1264 IEEE TRANSACTIONS ON MICROWAVE THEORY AND … · 1264 IEEE TRANSACTIONS ON MICROWAVE THEORY...

1264 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 6, JUNE 2007

Intermodulation Distortion of Third-Order NonlinearSystems With Memory Under Multisine Excitations

João Paulo Martins, Student Member, IEEE, Nuno Borges Carvalho, Senior Member, IEEE, andJosé Carlos Pedro, Fellow, IEEE

Abstract—This paper presents a methodology to compute thedistortion output of a class of third-order nonlinear dynamic sys-tems from only standard two-tone test results. Closed-form expres-sions are presented to compute the distortion output and metricsas adjacent channel power ratio and co-channel power ratio/noisepower ratio for an arbitrary multisine with tones. The impactof memory effects in a multisine excitation is also addressed, im-proving the design of RF components by a careful understandingof memory effects mechanism in real modulated signals. An exper-imental validation is presented to prove the proposed theory.

Index Terms—Measurements, memory effects, nonlinear sys-tems, waveform analysis.

I. INTRODUCTION

MEMORY effects have a strong impact on the design of RFsystem components for the new wireless scenarios. This

type of phenomenon can deeply impact any form of lineariza-tion mechanism, or even obviate its implementation in widebandsystems, due to the difficulty in controlling the correspondentwide baseband characteristics.

For these reasons, it is imperative that a deep study is un-dertaken on the memory effect’s mechanisms noticed in thetelecommunication systems when they are driven by real mod-ulated excitations.

Memory effects can be divided into short and long term, with“short” and “long” referring to the time constants involved inthe impulse response tail of the nonlinear dynamic system. Thelong-term memory time constants impact the signal’s envelope,while the short time constants affect the RF carrier. Since in acommunication system the information is carried by the enve-lope, the understanding of the long-term memory effect mecha-nisms is a fundamental topic for predicting the system’s perfor-mance degradation.

Most of the research on memory effects of nonlinear dynamicsystems was based on swept frequency two-tone tests [1]–[7],characterizing these dynamic responses through the variation ofamplitude and phase of the observed intermodulation distortion(IMD) products.

Manuscript received September 29, 2006; revised March 14, 2007. Thiswork was supported in part by the European Union under the Network of Ex-cellence TARGET Contract IS-1-507893-NoE and under Project ColteMepaiPOSC/EEA-ESE/55739/2004. The work of J. P. Martins was supported by thePortuguese Science Foundation, Fundação para a Ciência e Tecnologia underPh.D. Grant 22056/2006.

The authors are with the Instituto de Telecomunicações, Universi-dade de Aveiro, 3810-193 Aveiro, Portugal (e-mail: [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TMTT.2007.896794

However, little was done to extrapolate this knowledge tomore complex excitation forms such as multisine signals [8].In fact, the relation between two-tone and multisine signal testspresented in [9] was restricted to a memoryless third-order non-linearity. It was then extended in [10] for systems of fifth order,but still static.

In [11], we proved that, for a simplified model of an RFsystem describing the third-order behavior including memoryeffects, it is possible to relate two-tone IMD measurementswith nonlinear distortion evaluation under a multisine excita-tion. This was then demonstrated by simple simulations with afive-tone multisine signal.

Nevertheless, the presented results only stated the idea thatit is possible to infer a multisine characterization of a class ofnonlinear third-order dynamic systems from a set of two-tonetests, no quantification was given for these multisine responses.

Moreover, the results presented in [11] addressed, exclu-sively, spectral regrowth distortion, while co-channel distortionwas left uncovered.

In this paper, we expand the results presented in [9], derivingclosed-form expressions for the calculation of spectral regrowthand co-channel distortion of a class of third-order nonlinear dy-namic systems under multisine signal excitation from a set oftwo-tone test observations. Using this formulation, it is possibleto study and explain the memory effect mechanisms of thesetypes of systems, driving the RF design engineer to an optimizeddesign of baseband filtering.

Following this introduction, this paper presents a briefoverview of what was presented in [11] in order to put itssubject in context. In Section III, mathematical support for therelation between two-tone and multisine IMD components isgiven both for spectral regrowth and co-channel distortion.

