Enhancing Radio Frequency System Performance by Digital ...

Thesis for the degree of Licentiate of Engineering

Enhancing Radio Frequency System Performance byDigital Signal Processing

Charles Nader

Signal ProcessingSchool of Electrical Engineering

KTH (Royal Institute of Technology)

Stockholm 2010

Nader, CharlesEnhancing Radio Frequency System Performance by Digital Signal Processing

Copyright c©2010 Charles Nader except whereotherwise stated. All rights reserved.

TRITA-EE 2010:030ISSN 1653-5146

Signal ProcessingSchool of Eletrical EngineeringKTH (Royal Institute of Technology)SE-100 44 Stockholm, SwedenTelephone + 46 (0)26-648 850

Abstract

In this thesis measurement systems for the purpose of characterization of radio frequency power ampli-fiers are studied. Methods to increase the speed, accuracy, bandwidth, as well as to reduce the samplingrequirements and testing cost are presented. A method intended for signal shaping with respect to peak-to-average ratio reduction and its effects-improvements on the radio frequency front-end performanceis investigated.

A time domain measurement system intended for fast and accurate measurements and characteri-zation of radio frequency power amplifiers is discussed. An automated, fast and accurate technique forpower and frequency sweep measurements is presented. Multidimensional representation of measuredfigure of merits is evaluated for its importance on the production-testing phase of power amplifiers.

A technique to extend the digital bandwidth of a measurementsystem is discussed. It is based onthe Zhu-Frank generalized sampling theorem which decreases the requirements on the sampling rateof the measurement system. Its application for power amplifiers behavioral modeling is discussed andevaluated experimentally.

A general method for designing multitone for the purpose of out-of-band characterization of non-linear radio frequency modules using harmonic sampling is presented. It has an application with thevalidation of power amplifiers behavioral models in their out-of-band frequency spectral support whenextracted from undersampled data.

A method for unfolding the frequency spectrum of undersampled wideband signals is presented.It is of high relevance to state-of-the-art radio frequencymeasurement systems which capture repetitivewaveform based on a sampling rate that violates the Nyquist constraint. The method is presented ina compact form, it eliminates ambiguities caused by folded frequency spectra standing outside theNyquist band, and is relevant for calibration matters.

A convex optimization reduction-based method of peaks-to-average ratio of orthogonal frequencydivision multiplexing signals is presented and experimentally validated for a wireless local area networksystem. Improvements on the radio frequency power amplifierlevel are investigated with respect topower added efficiency, output power, in-band and out-of-band errors. The influence of the powerdistribution in the excitation signal on power amplifier performance was evaluated.

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Sammanfattning

I denna avhandling kommer mätsystem för karakterisering avhög frekvens förstärkare att behandlas.Metoder för att öka hastigheten, noggrannheten, bandbredden, minska samplings hastighet och reduc-erade test kostnader presenteras. En metod för signal formning med avsikt på peak-to-average och dessbeteende förbättring för prestandan på högfrekvens front-end är undersökt.

Ett tidsdomän mätsystem ämnat för snabba och noggranna mätningar och karakterisering avhögfrekvens effektförstärkare diskuteras. En automatiserad, snabb och noggrann mätmetod för effektoch frekvenssvep är presenterad. En fler dimensionell representation av parametrar för effektförstärkareär utvärderad för produktions testning.

En metod för att öka mätsystemets bandbred är diskuterad. Den är baserad på Zhu-Frank gen-eraliserad samplings teorem som minska samplingshastigheten för mätsystemet. Dess tillämpning påeffektförstärkare och dess beteende modulering är diskuterad och praktiskt utvärderad.

En generell metod för att skapa multiton signaler för karakterisering av icke linjariteter utanförkanalen för radio moduler baserat på undersampling är presenterad. Den kan tillämpas vi valideringav effektförstärkares beteende modeller för utanför kanalfrekvens spektrum vid när undersamplinganvänds.

En metod för återskapning av frekvensspektrum vid undersampling av bredbandssignaler är pre-senterad. Detta är viktigt för hög prestanda högfrekvens mätsystem som samplar en repetitiv signal sominte uppfyller Nyquist kriterium. Metoden presenteras i påett enkelt sätt att arbeta med. Den eliminerarfelaktigheter som uppkommer vid vikning av frekvenskomponenter som befinner sig ovanför Nyquistfrekvensen. Detta är relevant för kalibrering av mätsystemet.

