5.0 SIDELOBE CANCELLATION

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Frank Dixon
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1 5.0 SDELOBE CANCELLATON 5. NTRODUCTON Sidlob cnclltion (SLC) cn b conidrd n xtnion of STAP. t i ptil dptiv procing tchniqu tht i id t roving intrfrnc. n STAP, or or ccurtly SAP, th procor plc null in th ntnn pttrn t th ngulr loction of th intrfrnc ourc. n SLC, th procor ttpt to ubtrct th intrfrnc fro th ntnn output. n fct, SLC i lo lind to dptiv cnclltion tchniqu ud in couniction yt for roving intrfrnc nd ultipth ignl SLC i dignd to oprt gint ctiv lctronic ttc (EA) dvic (jr) nd not gint cluttr or piv intrfrnc uch chff. t i uully ud tht th EA ignl i noi-li ith bndidth tht xcd th intrdit frquncy (F) bndidth of th rdr rcivr. t i lo ud tht th EA ignl i ntring th rdr ntnn through on of it idlob. Th fct tht SLC cncl intrfrnc ntring th rdr through th ntnn idlob i blivd to b th origin of th tr idlob cnclltion. 5. BACKGROUND W ill driv th configurtion of idlob cncllr (bbrvitd SLC) by firt xining th intrfrnc cnclltion probl. To thi nd, Figur contin bloc digr of n intrfrnc cncllr. n thi figur, th top ntnn rprnt th in rdr ntnn nd th botto ntnn i n uxiliry ntnn ud to gthr infortion on th intrfrnc ignl. Th bloc ith t i th gin, or ight, (clr in thi c) nlogou to th ight in STAP nd th rro through th box dnot tht th ight i djutd bd on th rror voltg, v t. Th rror voltg i ford by ubtrcting ightd vrion of th uxiliry chnnl ignl, in chnnl ignl, t. Tht i v t, fro th v t v t t v t. ()

2 Figur ntrfrnc Cncllr Configurtion Th rulting rror ignl i nt to th rt of th rdr rcivr nd ignl procor. (Th intrfrnc cncllr i iplntd rly in th rcivr.) f th intrfrnc cncllr i oring corrctly v t ill contin only th trgt, or dird, ignl. ndd, uppo u tht th ignl fro th in ntnn conit of dird ignl, v t, nd n intrfrnc ignl, v t. Tht i v t K v t K v t. () Furthr, u th uxiliry chnnl ignl conit of only intrfrnc. Tht i 3 v t K v t. (3) No, uppo r bl to choo th ight, t, uch tht it i t K K3. (4) f cn choo thi t ould gt K v t v t t v t K v t K v t K v t K v t. (5) 3 K3 Thu, th rror ignl i, in fct, th dird ignl. Wht nd no i critrion for coputing t o tht it bhv indictd. Thi y not b good uption in o c but y b vlid in othr. Thi ill b dicud ltr

3 5.3 A METHOD FOR FNDNG t Bfor propo critrion nd to rcogniz tht proc. Bcu of thi t nd v t t, v t, v t v t i rndo t r lo rndo proc. Thi v t nd v t r rndo n tht, for prticulr vribl. Th ipliction of thi i tht ut u ttiticl thod to chrctriz th rndo vribl nd dvlop th critrion. With th bov rtriction th critrion ill u i to choo t o to iniiz th n-qurd vlu of v t. Mthticlly in t t in E v t E v t t v t. (6) t W u gnitud bcu rcogniz tht v t nd v t ut, in gnrl, b rprntd by coplx vribl o to cptur thir plitud nd ph. W u xpcttion bcu v t i rndo vribl, hich cu v t nd v t to b rndo vribl. W u th qur bcu it i firly y to or ith. Th critrion givn in Eqution (6) i trd lt n-qur (LMS) critrion. Fro our xprinc ith qudrtic, or qurd rror, iniiztion (rcll lt-qurd curv fitting), no tht cn lct t th t t hich th prtil drivtiv of th n-qurd rror ith rpct to t i zro. Tht i t E v t t v t t t 0 f prfor th indictd oprtion gt t t E v t t v t v t 0 (8) hich ld to th olution t t E t v t. (9) E v Although thi rult i intrting nt to if it ld to n rror v t v t, th dird ignl. W ill ignl tht i ronbl. Tht i, i (7) W u v t i dtrinitic (coplx) vribl. 3

