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2 IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, SEP 7 Cyclosaionary Analysis of Analog Leas Mean Square Loop for Self-Inerference Cancellaion in In-Band Full-Duplex Sysems Anh Tuyen Le, Le Chung Tran, Member, IEEE, and Xiaojing Huang, Senior Member, IEEE Absrac Analog leas mean square (ALMS) loop is a promising mechanism o suppress self-inerference (SI) in an in-band full-duplex (IBFD) sysem. In his leer, a general soluion for he weighing error funcion is derived o invesigae he performance of he ALMS loop employed in any IBFD sysem. The soluion is hen applied o IBFD sysems wih single carrier and muli-carrier signaling respecively. I is shown ha due o he cyclosaionary propery of he ransmied signal, he weighing error funcion in he muli-carrier sysem varies more significanly han ha in he single carrier one. Therefore, if he ALMS loop can perfecly mimic he SI channel, SI in he single carrier sysem can be suppressed o a much smaller level han ha in he muli-carrier counerpar. Index Terms In-Band Full-Duplex, self-inerference cancellaion, and ALMS loop. I. INTRODUCTION In-band full-duplex (IBFD) ransmission is a promising soluion o improve he efficiency of frequency specrum uilizaion. I is also regarded as a key echnology for he nex generaion mobile broadband neworks. To enable IBFD, he mos crucial issue is o miigae he self-inerference (SI) caused by he ransmier o is co-locaed receiver. Among various approaches proposed in he lieraure, a closed-loop muli-ap filer is proved as he mos effecive one in he radio frequency (RF) domain especially for wideband applicaions 3. This muli-ap adapive filer can be implemened in he analog domain by employing leas mean square (LMS) principle. However, convenional LMS loop requires an inegraor which is difficul o build a RF sage. As a resul, many exising SI cancellaion filers implemened he LMS loop a baseband sage wih addiional down-conversion circuis 4, 5. Some ohers even used dedicaed digial modules wih sophisicaed algorihms o conrol he weighing coefficiens 3, 6. Obviously, hese addiional blocks consume more power and produce furher noise and inerference o he receiver. An LMS loop purely implemened a RF sage, so called analog LMS (ALMS) loop, was proposed in 7 where a lowpass filer () was used o replace he ideal inegraor. The behaviors of he ALMS loop were invesigaed by examining he weighing error funcion. This is he firs paper considering he cyclosaionary properies of he ransmied signal in regards o he behavior of he cancellaion circui. However, he analysis was only conduced for a single carrier sysem, and he soluion for weighing error funcion was only derived for he specific roo raised cosine (RRC) pulse shaping. A. T. Le and X. Huang are wih Faculy of Engineering and IT, Universiy of Technology Sydney, Ausralia ( s: anhuyen.le@suden.us.edu.au and xiaojing.huang@us.edu.au). L.C. Tran is wih Faculy of Engineering and Informaion Sciences, Universiy of Wollongong, Ausralia ( lcran@uow.edu.au). In his leer, we firsly exend he soluion of he weighing error funcion in a general case. This general soluion can be applied o invesigae he performance of he ALMS loop employed in any IBFD sysem. I is revealed ha, due o he cyclosaionary effecs of he ransmied signal, he weighing error funcion canno converge o a sable value, bu i varies periodically. Consequenly, here always exiss an irreducible SI whose power depends on he variaion of he weighing error funcion. The soluion is hen applied o compare he performance of