I Cause an arbitrary waveform generator of the signal path to launch the predistorted signal in the time domain via the signal path

Transcription

1 (i9) United States (12) Patent Appliatin Publiatin Tufillar et al. US A1 (i) Pub. N.: US 2009/ Al (43) Pub. Date: Jun. 18, 2009 (54) METHDS AND APPARATUS FR CMPUTING AND USING A SPECTRAL MAP FR PERFRMING NNLINEAR CALIBRATIN F A SIGNAL PATH (76) Inventrs: Nihlas B. Tufillar, Crvallis, R (US); Rbert E. Jewett, Santa Clara, CA (US) Crrespndene Address: AGILENT TECHNLGIES INC. INTELLECTUAL PRPERTY ADMINISTRA- TIN,LEGAL DEPT., MS BLDG. E P.. BX 7599 LVELAND, C (US) (21) Appl.N.: (22) Filed: 12/004,442 De. 18, 2007 Publiatin Classifiatin (51) Int. CI. G01R 23/165 ( ) (52) U.S. CI 324/76.21 (57) ABSTRACT In ne embdiment, a spetral map fr perfrming nnlinear alibratin f a signal path is develped by 1) identifying a set f frequeny latins fr a set f partiular utput signal spurs that result frm applying ne-tne and tw-tne input signals vering a bandwidth f interest t the signal path; 2) develping, based n the set f frequeny latins, a spetral map fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain; and 3) saving the spetral map fr perfrming nnlinear alibratin f the signal path. 400 / Cmpute a disrete Furier transfrm (DFT) f an input signal t the signal path 402 V I 404 Apply a spetral map t the DFT f the input signal, t generate a predistrted signal in the frequeny dmain I Cmpute an inverse DFT (IDFT) f the predistrted signal in the frequeny dmain, t generate a predistrted signal in the time dmain I Cause an arbitrary wavefrm generatr f the signal path t launh the predistrted signal in the time dmain via the signal path J 406 J 408 J

2 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 1 f 31 US 2009/ Al r < V.

3 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 2 f 31 US 2009/ Al 200 Identify a set f frequeny latins fr a set f partiular utput signal spurs that result frm applying ne-tne and tw-tne input signals vering a bandwidth f interest t a signal path I Develp, based n the set f frequeny latins, a spetral map fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain I Save the spetral map fr perfrming nnlinear alibratin f the signal path 202 u 204 u 206 u FIG. 2

4 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 3 f 31 US 2009/ Al 300 Apply ne-tne input signals vering the bandwidth f interest t the signal path, and measure amplitudes and phases f single-tne spurs in the set f partiular utput signal spurs Use the measured amplitudes and phases f the single-tne spurs t nstrut a first-rder frequeny respnse mdel fr the signal path X 306 Cmpute a first-rder input amplitude spetrum in the frequeny dmain, restrited t the identified set f frequeny latins z Develp, frm the first-rder frequeny respnse mdel and the firstrder input amplitude spetrum, a first-rder spetral map fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain z Verify perfrmane f the first-rder spetral map fr varius predistrted signals applied t, r reeived frm, the signal path; and when the perfrmane f the first-rder spetral map is determined t prvide insuffiient mitigatin f utput signal spurs at the set f frequeny latins, identify a set f residual utput signal spurs and, Apply tw-tne input signals vering the bandwidth f interest t the signal path, and measure amplitudes and phases f the residual utput signal spurs I Use the measured amplitudes and phases f the residual utput signal spurs t nstrut a send-rder frequeny respnse mdel fr the signal path I Cmpute a send-rder input amplitude spetrum in the frequeny dmain, restrited t the identified set f frequeny latins 312 u 314 u 316 V u 308 U 370 U Develp, frm the send-rder frequeny respnse mdel and the send-rder input amplitude spetrum, a sendrder spetral map fr predistrting, in the frequeny dmain, signals that are input t the signal path in the time dmain 3j8 FIG. 3

5 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 4 f 31 US 2009/ Al 400 / Cmpute a disrete Furier transfrm (DFT) f an input signal t the signal path 402 u I 404 Apply a spetral map t the DFT f the input signal, t generate a predistrted signal in the frequeny dmain I Cmpute an inverse DFT (IDFT) f the predistrted signal in the frequeny dmain, t generate a predistrted signal in the time dmain I Cause an arbitrary wavefrm generatr f the signal path t launh the predistrted signal in the time dmain via the signal path J 406 J 408 J FIG. 4 r 500 ^ INPUT SIGNAL- ARBITRARY WAVEFRM GENERATR \: UTPUT - SIGNAL FIG. 5

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10 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 9 f 31 US 2009/ Al x KJ- 4 Send Harmni Amplitude Crretin (Vlts) Amp n, Pre ff, Filt ff, Full Range Data (Dts) r lypialmadej^j-ine) i i i i i i / : & /' / \ 2 1 I! 1!! 1 Freq (Hz) FIG. 8A y. 10 Send Harmni Amplitude Crretin (Radians) Amp n, Pre ff, Filt ff, Full Range Data (Dts), Typial Mdel (Line) 3 4 Freq (Hz) FIG. 8B x10

11 Patent Appliatin Publiatin Jun. 18, 2009 Sheet 10 f 31 US 2009/ Al *i- 4 7 Third Harmni Amplitude Crretin (Vlts) Amp n, Pre ff, Filt ff, Full Range Data (Dts), Typial Mdel (Line) 1 iii 6,5-6 - _>* ~ ~ ^ 5 > "i '* - %^ ^ 4 s ^ - 35 \ s ^ i 1 I Freq (Hz) FIG. 9A Third Harmni Amplitude Crretin (Radians) Amp n, Pre ff, Filt ff, Full Range Data (Dts), Typial Mdel (Line) i i i i i x10 '* t 03 a. 23 2G ""^ ^ -^ v 1: Freq (Hz) FIG. 9B x10

