Normally, when linearity behavior of an

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1 ICROWAVE JOURAL REVIEWED EDITORIAL BOARD TECHICAL FEATURE COPACT FORULAS TO RELATE ACPR AD PR TO TWO-TOE IR AD IP A set of compact formulas are presented to estimate modern multitone distortion figures of merit from well-established two-tone measurements. In addition, the ability of noise power ratio (PR) figures to predict in-band distortion of mildly nonlinear systems subject to real continuous spectra is discussed. It is shown that the usual procedure of eliminating one input tone to allow in-band distortion observation leads to an optimistic PR value higher than the real PR that would be obtained if the correspondent dense spectrum was used. Fig. Graphical definitions of (a) ACPR, (b) PR and (c) -IR. AITUDE (a) AITUDE (b) AITUDE (c) P D P S FREQUECY FREQUECY FREQUECY ormally, when linearity behavior of an amplifier must be evaluated, the twotone third-order intercept point (IP) is used as a standard. However, in new telecommunications circuits (such as the ones found in modern digital mobile communications), more complex signals must be analyzed and, thus, the two-tone IP standard is no longer a good figure of merit. Therefore, more robust distortion marks, such as P S adjacent-channel power ratio (ACPR), PR and multitone intermodulation ratio (-IR), must be considered. ACPR is the ratio between the total adjacent-channel integrated power PR and the power of the useful signal band. PR is the ratio between the in-band distortion and useful signal spectral densities when an in-band noise spectrum slice is removed. -IR is the ratio between a useful tone power and the highest distortion -IR tone power outside, but close to, the useful band. Figure shows graphical representations of these three parameters. ACPR P D + P D P D ACPR and -IR distortion measurements are simple extensions to the well-established two-tone intermodulation test since they relate the distortion power outside the useful band to the output signal power inside the useful band. PR is a different type of measurement since it is intended to assess the in-band distortion. Because devices linear output components mask in-band distortion, it is not possible to directly measure distortion products that follow on the fundamental signal unless a slice of the input signal spectrum is removed with a notch filter. A relationship between these multitone test results and the more easily measured two-tone intermodulation distortion (ID) would enable the full characterization of the device under multitone excitation using only the well-known two-tone intermodulation tests. Fortunately, two recently published papers, have demonstrated that these formulas can be easily deduced from some rigorous combinatory analysis. UO BORGES DE CARVALHO AD JOSÉ CARLOS PEDRO Universidade de Aveiro, Instituto de Telecomunicações Aveiro, Portugal

2 TECHICAL FEATURE which implies whichthat implies that TWO-TOE AALYSIS To begin with the well-established two-tone test and its figures of merit, IR and IP, it is assumed that the device under test (DUT) can be considered memoryless and under a small-signal excitation. This common restriction applies to almost every circuit a large signal-to-intermodulation ratio is desired. This is the case for the major portion of socalled linear components used in modern telecommunications systems when they are driven reasonably below gain compression. Under this assumption, all nonlinear devices of practical interest can be approximately modeled by Taylor series expansions up to the third degree. The device model representation is shown in Figure y k x + k x + k x () Usually, RF communications circuits are narrowband. Therefore, assuming only the in-band output is of interest and that x(t) x cos(ω t) + x cos(ω t) x x x yields IP implies that P ω P ω ω X P in ω DUT Fig. The nonlinear circuit representation. y in band t kx cos ωt cos ω t k x + { 9 cos( ω t ) + 9 cos( ω t ) + [ cos ( ω ω) t + cos ( ω ω ) t]} ( ) () By observing Equation, the expected linear components can be distinguished from the new distortion components. In fact, nine new mixing products appear at ω and ω and three appear at ω ω and ω ω. These numbers of newly generated distortion components can be generalized for any fixed mixing product using the previously published multinomial coefficient. Using Equations and, it is possible to calculate the two-tone IP and IR. The input two-tone IP is defined as the output power per tone required to produce equal distortion power P Do at ω ω (or ω ω ) and linear output power P Lo at ω (or ω ). To simplify calculations, consider the case of a normalized Ω load resistance. Then, - ( ) ( ) + ( ) Lo( ) Do( ) [ ] X ( ) ( ) Y () and so k ( ) () k The value of the two-tone IR normalized to the total output power P OT then is ACPR, -IR AD PR RELATIOAL FORULAS Following established theory,, formulas that relate general n-tone ACPR, -IR, PR and co-channel power ratio (CCPR) to the derived IR or IP are presented. These formulas enable a straightforward conversion between any pair of the distortion figures of merit. ACPR ACPR c IP k X IR P D k X 9 k IP k k X Pout 9 ( ω ) IP POT ( ω) IRc IP P c ( 5) ( ) + k X n k k X 9 X k X 9 k m Pouto( ω) k X k k OT m k X n n r r n IRc + log + ( ) (5)