he ALMS loop in a single carrier sysem o ha in a muli-carrier such as an orhogonal frequency division muliplexing (OFDM) sysem. We show ha he convergence speed of he weighing error funcion is he same in boh cases and depends on he loop gain. However, he weighing error funcion in he single carrier sysem has a smaller variaion han ha in he OFDM case. The irreducible inerference suppression raio lower bounds (ISRLBs) are also derived for he wo sysems respecively. Comparison beween hem shows ha more SI cancellaion can be ulimaely achieved for he single carrier sysem. Hence, he main conribuion of his leer is he general soluion for he weighing error funcion which can be used as a key o esimae he ulimae level of SI cancellaion obained by he ALMS loop. ISRLB is a very imporan performance meric in designing he whole IBFD sysem. The res of his leer is organized as follows. Secion II describes he mahemaical models of he ransmied signals and he srucure of he ALMS loop. In Secion III, we derive a general soluion of he weighing error funcion for he ALMS loop and compare is performance in single carrier and OFDM sysems. Finally, conclusions are drawn in Secion IV. II. SIGNAL AND SYSTEM DESCRIPTION A. Signal Models Consider an IBFD sysem including a baseband par which can operae in eiher single carrier or OFDM mode and an RF par employing an ALMS loop. The ransmied RF signal is expressed as x()=re{x()e πfc } where f c is he carrier frequency, and X() is he baseband equivalen which is furher denoed as X s () in he single carrier mode and X o () in he OFDM mode. Mahemaically, X s () and X o () can be expressed as X s ()= a i p( it s ) () and X o ()= i= N s/ n= m= k= N s/,k w n m T o T s a k,m e jπ k N (n mto Ts ) p( nt s ) ()

3 IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, SEP 7 respecively, where a i, and a k,m are he i-h daa symbol in he single carrier sysem and he daa symbol on he k- h sub-carrier of he m-h OFDM symbol respecively; T s is he symbol period of he single carrier sysem and also he sample period of he OFDM sysem; T o is he OFDM symbol period; N s is he oal number of daa subcarriers; N is he number of samples in one OFDM symbol excluding cyclic prefix; wn is he discree windowing funcion applied o an OFDM symbol; and p() is he pulse shaping funcion. The roo mean square ampliude of he ransmied T signal is defined as V X = T E{ X() }d, where E{.} sands for expecaion; T is he period of ransmied daa symbol, i.e., T s or T o. The complex daa symbols a i and a k,m are assumed o be independen o each oher in single carrier and OFDM sysems respecively. The auocorrelaion funcion of he ransmied baseband signal X() is defined as Φ(,τ)=E{X ()X( τ)}. Wih he symbol independence assumpion, he auocorrelaion funcions of single carrier and OFDM signals can be derived as Φ s (,τ)= p ( it s )p( τ it s ) (3) and Φ o (,τ)= i= respecively, where g(,τ)= N s/ l= l = k= N s/,k e jπ k N (l l) wlwl g( lt s,(l l)t s +τ) (4) m= p ( mt o )p( mt o τ). We see ha Φ s (,τ)=φ s (+T s,τ) and Φ o (,τ)=φ o (+T o,τ) for all and τ. Therefore, boh ransmied signals X s () and X o () can be reaed as wide-sense cyclosaionary processes. B. ALMS Loop The archiecure of he ALMS loop proposed in 7 is shown in Fig.. This is an L-sage muli-ap filer in which each ap has Tx x() PA Td Td From Up - Converer Im{w()} Re{w()} Re{wL-()} LNA LNA Gain µ Im{wL-()} y() Rx r() d()=r()-y() To Down - Converer Fig. : The ALMS loop srucure. a fixed delay T d. To avoid specral overlapping, T d is chosen as T d T s. The cancellaion signal y() is generaed o