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33 US 2009/ Al Jun. 18,2009 METHDS AND APPARATUS FR CMPUTING AND USING A SPECTRAL MAP FR PERFRMING NNLINEAR CALIBRATIN F A SIGNAL PATH BACKGRUND [0001] Eletrni instruments, suh as wide-band eletrni instruments used in the test and measurement f eletrni devies under test (DUTS), typially need t be alibrated. Examples f eletrni instruments are 1) an arbitrary wavefrm generatr (ARB), the re f whih typially mprises a digital-t-analg nverter (DAC), r 2) a reeiver suh as an sillspe, the re f whih typially mprises an analg-t-digital nverter (ADC). Arund the re funtinality f an eletrni instrument, there typially exist a number f ther devies, suh as nditining eletrnis (e.g., filters and amplifiers) and frequeny translatin devies (e.g., mixers fr up-nversin and dwn-nversin f signals f interest). [0002] A signal that is transmitted frm, t r between eletrni instruments an be pressed thrugh several stages in its transmissin r reeptin. Fr instane, in a typial appliatin, suh as the testing f a radi frequeny (RF) amplifier, an initial test signal uld be generated by a baseband ARB, filtered, upnverted by a mixer, and then subjeted t further amplifiatin and filtering befre its appliatin t a DUT. An utput test signal that is generated by the DUT, in respnse t the initial test signal, may then be reeived by a measurement instrument suh as an sillspe r spetrum analyzer. Due t nn-idealities in the eletrnis f the transmissin and reeptin paths, the signals transmitted t and frm the DUT may underg distrtins. Calibratin refers t the presses that are applied t physial r mathematial signal representatins in an attempt t remve r minimize these distrtins. [0003] Distrtin f a signal may be lassified as linear r nnlinear. Linear alibratin f an eletrni instrument is mmn, and is typially amplished by adjusting the phase and amplitude f an exitatin signal. This mpensatin is perhaps best understd in the frequeny dmain. That is, any signal an be represented in the frequeny dmain by nsidering a Furier depsitin (either a ntinuus dempsitin using the Furier Transfrm, r a disrete dempsitin using the s-alled disrete Furier transfrm (DFT) f the signal f interest. See, e.g., Julius. Smith III, "Mathematis f the Disrete Furier Transfrm (DFT), with Audi Appliatins Send Editin", W3K Publishing (2007). By a fundamental prperty f linear systems, linear distrtin results in a signal with exatly the same frequeny mpnents as the undistrted signal hwever, linear distrtin an ause a shift in the phase and amplitude f eah frequeny mpnent. When an input signal is represented (in the frequeny dmain) as a lletin f mplex numbers knwn as "phasrs," linear distrtin an be viewed as a transfrmatin that sales and rtates eah input phasr. In the time dmain, linear alibratin is typially amplished by the nstrutin f a finite impulse respnse (FIR) filter, whih is nstruted t satisfy frequeny dmain nstraints. In pratie, an 'impulse respnse' is typially used t estimate the linear distrtin f a signal, and an apprpriate FIR filter is used fr linear alibratin. Beause f the imprtane f linear alibratin predures, the Natinal Institute f Standards (NIST), has studied the linear alibratin prblem in detail and set frth standards fr linear alibratin presses and signals fr sme measurement instruments. See, e.g., William L. Gans, "Dynami Calibratin f Wavefrm Rerders and sillspes Using Pulse Standards", IEEE Transatins n Instrumentatin and Measurement, Vl. 39, N. 6, pp (Deember 1990). [0004] A hallmark f nnlinear distrtin is the reatin f energy at frequenies distint frm the frequeny f the input signal. In ne frm, nnlinear distrtin an result frm a s-alled squaring law. That is, if a signal has energy at a frequeny fj, then the squaring press in the time dmain y(t)=x(t) 2 leads t energy, r signal distrtin, at twie the input frequeny (i.e., at 2*^=^). Infrmally, nnlinear distrtin leads t s-alled "spurs" in the frequeny dmain. These spurs are easily viewed using a pwer spetrum analyzer. See, e.g., "Agilent PSA Perfrmane Spetrum Analyzer Series ptimizing Dynami Range fr Distrtin Measurements", Agilent Tehnlgies, In., (2000). [0005] The imprtane f spurs in limiting the perfrmane f eletrni instruments is well-knwn, and is mmnly quantified by the s alled "spurius free dynami range" (SFDR), usually prvided as a key metri fr measurement instrument perfrmane. SFDR is als the subjet f varius Institute f Eletrial and Eletrnis Engineers (IEEE) standards fr unifrm measurement. See, e.g., E. Balestrieri, et al., "Sme Critial Ntes n DAC Frequeny Dmain Speifiatins", XVIII Imek Wrld Cngress (2006). Cnsideratin f higher rder nnlinear interatins (e.g., third rder interatins) shws that nnlinear distrtin an als ause distrtin at r near the input exitatin frequenies. This distrtin is in additin t any linear distrtin, and is ften referred t as "intermdulatin distrtin", beause it arises frm signal mixing presses inherent in nnlinear signal interatins. Intermdulatin distrtin an mpliate signal transmissin and reeptin nsiderably, beause it needs t be untangled frm the underlying signal and its linear distrtin. [0006] Real-wrld eletrni systems are als subjet t a range f effets (e.g. sillatr feed thrugh, eletrni mpnent peratin, dependenies n temperature) that an ause a wide range f signal impairments and distrtins that ften need t be systematially aunted fr and rreted (r avided) during a alibratin press. A brief verview f sme f these effets and nsideratins fr ARBs is desribed by Mike Griffin, et al. in "Cnditining and Crretin f Arbitrary Wavefrms Part 1: Distrtin", High Frequeny Eletrnis, pp (August 2005) and in "Cnditining and Crretin f Arbitrary Wavefrms Part 2: ther Impairments", High Frequeny Eletrnis, pp (September 2005). BRIEF DESCRIPTIN F THE DRAWINGS [0007] Illustrative embdiments f the inventin are illustrated in the drawings, in whih: [0008] FIG. 1 illustrates an exemplary signal path mprising an ARB, an upnverter and an amplifier; [0009] FIG. 2 illustrates an exemplary methd fr develping a spetral map fr perfrming nnlinear alibratin f a signal path; [0010] FIG. 3 illustrates an exemplary way t develp the spetral map used by the methd shwn in FIG. 2; [0011] FIG. 4 illustrates an exemplary methd fr perfrming nnlinear alibratin f a signal path;