In Section IV, we present some results based on the devel-oped formulas, and identify the relation between the amplifierbaseband filtering characteristics and its nonlinear memory ef-fects.

Finally, in Section V, we demonstrate the proposed theorythrough experimental results.

This paper concludes in Section VI by summarizing the mainachievements of this study.

II. RELATIONSHIP BETWEEN TWO-TONE IMD AND

MULTISINE NONLINEAR DISTORTION IN A

THIRD-ORDER DYNAMIC NONLINEARITY

In [11], we discuss the relationship between the two-tone ex-citation IMD and multisine nonlinear distortion in a third-ordernonlinearity presenting memory. Here, we summarize the most

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MARTINS et al.: IMD OF THIRD-ORDER NONLINEAR SYSTEMS WITH MEMORY UNDER MULTISINE EXCITATIONS 1265

Fig. 1. Third-order dynamic nonlinearity model.

TABLE IFIVE-TONE THIRD-ORDER MIXING PRODUCTS

important results from [11] to contextualize the following sec-tions.

In order to model this phenomenon, we assumed that thenonlinear in-band response of a third-order nonlinear systempresenting memory to a narrowband signal can be decomposedas the sum of a cubic polynomial direct path response with anup-converted baseband component [13]. With such a model inmind, the baseband component is demodulated from the RFsignal in a second-order nonlinearity and then pressed withmemory in a low-pass filter that mimics the baseband responseof the nonlinear system (Fig. 1) [2], [12], [13].

According to this model, the in-band IMD transfer functionfor a two-tone signal is given by [2]

(1)

where is the third-order nonlinear transferfunction arising directly from the third-order static conversion,and and are the second-ordernonlinear transfer functions responsible for the baseband andsecond harmonic signal components that will then be remixedto fall onto the system’s first zone output.

If we now consider an uncorrelated multisine excitation, i.e.,when the tones do not share the same phase reference, the outputdistortion from a nonlinear dynamic system will be the vectoraddition of the several components whose amplitude and phasedepend on the tone spacing.

Table I presents these components obtained for a five-tonesignal (see also Fig. 2).

If the multisine excitation could be considered narrowband,i.e., if the system’s bandwidth is greater than the signal’s band-width, then would be approximately constantand equal to . The mixing product arising from ,where is at the second harmonic, could also be consid-ered constant ( ) since the relative bandwidth change with thetone spacing is very small.

Fig. 2. Five-tone multisine and correspondent IMD distortion.

Thus, for instance, we can see that the spectral regrowth toneidentified as in the output signal depends on

(2)

and

(3)

If we manage to characterize each of those terms individually,we could get all the long-term memory effects that we need fora multisine excitation.

In order to clearly identify each of those components, a two-tone test is performed and the result is computed according to(1).

Since the most important terms are the ones that vary withtone spacing, we start by first identifying the constant part ofthe expression. That is done from the asymptotic behavior of

at very low-frequency separations, i.e., in the limitwhen tends to zero Hz. Thus, the two-tone output dis-tortion becomes

(4)where is1

(5)

and is the remaining term that varies with tonespacing representing the memory contribution.

This way, by changing the tone spacing, the different compo-nents can be obtained. If the tone spacing is made sufficientlysmall, can also be extracted by continuity in zero separationfrequency if a smooth frequency response is assumed. However,since the terms in the multisine case are accountedin vector additions (2) and (3), we must have them characterizedin both amplitude and phase.

Thus, for each frequency component, we need to solve thefollowing equation:

(6)

The computation of the system nonlinear transfer functionsis achieved by using higher order statistics (HOS), considering

1This formula is presented in a compact way contrary to what is in (5) [11].In the current form, the complex value K represents the constant part of thesecond term of (5) [11]



Fig. 3. Co-channel and adjacent-channel spectral components.2

a two tone as the input test signal [14]. This way, we can obtainthe transfer function both in amplitude and phase.

These equations should be calculated for each tone spacing,having different tone spacing’s involved, and thus at leasta linear system of equations should be solved, correspondingto tone separations plus the constant .

