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Cartesian eedbac for RF er Ampliﬁer Linearization Jo el L. Da wson and Thomas H. Lee Cen ter for In tegrated Systems, Stanford Univ ersit jldawson@mtl.mit.edu Abstract discuss con trol problems that arise in con- nection with Cartesian feedbac radio-frequency er ampliﬁers. New solutions to oth problems are de- scrib ed, and the results of orking protot yp are presen ted. The protot yp e, in tegrated circuit (IC) fabricated in National Semiconductor’s 0.25 CMOS pro cess, represen ts the ﬁrst kno wn fully in tegrated im- plemen tation of the Cartesian feedbac concept. In tro duction Designers of radio-frequency (RF) er ampliﬁers (P A’s) for mo dern wireless systems are faced with diﬃcult tradeoﬀ. On one hand, the consumes the lion’s share of the er budget in most transceiv ers. It follo ws that in cellular phone, for example, battery lifetime is largely determined the er eﬃciency of the A. On the other hand, it ma desirable to ha high sp ectral eﬃciency—the abilit to transmit data at the highest ossible rate for giv en hannel bandwidth. The design conﬂict is that while sp ectral eﬃciency demands highly linear A, er eﬃciency is maximized when is run as constan t-en elop e, nonlinear elemen t. The curren state of the art is to design mo derately linear and emplo some lin- earization tec hnique. The ampliﬁer op erates as close to saturation as ossible, maximizing its er eﬃciency and the linearization system maximizes the sp ectral ef- ﬁciency in this near-saturated region. There are man diﬀeren linearization tec hniques. Our ork fo cuses on Cartesian feedbac systems for main reasons. First, ecause they emplo analog feed- bac k, the requiremen for detailed nonlinear mo del of the is greatly relaxed. This is an extremely com- elling adv an tage, as RF A’s are orly understo and notoriously diﬃcult to mo del. Second, Cartesian feedbac systems automatically and elegan tly comp en- sate for pro cess ariations, temp erature ﬂuctuations, and aging. Nev ertheless, historically the tec hnique has suﬀered the shortcoming of relying on sync hronous do wncon ersion, whic has een diﬃcult to realize without man ual trimming. This problem, com bined with the recen trend to ard fully monolithic systems, has caused Cartesian feedbac to languish for ears as little more than an academic curiosit approac the sync hronous do wncon ersion, or phase alignmen t, problem from directions. De- tailed analysis of Cartesian feedbac system is er- formed, and it is sho wn to suggest means of com- ensating the system for robustness to phase misalign- men t. Alternativ ely describ and analyze non- linear, analog phase alignmen regulator[1 ]. est re- sults for fabricated IC, designed as testb ed for these ideas, are then presen ted and analyzed. Cartesian eedbac The idea of using Cartesian feedbac to linearize er ampliﬁers has een discussed at least as early as the 1970’s[2 ]. It is called Cartesian feedbac ecause the feedbac is based on the Cartesian co ordinates of the baseband sym ol, and as opp osed to the o- lar co ordinates. The ypical system is illustrated in ﬁgure 1. [2 ]. undamen tally the concept ehind this PA PSfrag replacemen ts sin cos sin( cos( Figure 1: ypical Cartesian feedbac system. Ideally 0. system is negativ feedbac k. couple of factors com- plicate its expression in the con text of an RF transmit- ter, ho ev er. The ﬁrst is the extremely high frequency of man RF carriers, with mo dern standards calling for frequencies on the order of few gigahertz. this time, it is virtually imp ossible to build high-gain, sta- ble analog feedbac lo op with crosso er frequency in p.

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that range. The second factor is the recognition that in mo dulating an RF carrier, are not shaping oltage eform in its en tiret Instead, are shaping indep enden haracteristics of that carrier. Cartesian feedbac k’s of dealing with the ﬁrst factor is the inclusion of frequency translation step in the feedbac path, sho wn as do wncon ersion mixer in ﬁgure 1. The lo op is then closed at baseband, rather than at the carrier frequency The system consequence is to linearize only in narro band of the sp ectrum cen tered ab out the carrier, rather than from DC to the carrier. This is an ingenious to exploit the narro wband nature of most RF signals. The second factor manifests as the “double lo op struc- ture of the system. There are degrees of freedom in shaping, or mo dulating, an otherwise free-running RF carrier, and at least hoices of co ordinate systems that fully describ the mo dulation. or olar feedbac the hoice made is to consider an RF carrier as ha ving an amplitude and phase. The structure of olar feedbac system reﬂects this hoice, ha ving one con trol lo op for the amplitude, and another for the phase. An equiv alen hoice of co ordinates is the Cartesian com- onen ts, in whic consider the mo dulated carrier as the sum sin )) sin cos t, where cos and sin It is seen that Cartesian feedbac treats the degrees of freedom in symmetrical allo wing the structure of the system to tak the form of iden tical lo ops. This is in direct con trast to olar feedbac k, where the degrees of freedom ust treated ery diﬀeren tly Consequences of phase misalignmen in Cartesian feedbac systems Figure sho ws ypical Cartesian feedbac system. The system blo represen ts the lo op driv er am- pliﬁers, whic pro vide the lo op gain as ell as the dy- namics in tro duced the comp ensation strategy The lo op driv ers feed the baseband inputs of the up con er- sion mixer, whic in turn driv es the er ampliﬁer. Some means of coupling the output of the er am- pliﬁer to the do wncon ersion mixer is emplo ed, and the output of this mixer is used to close the feedbac system. 