34 US 2009/ Al Jun. 18,2009 [0012] FIG. 5 illustrates an exemplary signal path mprised f a wide-band arbitrary wavefrm generatr and a (2x) amplifier. [0013] FIG. 6 illustrates a tw-tne input signal; [0014] FIG. 7 illustrates exemplary measurements f the phase and amplitude f a single-tne spur; [0015] FIGS. 8A & 8B illustrate exemplary send rder spur distrtin data, and the fit f a lw-rder plynmial t the data; [0016] FIGS. 9A & 9B illustrate exemplary third rder spur distrtin data, and the fit f a lw-rder plynmial t the data; [0017] FIG. 10 illustrates an exemplary 'frequeny set' fr the apparatus shwn in FIG. 5; [0018] FIGS. 11A & 11B illustrate the exemplary appliatin f a multi-tne signal t the apparatus shwn in FIG. 5; [0019] FIGS. 12A & 12B illustrate the exemplary appliatin f a pseud-randm signal t the apparatus shwn in FIG. 5; [0020] FIGS. 13 & 14 shw lse-ups f the utput f the FIG. 5 signal path, with and withut predistrtin, fr an exemplary mplex input signal; and [0021] FIG. 15 illustrates an exemplary redutin f intermdulatin distrtin using the methds illustrated in FIGS. 2&3. DETAILED DESCRIPTIN [0022] Linear alibratin, temperature alibratins and ther signal distrtin rretins are mmnly perfrmed fr eletrni instruments. T date, methds fr perfrming nnlinear alibratin f eletrni instruments have nt been as mmn. [0023] Desribed herein are methds and apparatus fr perfrming nnlinear alibratin f a signal path, inluding a signal path thrugh ne r mre eletrni instruments r devies. The perfrmane f an eletrni instrument r devie, as measured by its spurius free dynami range (SFDR), is typially limited by herent nnlinear effets that are signifiantly abve the inherent nise flr f the instrument r devie. Nnlinear alibratin prvides a methd that rrets fr these effets, thereby extending the useable SFDR f the instrument r devie. [0024] Fr instane, and as will be desribed in mre detail later in this desriptin, an eletrni instrument nsisting f an arbitrary wavefrm generatr (ARB) and an amplifier uld have a SFDR f 65 db befre nnlinear alibratin, but have an SFDR f 80 db after nnlinear alibratin. This inreased dynami range has many ptential uses. Fr instane, if the ARB is being used t test an ADC with an SFDR f 70 db, then inreasing the SFDR an enable mre aurate testing f the ADC. Any alibratin that inreases the dynami range, bandwidth, r fidelity f a transmitter r reeiver typially translates diretly int tangible useful results fr test and measurement systems. [0025] The nvel methds and apparatus desribed herein use a 'nulling methd' and predistrtin t remve nnlinear spurs in an eletrni instrument's utput. Althugh the use f nulling methds are well-knwn, they are nt mmnly emplyed, beause the number f spurs generated by an arbitrary test signal is typially t mplex t be measured in a reasnable time. That is, there are typially just t many spurs fr a nulling methd t be pratial fr anything but ne r a few peridi exitatin tnes. Hwever, the methds and apparatus desribed belw slve this prblem by prviding a alibratin methd that is based n a behaviral mdel. The behaviral mdel maps input signals (in the frequeny dmain) t utput signals (inluding all spurs abve a prespeified signal level). A behaviral mdel is very useful fr alibratin beause it des nt require detailed measurements f eah spur, but rather generates infrmatin f 'suffiient' auray abut the frequeny, phase, and amplitude f eah spur, allwing the reatin f a nulling signal t be reated fr nnlinear distrtin. [0026] Befre desribing nvel methds and apparatus fr perfrming nnlinear alibratin f a signal path in detail, it is useful t nsider sme f the differenes between existing linear and nnlinear alibratin methds, and t nsider sme f the defiienies f existing nnlinear alibratin methds. [0027] Linear alibratin methds are greatly aided by the thery and pratial experiene in estimating the linear 'transfer funtin' f an eletrni devie. Indeed, the use f 'impulse respnse testing' t gauge linear distrtin is a diret expressin f the well-studied thery f linear transfrms, Dira delta funtins, and the general slutin (via the Laplae transfrm) f linear systems. See, Gans, supra. A similar thery des nt exist fr nnlinear systems hene the diffiulty in perfrming nnlinear alibratin. The lsest thery, the s-alled Vlterra thery, is desribed in detail, fr instane, by Stephen P. Byd, "Vlterra Series, Engineering Fundamentals", Dissertatin, U. C. Berkeley (1985). Hwever, despite many years f researh, its pratial appliatin, t date, has been smewhat limited (unlike the linear transfer funtin thery). Appliatin f the Vlterra thery has been limited fr several reasns, suh as, beause the thery is nly appliable t 'weak'nnlinearities, and beause, frm a pratial pint f view, the estimatin f the full Vlterra kernels has nt prven pratial sine it invlves an enrmus number f measurements t apture wide-band frequeny dmain effets. Thus, thugh very insightful frm a theretial pint f view, Vlterra series have generally nt prven pratial fr typial engineering appliatins invlving signal rretins (that is, withut signifiant simplifiatins). These inherent prblems are nly aentuated fr alibratin prblems, where the speed at whih measurements are taken and alibratin is perfrmed are usually at the frefrnt. [0028] In mparisn t mplex alibratin methds based n the Vlterra thery, ther alibratin methds are smetimes based n 'Lk Up Tables' (LUTs). Calibratin methds based n LUTs mpare the measured perfrmane f an instrument r devie t an expeted 'ideal' perfrmane and are a mmn engineering slutin. Hwever, LUTs are nt a pratial slutin fr typial nnlinear alibratin, beause the utput signal f an instrument typially depends in a mplex fashin n an input signal, and it is generally nt feasible t measure the atual instrument respnse t a mplete sample f perating nditins and input signals. In mre pratial terms effetive nnlinear alibratin invlves the latin and estimatin (in bth phase and magnitude) f eah signifiant utput signal spur. Fr knwn test signals, a alibratin based n LUTs might be appliable. Calibratin based n LUTs is a mmn staple f alibratin predures fr sme auses f signal distrtin (e.g. temperature alibratin). Hwever, ne f the harateristis f a nnlinear system is that its utput is a funtin f an input signal, and there is n general way t denvlve the system's respnse frm a wide range f different input signals.