En konvex optimerings metod för att reducera peak-to-average förhållande av orthogonal fre-quency division multiplexing signaler är presenterad. Denär experimentellt validerad för trådlöst LAN.Förbättringar på högfrekvens effektförstärkare är undersökta med avseende på effektivitet, uteffekt, ikanalen och utanför kanalen fel. Påverkan av effekt fördelningen av insignal till effektförstärkarensprestanda har utvärderats

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Acknowledgements

First and foremost, my deepest gratitude goes to my supervisors Professor PeterHändel from the Royal Institute of Technology, and Dr. Niclas Björsell from theUniversity of Gävle, for their relevant guidance, creativeideas and tremendoussupport. Special thanks also go to Dr. Wendy Van Moer from Vrije UniversiteitBrussel and Dr. Olof Bengtsson from Ferdinand Braun Institute, for our inspiringdiscussions.

I would like to thank Dr. Niclas Keskitalo from Ericsson AB for being a sup-portive project manager, as well as the funders of my research: Ericsson AB,Freescale Semiconductor Nordic AB, Infineon Technologies Nordic AB, Knowl-edge Foundation, NOTE AB, Rohde & Schwarz AB and Syntronic AB

I would like to thank my colleagues at the RF center for measurement tech-nology in Gävle and TB/Electronics department at the University of Gävle forintroducing some fun in my last two years of research and making the place asa friendly research environment. Special thanks to Javier Ferrer Coll, SathyaveerPrasad, Carl Elofsson, Carl Karlsson, Helena Eriksson for their pleasant humor,and Per Landin for our research discussions-argues. I wouldlike to thank also mycolleagues in the ”4th floor” at the signal processing Lab-Royal Institute of Tech-nology for making my visits interesting and enjoyable. Special thanks to SamerMedawar for being a good help in Latex.

Last but not least, gratitude goes to my family for believingin me, support-ing me, and motivating me for a better outcome: my father Tonyand my motherGeorgette, my sisters Claudia and Roula, and my brother Joseph.

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Contents

Abstract i

Sammanfattning iii

Acknowledgements v

Contents vii

I Introduction 1

Introduction 11 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . 42 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Concluding remarks and future research . . . . . . . . . . . . . . 8

II Included papers 11

A Automated Multidimensional Characterization of Power Amplifierfor Design and Production A11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A12 Measurement system . . . . . . . . . . . . . . . . . . . . . . . . A2

2.1 Test-bed . . . . . . . . . . . . . . . . . . . . . . . . . . . A22.2 Measurement method . . . . . . . . . . . . . . . . . . . . A32.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . A5

3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A64 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A8

4.1 Measurement Speed and Accuracy . . . . . . . . . . . . . A94.2 Faster Approach . . . . . . . . . . . . . . . . . . . . . . A10

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A11

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B Wideband Characterization of Power Amplifiers Using Undersam-pling B11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B12 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B23 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . B24 Experimental and Results . . . . . . . . . . . . . . . . . . . . . . B35 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . B4

5.1 Multitone . . . . . . . . . . . . . . . . . . . . . . . . . . B55.2 Spectrum Scan . . . . . . . . . . . . . . . . . . . . . . . B6

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9

C Multi-tone design for out-of-band characterization of nonlinear RFmodules using harmonic sampling C11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C12 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C23 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . C44 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C65 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C8

D Unfolding the Frequency Spectrum for Undersampled WidebandData D11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D12 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . D33 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D54 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D8

E Peak-to-Average Power Reduction of OFDM Signals by ConvexOpti-mization: Experimental Validation and Performance Optimization E11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E12 OFDM PAR reduction by convex optimization . . . . . . . . . . . E2

2.1 PAR reduction by convex optimization . . . . . . . . . . E22.2 Channel leakage and an extended method . . . . . . . . . E42.3 Algorithmic details . . . . . . . . . . . . . . . . . . . . . E4

3 Measurement setup and device under test . . . . . . . . . . . . . E53.1 Measurement set-up . . . . . . . . . . . . . . . . . . . . E53.2 Device under test . . . . . . . . . . . . . . . . . . . . . . E6