4 u tht v proc ith vrinc of W lt rit t i zro-n, id-n ttionry (WSS), rndo P E v t. (0) t nd v t t b dfind in Eqution () nd (3). With thi cn 3 E v E K v t K v t K v t t v t E v t E K v t K K P K K P K3 3 K hich i th rult id ndd to obtin v t v t, () (S Eqution (4)). Thu, conclud tht if u th LMS thodology to dtrin th ight (gin), t, th rror ignl out of th intrfrnc cncllr ill indd contin v t nd not v t t lt in thory. 5.4 PRACTCAL MPLEMENTATON CONSDERATONS Dirct iplnttion of th LMS cncllr dicud thu fr rquir priori noldg of P E v t. n gnrl, thi i not prcticl uption in tht it rquir noldg of th intrfrnc ourc, hich i not gnrlly vilbl. Bcu of thi titd bd on urnt of E v t nd E v tv t t nd v t. ut b Strictly ping, th xpctd vlu r nbl vrg nd cn t b vlutd fro ingl t of v t urnt. To b vlid on t nd ut vrg cro ny rdr, trgt, nvironnt nd intrfrnc ourc (ll of th typ nd in th loction) to obtin tru nbl vrg. Clrly thi i not poibl inc hv only on rdr, tc. To gt round thi probl invo th concpt of rgodicity. Thi concpt tt tht, if rndo proc i rgodic, nbl vrg cn b rplcd by ti vrg. Proving tht proc i rgodic i vry difficult, if not ipoibl. On of th y chrctritic of rgodic proc i tht thir n nd vrinc r contnt. Thu dignr uully try to rtionliz n uption of contnt n nd vrinc nd invo, ithout jutifiction, rgodicity. Although not trictly lgitit fro rndo proc prpctiv, thi pproch or ll. 4

5 Thu, if u T nd hr 3, cn rit t nd v t r WSS ovr th urnt intrvl, C E tv t tv tdt T () T C E v t v t dt (3) T T T n vlut th intgrl ovr o intrvl T T ut b lrg rltiv to th rciprocl of th bndidth of t. Thi i ncry to obtin good vrg. v f u digitl procing gt. n th bov t nd nd N N 0 C E v v v v (4) N C E v v. (5) N 0 Th pcing btn pl, rciprocl of th bndidth of t, hould b grtr thn or qul to th v t nd N hould b uch tht v t nd i lrg rltiv to th rciprocl of th bndidth of v N t T t. Givn th bov, on cn coput C fro t nd (6) C nd iplnt th cncllr v t v t v t. (7) n th bov, th nottion ud in plc of to indict tht th ight i on pproxitly iu (bcu of hving to tit th xpctd vlu). 3 f t nd v T. t r WSS ovr T, by dfinition thir n nd vrinc r contnt ovr 5

6 n typicl ppliction, on ould priodiclly coput C, C nd, nd u if for hil. Th ti btn updt ould b dtrind by ho oftn on could u E v t nd E v tv t rin contnt. A good rul of thub i tht thy hould b updtd onc pr PR in lo PRF rdr or onc pr cohrnt dll in puld-dopplr rdr. Th bov iplnttion i fibl if th rdr h digitl coputr ft nough to coput C, C nd, nd pply it vi Eqution (7). With odrn rdr tht u DSP thi y b fibl pproch. Hovr, in oldr nlog rdr it not. Thrfor, th lgorith hd to b odifid to or in uch rdr. Th rult th SLC. 5.4 SDELOBE CANCELLER Th ight clcultion tchniqu upon hich th SLC i bd i trd grdint rch tchniqu. Th grdint tchniqu itrtivly coput ight o to vntully iniiz th n-qurd rror E v t. (8) n th iplnttion of th tchniqu cn t rlly vlut th xpctd vlu o pproxit it by th ipl qurd rror, or t v t. (9) Th grdint lgorith i givn by th qution. (0) n Eqution (0), i th grdint of th rror vlutd t v t v t v t v t v t v t v t nd i givn by. () Biclly, th grdint updt th ltt tit by dding corrction tht i proportionl to th ngtiv of th lop, or grdint, of th rror vlutd t th ltt tit. Thi i illutrtd in Figur. n thi figur nd not tht th lop i poitiv. W lo not tht nt to b l thn if r to ov tord. Thu, tht nt to ov in dirction tht i oppoit to th ign of th lop. With o thought, lo not tht if i fr y fro ould li to chng by lrg ount; hr if i clo to nt to chng by ll ount. 6