35 US 2009/ Al Jun. 18,2009 [0029] Frm a metrlgy pint f view, nnlinear alibratin requires an aurate and detailed knwledge f bth the amplitude and phase f eah nnlinear spur f interest. Aurate amplitude measurements are typially straight-frward using a pwer spetrum analyzer. Phase measurements are als pssible, assuming that the signal f interest an be digitized (i.e., the full time dmain signal an be aptured), and that the digitized signal has suffiient dynami range t measure signal distrtins f interest. Unfrtunately, typially ne r bth f these assumptins fails fr high dynami range measurement instruments r devies, and this requires the develpment f alternative metrlgy tehniques t rever the phase and amplitude infrmatin f nnlinear distrtin. This fres the use f sme srt f 'frequeny dmain measurement instrumentatin' fr the revery f phase infrmatin f lw level signal distrtins. [0030] The Natinal Institute f Standards and Tehnlgy (NIST) has lked at sme preliminary methds fr nnlinear alibratin, whih allw the aurate revery f the phase f nnlinear instruments r devies. These methds inlude the 'nse-t-nse' methd, as well as measurement systems utilizing wide-bandwidth, aurate phase referene generatrs. See, e.g., Tray S. Clement, et al., "Calibratin f Sampling sillspes With High-Speed Phtdides", IEEE Transatins n Mirwave Thery and Tehniques, Vl. 54, N. 8, pp (August 2006). Suh systems demnstrate the fundamental metrlgy methds needed fr nnlinear alibratin, but neither the measurement systems, nr the methds, have prven pratial in terms f st and time fr wide-spread use. [0031] Lastly, it is mmn pratie in eletrni design t minimize the effet f nnlinear spurs as part f the initial design press fr an eletrni hip. Fr example, several tehniques are emplyed t minimize any herent nnlinear signals. See, e.g., Russ Radke, et al., "A Spurius-Free Delta- Sigma DAC Using Rtated Data Weighted Averaging", IEEE Custm Integrated Ciruits Cnferene, pp (1999). Hwever, despite the best design tehniques, signifiant nnlinear spurs ften still appear in the hip's utput(s). [0032] The nvel methds and apparatus desribed belw utilize signal predistrtin t amplish nnlinear alibratin f a signal path. There is a great deal f researh n predistrtin and its appliatins t slve varius eletrni signal integrity issues. Perhaps the mst mmn appliatin f predistrtin is t inrease the fidelity f eletrni transmissin and reeptin systems, in partiular ellular phne base statins and handset eletrnis. See, e.g., Rahul Gupta, et al., "Adaptive Digital Baseband Predistrtin fr RF Pwer Amplifier Linearizatin", High Frequeny Eletrnis, pp (September 2006). Hwever there have been far fewer appliatins f predistrtin t nnlinear alibratin f eletrni instruments. [0033] The nvel methds and apparatus desribed herein differ signifiantly frm a Vlterra apprah. ne reasn is beause the methds and apparatus are nstrained early n t a fixed "frequeny set" (as will be desribed in mre detail later). Put mre simply, the nvel methds and apparatus dislsed herein fus n prediting and remving partiular utput signal spurs in the frequeny dmain, instead f building the large mdels presribed by the Vlterra thery and attempting t remve all spurs. [0034] The nvel methds and apparatus dislsed-herein amplish predistrtin using a spetral map. The nept f a spetral map is straight-frward, thugh its details an beme enrmusly umbersme. This is why the methds and apparatus dislsed herein utilize a behaviral mdeling apprah fr apprximating (r estimating) the apprpriate ntent fr a spetral map. [0035] An input signal t a devie (r DUT) may be represented in the frequeny dmain by means f a disrete Furier transfrm (DFT). Let u(t) be the input signal and U(k) be its disrete Furier Transfrm. See, e.g., Smith III, supra. If u(t) is a multitne signal, then: N / k (1) H(0 = Yt UWexp[jir-/ ml where U(k)=U*(-k)=IU(k)lexp(j( ) i ), and where f max is the maximum frequeny f exitatin. Fatring ut the time dependene, the abve representatin f the signal is typially alled the 'phasr' representatin, beause a real signal is represented gemetrially as mplex numbers with phase k and amplitude A i =IU(k)l. Using the Euler identity, it an be written that /\s(wr + (5) = A - [exp(wr + 6) + exp(-wr - (5)], thereby making it easy t see the input signal's tw-sided DFT spetrum. Representing the input signal U(k) in the frequeny dmain, the nnlinear dependene f an utput signal (y(t) in the time dmain, and Y(k) in the frequeny dmain) n the input signal an be desribed in the frequeny dmain, fr either a range f frequenies r a fixed utput frequeny, as: Y(k)=G[U(k)l (2) where G is a nnlinear peratr. G may als be referred t as a "spetral map", beause it effetively maps input phasrs in the frequeny dmain t utput phasrs in the frequeny dmain. An inverse disrete Furier transfrm (IDFT) may be used t rever the time dmain signals u(t) and y(t). The idea f examining spetral maps fr the purpse f mdeling nnlinear systems is essentially the idea behind the 'desribing funtin' thery fr apprximating nnlinear systems. See, e.g., James H. Taylr, "Desribing Funtins", Eletrial Engineering Enylpedia (1999). Spetral maps als play a key rle in eletrni simulatin tehnlgies suh as 'harmni balane methds' and the 'methds f spetral balane'. See, e.g., Nun Brges de Carvalh, et al., "Simulatin f Multi-Tne IMD Distrtin and Spetral Regrwth Using Spetral Balane", IEEE MTT-S Digest, pp (1998). [0036] The idea f a spetral map an be better understd with sme examples. Cnsider, fr example, a spetral map up t third rder (NL3) with delay (memry) terms. In the time dmain, this spetral map has the frm: ynli(t)=alx(t-%l)+a1x 2 (t-%2)+a:f i (t-%i). (3) [0037] Nw nsider the utput f this spetral map, subjet t the tw-tne signal x(t)=s((b 1 t)+s((b 2 t). Fr simpliity sake, assume unit magnitude fr eah amplitude and initially zer phase ffset (althugh when building real