3 TECHICAL FEATURE Usual PR n mod r r n + r mod n r n n n + 8 r r n r + n x mod the division remainder of x by two ( 7) PRu c n b b + + b( n b ) ( ) n b b k X n k X n ( ) 9 + n IRc + log + ( 9) ( ) -IR -IR c k X n n r n r + k X n ( ) 9 + n IRc + log + ( 8) In this case, and are calculated for r, that is, the highest distortion tone power outside the useful band is located. n + b mod b + mod This case is called the usual PR since it is the measurement usually performed in the laboratory: The circuit is excited with a multitone signal with one middle tone shut down in order to reveal the distortion components present. CCPR The CCPR test is what ideally could be done if the input signal consisted of a multitone spectrum without any tone shut down (a situation much closer to the normal nonlinear system s operation). Contrary to the previously discussed PR, the distortion on that particular position is measured in the presence of the corresponding input spectral line.

4 TECHICAL FEATURE I 8 pf pf Q Ω L ch V gg pf pf pf L ch V dd Fig. The class A power amplifier circuit schematic used in the application examples. Fig. Harmonic balance-simulated two-tone P out and ID, and extrapolated IP. P out (ω ω) P out (ω) SLOPE SLOPE POUT (m) IP 8. IR 58. c 8 5 P in (m) OUT ACPR CCPR -IR P out CCPR 55.5 c. c 5. c FREQUECY Fig. 5 Output distorted spectrum and CCPR measurement. Pout (m) Fig. Output distorted spectrum and traditional PR measurement. ACPR 55.5 c -IR 5. c 8 PR 5. c FREQUECY Pout (m) CCPRr S c n b b + + b( n b ) + b ( ) n b b ( ) n ( 5) ( ) k X n k X n + + S 9 9 n IRc + log + + S 9 ( ) A comparison between Equations through 5 and 9 through reveals that significantly more mixing terms now appear. Therefore, the distortion power in this case is indeed much worse than that obtained from the usual PR. This result is a clear indication that the traditional way of measuring PR only provides a first and optimistic estimate of the real in-band distortion generated by the nonlinear component subject to a dense input spectrum., In fact, any simple combinatory analysis will show that the intuitive thought that shutting down only one TABLE I SIULATED AD CALCULATED RESULTS ACPR -IR PR CCPR (c) (c) (c) (c) Simulated Calculated tone from a large number of input spectral lines will have a negligible impact on the measurement is indeed false. Actually, when the single tone ω i is shut down, there will be at least (n ) mixing products of the form ω i + ω j ω j and three of the form ω i + ω i ω i eliminated., APICATIO EXAES Consider the microwave power amplifier shown in Figure, biased for class A operation. This power amplifier was excited by a two-tone test. The simulated output power at ω and in-band ID at ω ω with an inhouse harmonic balance engine are shown in Figure. IP can be obtained as the interception of the extrapolated ID slope line and the linear output power line of slope. In this case, IP 8. m. The power amplifier was then excited with a 5-tone input spectrum at a total input power level of 8. m, equal to the equivalent two-tone input power for an IR 58. c. The amplifier s simulated output response is shown in Figure 5. Figure shows results that are similar but now correspondent to a traditional PR test three input tones were shut down. Using the ACPR, -IR, PR and CCPR values derived from the graphs and the calculated results obtained with the expressions presented in previous sections, Table lists the data for comparison. As can be seen, fast and accurate results for multitone figures of merit can be easily obtained using the presented formulas. For example, for the case just presented, the required simulation time was five hours against immediate results provided by the formulas.