7 Thu th ount of chng i rltd to th gnitud of th lop. Thi i ht th lgorith of Eqution (0) do. Figur llutrtion of Grdint Tchniqu Th prtr control th rt t hich th tit pproch. f i ll ill pproch in ll tp nd if i lrg ill pproch in lrg tp. f i too ll, convrgnc ill b vry lo. On th othr hnd, if i too lrg th olution could divrg. Thu, chooing i on of th iportnt prt of iplnting SLC. W rcogniz tht Eqution (0) i diffrnc qution. Hovr, SLC r uully iplntd in th continuou ti doin. Fro our xprinc ith diffrntil nd diffrnc qution, cn convrt Eqution (0) to diffrntil qution of th for d t dt v t v t () hr hv d u of Eqution (). Not tht th prtr h bn chngd to. W did thi to not tht th cling contnt ill b diffrnt for dicrt-ti nd continuou-ti iplnttion. Eqution () lo contin nothr, ubtl, chng ovr Eqution (0). Spcificlly, in Eqution () llo th rror voltg nd uxiliry chnnl voltg to chng ith ti th ight i bing updtd. n Eqution (0), ud on pl of th rror voltg nd uxiliry chnnl voltg to itrt on th ight. Alloing th voltg to chng ill incorport vrging into th SLC to hlp tbiliz th prfornc. f rprnt Eqution () bloc digr hv th clicl Holl-Applbu SLC. Thi i hon in Figur 3. 7

8 Figur 3 Holl-Applbu SLC 5.6 PRACTCAL CONSDERATONS Th Holl-Applbu loop i uully iplntd t o intrdit frquncy (F). A uch, th lor ultipliction of Figur 3 i gnrlly prford by ixr hr th uppr ultiply i vribl gin plifir. Th t i t, v t nd v t voltg r F ignl hil bbnd ignl. Th conjugtion t th botto i iplntd 90º ph hift. Th bloc digr of Figur 3 u coplx ignl nottion. n n ctul iplnttion, on nd to u qudrtur ignl to produc n F iplnttion tht cptur th oprtion iplid by th coplx ignl nottion. An xpl bloc digr for n F iplnttion i contind in Figur 4. n thi figur, th circl ith cro r ixr nd th qur ith cro r vribl gin plifir. Th gin i bipolr. Tht i, th ight cn vry th plifir gin nd th ign of th product dpnding upon th ign of th input. Th bloc ith intgrl ign in th r typiclly iplntd uing lo-p filtr hr th bndidth of th lo-p filtr i t oht lor thn th rciprocl of th intgrtion ti of th SLC. 8

Figur 4 F plnttion of SLC Erlir, it indictd tht SLC r ud to cncl intrfrnc ntring through th idlob of th in ntnn. n thory, SLC cn lo cncl ignl ntring through th inlob of th in ntnn.

9 Figur 4 F plnttion of SLC Erlir, it indictd tht SLC r ud to cncl intrfrnc ntring through th idlob of th in ntnn. n thory, SLC cn lo cncl ignl ntring through th inlob of th in ntnn. Th rtriction to idlob i n iplnttion contrint. Gnrlly, th rltiv gin of ignl ntring through th in lob nd idlob diffr by 30 to 50 db. Furthr, th gin of th uxiliry ntnn i uully only 3 to 0 db highr thn th gin through th idlob of th in ntnn. f th SLC r to cncl intrfrnc ntring through th inlob, th gin of th vribl gin plifir ould nd to b vribl btn 0 nd 47 db (30-0 nd 50-3). Furthr, if th SLC to lo b bl to ccoodt intrfrnc ntring through th idlob of th in ntnn, th gin of th vribl gin plifir ould nd to b btn roughly -3 nd -40 or -50 db. Thu, th vribl gin plifir ould nd cpbility of providing vribl gin btn bout -50 nd +50 db, nd thy ould nd to rin firly linr. Thi i quit tringnt t of rquirnt to plc on th plifir. Thu, th dignr ut uully choo btn cncling inb or idlob intrfrnc. Cncling idlob intrfrnc i th ot coon pproch. Whn r driving th SLC lgorith d th uption tht v t ( t WSS. Thi i gnrlly not good uption bcu Eqution ()) i not ttionry. On thod of voiding thi probl i to hv th SLC coput t during ti hn only th intrfrnc i prnt nd hold th ight during th rt of th PR. On pproch ight b to ctivt th dptiv prt of th SLC f 0 of µ bfor th trnit pul nd thn hold th ight through th PR. Tht y, during th dpttion ti, v t 9