36 US 2009/ Al Jun. 18,2009 behaviral mdels, these assumptins are nt made). Using Euler's identity, the signal x(t) an be written in terms f 'phasrs' as: x(r - ri) = -(exp(iwir- iri) + exp(-iwir + iri)) H 1 -(exp(itj2l- itzi) + exp(-itj2t + itzi)). and the spetral map f first, send, and third rder (in n partiular herent arrangement) an be written as: (4) ne anther. Thus, slving analytially fr a spetral map f a multidimensinal time-dependent nnlinear system is beynd the apability f mdem siene thugh its simulatin, given a fixed mdel, is straightfrward (if smewhat umbersme). [0040] A brief review f Vlterra thery is helpful t understand the full mplexity f the prblem f develping spetral maps. Fr a lass f nn-linear ausal time-invariant system (rughly, systems fr whih there exist unique steady states), the utput f suh a system an be frmally represented (meaning that it is mathematially true, but perhaps nt f muh pratial use) by a nn-linear peratr: ym = a l [i C ir i-"-i + i»-i--i + I -i-»-2 + I»-2--i j (5) y(t) = F[u(t)] = Y j y (t), (7) 4 4 _ 2 ' r 2-2 «" u 2 + _e2it2-it0jl-il0j Wt = 12 l l l where y (t)=j... Jh^T^,..., x )u(t-x 1 )... u(t-x )dx 1 dx 2... dx, and where h is alled the nth-rder (time dmain) Vlterra kernel (see Byd, supra). The details are mitted here, but the abve (frmal) slutin an be transfrmed t ne with frequeny dmain Vlterra kernels (see Byd, supra), yielding: yst = a *3-3^ _jt3-itil _ it^-ir, + _ 3 '" u l- 3 ' r 3 + -e 3 '^- 3 "'"^. yn(t) zz (8) _ 3iT}-ilwi-2tW2. _ it}+ilwi-2aw2. _etri-ilw _ 3 «i"3-2 «""l-««"2 + _-«i"3+ 2 «""l-««"2 + _'«u 2-' r 3. Eailai2 aillh {j(il,j(i2,..., jajiri )exp j^ a>ik t. 3 3 _ tr3-2trw^+trw2 j -3tr3+2trw^+trw e~' T 3~" w l +2 " w 2 + _- 3 >r3+>«"l+2»^2 where.f- 3 '"^- 3 '^) "(0 = ^ a,- [0038] Equatin (5) is useful fr reading frequeny dmain infrmatin. Fr instane, all terms at the frequeny 2(B 1 +(B 2 an be gruped t find ntributins f the map t this partiular frequeny bin, as: y((2a>i +0)2)0= -[e ii2j i +CJ 2><-,iT 3 + -«M+^2>'+»3] ( ' s((2wi +W2)r-3r3) 4 [0039] S, even in the simplest ase f a plynmial, sme f the mplexity f a spetral map is revealed. nly a few ases, suh as (stati) pwer laws, are apable f lsed frm analyti slutins. See, e.g., Mihael B. Steer, et al., "An Algebrai Frmula fr the utput f a System with Large- Signal, Multifrequeny Exitatin", Preedings f the IEEE, Vl. 71, N. l,pp (January 1983). In general, the utput t a given frequeny mpnent, fr a fixed mdel, depends n a weighted sum f the input phasrs determined by a (mdel dependent) set f effiients, as well as the riginal amplitude and phases f the input tnes relative t expqa^t) it is an input test signal. The reasn this frm f the Vlterra slutin is f interest is that it makes expliit the fat that any (but the mst trivial) spetral map will have the prperty that the utput signal depends n the amplitude and phase f the input signal. Put anther way, any behaviral mdel f a system shuld 1) keep trak f the (relative) phases and amplitudes f mpnents f the input signal, and 2) expliitly use this infrmatin in mdeling the system. That is, an amplitude mdel alne will usually be inadequate. Further exerises shw that the dependene n phase an be quite sensitive. In fat, fr many input signals, errrs f 0.01 radians in the phase f the input signal an translate int amplitude errrs f a fatr f tw in the utput signal. Thus, frm a metrlgy standpint, nnlinear behaviral mdeling requires preise measurement f the (relative) phases f harmni mpnents, in rder t develp and verify a behaviral mdel fr use in nnlinear alibratin. [0041] T finish this disussin f Vlterra thery, it is nted that the spetral map fr a (frmal) Vlterra slutin, up t third rder, an be written as:

37 US 2009/ Al Jun. 18,2009 Y(k) = Y il \k) + Y i2 \k) + Y i3 \k) (9) N = G l k-u(k)+ ^ G 2 kvk_ k _ l -U(k l )-U(k-k l ) + k1=-n N N 2 2 Gl^-K^-mk-m^-ua-ki-h), where k=l,2,3,,3n, and where U(k) and Y(k) are the Furier effiients f the input and utput signals. Nte that fatrizatin f the frequeny dmain Vlterra kernel int 'prduts' (r a sequene f ne-dimensinal nvlutins), similar t thse in the time dmain, is a nn-trivial exerise. See, e.g., Gil M. Raz, et al., "Baseband Vlterra Filters fr Implementing Carrier Based Nnlinearities", IEEE Transatins n Signal Pressing, Vl. 46, N. 1, pp (January 1998). Als nte that the G^1 is the frequeny dmain representatin f the linear transfer funtin, smetimes als alled the 'frequeny respnse funtin.' The G's an be thught f as ntaining all f the infrmatin abut the 'distrtin' prduts, beause they an be frmally fatred in the Vlterra frmulatin, frm the input signal, just like the linear transfrm funtin. The diffiulty, thugh, in using this frmulatin t build behaviral mdels (besides the inherent limits f its mathematial assumptins suh as 'weaknnlinearity') is the enrmus number f measurements needed t estimate the frequeny dmain Vlterra kernels, whih preeds like 0{N}+0{N 2 }+0{N 3 } in the third rder ase. Fr example, assuming a 1 Mhz spaing, a 1 Ghz wide linear transfer funtin an be estimated with 1000 single tne measurements, while G 2 requires measurements, and G 3 requires measurements. An empirial 'reipe' fr mre quikly building a mdel fr a spetral map is therefre desribed belw. [0042] FIG. 1 illustrates an exemplary signal path 100. The signal path 100 mprises an ARB 106, an ptinal upnverter 108 upled t the utput f the ARB 106, an amplifier 110 upled t the utput f the upnverter 108, and a DUT upled t an utput f the amplifier 110. An input signal 102 is applied t the signal path 100 by the ARB 106, and an utput f the signal path 100 generates an utput signal 104 fr appliatin t a DUT 112. A signal analyzer 114 samples the utput signal 104; and a distrtin rretin engine 116 predistrts the input signal 102 in the frequeny dmain, in ard with a spetral map. f nte, the graphs and sreenshts shwn in the drawings, and referened herein, prvide exemplary data fr a signal path 100 that des nt inlude the ptinal upnverter 108. [0043] FIG. 2 illustrates an exemplary methd 200 fr develping a spetral map fr perfrming nnlinear alibratin f a signal path 100. The methd 200 begins with the identifiatin f a set f frequeny latins fr a set f partiular utput signal spurs that result frm applying netne and tw-tne input signals vering a bandwidth f interest t the signal path (at step 202). The set f frequeny latins, r "frequeny set", may be identified by perfrming a pwer spetral san ver the bandwidth f interest. [0044] After identifying the set f frequeny latins in step 202, a spetral map is develped based n the set f frequeny latins (at step 204). The spetral map is devel- ped fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain. [0045] Ultimately, the spetral map develped in step 204 may be saved fr perfrming nnlinear alibratin f the signal path 100. [0046] FIG. 3 illustrates an exemplary way t develp the spetral map referened by the methd 200 (FIG. 2). The methd 300 begins with 1) the appliatin f ne-tne input signals vering the bandwidth f interest t the signal path 100, and 2) the measurement f amplitudes and phases f single-tne spurs in the set f partiular utput signal spurs (at step 302). The measured amplitudes and phases f the singletne spurs are used t nstrut a first-rder frequeny respnse mdel fr the signal path 100 (at step 304). In sme ases, the mdel may be extended t arbitrary input tnes by applying apprpriate pwer law, amplitude and phase saling relatinships. [0047] The methd 300 ntinues with the mputatin (e.g., by frmula r algrithm) f a first-rder input amplitude spetrum in the frequeny dmain (at step 306; see input amplitude spetrum U(k), Eq. 2). The input amplitude spetrum is restrited t the set f frequeny latins identified in step 302. [0048] Frm the first-rder frequeny respnse mdel and the first-rder input amplitude spetrum, a first-rder spetral map is develped fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path 100 in the time dmain (at step 308). This may be dne by applying the first-rder frequeny respnse mdel nstruted in step 304 t the input amplitude spetrum mpute mputed in step 306. [0049] After develping and saving the first-rder spetral map, the perfrmane f the first-rder spetral map may be verified fr varius predistrted signals that are applied t, r reeived frm, the signal path 100 (at step 310). The signals that are applied t, r reeived frm, the signal path 100 during verifiatin may mprise ne-tne, tw-tne, multitne and pseud-randm signals. [0050] When the perfrmane f the first-rder spetral map is determined t prvide insuffiient mitigatin f utput signal spurs at the set f frequeny latins determined in step 202, a set f residual utput signal spurs may be identified (at step 310). Then, tw-tne input signals vering the bandwidth f interest may then be applied t the signal path, and amplitudes and phases f the residual utput signal spurs may be measured (at step 312). The measured amplitudes and phases f the residual utput signal spurs may be used t nstrut a send-rder frequeny respnse mdel fr the signal path (at step 314); and a send-rder input amplitude spetrum ((U(k 1 )-(U(k-k 1 )) in Eq. 2) may be mputed in the frequeny dmain (at step 316). The input amplitude spetrum is restrited t the set f frequeny latins identified in step 302. Frm the send-rder frequeny respnse mdel and the send-rder input amplitude spetrum, a sendrder spetral map may be develped fr predistrting, in the frequeny dmain, signals that are input t the signal path in the time dmain (at step 318). [0051] After develping and saving the send-rder spetral map, the mbined perfrmane f the first-rder spetral map and the send-rder spetral map may be verified fr varius predistrted signals that are applied t, r reeived frm, the signal path 100. The signals that are applied t, r reeived frm, the signal path 100 during verifiatin may