This system of equations is built by measuring the HOS foreach two-tone signal at each different tone spacing. In a test withan arbitrary number of tones, the system to be solved can berepresented in matrix form as

(7)

where is the third-order statistic for an tonespacing, is the second-order transfer function for an

tone spacing, and is the number of tone spacingsconsidered.

In Section III, we show how to use this matrix for the calcu-lation of an overall nonlinear distortion response to an -tonemultisine excitation.

III. IMD COMPUTATION- GENERALIZATION

TO AN ARBITRARY NUMBER OF TONES

In Sections I and II, we discussed the inherent relation be-tween multisine and two-tone responses in a nonlinear third-order system presenting memory. Now we will discuss how toquantify this relation deriving analytical results amenable to pre-dict the multisine characterization outcome.

In order to do so, we need to account for every nonlinearmixing product that falls on each output spectral regrowth com-ponent (Fig. 3) .

This task is somehow complex since it will demand an ex-tensive account of every combination of three frequencies that

2The N , b, and k symbols denote the number of tones of the input multisine,half the number of tones, and the index of a specific component, respectively.

will land on a specific position of the spectral regrowth. Never-theless, this study is somehow simplified considering an equallyspaced multisine since, in that case, any .

The main objective will thus be to obtain, in an automaticway, every mixing product for an arbitrary number of tonesfor a specific spectral regrowth mixing position. This tool willallow the computation of the multisine nonlinear distortion veryefficiently by exclusively calculating the number of basebandcomponents to be analyzed.

In order to understand how to apply the proposed formula-tion, consider an equally spaced -tone input signal (Fig. 3) de-scribed by , . The response of athird-order nonlinear system to this input excitation is obtainedcombining all three tones of the input signalwith , , and representing the input spectral positions.

The output will be given by

(8)

where

(9)

as was previously seen and as in [11].Despite the large number of combinations that fall in for

a multisine excitation, a precise account of those combinationsis needed before proceeding with the overall power calculationof the mixing product.

In [7], the number of different combinations that fall on eachposition was calculated for a memoryless nonlinearity. The dif-ferent arrangements were then considered for the ones fallingwithin the input signal components and, in that case, there weresome contributions that were correlated with the input, and someothers uncorrelated. Beyond these, there were some others thatfall at the adjacent channel, where two types of mixing productswere also considered: the ones presenting equal andthe ones where .

However, in that case, due to memoryless nonlinearity, noimportance was given to the fact that each is affected by adifferent . This coefficient is dependent on abinomial function that depends on two baseband values (and ), responsible for the nonlinear system’s memoryeffects. In fact, we are considering that can beexpressed as

(10)

where

Thus, within this model, accounting for the system’s non-linear dynamic effects means to calculate each different and

values that falls on each position for an arbitrary numberof tones .

In order to obtain closed formulas to calculate these differentarrangements, we will divide the nonlinear mixing products inthree different classes, i.e.: 1) the ones falling onto the adjacent-



Fig. 4. Different matrices of spectral regrowth mixing products that fall on thefrequency position ! for a five-tone excitation.

channel; 2) the co-channel products that are correlated to thesignal; and 3) the co-channel uncorrelated products.

The measurement procedure starts with a set of sweptfrequency two-tone tests covering tone spacings, andrecording the amplitude and phase values of each IMD com-ponent [14], [15]. Using (7), the function of the tone spacing

is then computed. With these functions, it is thenpossible to calculate each distortion component that falls at aspecific frequency position according to (9). Finally, the variouscomponents are added together, according to the characteristicsof each mixing product. The power of the correlated distortionproducts must be accounted for as the average power of the ad-dition of phasor quantities, while the power of the uncorrelatedones can be computed by the direct addition of the powers ofthe individual components.

A. Adjacent-Channel Mixing Products

For a giving adjacent spectral position , we must accountfor all different values of and . In that respect, and in orderto simplify the calculation, we have developed a matrix scheme,shown in Fig. 4, where for each spectral regrowth, (in the il-lustrated case for a five-tone multisine excitation), the binomialcombinations were calculated— and pairs. The combina-tions marked with an “ ” correspond to different and values,while the ones marked with an “o” correspond to equal and

. This separation is important since the amplitude of mixingproducts with equal and must be multiplied by the multino-mial coefficient 3/8, while the ones with different and mustbe scaled by 3/4.