4 Results and Evaluation . . . . . . . . . . . . . . . . . . . . . . . E64.1 Power added efficiency . . . . . . . . . . . . . . . . . . . E64.2 In-band errors . . . . . . . . . . . . . . . . . . . . . . . . E74.3 Spectral mask and out-of-band errors . . . . . . . . . . . E94.4 Amplifier saturation . . . . . . . . . . . . . . . . . . . . E12

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5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E16

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Part I

Introduction

Introduction

This thesis is devoted to the subject of radio frequency (RF)measurement systemsfor testing and characterization of nonlinear microwave products, with a focus onRF power amplifiers (PAs). It also investigates shaping of signals for improvingoverall RF system performance.

RF measurement systems occupy a hot spot in the instrumentation and mea-surement society, as they are the key player in measuring transmitted signals, ex-tracting information data, and testing/evaluating developed microwave productsfrom a simple cable to PAs by the use of repetitive excitationscenarios. Such mi-crowave products, mainly the PAs, are classified as nonlinear devices, and havea large number of tradeoffs between different parameters that are highly relevantfor the overall RF system performance. Characterizing accurately those param-eters in the production-testing phase is crucial for an errorless system operation;however it should be performed in a fast and costless way. Thefirst part of thisthesis presents a test setup solution for an automated, accurate, fast, and multidi-mensional characterization of PAs.

With wireless communication systems moving towards wideband signals, from3GPP long term evolution (LTE) with scalable bandwidths up to 20MHz, world-wide interoperability for microwave access (WiMAX) supporting bandwidthsup to 28MHz, International Mobile Telecommunications (IMT) advanced usingbroader band signals, to ultra-wide band (UWB) signals surpassing the 500MHzfrequency space, high interest is put on characterizing thenonlinear behavior ofthe PA. In fact, combining such wideband signals with the input-output nonlinearbehavior of PAs, the spectral support of its output does not only cover the spectralsupport of the input, but also the adjacent channels, spreading over bandwidthsin the scale of hundreds of Megahertz. Measuring such wideband output signalsand characterizing the input-output relation of the deviceunder test (DUT) is dif-ficult to achieve by normal instrumentation and measurementsystems, as largeanalog baseband bandwidth with high speed analog to digitalconverters (ADCs)are required. One possible solution to characterize such nonlinear behavior is touse undersampled data by violating the Nyquist-Shannon sampling theorem, asdescribed by Zhu-Frank generalized sampling theorem (ZFGST). In short, Zhu-Franks’s work implies that if there is a static invertable function that compressesthe spectral support of an analog signal, it is sufficient to sample it with a speed

2INTRODUCTION

corresponding to twice the bandwidth of the compressed signal. In other words,for modeling purpose, it is enough to sample the output of thePA with a samplingrate twice its input excitation bandwidth. As ZFGST does only decrease the re-quirements on sampling rate and not the analogue bandwidth of the system, onecannot rely on the vector signal analyzers on the market today since the interme-diate frequency (IF) bandwidth preceding the ADC sampler front-end is limited toincrease the dynamic range by avoiding noise folding from the broadband noise.Thus, a specially designed test system that downconverts data from RF to an in-termediate frequency (IF) scale followed by a 12-bits pipeline analogue to digitalconverter (ADC) was used. Wideband characterization of thebehavioral modelof PAs based on ZFGST and the specially designed test setup isthe focus of thesecond part of this thesis.

While extracting behavioral model parameters is of high interest, validatingthem is of extreme importance, certainly in the out-of-bandspectral region whereneighbor users are usually operating. Adjacent channel error power ratio (ACEPR)is the figure of merit judged to be the best fit for such validation process. ACEPRstands for the ratio between the spectral power leaking to adjacent channels, out-of-band, compared to the power in-band. A major problem thatfronts such evalu-ation is the frequency spectral overlapping caused by aliasing of frequency tonesstanding outside the Nyquist band, that is half the samplingrate. Due to that, tech-niques for extracting original information from their overlapped part are of highinterest.