cancel he SI z() included in he received signal r()=z()+s()+n() where s() is he received signal from a remoe ransmier, and n() is he addiional Gaussian noise. The residual signal d() is amplified by he low noise amplifier (LNA) and muliplied using he I/Q demodulaion archiecure wih he delayed versions of he ransmied signal x(). The oupus of he l-h I/Q demodulaor are filered by respecive Resisor- Capacior (RC) s wih consan α (α=/rc) o generae he complex weigh coefficien w l () which is derived in 7 as w l ()= µα e α( τ) r(τ) y(τ) K K X(τ lt d )e jπfc(τ ltd) dτ (5) where K and K are he dimensional consans of mulipliers in he I/Q demodulaor and I/Q modulaor respecively; and µ is he gain of he LNA. Assume ha he SI channel is modeled as an L-sage muli-ap filer where each ap has a coefficien h l and delay T d, and hence he baseband equivalen of he SI z() can be expressed as Z()= l= h l X( lt d). Since { he cancellaion signal y() is consruced as y()= } Re l= w l ()X( lt d)e jπfc( lt d), he performance of he ALMS loop can be deermined by he weighing error funcion u l ()=h l w l ()e jπfclt d. Is expeced value ū l () is derived in 7 as ū l ()=h l µα e α( τ) ū l (τ)φ(τ,(l l )T d )dτ. (6) K K l = This equaion shows ha he weighing error funcion no only depends on he loop parameers α, µ and K K bu also relaes o he auocorrelaion funcion of he ransmied signal, and hus he cyclosaionary properies will have significan impac on he ALMS loop performance. III. CYCLOSTATIONARY ANALYSIS A. General Soluion of Weighing Error Funcion I is very difficul o solve (6) in a general case. However, if he auocorrelaion funcion of he ransmied signal saisfies ha { VX Φ(,τ)= Φ(,), for τ= (7) for τ=ineger muliples of T d where Φ(,) is he normalized auocorrelaion funcion, (6) can be simplified as ū l ()=h l αµa e α( τ) ū l (τ) Φ(τ,)dτ (8) where A =VX /KK. Taking he differeniaion wih respec o on boh sides of (8), we have dū l () =µa α e α( τ) ū l (τ) Φ(τ,)dτ µαa ū l () Φ(,) d =α h l ū l () µαa ū l () Φ(,) (9) which can be furher rearranged in he form of he ordinary differenial equaion (ODE), i.e., dū l () +α Φ(,) ū l ()=αh l. () d The soluion for he homogeneous form of he ODE, i.e., U ()+α Φ(,) U()= can be found by rearranging i as U () U() = α Φ(,). () Inegraing boh sides from o, we ge lnu()= α + µa Φ(τ,) dτ+lnu() so ha Φ(τ,) U()=U()e α dτ. () Replacing U() by a funcion f(), ū l ()= f()e α Φ(τ,) dτ is he soluion for he non-homogeneous form of he ODE. Taking he

4 IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, SEP 7 3 differeniaion of ū l () and subsiuing i ino () we ge f ()=αh l e α Φ(τ,) dτ. Therefore, τ f()=αh Φ(v,) l eα dv dτ+c where C is any consan. Thesoluion for ū l () is hus ū l ()= αh l e α τ Φ(v,) dv dτ+c e α Φ(τ,) dτ = αh l e α( )( τ) e αµa τ +Ce α( ) e αµa Φ(τ,) dτ. Φ(v,) dv dτ (3) When αµa τ Φ(v,) dv and /α( ), )( τ) e α( e αµa τ Φ(v,) dv dτ α( ). Therefore, ū l () hl +Ce ( ) e µa α Φ(τ,) dτ. µa From he iniial condiion ha ū l ()=h l we have C=h l, and hence he final soluion is ū l ()= h e α( ) l e µa αq() (4) where q()= Φ(τ,) dτ. Due o he cyclosaionary properies of Φ(τ,), we see ha e µαaq() is a periodical funcion so ha ū l () varies periodically. If here was no cyclosaionary effec, he weighing error funcion, denoed as ŭ l (), would have he expecaion E{ŭ l ()}=h e α( ) l, which would converge o a sable value h l when /α( ). In his case he residual SI could be furher removed in he digial domain. However, he presence of cyclosaionary effec in he residual SI makes i impossible o be compleely removed in digial