38 US 2009/ Al Jun. 18,2009 mprise ne-tne, tw-tne, multitne and pseud-randm signals. Typially (band-limited) pseud-randm signals inlude thse generated by digitial mmuniatins frmats (suh as de divisin multiple aess (MA) signals) and (band-limited) pseudnise (PN) sequenes fund in spread spetrum radar appliatins. [0052] When the mbined perfrmane f the first-rder spetral map and the send-rder spetral map is determined t prvide insuffiient mitigatin f utput signal spurs at the set f frequeny latins, an additinal set f residual utput signal spurs may be identified, and any f methd steps may be iteratively repeated t update the first-rder spetral map r the send-rder spetral map. Alternately (r additinally), a third r higher rder spetral map may be develped in a manner similar t hw the first and sendrder spetral maps were develped. [0053] The abve methds 200,300 develp a spetral map using an iterative apprah. The iterative apprah is guided by measurement data at tw levels. First, the frequeny respnse mdels f the signal path 100 are nt nstruted using the full frequeny dmain infrmatin speified in the Vlterra mdel. Rather, the methds 200, 300 start with (a minimal set f wide-band) measurements f ne and twtne tests and attempt t apply these rretins t a wider set f input tnes. Send, any residual errrs (residual utput signal spurs) in a send-rder mdel are again based n a restrited set f tw-tne measurements. In bth ases, the detailed phase and amplitude infrmatin f an input signal is traked, and is used as the input t any empirial spetral maps. The methds 200 and 300 are empirial, and need t be guided by and verified by experiments. Thus, the general utility f the methds 200 and 300 need t be examined n a ase-by-ase basis. In experiments, the methds 200 and 300 have prvided gd results when alibrating wide-band arbitrary wavefrm generatrs and surrunding nditining eletrnis. [0054] Intuitively, the methds 200 and 300 are guided by an attempt t use measurement data t whittle dwn the (frequeny dmain) mdel t a minimal set f tratable data and measurements. The methds 200 and 300 are based n an intuitive (simplifying) assumptin that 'energy' in a given frequeny band (nstruted by mputing an apprpriate U(k) input signal spetrum) will get mapped in a similar way as the energy in ne r tw-tne mappings. That is, in the mdeling press we start by building the simplest mdel pssible, and then add refinements t this mdel based n residual utput signal spurs. [0055] ne a spetral map fr perfrming nnlinear alibratin f a signal path is determined, then it an be used t predistrt arbitrary input signals. As shwn by the methd 400 illustrated in FIG. 4, the DFT f an input signal 102 may be mputed (at step 402). The spetral map develped using methd 300 r 400 is then applied t the DFT f the input signal, t generate a predistrted signal in the frequeny dmain (at step 404). The IDFT f the predistrted signal (in the frequeny dmain) is then mputed t generate a predistrted signal in the time dmain (at step 406). An arbitrary wavefrm generatr is then aused t launh the predistrted signal (in the time dmain) via the signal path 100 (at step 408). The input signal 102 that is subjet t predistrtin may be a peridi, multiperidi r arbitrary wavefrm. [0056] Further details and an exemplary appliatin f the methds 200 and 300 will nw be desribed. The exemplary appliatin is nnlinear alibratin f a signal path 500, thrugh a wide-band arbitrary wavefrm generatr 502 fllwed by a (2x) amplifier 504. See, FIG. 5. [0057] In step 202 f the methd 200 (FIG. 2), the 'frequeny set' may be identified by perfrming a pwer spetral san ver the bandwidth f interest. ne and tw tne test signals are used. In the ntext f the signal path shwn in FIG. 5, three sures f spurs where examined: (i) higher rder harmnis as predited by a stati plynmial mdel (see Steer, supra), (ii) higher Nyquist zne harmnis that fld dwn t the first Nyquist band (see, e.g., Ryan Gruix, et al., "Minimizatin f DDS Spurius Cntent in Multi-Channel Systems", High Frequeny Eletrnis, pp (tber 2006)), and (iii) mixing with any internal spurs suh as internal sillatr feed thrugh (see Griffin, supra). In this example, the amplifier nnlinearities dminated, and the frequeny latins f all spurs between 65 dbm and 90 dbm (and frm Mhz) fr ne and tw tne tests were aurately predited based n 2nd and 3rd rder harmnis generated frm a third rder plynmial mdel, as hypthesized by distrtin sure (i). S, in this partiular ase, the latins f spurs an be predited either by an analyti frmula, r by simulatin, based n a lw-rder plynmial mdel. [0058] Fr example, FIG. 6 shws a tw-tne input signal having tnes at f 1 and f 2 =f 1 +Af. All spurs abve 90 dbm an be aunted by the sum and differene frequenies fr a third-rder mdel. A system withut the amplifier requires inluding a frmula fr the higher Nyquist spurs as well (see Gruix, supra). [0059] Step 302 f the methd 300 (FIG. 3) requires the measurement in bth phase and amplitude f the (nnlinear) distrtin f a ne-tne input signal, fr the dminant terms in the frequeny set identified in step 202. In the example nsidered in FIG. 5, this requires measuring, as illustrated in FIG. 7, bth the phase and amplitude f single-tne spurs. A 'miller' r nulling signal may be reated by adding a spur signal f the same magnitude but ppsite phase t the input f the arbitrary wavefrm generatr. This type f signal rretin press is alled predistrtin. This first level f mdeling is similar t traditinal AM/AM r AM/PM mdeling, whih measures the distrtin f the input tne. These data are AM/AM and AM/PM Tike' mdels fr the spurs generated by the input tne. See, e.g., Fadhel M. Ghannuhi, et al., "AM-AM and AM-PM Distrtin Charaterizatin f Satellite Transpnders/Base Statin Transmitters Using Spetrum Measurements", IEEE Cnferene Preedings f Reent Advanes in Spae Tehnlgies, pp (2003). The mdels are therefre alled the' send-rder' and 'thirdrder'am/am, AM/PM mdels respetively. FIGS. 8A & 8B shw the send rder spur distrtin data, and FIGS. 9A & 9B shw the third rder spur distrtin data. T reate a 'mdel,' a lw-rder plynmial may be fitted t the phase and amplitude distrtin data. This is suffiient beause the urves have few lal ritial pints, and interplatin an be used t find the phase and amplitude respnse at a frequeny within the upper and lwer frequeny bunds presented by the experimental data. Nte that beause a mdel is reated in the frequeny dmain, the mdel at least fr single tne exitatins has n diffiulty vering a wide-bandwidth. The reatin f a rrespnding time dmain mdel with a similar wide-bandwidth distrtin harateristi wuld be mre hallenging. This is ne f the mtivatins fr keeping the measurement and mdeling in the frequeny dmain (and hene, the use f 'spetral maps').