3.1 Impact of phase misalignmen on stabilit Ideally Cartesian feedbac system functions as iden tical, decoupled feedbac lo ops: one for the com- onen t, and one for the comp onen t. This corre- sp onds to the case of in ﬁgure 1. In practice, ho ev er, this state of aﬀairs ust activ ely enforced. Dela through the er ampliﬁer, phase shifts of the RF carrier due to the reactiv load of the an tenna, and mismatc hed in terconnect lengths et een the lo cal os- cillator (LO) source and the mixers all manifest as an eﬀectiv nonzero orse, the exact alue of aries with temp erature, pro cess ariations, output er, and carrier frequency Cartesian feedbac system in whic is nonzero is said to ha phase mis- alignmen t. In this state the feedbac lo ops are cou- pled, and the stabilit of the system is compromised. The impact of phase misalignmen on system stabil- it can seen mathematically start observing that the demo dulated sym ol is rotated relativ to an amoun equal to the phase misalignmen see this, write Cartesian comp onen ts of the demo d- ulated sym ol sin cos sin sin cos cos where is the carrier frequency Using trigonometric iden tities and assuming frequency comp onen ts at are ﬁltered out, arriv at cos sin (1) sin cos (2) see that for 0, an excitation on the input of the mo dulator results in signal on the do wncon- erter output (and similarly for and ). Accordingly sa that the lo ops are coupled. One metho of stabilit analysis is to consider the error signals and sho wn in Figure 1. Recall that for single feedbac lo op, the error signal is written where is the command input and is the lo op transmission. In the presen case, let the phase mis- alignmen urthermore, set without loss of generalit The error expressions, as function of the single input ), are written cos sin sin cos φ, ec hnically is also an uncoupled case. Ho ev er, there is no an in ersion in oth lo ops, resulting in ositiv feedbac instead of the desired negativ feedbac k. do not lose generalit as long as sta with linear analysis. p.

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where includes the dynamics of the lo op comp en- sation sc heme and the (linearized) dynamics in- tro duced the mo dulator, er ampliﬁer, and de- mo dulator. rom here, it is straigh tforw ard to sho that cos sin 1+ cos This reduction of the system to single-input problem no yields considerable insigh t. iden tify an eﬀec- tiv lo op transmission, e s, ), as follo ws: e s, cos sin cos (3) or erfect alignmen t, and e is simply ). The orst alignmen is for whic e )] and so the lo op dynamics are cascade of the dynamics in the uncoupled case. Unless designed with this os- sibilit in mind, most hoices of yield unstable eha vior in this second case. Equation sho ws that traditional measures of stabilit degrade con tin uously as sw eeps from to fact demonstrated exp eri- men tally Briﬀa and aulkner [4 ]. 3.2 Comp ensating the system for robustness to phase misalignmen Equation oﬀers great deal of insigh in to what hap- ens in phase-misaligned Cartesian feedbac system. Ph ysically the fully coupled case eha es as depicted in ﬁgure 2, where represen ts the dynam- ics that the up con ersion mixer, er ampliﬁer, and do wncon ersion mixer con tribute to the lo op transmis- sion. In the literature, all eﬀorts with regards to the PSfrag replacemen ts Figure 2: Cartesian feedbac under 90-degree misalign- men t. phase alignmen problem ha fo cused, naturally on ensuring phase alignmen t. But there is at least one other approac that deserv es consideration: is it ossi- ble to ho ose suc that it is stable for large phase misalignmen ts? The answ er dep ends in part on what one means “large. Considering misalignmen of for instance, is discouraging. In this case e ), and there is simply no comp ensation strategy that is indiﬀeren to the sign of the lo op transmission. Carte- sian feedbac in fact do es ecome ositiv feedbac system for misalignmen ts in the op en in terv al ), where the exact oin of transition from negativ to ositiv feedbac dep ends on the details of ). oid considering ositiv feedbac cases, then, it is sensible to restrict the range of misalignmen ts to the closed in terv al ]. That stabilit margins degrade ontinuously with suggests that ﬁnding comp ensation strategy that orks in the limiting cases of and will solv the problem for the whole in terv al. Assuming the dynamics of the lo op are dominated ), com- ensation strategy that emerges is where 1. Suc “slo w-rollo functions, while not truly realizable with lump ed-elemen net ork, can appro ximated alternating oles and zeros suc that the erage slop of H(s) is the appropriate dB-p er-decade[5 ]. In the case of 5, for instance, stabilit as measured phase margin ould excel- len t: 135 degrees in the aligned case, and 90 degrees in the misaligned case. Ro ot lo cus analysis conﬁrms that slo w-rollo com- ensation is viable approac to designing for large misalignmen ts. Figure sho ws the ro ot lo ci for the PSfrag replacemen ts Dominan t-p ole Slo w-rollo comp ensation comp ensation )] Figure 3: Ro ot lo cus plots for dominan t-p ole and slo w- rollo comp ensation. dominan t-p ole and comp ensation strategies. It can seen that ev en in the case of the dominan t-p ole, 90- degree misalignmen do esn’t necessarily lead to righ t- half-plane oles. est, ho ev er, the result is ligh tly damp ed, complex pair of oles. orst, high- frequency oles not sho wn here (or not mo deled) push this complex pair in to the righ t-half plane. By con- trast, the slo w-rollo comp ensation is seen to lead to hea vily-damp ed complex ole pairs, and one exp ects corresp onding reduction in ersho ot and ringing in resp onse to an input step. One also exp ects the lo w- frequency zero-p ole doublets of the ro ot lo ci to man- p.