By expanding these matrices for an increasing number oftones, from 2 to , and then adding up the correspondingmixing products, we were able to derive a formula that au-tomatically generates the output power of the component at

and for each pair of and values. The obtainedformula is

(11)

where corresponds to the multinomial coefficient [6] thatvalues 3/8 if and 3/4 if , and is the spec-tral regrowth tone accordingly to Fig. 3. The amplitude of eachmultisine component is represented by and is a constant.

B. Co-Channel Signal-Correlated Mixing Products

As was explained earlier, the co-channel mixing products aredivided into signal-correlated and uncorrelated components.The correlated ones are the products that obey the constraint

when two involved frequencies areequal.

Now at all times, and spans from 1 to . Thus,the output power at each co-channel mixing product becomes

(12)

where is the frequency position of the desired mixing productaccording to Fig. 3. In this case, each contribution is accountedfor in a vector addition since the distortion products are all cor-related to each other.

C. Co-Channel Signal-Uncorrelated Mixing Products

Since we are assuming that there is no phase correlationbetween any of the input tones, the generation mechanism ofsignal-uncorrelated co-channel mixing products is similar tothe one responsible for the ones falling at the adjacent channel.

Indeed, as any input frequency combi-nation has a phase that is different from one ofany other combination (as long as is different from bothand ), the mixing products that fall at a certain will allbe uncorrelated. Therefore, the final result of the power at each

position is, on average, equivalent to the addition of each in-dividual component’ power. This implies that, while in (12) theoutput power was given as the square of the voltage sum,it must now be given as the sum of the squares of voltage

if

if vertice

if vertice if(13)

where

vertice (14)

and is a function that rounds the elements of to thenearest integers towards minus infinity.

Thus, if the overall output nonlinear distortion is to be cal-culated from a two-tone measurement, the procedure is the onepresented below. First, a two-tone measurement is made and thecorresponding is calculated for all values ofspanning from 0 to . This will then be stored into a



Fig. 5. Tone spacing contributions for the adjacent-channel power.

matrix. The next step involves the calculation offor all binomials at each position, and, using and , ob-taining the value of the function from the previousmatrices. The solution of the nonlinear distortion to a multisineexcitation then becomes straightforward.

IV. IMPACT OF MEMORY IN THE IMD OF MULTISINE SIGNALS

Here, we aim to analyze the impact of memory effects on theoutput of a nonlinear system presenting memory when drivenby a random multisine signal.

For performing this task, the formulation presented abovewill be very important since by accounting for the values ofand , we can infer information of what are the most importantcharacteristics of the baseband that impact the overall nonlineardistortion.

Beginning with the adjacent-channel scenario, we calculatethe number of times that each binomial functionappears for each . This result is visible in Fig. 5.

An interesting observation from this analysis is that the maincontribution to the adjacent-channel power is at the middle ofthe figure, which means that the binomial and

, is the most important term tobe considered in order to minimize the adjacent channel powerratio (ACPR) figure-of-merit [6].

The same analysis was done for the co-channel distortioncomponents, as is presented in Fig. 6. In this case, the mostimportant components, i.e., having stronger impact in theco-channel distortion, are the baseband components of lowerfrequency.

Considering now the correlated co-channel case (Fig. 7), wecan conclude that the most important components are again thebaseband components of lower frequency. Another interestingaspect is that all the correlated distortion components have atleast one dependence with zero tone spacing, corresponding tothe dc value of the baseband filter.

In order to validate these hypotheses a computer-aided design(CAD)/computer-aided engineering (CAE) simulation was runby using different baseband filters.

The operating bandwidth of 1.7 MHz was split into threebands with equal bandwidth, as shown in Fig. 8.

Fig. 6. Tone spacing contributions for the signal-uncorrelated co-channelpower.

Fig. 7. Tone spacing contributions for the signal-correlated co-channel power.

Fig. 8. Baseband filters for memory effect generation.

The three filters considered are: 1) a low-pass filter for thebaseband components; 2) a filter reinforcing the middle of thebaseband; and finally 3) a filter accounting for the remainingpart of the baseband.



Fig. 9. System output and predicted output for a baseband low-pass filter.