domain. Thus here always exiss an irreducible inerference whose power P II is deermined by he variaion beween u l () and ŭ l (). The expeced value of his variaion is denoed as ũ l ()=E{u l () ŭ l ()}=h l (e µaαq() ) when /α( ). Since E{ X } E{X} for any random process X, he ime averaged P II is P II = A T E{ u l () ŭ l () } d T l= T T A { } A E u l () ŭ l () d= ũ() d T l= T l= T d PI =P q() T I ) T (e αµa αq() d T (5) where P I = A l= h l is he normalized inerference power. Based on he lower bound of P II, he irreducible inerference suppression lower bound (ISRLB) defined as ISRLB= P I T T αq() d = T αq() d (6) P I T can be used as a measure o compare he performance of he ALMS loop for differen ypes of he ransmied signal. Therefore, ISRLB is an imporan figure o be considered in he cancellaion design process. In he following secion, we compare he performance of he ALMS loop in a single carrier sysem wih ha in a muli-carrier one o show he impac of cyclosaionary properies. g(, τ) /T s τ/t s Fig. : One period of g(,τ) wih T o =8T s. B. Single Carrier Versus OFDM To apply he above soluion of he weighing error funcion of he ALMS loop o he wo sysems, we firsly examine heir respecive auocorrelaion funcions. For a single carrier sysem wih RRC pulse shaping funcion, i is shown in 7 ha Φ s (,τ) { saisfies ( (7) wih ) a closed-form as V βs Φ s (,τ) X π cosπ T s +, for τ=, for τ=ineger muliples of T s (7) where is he roll-off facor of he RRC pulse shaping funcion. Hence, q() for he single carrier sysem is derived β as q s ()=T s s 7, Eq.(7). In case of he muli-carrier sysem, an IEEE8.a baseband is aken as an example. We firsly examine he auocorrelaion g(,τ) of he pulse shaping funcion p() inroduced in (4), which is a periodical funcion of he period T o. One period of g(,τ) wih he power of p() normalized o is shown in Fig.. Obviously, g(,τ) when τ is any ineger muliple of T s, and hence he auocorrelaion funcion of his OFDM signal a τ = becomes Φ o (,)=N s l= w lg( lt s,) wih he period T o. For simpliciy, one period of he convoluion of w l and g(,) can be furher approximaed as a coninuous window w (). Therefore, he auocorrelaion funcion of he OFDM signal has a closed form of a periodical funcion of whose period π sin π T s conains he coninuous { window w (), i.e., Φ o (,(l l V )T d ) X m= w ( mt o ), for l=l, for l l (8) where w(), T o, is he normalized coninuous window- ing funcion such ha To T o w ()d=. For he discree windowing funcion recommended in he IEEE8.a sandard 8, afer conversion o he coninuous funcion and normalizaion, we have expression of he windowing funcion w() as 4(+ ) sin ( π ( T )) <T w()= T 4 β <T o (9) sin ( π (To T )) T <T o where T = T o /(+ ) and T =T o /(+ ) wih as he roll-off facor of he windowing funcion. Applying he above soluion, we find he OFDM version of he q() funcion as 5( ) βoto (4 ) (4 )π sin(π T )+ T o 4π(4 ) sin(π T ) <T 5β q o ()= o 4 ( T o /) T <T 5( ) (4 ) ( T o)+ βoto (4 )π sin(π(to ) T ) βoto 4π(4 ) sin(π(to ) T ) T <T o. () 4

5 IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. XX, SEP 7 4 ūl()/hl OFDM Single Carrier /T s Fig. 3: Normalized weighing error funcions wih he loop gain µa =, αt s =.3, T o =8T s, and = =.5. ũl()/hl OFDM Single Carrier /T s Fig. 4: Normalized weighing error variaion wih he loop gain µa =, αt s =.3, T o =8T s, and = =.5. From q s (), q o (), and (4) we can obain he weighing error funcions for he single carrier and OFDM sysems as ū l,s () and ū l,o () respecively. To compare he performance of he ALMS loop in he wo sysems, he convergence curve