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ifest themselv es as slo w-settling “tails in the step re- sp onse [5 ]. Exp erimen ts carried out on the ﬁnal IC in accordance with this comp ensation discussion alidate these exp ec- tations. As seen in section 5.2, the slo w-rollo tec h- nique stabilizes the system for all misalignmen ts up to 90 degrees. In addition to shedding ligh on comp en- sation strategies for Cartesian feedbac systems, the imp ortance of these exp erimen ts is that they conﬁrm the understanding dev elop ed in section 3.1. nonlinear regulator for main taining phase alignmen Occasionally con tin uous regulation of the phase align- men is not needed, and it suﬃces to in tro duce man u- ally adjustable dela et een the LO source, and, sa the demo dulator. This approac is only feasible, ho w- ev er, if the system is not sub ject to ariations in tem- erature, carrier frequency pro cess parameters, or, in some cases, output er. or cases in whic the align- men ust regulated, arious metho ds ha een prop osed in the literature; see, for example [6 ]. presen our con trol concept as compact, truly con- tin uous solution to the problem of LO phase alignmen t. It is truly con tin uous ecause it do es not, for example, rely on the app earance of sp eciﬁc sym ol or pattern in the outgoing data stream. It is compact ecause it is easily implemen ted without digital signal pro cessing, as presen ted here. This is particularly comp elling ad- an tage, as the signals in Cartesian feedbac system are necessarily in analog form. And emphasize that, ecause the concept is based on the pro cessing of base- band sym ols, its realization is indep enden of carrier frequency 4.1 Nonlinear dynamical system Figure represen ts baseband sym ol at the inputs of the mo dulator and at the outputs of the demo du- lator of Cartesian feedbac system. Mathematically the ectors are describ ed in oth Cartesian and o- lar co ordinates, with primed co ordinates denoting the demo dulated er ampliﬁer output and unprimed co- ordinates denoting the mo dulator input. In addition to undergoing distortion in magnitude, the demo du- lated sym ol is rotated an amoun exactly equal to the phase misalignmen (see equations and 2). start to the design of phase alignmen regulator is to observ that the signals and tak en to- gether, represen enough information to determine the phase misalignmen t. urther, they are easily accessible within the system. seek to com bine these ariables suc that, er suitable range, the deriv ed signal is monotonic in the phase misalignmen t. PSfrag replacemen ts Figure 4: Rotation of the baseband sym ol due to phase misalignmen t. One suc com bining of the ariables is the sum of pro d- ucts QI Recognizing that sin and cos and using trigonometric iden tities, write the ey relation QI sin( (4) see that using ultipliers and subtractor, op- erations easily realizable in circuit form, one can deriv con trol signal that is indeed monotonic in the phase misalignmen er the range Figure details nonlinear dynamical con troller built around equation 4. Using the notation an PSfrag replacemen ts phase LO LO RF RF QI d dt dt cos cos sin sin d dt QI Gr sin( Figure 5: Phase alignmen concept. implemen tation can understo as mec hanizing the equation d dt )] sin ( (5) where is constan of prop ortionalit and gain is asso ciated with the in tegrator. Equation presupp oses the abilit to correct the phase shift hanging The original protot yp describ ed in[7 realizes the required rotation directly phase shifting the mo dulator LO. Ho ev er, substan tial er sa vings result from doing sym ol rotation at baseband as sho wn in ﬁgure 5. Regardless, rotation should p.

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erformed in the forw ard path of the Cartesian feed- bac system, where the una oidable artifacts of imp er- fect rotation are suppressed. 4.2 Stabilit concerns Our con trol solution for the phase alignmen problem is the simplest of nonlinear dynamical systems. It is seen from equation to ha equilibrium oin ts. The ﬁrst, for whic the sym ols are aligned, is stable The second, for whic the sym ols are misaligned radians, is unstable. or the ideal system represen ted equation 5, this is the exten of rigorous stabilit analysis. The real-w orld situation can complicated dynam- ics asso ciated with the phase shifter (and, ossibly the subtractor). If pro visionally consider mo d- ulation sc heme in whic the magnitude of transmitted sym ols is held constan t, in equation loses its time dep endence. Linearizing for small phase misalign- men ts, and including the dynamics of the phase shifter as ), can represen the system as sho wn in ﬁg- ure 6. Dra wing the system this requires some ma- PSfrag replacemen ts Gk Drift Phase Distortion Figure 6: Linearized phase regulation system. ’M is the desired misalignmen t, whic is nominally zero. nipulation. The output of the phase shifter is not really but rather an additiv art of that gets com bined with the olar angle of the sym ol eing transmitted. Ho ev er, in the absence of phase distortion and drift, the sym ol-b y-sym ol hanges of the olar angle are trac ed iden tical hanges in These sym ol-rate hanges are th us in visible to an alignmen system, and it is appropriate to lab el the output of as can then include the eﬀects of phase distortion and phase alignmen drift as the additiv disturbances of ﬁgure 6. One can ensure stabilit ho osing suc that, for the largest sym ol magnitude, lo op crosso er ccurs e- fore non-dominan oles ecome an issue. ortunately the drift disturbance will normally ccur on time scales asso ciated with temp