Fig. 10. System output and predicted output for the middle passband filter.

The input signal used in the simulation is an uncorrelatedmultisine of 100 tones. To simulate a system with memory, adirect implementation of the model of Fig. 1 was done.

Fig. 9 presents the obtained results for the multisine outputand the output obtained by our formulation. The ACPR andco-channel power ratio (CCPR) [6], were computed and the cor-responding values are 41.2 and 24.7 dB. A good agreementis visible at the spectral regrowth.

Fig. 10 presents the obtained results for a passband basebandfilter and with all the other configurations remaining unchanged.The obtained ACPR and CCPR results are 38.2 and 32.6 dB,respectively.

Finally, a last test was run with the higher frequency passbandfilter. In this case, the ACPR and CCPR results are 40.9 and

34.9 dB (Fig. 11).Based on previous observations, we can conclude that the

low-pass and higher band bandpass filters impose a similarACPR. On the contrary, the passband filter has a higher contri-bution to the adjacent distortion degrading the ACPR values.This validates our previous hypothesis based on the formulationdeveloped.

The same analysis was done for the co-channel distortion.In this case, the most important components are the ones withlower tone spacing, as can be seen in the lower CCPR values forthe low-pass filter, and a decrease for the other two subsequentfilters.

Fig. 11. System output and predicted output for the higher band bandpass filter.

Fig. 12. System output and predicted output for a memoryless system.

Fig. 12 is a reference test. It presents the value of the outputspectrum for a memoryless nonlinearity, which, compared tothe results presented in [8], guarantees that the formulation nowobtained is consistent with previous results.

In order to deeply test our technique, we have also simulateda system where the third-order direct path is near zero, and thesecond harmonic filter presents a flat, but complex, response.In this case, some spectral regrowth asymmetry is clearly vis-ible from Fig. 13, and a different dB and

dB is obtained.In all the previous results, we have showed superimposed

both the multisine output arising for an intensive CAD/CAEsimulation and the results arising from our proposed two-toneapproach; a remarkable match is visible.

In summary, our analysis points to several conclusions, whichare: 1) the middle frequency components of the baseband aredeterminant to the behavior of the adjacent-channel distortionand, thus, to the ACPR and 2) for the in-channel distortion, themost important baseband contributions are the ones arising fromcomponents located around dc.

V. EXPERIMENTAL RESULTS

An experimental test was also run in order to validate thetheory presented above. The setup comprises a low-power



Fig. 13. System output and predicted output for a system presenting notoriousasymmetry.

Fig. 14. Input impedance of the drain biasing network.

amplifier based on the ATF 55143 pseudomorphic HEMT(pHEMT) device biased in classes A and B.

A drain bias network was designed to present a varying base-band response in order to press memory effects in the outputresponse.

A memoryless network was also designed to prove the va-lidity of the presented algorithm in the standard memoryless sit-uation.

The test was run at a central frequency of 900 MHz with atone spacing of 20 kHz. The input signal is composed by 20tones leading to a 400-kHz bandwidth signal obtained from anarbitrary waveform generator (AWG). A record of 1000 wave-form segments with random phase was used as a random multi-sine signal to predict the output of the system to a narrowbandGaussian noise input. The measurement procedure comprisedthe synchronous acquisition of both the input and output signalsby a high-speed sampler and the post-processing of the data inorder to get system output [14].

In Fig. 14, we present the of the biasing network thatwas applied to the drain of the device. As can be seen, theimpedance of the memoryless bias is almost constant and closeto a short circuit. The bias presenting memory has varyingimpedance with a resonance in the middle of the band.

A first test was done in the class A operation with the biasnetwork that presents a varying baseband characteristic.

Fig. 15. Measured output and predicted response for the amplifier withmemory.

Fig. 16. Measured output and predicted response for the memoryless amplifier.

Fig. 17. Measured output and predicted response for the amplifier presentingmemory and asymmetry.

Fig. 15 presents the computed output distortion obtained byour method and the measured results. A good agreement canbe noticed between the computed values and measured output.The increasing error observed in the low-power tones was at-tributed to the fifth-order distortion that is always present in areal system, but was not accounted for in our third-order anal-ysis.