of he normalized weighing error funcion ū l ()/h l for he wo cases under loop gain µa =, αt s =.3 and T o =8T s are ploed in Fig. 3. The normalized variaion ũ l ()/h l is presened in Fig. 4. The inses in Fig. 3 and Fig. 4 show a closer look for he ū l,s () and ũ l,s () respecively. From Fig. 3, i can be concluded ha wih he same loop gain µa and he RC consan α, he convergence speeds of ū l,s () and ū l,o () are he same for boh cases of he ransmied signals. Moreover, boh ū l,s () and ū l,o () do no converge o a sable value, bu hey vary wih periods T s and T o respecively. In erms of variaion, as shown in Fig. 4, ũ l,s () varies in a smaller range han ũ l,o () does. Thus, he ISRLB of he single carrier sysem is expeced o be smaller han ha of he OFDM counerpar. Subsiuing q s () and q o () ino (6), we obain he ISRLB for he single carrier and OFDM sysems as ISRLB s = ( αts π ), and ISRLB o = { α T o 5 (4 ) (+) } 6π (8 55β o ) ( ) + 5βo respecively. Puing he ISRLB s and ISRLB o ogeher in Fig. 5 as funcions of αt s and various values of he roll-off facors,, we see ha, wih he same value of and ISRLB (db) =. =.5 =.5 =.75 = αt s =. =.5 =.5 =.75 = Fig. 5: ISRLB of he wo sysems wih various values of and. excep for = and =, ISRLB s is much smaller han ISRLB o. I means ha when he ALMS loop has exacly he same ap spacing as he SI channel, he SI in he single carrier sysem can be suppressed o a much lower level han ha in he ODFM sysem. The reason is ha he weighing coefficiens of he ALMS loop are affeced by he auocorrelaion funcion of he ransmied signal as we have analyzed. As he period of an OFDM symbol is much longer han ha of a daa symbol in he single carrier sysem, he weigh coefficiens in he OFDM sysem vary more significanly. IV. CONCLUSIONS The general soluion for he weighing error funcion is derived o reveal he significan impacs of cyclosaionary properies of he ransmied signal on he performance of he ALMS loop. Applying his soluion o boh single carrier and OFDM IBFD sysems, we show ha, given he same loop gain and oher parameers, he SI can be poenially canceled more effecively o a smaller level of ISRLB in he single carrier sysem han ha in he OFDM sysem due o he differen cyclosaionary properies of he ransmied signals. Deermining he ISRLB is an imporan consideraion in he SI cancellaion design process. ACKNOWLEDGMENT This work was suppored by he Ausralian Research Council (DP6693). REFERENCES A. Sabharwal e al., In-band full-duplex wireless: Challenges and opporuniies, IEEE J. Sel. Areas Commun, vol. 3, no. 9, pp , 4. S. Han e al., Full duplex: Coming ino realiy in? in 4 IEEE Global Communicaions Conference, Dec 4, pp D. Korpi e al., Full-duplex mobile device: Pushing he limis, IEEE Communicaions Magazine, vol. 54, no. 9, pp. 8 87, Sep 6. 4 T. Huusari e al., Wideband self-adapive RF cancellaion circui for full-duplex radio: Operaing principle and measuremens, in IEEE 8s Vehicular Technology Conference (VTC Spring), May 5. 5 J. Kim e al., Full-duplex Radios in 5G: Fundamenals, Design and Prooyping, in Signal Processing for 5G. Chicheser, UK: John Wiley & Sons, Ld, Aug 6. 6 K. E. Kolodziej, J. G. McMichael, and B. T. Perry, Muliap RF canceller for in-band full-duplex wireless communicaions, IEEE Trans. Wireless Commun., vol. 5, no. 6, pp , June 6. 7 X. Huang and Y. J. Guo, Radio frequency self-inerference cancellaion wih analog leas mean-square loop, IEEE Trans. Microw. Theory Tech., vol. 65, no. 9, pp , Sep 7. 8 IEEE Wireless LAN Medium Access Conrol (MAC) and Physical Layer (PHY) Specificaions, IEEE Sd , 997.