erature drift and aging [2 ]. Sup- pression of the phase distortion is the domain of the Cartesian feedbac itself. It follo ws that for man sys- tems, little of the design eﬀort need fo cused on fast Unlik ely when using Cartesian feedbac k, of course. em- orarily making this assumption, ho ev er, yields insigh that is broadly relev an to the stabilit analysis. phase alignmen t. Exp erimen tal results What the new phase alignmen regulator enables is the building of highly in tegrated Cartesian feedbac sys- tems. This is comp elling design goal, as it ma al- lo this linearization tec hnique to used for mo dern, handheld wireless devices. As demonstration, fully monolithic protot yp e, fabricated in National Semicon- ductor’s 0.25 CMOS tec hnology as designed and tested. our kno wledge, this is the ﬁrst successful in tegration of er ampliﬁer, phase alignmen sys- tem, and Cartesian feedbac linearization circuitry all on the same die. 5.1 Phase alignmen system Figure is trace capture of the yp of exp erimen used to haracterize the erformance of the phase align- men system. The Cartesian feedbac lo op is op en, PSfrag replacemen ts align off align off align on align on 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 -0.02 -0.02 -0.02 -0.02 1.4 1.4 1.4 1.4 1.45 1.45 1.45 1.45 1.5 1.5 1.5 1.5 1.55 1.55 1.55 1.55 1.6 1.6 1.6 1.6 1.65 1.65 1.65 1.65 1.7 1.7 1.7 1.7 1.75 1.75 1.75 1.75 1.8 1.8 1.8 1.8 1.85 1.85 1.85 1.85 Time(msec) Figure 7: race capture of phase alignmen exp erimen t. The Cartesian feedbac lo op is op en. 500mV amplitude, 10kHz square driv es the hannel, and the hannel is grounded. The top traces sho that, initially the misalignmen is man u- ally set to 45 degrees. The ottom traces sho the result of turning on the phase alignmen system (releasing it from the “reset mo de). Ov er the full, 90 degree range that the sym ol rotator ermitted, the regulator ept the phase misalignmen elo de- grees. Figure serv es to illustrate the impact of phase mis- alignmen on the stabilit margins of the closed-lo op The oltage dro op on what is normally the ﬂat part of the square es is due to the fact that, at the oard-lev el, the pre- distortion inputs ha een C-coupled. p.

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PSfrag replacemen ts align off align off align on align on 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 -0.02 -0.02 -0.02 -0.02 -0.5 -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 Time(msec) Figure 8: Illustration of phase alignmen stabilizing the closed-lo op CFB system. CFB system. Dominan t-p ole comp ensation is used in the CFB lo op, and for the upp er traces the mis- alignmen is man ually set to 74 degrees. Ov ersho ot and ringing is eviden on these eforms, and further mis- alignmen causes outrigh oscillation. or the ottom traces the phase alignmen system is turned on, and one sees the classic ﬁrst-order step resp onses that are exp ected when using dominan t-p ole comp ensation. 5.2 Comp ensation exp erimen ts The protot yp as designed suc that the lo op trans- mission could aried. Among ossible hoices of lo op comp ensation, the slo w-rollo net ork is of particular in terest as demonstration of the ideas dev elop ed in section 3. Our slo w-rollo net ork realized three oles and zeros. Figure pro vides dramatic comparison of the system under dominan t-p ole ersus slo w-rollo comp ensation. or this exp erimen t, the phase misalignmen of the sys- tem is man ually set to 90 degrees. The top traces sho the dominan t-p ole comp ensated system under 90- degree misalignmen t. Substan tial ersho ot and ring- ing is visible, indicativ of ligh tly damp ed, complex ole pair. The ottom traces sho the system under slo w-rollo comp ensation. The step resp onse is remark ably similar to that of single ole system. Conclusion The problem of phase alignmen has sto as the pri- mary barrier to the widespread use Cartesian feedbac k. In this pap er describ new analysis, and use the resulting insigh to design system that is toleran to 90 degrees of misalignmen t. new phase alignmen PSfrag replacemen ts 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 -0.02 -0.02 -0.02 -0.02 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 4.5 4.5 4.5 4.5 /s /s Time (msec) Figure 9: Step resp onse comparison et een dominan t- ole and slo w-rollo comp ensated systems for 90-degree misalignmen t. regulator is also review ed. ak en together, these re- sults considerably lo er the barrier to implemen ting Cartesian feedbac in mo dern wireless transceiv ers. References [1] J.L. Da wson and T.H. Lee. Automatic phase alignmen for fully in tegrated Cartesian feedbac er ampliﬁer system. IEEE Journal of Solid-State Cir cuits 38:2269–2279, Decem er 2003. [2] B. Raza vi. RF Micr ele ctr onics Pren tice-Hall, Inc., Upp er Saddle Riv er, NJ, 1998. [3] D. Co x. Linear ampliﬁcation sampling tec h- niques: new application for delta co ders. IEEE ansactions on Communic ations COM-23:793–798, 1975. [4] M. Briﬀa and M. aulkner. Gain and phase mar- gins of Cartesian feedbac RF ampliﬁer linearisation. Journal of Ele ctric al and Ele ctr onics Engine ering, us- tr alia 14:283–289, 1994. [5] J. K. Rob erge. Op er ational mpliﬁers: The ory and Pr actic John Wiley and Sons, Inc., New ork, New ork, 1975. [6] Y. Ohishi, M. Mino a, E. ukuda, and T. ak ano. Cartesian feedbac ampliﬁer with soft landing. In 3r IEEE International Symp osium on Personal, Indo or, and Wirless Communic ations pages 402–406, 1992. [7] J.L. Da wson and T.H. Lee. Automatic phase alignmen for high bandwidth Cartesian feedbac er ampliﬁers. In IEEE adio and Wir eless Con- fer enc pages 71–74, 2000. p.