10 ill (hopfully) contin only th intrfrnc ignl, hich cn ronbly b ud to b WSS ovr th ti th SLC i dpting th ight. For th c of puld-dopplr ignl, th SLC ould dpt bfor th cohrnt dll, during dd ti, nd intin th SLC ight during th cohrnt dll. n phd rry rdr, th SLC ight hould b coputd ftr th b h bn trd to th dird ngl nd th b hould not b rtrd during th ti th SLC i ttpting to cncl th intrfrnc. n cnning ntnn, th SLC dignr ut ccount for ntnn otion in digning nd pcifying th prfornc of th SLC inc b otion during ight coputtion, nd th cnclltion intrvl, ill dgrd prfornc. Thi i du to th fct tht th chrctritic of th intrfrnc ntring through th idlob ill chng if th ntnn ov. Whn dvlopd th cnclltion lgorith ud tht th uxiliry chnnl ignl did not contin th trgt ignl. f th ight clcultion i prford indictd in th prviou dicuion, thi i good uption inc th ight i clcultd hn no trgt ignl i xpctd in ithr th in or uxiliry chnnl. Hovr, during th cnclltion intrvl, both trgt nd intrfrnc ignl ill b in both th in nd uxiliry chnnl ignl. Lt u uppo tht th ignl-to-intrfrnc rtio i SR. Lt u furthr u, for iplicity, tht th in ntnn offr gin of G to th trgt ignl nd unity gin to th intrfrnc ignl. W u tht th uxiliry ntnn offr (por) gin of K to both th trgt nd intrfrnc ignl. f rlt thi to Eqution () nd (3) ould rit nd v t G SRv t v t (3) 3 v t K v t SRv t. (4) W cn ho tht th iu ight i 3 3. (5) K With thi, th ignl out of th SLC i v t v t v t G SRv t v t K v t SRv t SRv t G 3 K3. (6) f th uxiliry chnnl did not contin trgt ignl coponnt, th output of th cncllr ould b 0

11 v t v t v t G SRv t v t K v t SRv t G 3 K3. (7) t ill b notd tht th prnc of th trgt in th uxiliry chnnl dgrd th output of th cncllr, but only lightly uch l thn db. Th SLC of Figur 3 nd 4 r trd ingl-loop SLC in tht thy r cpbl of cncling only on intrfrnc ourc. f to or or intrfrnc ourc r prnt, th SLC ill ttpt to prtilly cncl ll of th ourc. Hovr, it ill gnrlly not cncl ny of th vry ll. f on r to dd nothr uxiliry ntnn nd build duplict loop, but p th ur in Figur 3 nd 4, on ould hv to-loop cncllr tht ould b cpbl of cncling to intrfrnc ourc. Adding or uxiliry ntnn nd loop incr th potntil of cncling ultipl ourc. Th rul of thub i tht N SLC loop ill cncl N intrfrnc ourc. Thr i o prcticl liit to th nubr of loop nd intrfrnc ourc tht cn b cncld but not ur ht th liit i. Alo, thr i o concrn to hthr th vriou loop ill fight ch othr. v n thi dicud in th litrtur, but v not hrd of it bing ncountrd in ctul rdr.

The Z transform techniques

The Z transform techniques h Z trnfor tchniqu h Z trnfor h th rol in dicrt yt tht th Lplc trnfor h in nlyi of continuou yt. h Z trnfor i th principl nlyticl tool for ingl-loop dicrt-ti yt. h Z trnfor h Z trnfor i to dicrt-ti yt