5 TECHICAL FEATURE AITUDE () 8 Fig. 7 ormalized relationship between multitone ACPR, -IR, PR, CCPR and IR vs. the number of input tones. Example ow consider the same power amplifier (IP 8. m and gain G ), a design specification of c ACPR under a -tone excitation is required. The design goal is the input power level per tone in order to meet the ACPR specifications. Using Equations 5 and 7, and for tones become 875 and 5. Thus, ACPR c IR + log IR. ( IP POT ) POT. 5 m PinT POT Pin( tone) n ng POTm log ( n) G m If an alternative specification of -IR c were the goal, then using Equations 5 and 8 yields 5 and 5. - IR c IR + log + ( IP POT ) + +. P. 9 m P in ACPR-IR -IR-IR UBER OF TOES (K) tone OT POT ng ( ) PR-IR CCPR-IR 9. m ow consider the co-channel interference problem. If the amplifier were specified for the same c using a traditional PR test on the 5st tone, then according to Equations 9 through, 577 and 9. PRu c IR + log + ( IP POT ). POT. 7 m P ( tone). 8 m in If, on the other hand, a similar CCPR of c were considered but the 5st tone is no longer shut down, the input power using Equations through 5 would now have to be, 9 and S 597. CCPRa c IR + log + + S 9 ( IP POT ) 7. POT 5. m P ( tone). 5 m in As can be seen, there is a difference of more than in input power when the CCPR or usual PR is considered. Thus, if the circuit is excited with m per tone (as would have been done if the usual PR approach was considered), CCPR.9 c would only be produced under normal operation, which is more than 7 below the c goal. Example Consider now a plot of ACPR-IR, -IR-IR, PR-IR and CCPR-IR vs. the number of input tones n. For n > a good approximation of a continuous spectrum is obtained. Thus, the multitone input spectrum with constant total power becomes a good discretized version of the real signal power spectral density. Figure 7 shows that the usual PR is approximately below the CCPR. Therefore, if the usual PR measurement procedure is to be used to estimate distortion arising from an input-dense spectrum, a offset in distortion must be added to the observed measurement results. The values presented here for -IR have already been demonstrated by measurements in previously published work. Calculating the limits of Equations through 5 when n tends toward infinity gives ACPR- IR., -IR-IR., PR-IR 7.8 and CCPR-IR.. Comparing the results of CCPR-IR and PR-IR, a 5. increase in the middle of the input band must be considered when a usual PR is measured. COCLUSIO A rigorous analysis of the distortion behavior of a thirdorder memoryless nonlinear system subject to a multitone spectrum has been presented. The derived expressions al-

6 low closed-form relationships between well-established two-tone results and modern multitone figures of merit. This analysis also showed that the usual PR measurement procedure provides an optimistic estimate of the distortion that arises in a real nonlinear system driven by a multitone or even continuous spectrum. However, it also can be shown that, under the considered conditions, the addition of a constant correction factor of is sufficient to produce accurate results. References..B. Carvalho and J.C. Pedro, ulti-tone Intermodulation Distortion Performance of rd Order icrowave Circuits, 999 IEEE International icrowave Theory and Techniques Symposium Digest, Anaheim, CA, June 999, pp J.C. Pedro and. Carvalho, On the Use of ulti-tone Techniques for Assessing RF Components Intermodulation Distortion, IEEE Transactions on icrowave Theory and Techniques, Vol. 7, o., December S. aas, onlinear icrowave Circuits, Artech House, orwood, A, B. Carvalho and J.C. Pedro, ulti-tone Frequency Domain Simulation of onlinear Circuits in Large and Small Signal Regimes, IEEE Transactions on icrowave Theory and Techniques, Vol., o., December 998, pp Leffel, Intermodulation Distortion in a ulti-signal Environment, RF Design, June 995, pp R. Hajji, F. Beauregard and F.. Gannouchi, ultitone Power and Intermodulation Load-pull Characterization of icrowave Transistors Suitable for Linear SSPA s Design, IEEE Transactions on icrowave Theory and Techniques, Vol. 5, o. 7, July 997, pp TECHICAL FEATURE uno Borges de Carvalho received his diploma degree in electronics and telecommunications engineering from the University of Aveiro, Portugal in 995. Since 995, he has been a PhD student at the University of Aveiro with a Praxis XXI grant. In 997, he was appointed assistant lecturer at the University. Carvalho received the 995 University of Aveiro and Portuguese Engineering Association Prize for the best 995 student at the University of Aveiro. He also was awarded third place in the student paper competition at the 998 IEEE International icrowave Symposium. His main research interests include CAD for nonlinear circuits and active device modeling and design. He is a member of the Portuguese Engineering Association and an IEEE student member. Carvalho can be reached via at nborges@ieee.org. José Carlos Pedro received his diploma and doctoral degrees in electronics and telecommunications engineering from the University of Aveiro, Portugal in 985 and 99, respectively. From 985 to 99, he was an assistant lecturer at the University of Aveiro. Currently, Pedro is an associate professor at the University and a senior research scientist at the Telecommunications Institute. His main scientific interests include active device modeling and analysis and design of various nonlinear microwave and optoelectronics circuits, in particular, the design of highly linear multicarrier power amplifiers. Pedro received the arconi Young Scientist Award in 99. He can be reached via at jcpedro@ieee.org.