Da wson and Thomas H Lee Cen ter for In tegrated Systems Stanford Univ ersit jldawsonmtlmitedu Abstract discuss con trol problems that arise in con nection with Cartesian feedbac radiofrequency er ampli64257ers New solutions to oth problems are de s ID: 22733

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Cartesian eedbac for RF er Ampliﬁer Linearization Jo el L. Da wson and Thomas H. Lee Cen ter for In tegrated Systems, Stanford Univ ersit jldawson@mtl.mit.edu Abstract discuss con trol problems that arise in con- nection with Cartesian feedbac radio-frequency er ampliﬁers. New solutions to oth problems are de- scrib ed, and the results of orking protot yp are presen ted. The protot yp e, in tegrated circuit (IC) fabricated in National Semiconductor’s 0.25 CMOS pro cess, represen ts the ﬁrst kno wn fully in tegrated im- plemen tation of the Cartesian feedbac concept. In tro duction Designers of radio-frequency (RF) er ampliﬁers (P A’s) for mo dern wireless systems are faced with diﬃcult tradeoﬀ. On one hand, the consumes the lion’s share of the er budget in most transceiv ers. It follo ws that in cellular phone, for example, battery lifetime is largely determined the er eﬃciency of the A. On the other hand, it ma desirable to ha high sp ectral eﬃciency—the abilit to transmit data at the highest ossible rate for giv en hannel bandwidth. The design conﬂict is that while sp ectral eﬃciency demands highly linear A, er eﬃciency is maximized when is run as constan t-en elop e, nonlinear elemen t. The curren state of the art is to design mo derately linear and emplo some lin- earization tec hnique. The ampliﬁer op erates as close to saturation as ossible, maximizing its er eﬃciency and the linearization system maximizes the sp ectral ef- ﬁciency in this near-saturated region. There are man diﬀeren linearization tec hniques. Our ork fo cuses on Cartesian feedbac systems for main reasons. First, ecause they emplo analog feed- bac k, the requiremen for detailed nonlinear mo del of the is greatly relaxed. This is an extremely com- elling adv an tage, as RF A’s are orly understo and notoriously diﬃcult to mo del. Second, Cartesian feedbac systems automatically and elegan tly comp en- sate for pro cess ariations, temp erature ﬂuctuations, and aging. Nev ertheless, historically the tec hnique has suﬀered the shortcoming of relying on sync hronous do wncon ersion, whic has een diﬃcult to realize without man ual trimming. This problem, com bined with the recen trend to ard fully monolithic systems, has caused Cartesian feedbac to languish for ears as little more than an academic curiosit approac the sync hronous do wncon ersion, or phase alignmen t, problem from directions. De- tailed analysis of Cartesian feedbac system is er- formed, and it is sho wn to suggest means of com- ensating the system for robustness to phase misalign- men t. Alternativ ely describ and analyze non- linear, analog phase alignmen regulator[1 ]. est re- sults for fabricated IC, designed as testb ed for these ideas, are then presen ted and analyzed. Cartesian eedbac The idea of using Cartesian feedbac to linearize er ampliﬁers has een discussed at least as early as the 1970’s[2 ]. It is called Cartesian feedbac ecause the feedbac is based on the Cartesian co ordinates of the baseband sym ol, and as opp osed to the o- lar co ordinates. The ypical system is illustrated in ﬁgure 1. [2 ]. undamen tally the concept ehind this PA PSfrag replacemen ts sin cos sin( cos( Figure 1: ypical Cartesian feedbac system. Ideally 0. system is negativ feedbac k. couple of factors com- plicate its expression in the con text of an RF transmit- ter, ho ev er. The ﬁrst is the extremely high frequency of man RF carriers, with mo dern standards calling for frequencies on the order of few gigahertz. this time, it is virtually imp ossible to build high-gain, sta- ble analog feedbac lo op with crosso er frequency in p.

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that range. The second factor is the recognition that in mo dulating an RF carrier, are not shaping oltage eform in its en tiret Instead, are shaping indep enden haracteristics of that carrier. Cartesian feedbac k’s of dealing with the ﬁrst factor is the inclusion of frequency translation step in the feedbac path, sho wn as do wncon ersion mixer in ﬁgure 1. The lo op is then closed at baseband, rather than at the carrier frequency The system consequence is to linearize only in narro band of the sp ectrum cen tered ab out the carrier, rather than from DC to the carrier. This is an ingenious to exploit the narro wband nature of most RF signals. The second factor manifests as the “double lo op struc- ture of the system. There are degrees of freedom in shaping, or mo dulating, an otherwise free-running RF carrier, and at least hoices of co ordinate systems that fully describ the mo dulation. or olar feedbac the hoice made is to consider an RF carrier as ha ving an amplitude and phase. The structure of olar feedbac system reﬂects this hoice, ha ving one con trol lo op for the amplitude, and another for the phase. An equiv alen hoice of co ordinates is the Cartesian com- onen ts, in whic consider the mo dulated carrier as the sum sin )) sin cos t, where cos and sin It is seen that Cartesian feedbac treats the degrees of freedom in symmetrical allo wing the structure of the system to tak the form of iden tical lo ops. This is in direct con trast to olar feedbac k, where the degrees of freedom ust treated ery diﬀeren tly Consequences of phase misalignmen in Cartesian feedbac systems Figure sho ws ypical Cartesian feedbac system. The system blo represen ts the lo op driv er am- pliﬁers, whic pro vide the lo op gain as ell as the dy- namics in tro duced the comp ensation strategy The lo op driv ers feed the baseband inputs of the up con er- sion mixer, whic in turn driv es the er ampliﬁer. Some means of coupling the output of the er am- pliﬁer to the do wncon ersion mixer is emplo ed, and the output of this mixer is used to close the feedbac system. 