Not this roximt SFG rovids rcisly th rsults of th thory of smll rflctions! Q: Why is tht? A: Th roximt thory of smll rflctions SFG Conts ll of th significnt hysicl rogtion mchnisms of th two forwrd ths, nd only th two significnt rogtion mchnisms of th two forwrd ths. Nmly:. Th rflction t th connctor (i.., ).. Th rogtion down th qurtr-wv trnsmission l ( β ), th rflction off th lod ( ), nd th rogtion ck u th qurtr-wv trnsmission l ( β ). R R Th roximt SFG whn lyg th thory of smll rflctions!

8 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 8/9 From sris rul From rlll rul + Q: But wit! Th qurtr-wv trnsformr is mtchg ntwork, thrfor =. Th thory of smll rflctions, howvr, rovids th roximt rsult: + β Is this roximtion vry ccurt? How clos is this roximt vlu to th corrct nswr of =? A: t s fd out! Rcll tht = for rorly dsignd qurtr-wv mtchg ntwork, nd so: + = + ( ) ikwis, = λ 4 (ut only t th dsign frquncy!) so tht: whr you of cours rcll tht π λ β = = π λ 4 β = π λ!

9 Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions 9/9 Thus: + = + ( β ) ( π ) ( ) = =!!! Q: Wow! Th thory of smll rflctions rs to rfct roximtion no rror t ll!?! A: Not so fst. Th thory of smll rflctions most dfitly rovids n roximt solution (.g., it ignors most of th trms of th rogtion sris, nd it roximts connctor trnsmission s =, whn fct ). As rsult, th solutions drivd usg th thory of smll rflctions will gnrlly skg xhiit som (hofully smll) rror. W ust got it lucky for th qurtr-wv mtchg ntwork; th roximt rsult = ws xct for this on cs! Th thory of smll rflctions is n roximt nlysis tool!