39 US 2009/ Al Jun. 18,2009 [0060] As desribed in step 304, FIGS. 8 and 9 als shw the fits t lw-rder plynmials. The phase and amplitude dependene f the single-tne spur distrtin shuld als be heked against any hange in input amplitude, and this dependene may be added as an additinal mdel parameter (if it is nt predited by simple saling laws). FIGS. 8A, 8B, 9A and 9B are taken at the full amplitude input range A max and the data was heked fr less than full amplitude with n signifiant disrepanies frm a simple saling preditin: namely, (A I /A max ) 2 fr the send harmni, and (A,- / A max ) 3 fr the third harmni. [0061] In step 306, U(k) may be mputed analytially fr a limited set f mdels (see Steer, supra) r by simulatin. The simulatin is straightfrward. The funtin y(t) is mputed in the time dmain, the fast Furier transfrm (FFT) f y(t) is taken, and the amplitudes and frequenies f any spurs abve a predetermined amplitude level are rerded. The resulting set f phasrs is alled the 'frequeny set', and all further mputatins are limited t mputatins r simulatins n this restrited set f phasrs, similar t what is dne in harmni balane methds f iruit simulatrs. See, e.g., Bris Tryanvsky, "Frequeny Dmain Algrithms fr Simulating Large Signal Distrtin in Semindutr Devies", Dissertatin, Stanfrd University (1997). FIG. 10 shws the frequeny set fr the exemplary signal path shwn in FIG. 5. The example shws the mputatin f the utput signal based n a fur-tne input signal entered arund 50 Megahertz (MHz). Different input signals will generate different frequeny sets. [0062] The utput f step 308 is als shwn in FIG. 10, n the right. The utput spetrum limited t the 'frequeny set' is multiplied (in the frequeny dmain) by the mplex number frm the rrespnding alibratin urves shwn in FIGS. 8 & 9. This simply amunts t a resealing and rtatin f the phasrs in the frequeny set. The ad h assumptin being made here is that, t 'first-rder,' the utput frm a single tne test will be lse t the utput frm a multitne test, where the input is nt a single tne, but rather the phasr at U(k) whse amplitude and phase are mputed as a weighted sum f the input tnes. Put mre simply, energy within the bin k m is getting mapped t bin k, at least t first-rder, in the same way that a single tne gets mapped. [0063] Step 310 f the methd 300 examines the auray f the first-rder mdel. See, FIG. 3. That is, the predited spurs (bth their phase and magnitude) are mputed based n the input signal (in the frequeny dmain), and then a it radians ut f phase versin f this signal is added t the input f the arbitrary wavefrm generatr. This rretin, even fr a few input tnes, nsists f many terms. FIGS. 11A, 11B, 12A & 12B shw results fr the signal path shwn in FIG. 5, fr bth multi-tne (FIGS. HA & 11B) and pseud-randm (FIGS. 12A & 12B) signals, respetively. In bth ases, the methds 200 and 300 shw gd perfrmane fr remving 'ut-f-band' distrtin. The test signal fr FIG. 13 is a pseud-randm test sequene whih is passed thrugh a very steep (100 db) band pass filter. FIGS. 13 & 14 shw lseups f the utput f the FIG. 5 signal path, with and withut predistrtin, fr an exemplary mplex input signal. [0064] Thugh a typial redutin f 4 dbm is seen fr intermdulatin terms, there als appears t be signifiant residual spurs at (r near t) the frequeny band f the input (stimulus) signal. Steps address this by nstruting a tw-tne alibratin urve. In the example studied this is nstruted at 1 Mhz steps, with a fixed ffset tne f 1 Mhz. Variable ffset mdels may als need t be nsidered, whih inrease mdel mplexity. With the additin f this send rder rretin, r 'send-rder mdel,' applied t the residual spur spetrum, a further redutin in intermdulatin distrtin an be arhived as illustrated in FIG. 15. [0065] The nvel methds 200,300,400 and apparatus 116 dislsed herein an prvide varius advantages. First, they wrk with a mplete eletrni instrument r devie and an therefre be applied t systems after their design and manufaturing is mplete. That is, alibratin an be applied after ther system and design ptimizatins are mplete. [0066] Send, the methds and apparatus are based nly n measured signal prperties, and nt n the detailed knwledge f eletrni instruments r devies that is required by many alibratin tehniques. This means that the methds and apparatus an be applied t individual instruments r devies, r t systems mpsed f tw r mre nneted instruments r devies. Als, the methds and apparatus an be used t alibrate 'distributed' eletrni instruments and devies. [0067] Third, the nvel methd and apparatus dislsed herein are iterative, and are based n a fixed sequene f relatively standard measurements, whih unlike estimatin f Vlterra kernels, an be amplished with fewer measurements. This is beause the methds and apparatus desribed herein make use f measurement data in the initial reatin f a mdel. The mdel is then whittled dwn t fus it n the nnlinear spurs that are atually measured at the start f the press. This is in ntrast t methds based n a straightfrward implementatin f Vlterra thery, whih need t estimate all Kernel elements, withut using prir knwledge btained by experiments t nstrain the Vlterra mdel. Plaing early nstraints n the mdeling press, at least fr alibratin, enables an effiient number f measurements t be made. This tradeff between mdel generality is well wrth the simplifiatin that results in fewer and simpler measurements then needed fr the full identifiatin f Vlterra Kernels f a fixed rder. [0068] It is als nted that the methds and apparatus desribed herein differ fundamentally frm the methd prpsed by Byd, supra. Byd's methd nly applies t systems where the send rder Vlterra kernel, H^Sj,s 2 ), is nn-zer fr bth s 1 and s 2 that is, data frm a tw-tne test is required, and there is n desriptin f the use f data frm a ne-tne test mdel. In ntrast, the methds and apparatus desribed abve expliitly measure and use data frm a netne test as the primary data fr the first-rder frequeny respnse mdel. And, as previusly mentined, any Vlterra series methd nly applies in ases where speifi mathematial assumptins hld, namely the system is weakly linear, has 'fading memry,' and additinal mathematial assumptins desribed by Byd, supra. These harateristis are nt required in the methds and apparatus desribed abve. What is laimed is: 1. A methd f develping a spetral map fr perfrming nnlinear alibratin f a signal path, mprising: identifying a set f frequeny latins fr a set f partiular utput signal spurs that result frm applying ne-tne and tw-tne input signals vering a bandwidth f interest t the signal path;