3.1 Impact of phase misalignmen on stabilit Ideally Cartesian feedbac system functions as iden tical, decoupled feedbac lo ops: one for the com- onen t, and one for the comp onen t. This corre- sp onds to the case of in ﬁgure 1. In practice, ho ev er, this state of aﬀairs ust activ ely enforced. Dela through the er ampliﬁer, phase shifts of the RF carrier due to the reactiv load of the an tenna, and mismatc hed in terconnect lengths et een the lo cal os- cillator (LO) source and the mixers all manifest as an eﬀectiv nonzero orse, the exact alue of aries with temp erature, pro cess ariations, output er, and carrier frequency Cartesian feedbac system in whic is nonzero is said to ha phase mis- alignmen t. In this state the feedbac lo ops are cou- pled, and the stabilit of the system is compromised. The impact of phase misalignmen on system stabil- it can seen mathematically start observing that the demo dulated sym ol is rotated relativ to an amoun equal to the phase misalignmen see this, write Cartesian comp onen ts of the demo d- ulated sym ol sin cos sin sin cos cos where is the carrier frequency Using trigonometric iden tities and assuming frequency comp onen ts at are ﬁltered out, arriv at cos sin (1) sin cos (2) see that for 0, an excitation on the input of the mo dulator results in signal on the do wncon- erter output (and similarly for and ). Accordingly sa that the lo ops are coupled. One metho of stabilit analysis is to consider the error signals and sho wn in Figure 1. Recall that for single feedbac lo op, the error signal is written where is the command input and is the lo op transmission. In the presen case, let the phase mis- alignmen urthermore, set without loss of generalit The error expressions, as function of the single input ), are written cos sin sin cos φ, ec hnically is also an uncoupled case. Ho ev er, there is no an in ersion in oth lo ops, resulting in ositiv feedbac instead of the desired negativ feedbac k. do not lose generalit as long as sta with linear analysis. p.

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where includes the dynamics of the lo op comp en- sation sc heme and the (linearized) dynamics in- tro duced the mo dulator, er ampliﬁer, and de- mo dulator. rom here, it is straigh tforw ard to sho that cos sin 1+ cos This reduction of the system to single-input problem no yields considerable insigh t. iden tify an eﬀec- tiv lo op transmission, e s, ), as follo ws: e s, cos sin cos (3) or erfect alignmen t, and e is simply ). The orst alignmen is for whic e )] and so the lo op dynamics are cascade of the dynamics in the uncoupled case. Unless designed with this os- sibilit in mind, most hoices of yield unstable eha vior in this second case. Equation sho ws that traditional measures of stabilit degrade con tin uously as sw eeps from to fact demonstrated exp eri- men tally Briﬀa and aulkner [4 ]. 3.2 Comp ensating the system for robustness to phase misalignmen Equation oﬀers great deal of insigh in to what hap- ens in phase-misaligned Cartesian feedbac system. Ph ysically the fully coupled case eha es as depicted in ﬁgure 2, where represen ts the dynam- ics that the up con ersion mixer, er ampliﬁer, and do wncon ersion mixer con tribute to the lo op transmis- sion. In the literature, all eﬀorts with regards to the PSfrag replacemen ts Figure 2: Cartesian feedbac under 90-degree misalign- men t. phase alignmen problem ha fo cused, naturally on ensuring phase alignmen t. But there is at least one other approac that deserv es consideration: is it ossi- ble to ho ose suc that it is stable for large phase misalignmen ts? The answ er dep ends in part on what one means “large. Considering misalignmen of for instance, is discouraging. In this case e ), and there is simply no comp ensation strategy that is indiﬀeren to the sign of the lo op transmission. Carte- sian feedbac in fact do es ecome ositiv feedbac system for misalignmen ts in the op en in terv al ), where the exact oin of transition from negativ to ositiv feedbac dep ends on the details of ). oid considering ositiv feedbac cases, then, it is sensible to restrict the range of misalignmen ts to the closed in terv al ]. That stabilit margins degrade ontinuously with suggests that ﬁnding comp ensation strategy that orks in the limiting cases of and will solv the problem for the whole in terv al. Assuming the dynamics of the lo op are dominated ), com- ensation strategy that emerges is where 1. Suc “slo w-rollo functions, while not truly realizable with lump ed-elemen net ork, can appro ximated alternating oles and zeros suc that the erage slop of H(s) is the appropriate dB-p er-decade[5 ]. In the case of 5, for instance, stabilit as measured phase margin ould excel- len t: 135 degrees in the aligned case, and 90 degrees in the misaligned case. Ro ot lo cus analysis conﬁrms that slo w-rollo com- ensation is viable approac to designing for large misalignmen ts. Figure sho ws the ro ot lo ci for the PSfrag replacemen ts Dominan t-p ole Slo w-rollo comp ensation comp ensation )] Figure 3: Ro ot lo cus plots for dominan t-p ole and slo w- rollo comp ensation. dominan t-p ole and comp ensation strategies. It can seen that ev en in the case of the dominan t-p ole, 90- degree misalignmen do esn’t necessarily lead to righ t- half-plane oles. est, ho ev er, the result is ligh tly damp ed, complex pair of oles. orst, high- frequency oles not sho wn here (or not mo deled) push this complex pair in to the righ t-half plane. By con- trast, the slo w-rollo comp ensation is seen to lead to hea vily-damp ed complex ole pairs, and one exp ects corresp onding reduction in ersho ot and ringing in resp onse to an input step. One also exp ects the lo w- frequency zero-p ole doublets of the ro ot lo ci to man- p.