Fig. 16 presents the obtained results for the memoryless case.A good agreement can be noticed, even in this ideal case, indi-cating the applicability of the method.



To test the practical ability to predict asymmetry (strong ev-idence of long-term memory effects), the amplifier was biasedin class B. The output is presented in Fig. 17.

As can be seen, the predicted response and measured outputare in perfect agreement. The memory effects can also be no-ticed in the shape of the correlated and uncorrelated distortioncomponents falling in the co-channel band.

VI. CONCLUSIONS

This paper has presented an analytical methodology to com-pute the output distortion of a certain class of dynamic third-order systems to a multisine excitation using only two-tone tests.A qualitative explanation of the impact of the baseband filterin the co-channel and adjacent channel has also given, permit-ting a better bias network design for power amplifiers. The pre-sented methodology has been tested against simulated and lab-oratory experiments, demonstrating its validity even in the caseof adjacent-channel asymmetric responses. The computed re-sults enable the computation of multitone figures-of-merit asACPR and CCPR from only a small set of two-tone measure-ments, which are standard in every RF laboratory. This study isa step forward toward the understanding of the memory gener-ation mechanisms and in the extrapolation of the usual standardRF test results to the prediction of the dynamic system’s outputto a multisine signal excitation.

REFERENCES

[1] W. Bosch and G. Gatti, “Measurement and simulation of memory ef-fects in predistortion linearizers,” IEEE Trans. Microw. Theory Tech.,vol. 37, no. 12, pp. 1885–1890, Dec. 1989.

[2] K. J. Vuolevi and T. Rahkonen, Distortion in RF Power Amplifiers.Norwood, MA: Artech House, 2003.

[3] K. Remley, D. Williams, D. Schreurs, and J. Wood, “Simplifying andinterpreting two-tone measurements,” IEEE Trans. Microw. TheoryTech., vol. 52, no. 11, pp. 2576–2584, Nov. 2004.

[4] H. Ku and J. Kenney, “Behavioral modeling of nonlinear RF poweramplifiers considering memory effects,” IEEE Trans. Microw. TheoryTech., vol. 51, no. 12, pp. 2495–2504, Dec. 2003.

[5] N. B. de Carvalho and J. C. Pedro, “A comprehensive explanation ofdistortion sideband asymmetries,” IEEE Trans. Microw. Theory Tech.,vol. 50, no. 9, pp. 2090–2101, Sep. 2002.

[6] J. Pedro and N. Carvalho, Intermodulation Distortion in Microwaveand Wireless Circuits. Norwood, MA: Artech House, 2003.

[7] A. Soury, E. Ngoya, J. M. Nebus, and T. Reveyrand, “Measurementbased modeling of power amplifiers for reliable design of moderncommunication systems,” in IEEE MTT-S Int. Microw. Symp. Dig.,Philadelphia, PA, Jun. 2003, pp. 795–798.

[8] D. Schreurs, M. Myslinski, and K. A. Remley, “RF behaviouralmodelling from multisine measurements: Influence of excitationtype,” in 33th Eur. Microw. Conf., Munich, Germany, Sep. 2003, pp.1011–1014.

[9] J. Pedro and N. Carvalho, “On the use of multitone techniques for as-sessing RF components’ intermodulation distortion,” IEEE Trans. Mi-crow. Theory Tech., vol. 47, no. 12, pp. 2393–2402, Dec. 1999.

[10] N. Boulejfen, A. Harguem, and F. M. Ghannouchi, “New closed-formexpression for the prediction of multitone intermodulation distortionin fifth-order nonlinear RF circuits/systems,” IEEE Trans. Microw.Theory Tech., vol. 52, no. 1, pp. 121–132, Jan. 2004.

[11] J. P. Martins, N. B. Carvalho, and J. C. Pedro, “Multi-sine responseof third order nonlinear systems with memory based on two-tone mea-surements,” in 36th Eur. Microw. Conf., Manchester, U.K., Sep. 2006,pp. 263–268.

[12] J. C. Pedro, N. B. Carvalho, and P. M. Lavrador, “Modeling nonlinearbehavior of bandpass memoryless and dynamic systems,” in IEEEMTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp.2133–2136.