40 US 2009/ Al Jun. 18,2009 develping, based n the set f frequeny latins, a spetral map fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain; and saving the spetral map fr perfrming nnlinear alibratin f the signal path. 2. The methd f laim 1, wherein the set f frequeny latins fr the set f partiular utput signal spurs is identified by perfrming a pwer spetral san ver the bandwidth f interest. 3. The methd f laim 1, wherein the set f frequeny latins fr the set f partiular utput signal spurs is identified using at least ne f: an analyti frmula, and a simulatin. 4. The methd f laim 1, wherein develping the spetral map mprises: applying ne-tne input signals vering the bandwidth f interest t the signal path, and measuring amplitudes and phases f single-tne spurs in the set f partiular utput signal spurs; using the measured amplitudes and phases f the singletne spurs t nstrut a first-rder frequeny respnse mdel fr the signal path; mputing a first-rder input amplitude spetrum in the frequeny dmain, restrited t the identified set f frequeny latins; and develping, frm the first-rder frequeny respnse mdel and the first-rder input amplitude spetrum, a firstrder spetral map fr predistrting, in the frequeny dmain, signals that are applied t, r reeived frm, the signal path in the time dmain; wherein saving the spetral map mprises saving the firstrder spetral map. 5. The methd f laim 4, further mprising: verifying perfrmane f the first-rder spetral map fr varius predistrted signals applied t, r reeived frm, the signal path; when the perfrmane f the first-rder spetral map is determined t prvide insuffiient mitigatin f utput signal spurs at the set f frequeny latins, identifying a set f residual utput signal spurs and, applying tw-tne input signals vering the bandwidth f interest t the signal path, and measuring amplitudes and phases f the residual utput signal spurs; using the measured amplitudes and phases f the residual utput signal spurs t nstrut a sendrder frequeny respnse mdel fr the signal path; mputing a send-rder input amplitude spetrum in the frequeny dmain, restrited t the identified set f frequeny latins; and develping, frm the send-rder frequeny respnse mdel and the send-rder input amplitude spetrum, a send-rder spetral map frpredistrting, in the frequeny dmain, signals that are input t the signal path in the time dmain; wherein saving the spetral map mprises saving the send-rder spetral map. 6. The methd f laim 5, further mprising: verifying a mbined perfrmane f the first-rder spetral map and the send-rder spetral map fr varius predistrted signals applied t, r reeived frm, the signal path; when the mbined perfrmane f the first-rder spetral map and the send-rder spetral map is determined t prvide insuffiient mitigatin f utput signal spurs at the set f frequeny latins, i) identifying an additinal set f residual utput signal spurs, and ii) iteratively repeating the applying, using r mputing steps t update the first-rder spetral map r the send-rder spetral map. 7. The methd f laim 4, wherein the first-rder frequeny respnse mdel fr the signal path is nstruted by fitting lw-rder plynmials t the measured amplitudes and phases f the single-tne spurs. 8. The methd f laim 4, further mprising, adding an input amplitude saling parameter t the first-rder frequeny mdel. 9. The methd f laim 4, further mprising: varying the input amplitudes f the ne-tne input signals applied t, r reeived frm, the signal path, and measuring an input amplitude dependene f the measured amplitudes and phases f the single-tne spurs in the set f partiular utput signal spurs. 10. A methd f perfrming nnlinear alibratin f a signal path, mprising: mputing a disrete Furier transfrm (DFT) f an input signal t the signal path; applying a spetral map t the DFT f the input signal, t generate a predistrted signal in the frequeny dmain, the spetral map being based n i) a set f frequeny latins fr a set f partiular utput signal spurs that result frm applying ne-tne and tw-tne input signals vering a bandwidth f interest t the signal path, and ii) measured amplitudes and phases f single-tne spurs resulting frm an appliatin f ne-tne input signals t the signal path; mputing an inverse DFT (IDFT) f the predistrted signal in the frequeny dmain, t generate a predistrted signal in the time dmain; and ausing an arbitrary wavefrm generatr f the signal path t launh the predistrted signal in the time dmain via the signal path. 11. The methd f laim 10, wherein the input signal is a peridi wavefrm. 12. The methd f laim 10, wherein the input signal is a multiperidi wavefrm. 13. The methd f laim 10, wherein the input signal is an arbitrary wavefrm. 14. The methd f laim 10, wherein the signal path further mprises ne f: an amplifier, and an up-nverter. 15. The methd f laim 10, wherein the spetral map is further based n measured amplitudes and phases f multiple-tne spurs resulting frm an appliatin f tw-tne input signals t the signal path. 16. Apparatus fr alibrating a signal path, mprising: an arbitrary wavefrm generatr, upled in an input-end f the signal path; and a distrtin rretin engine nfigured t, reeive an input signal fr the arbitrary wavefrm generatr; mpute a disrete Furier transfrm (DFT) f the input signal; apply a spetral map t the DFT f the input signal, t generate a predistrted signal in the frequeny dmain, the spetral map being based n i) a set f frequeny latins fr a set f partiular utput sig-