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ifest themselv es as slo w-settling “tails in the step re- sp onse [5 ]. Exp erimen ts carried out on the ﬁnal IC in accordance with this comp ensation discussion alidate these exp ec- tations. As seen in section 5.2, the slo w-rollo tec h- nique stabilizes the system for all misalignmen ts up to 90 degrees. In addition to shedding ligh on comp en- sation strategies for Cartesian feedbac systems, the imp ortance of these exp erimen ts is that they conﬁrm the understanding dev elop ed in section 3.1. nonlinear regulator for main taining phase alignmen Occasionally con tin uous regulation of the phase align- men is not needed, and it suﬃces to in tro duce man u- ally adjustable dela et een the LO source, and, sa the demo dulator. This approac is only feasible, ho w- ev er, if the system is not sub ject to ariations in tem- erature, carrier frequency pro cess parameters, or, in some cases, output er. or cases in whic the align- men ust regulated, arious metho ds ha een prop osed in the literature; see, for example [6 ]. presen our con trol concept as compact, truly con- tin uous solution to the problem of LO phase alignmen t. It is truly con tin uous ecause it do es not, for example, rely on the app earance of sp eciﬁc sym ol or pattern in the outgoing data stream. It is compact ecause it is easily implemen ted without digital signal pro cessing, as presen ted here. This is particularly comp elling ad- an tage, as the signals in Cartesian feedbac system are necessarily in analog form. And emphasize that, ecause the concept is based on the pro cessing of base- band sym ols, its realization is indep enden of carrier frequency 4.1 Nonlinear dynamical system Figure represen ts baseband sym ol at the inputs of the mo dulator and at the outputs of the demo du- lator of Cartesian feedbac system. Mathematically the ectors are describ ed in oth Cartesian and o- lar co ordinates, with primed co ordinates denoting the demo dulated er ampliﬁer output and unprimed co- ordinates denoting the mo dulator input. In addition to undergoing distortion in magnitude, the demo du- lated sym ol is rotated an amoun exactly equal to the phase misalignmen (see equations and 2). start to the design of phase alignmen regulator is to observ that the signals and tak en to- gether, represen enough information to determine the phase misalignmen t. urther, they are easily accessible within the system. seek to com bine these ariables suc that, er suitable range, the deriv ed signal is monotonic in the phase misalignmen t. PSfrag replacemen ts Figure 4: Rotation of the baseband sym ol due to phase misalignmen t. One suc com bining of the ariables is the sum of pro d- ucts QI Recognizing that sin and cos and using trigonometric iden tities, write the ey relation QI sin( (4) see that using ultipliers and subtractor, op- erations easily realizable in circuit form, one can deriv con trol signal that is indeed monotonic in the phase misalignmen er the range Figure details nonlinear dynamical con troller built around equation 4. Using the notation an PSfrag replacemen ts phase LO LO RF RF QI d dt dt cos cos sin sin d dt QI Gr sin( Figure 5: Phase alignmen concept. implemen tation can understo as mec hanizing the equation d dt )] sin ( (5) where is constan of prop ortionalit and gain is asso ciated with the in tegrator. Equation presupp oses the abilit to correct the phase shift hanging The original protot yp describ ed in[7 realizes the required rotation directly phase shifting the mo dulator LO. Ho ev er, substan tial er sa vings result from doing sym ol rotation at baseband as sho wn in ﬁgure 5. Regardless, rotation should p.

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erformed in the forw ard path of the Cartesian feed- bac system, where the una oidable artifacts of imp er- fect rotation are suppressed. 4.2 Stabilit concerns Our con trol solution for the phase alignmen problem is the simplest of nonlinear dynamical systems. It is seen from equation to ha equilibrium oin ts. The ﬁrst, for whic the sym ols are aligned, is stable The second, for whic the sym ols are misaligned radians, is unstable. or the ideal system represen ted equation 5, this is the exten of rigorous stabilit analysis. The real-w orld situation can complicated dynam- ics asso ciated with the phase shifter (and, ossibly the subtractor). If pro visionally consider mo d- ulation sc heme in whic the magnitude of transmitted sym ols is held constan t, in equation loses its time dep endence. Linearizing for small phase misalign- men ts, and including the dynamics of the phase shifter as ), can represen the system as sho wn in ﬁg- ure 6. Dra wing the system this requires some ma- PSfrag replacemen ts Gk Drift Phase Distortion Figure 6: Linearized phase regulation system. ’M is the desired misalignmen t, whic is nominally zero. nipulation. The output of the phase shifter is not really but rather an additiv art of that gets com bined with the olar angle of the sym ol eing transmitted. Ho ev er, in the absence of phase distortion and drift, the sym ol-b y-sym ol hanges of the olar angle are trac ed iden tical hanges in These sym ol-rate hanges are th us in visible to an alignmen system, and it is appropriate to lab el the output of as can then include the eﬀects of phase distortion and phase alignmen drift as the additiv disturbances of ﬁgure 6. One can ensure stabilit ho osing suc that, for the largest sym ol magnitude, lo op crosso er ccurs e- fore non-dominan oles ecome an issue. ortunately the drift disturbance will normally ccur on time scales asso ciated with temp erature drift and aging [2 ]. Sup- pression