[13] A. Walker, M. Steer, K. Gard, and K. Gharaibeh, “Multi-slice behav-ioral model of RF systems and devices,” in Radio Wireless Conf., At-lanta, GA, Sep. 2004, pp. 71–74.

[14] J. C. Pedro and J. P. Martins, “Amplitude and phase characterizationof nonlinear mixing products,” IEEE Trans. Microw. Theory Tech., vol.52, no. 8, pp. 3237–3245, Aug. 2006.

[15] J. P. Martins, P. M. Cabral, N. B. Carvalho, and J. C. Pedro, “A metricfor the quantification of memory effects in power amplifiers,” IEEETrans. Microw. Theory Tech., vol. 54, no. 12, pp. 4432–4439, Dec.2006.

João Paulo Martins (S’06) was born in Sever doVouga, Portugal, on May 13, 1973. He received theB.Sc. and M.Sc. degrees from the Universidade deAveiro, Aveiro, Portugal, in 2001 and 2004, respec-tively, and is currently working toward the Ph.D. de-gree at in memory effects in nonlinear systems at theUniversidade de Aveiro.

From 2001 to 2005, he was a Researcher withthe Instituto de Telecomunicações, Universidade deAveiro. Since 2006, he has been with Chipidea Mi-croelectrónica, Lisbon, Portugal. His main research

interests are wireless systems and nonlinear microwave circuit design.

Nuno Borges Carvalho (S’92–M’00–SM’05), wasborn in Luanda, Angola, in 1972. He received theDiploma and Doctoral degrees in electronics andtelecommunications engineering from the Universi-dade de Aveiro, Aveiro, Portugal, in 1995 and 2000,respectively.

From 1997 to 2000, he was an Assistant Lecturerwith the Universidade de Aveiro, in 2000 a Professor,and is currently an Associate Professor. He is alsoa Senior Research Scientist with the Instituto deTelecomunicações, Universidade de Aveiro. He was

a Scientist Researcher with the Instituto de Telecomunicações, during whichtime he was engaged in different projects on nonlinear CAD and circuits and RFsystem integration. He coauthored Intermodulation in Microwave and WirelessCircuits (Artech House, 2003). He has been a reviewer for several magazines.His main research interests include CAD for nonlinear circuits, design of highlylinear RF-microwave power amplifiers (PAs), and measurement of nonlinearcircuits/systems.

Dr. Borges Carvalho is a member of the Portuguese Engineering Association.He is a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND

TECHNIQUES and an IEEE MTT-11 Technical Committee member. He was therecipient of the 1995 Universidade de Aveiro and the Portuguese EngineeringAssociation Prize for the best 1995 student at the Universidade de Aveiro, the1998 Student Paper Competition (third place) presented at the IEEE MicrowaveTheory and Techniques Society (IEEE MTT-S) International Microwave Sym-posium (IMS), and the 2000 Institution of Electrical Engineers (IEE), U.K.,Measurement Prize.

José Carlos Pedro (S’90–M’95–SM’99–F’07) wasborn in Espinho, Portugal, in 1962. He received theDiploma and Doctoral degrees in electronics andtelecommunications engineering from the Universi-dade de Aveiro, Aveiro, Portugal, in 1985 and 1993,respectively.

From 1985 to 1993, he was an Assistant Lecturerwith the Universidade de Aveiro, and a Professorsince 1993. He is currently a Senior ResearchScientist with the Instituto de Telecomunicações,Universidade de Aveiro, as well as a Full Professor.

He coauthored Intermodulation Distortion in Microwave and Wireless Circuits(Artech House, 2003) and has authored or coauthored several papers appearingin international journals and symposia. His main scientific interests includeactive device modeling and the analysis and design of various nonlinearmicrowave and opto-electronics circuits, in particular, the design of highlylinear multicarrier PAs and mixers.

Dr. Pedro is an associate editor for the IEEE TRANSACTIONS ON MICROWAVE

THEORY AND TECHNIQUES and is a reviewer for the IEEE Microwave Theoryand Techniques Society (IEEE MTT-S) International Microwave Symposium(IMS). He was the recipient of the 1993 Marconi Young Scientist Award andthe 2000 Institution of Electrical Engineers (IEE) Measurement Prize.


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