of the phase distortion is the domain of the Cartesian feedbac itself. It follo ws that for man sys- tems, little of the design eﬀort need fo cused on fast Unlik ely when using Cartesian feedbac k, of course. em- orarily making this assumption, ho ev er, yields insigh that is broadly relev an to the stabilit analysis. phase alignmen t. Exp erimen tal results What the new phase alignmen regulator enables is the building of highly in tegrated Cartesian feedbac sys- tems. This is comp elling design goal, as it ma al- lo this linearization tec hnique to used for mo dern, handheld wireless devices. As demonstration, fully monolithic protot yp e, fabricated in National Semicon- ductor’s 0.25 CMOS tec hnology as designed and tested. our kno wledge, this is the ﬁrst successful in tegration of er ampliﬁer, phase alignmen sys- tem, and Cartesian feedbac linearization circuitry all on the same die. 5.1 Phase alignmen system Figure is trace capture of the yp of exp erimen used to haracterize the erformance of the phase align- men system. The Cartesian feedbac lo op is op en, PSfrag replacemen ts align off align off align on align on 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 -0.02 -0.02 -0.02 -0.02 1.4 1.4 1.4 1.4 1.45 1.45 1.45 1.45 1.5 1.5 1.5 1.5 1.55 1.55 1.55 1.55 1.6 1.6 1.6 1.6 1.65 1.65 1.65 1.65 1.7 1.7 1.7 1.7 1.75 1.75 1.75 1.75 1.8 1.8 1.8 1.8 1.85 1.85 1.85 1.85 Time(msec) Figure 7: race capture of phase alignmen exp erimen t. The Cartesian feedbac lo op is op en. 500mV amplitude, 10kHz square driv es the hannel, and the hannel is grounded. The top traces sho that, initially the misalignmen is man u- ally set to 45 degrees. The ottom traces sho the result of turning on the phase alignmen system (releasing it from the “reset mo de). Ov er the full, 90 degree range that the sym ol rotator ermitted, the regulator ept the phase misalignmen elo de- grees. Figure serv es to illustrate the impact of phase mis- alignmen on the stabilit margins of the closed-lo op The oltage dro op on what is normally the ﬂat part of the square es is due to the fact that, at the oard-lev el, the pre- distortion inputs ha een C-coupled. p.

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PSfrag replacemen ts align off align off align on align on 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 -0.02 -0.02 -0.02 -0.02 -0.5 -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 Time(msec) Figure 8: Illustration of phase alignmen stabilizing the closed-lo op CFB system. CFB system. Dominan t-p ole comp ensation is used in the CFB lo op, and for the upp er traces the mis- alignmen is man ually set to 74 degrees. Ov ersho ot and ringing is eviden on these eforms, and further mis- alignmen causes outrigh oscillation. or the ottom traces the phase alignmen system is turned on, and one sees the classic ﬁrst-order step resp onses that are exp ected when using dominan t-p ole comp ensation. 5.2 Comp ensation exp erimen ts The protot yp as designed suc that the lo op trans- mission could aried. Among ossible hoices of lo op comp ensation, the slo w-rollo net ork is of particular in terest as demonstration of the ideas dev elop ed in section 3. Our slo w-rollo net ork realized three oles and zeros. Figure pro vides dramatic comparison of the system under dominan t-p ole ersus slo w-rollo comp ensation. or this exp erimen t, the phase misalignmen of the sys- tem is man ually set to 90 degrees. The top traces sho the dominan t-p ole comp ensated system under 90- degree misalignmen t. Substan tial ersho ot and ring- ing is visible, indicativ of ligh tly damp ed, complex ole pair. The ottom traces sho the system under slo w-rollo comp ensation. The step resp onse is remark ably similar to that of single ole system. Conclusion The problem of phase alignmen has sto as the pri- mary barrier to the widespread use Cartesian feedbac k. In this pap er describ new analysis, and use the resulting insigh to design system that is toleran to 90 degrees of misalignmen t. new phase alignmen PSfrag replacemen ts 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 -0.02 -0.02 -0.02 -0.02 0.5 0.5 0.5 0.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5 3.5 4.5 4.5 4.5 4.5 /s /s Time (msec) Figure 9: Step resp onse comparison et een dominan t- ole and slo w-rollo comp ensated systems for 90-degree misalignmen t. regulator is also review ed. ak en together, these re- sults considerably lo er the barrier to implemen ting Cartesian feedbac in mo dern wireless transceiv ers. References [1] J.L. Da wson and T.H. Lee. Automatic phase alignmen for fully in tegrated Cartesian feedbac er ampliﬁer system. IEEE Journal of Solid-State Cir cuits 38:2269–2279, Decem er 2003. [2] B. Raza vi. RF Micr ele ctr onics Pren tice-Hall, Inc., Upp er Saddle Riv er, NJ, 1998. [3] D. Co x. Linear ampliﬁcation sampling tec h- niques: new application for delta co ders. IEEE ansactions on Communic ations COM-23:793–798, 1975. [4] M. Briﬀa and M. aulkner. Gain and phase mar- gins of Cartesian feedbac RF ampliﬁer linearisation. Journal of Ele ctric al and Ele ctr onics Engine ering, us- tr alia 14:283–289, 1994. [5] J. K. Rob erge. Op er ational mpliﬁers: The ory and Pr actic John Wiley and Sons, Inc., New ork, New ork, 1975. [6] Y. Ohishi, M. Mino a, E. ukuda, and T. ak ano. Cartesian feedbac ampliﬁer with soft landing. In 3r IEEE International Symp osium on Personal, Indo or, and Wirless Communic ations pages 402–406, 1992. [7] J.L. Da wson and T.H. Lee. Automatic phase alignmen for high bandwidth Cartesian feedbac er ampliﬁers. In IEEE adio and Wir eless Con- fer enc pages 71–74, 2000. p.

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