Two techniques that can be applied for the out-of-band validation process arepresented in the third and fourth part of the thesis respectively. The third part eval-uates the generation of a multitone set for characterizing the behavior of nonlinearradio frequency (RF) modules in its out-of-band when harmonic sampling is usedas a digitizing technique. It provides the reader with a toolto select proper fre-quencies and record length for a given application and test-bed. Such techniqueis highly relevant for improving the performance of previous stat-of-art nonlineartest equipment.

In the fourth part, an attention is given for reconstructingthe frequency spec-trum of an undersampled waveform, which might be useful for validating the out-of-band nonlinearities of PAs, and is of relevance to RF measurement systems thatcapture repetitive waveform based on a sampling rate that violates the Nyquistconstraint. The unfolding problem is solved by a frequency based approach basedon local oscillator stepping at the RF stage. The problem is presented in a com-pact form by the inclusion of a complex operator called the CNoperator. Theease-of-use problem formulation eliminates the ambiguitycaused by folded fre-quency spectra, in particular those with lines standing on multiples of the Nyquistfrequency that are captured with erroneous amplitude and phase values. The pre-sented approach is highlighted by a formalism for handling issues such as calibra-tion.

The last part of this thesis is dedicated for shaping signalsused in wireless sys-tems, mainly orthogonal frequency division multiplexing (OFDM) based systems,

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and evaluating the impact and improvements on the RF front-end.OFDM is a widely used modulation scheme because of its high bandwidth

efficiency and robustness against frequency fading due to multipath propagation.It is adopted in many Wireless systems from Wireless local area network (WLAN),to WiMAX, LTE, and their derivations. However, a major drawback of OFDM isthe generally high peak to average ratio (PAR) of the RF signal entering the PA,which causes early clipping of the signal due to amplifier saturation and results innonlinear distortions presented in the frequency domain onthe shape of unwantedintermodulation products, spectral regrowth, and harmonics, which cause spectralinterference to neighbor channels. Due to that, the input power of the amplifierhas to be reduced; that is, a large number of dBs have to be backed-off to keepthe amplifier in linear operation. Such a back-off drastically reduces the poweradded efficiency (PAE) of the amplifier because a large amountof power (i.e.,heat) must be dissipated. Due to that, a strong need rises forreducing the PARof OFDM signals prior to the conversion to RF. One interesting method/algorithmis based on the merging convex optimization technique. The method reshapes thepower spectral density of the signal by minimizing the time domain peak power,subject to some constraints on the error vector magnitude (EVM), and the spectralpower of the signal. A study and experimental evaluation of that PAR optimizationmethod, applied to a WLAN system, is presented in the last part of this thesis.

That was a short introduction to the work investigated in this thesis. It is pre-sented in five papers.

Paper A: C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keski-talo and H. De La Rosa, "Automated Multidimensional Characterizationof Power Amplifier for Design and Production", inProceedings IEEEInternational Instrumentation and Measurement Technology conference,I2MTC2009, pp. 144-148, Singapore, May 2009.

Paper B: P. Landin, C. Nader, N. Björsell, M. Isaksson, D. Wisell, P. Händel,O. Andersen and N. Keskitalo, "Wideband Characterization of Power Am-plifiers Using Undersampling", inProceedings IEEE MTT-S InternationalMicrowave Symposium, IMS 2009,pp. 1365-1368, Boston, June 2009.

Paper C: N. Björsell, C. Nader and P. Händel, "Multi-tone design for out-of-band characterization of nonlinear RF modules using harmonic sampling",to appear inProceedings IEEE International Instrumentation and Measure-ment Technology conference,I2MTC2010, Austin, Texas, United States ofAmerica, May 2010.

Paper D: C. Nader, N. Björsell and P. Händel, "Unfolding the Frequency Spec-trum for Undersampled Wideband Data",EURASIP Journal on Signal Pro-cessing: Fast Communication, submitted, June 2010.

Paper E: C. Nader, P. Händel and N. Björsell, "Peak-to-Average PowerReduc-tion of OFDM Signals by Convex Optimization: Experimental Validation

4INTRODUCTION

and Performance Optimization",IEEE Transactions on Instrumentation andMeasurements,doi: 10.1109/TIM.2010.2050360, 2010.

1 Contributions of the Thesis

The main contributions of this thesis are presented in five scientific papers. PaperA is devoted for the design of a fast and accurate measurementsetup to be usedin the production and testing phase of power amplifiers. Due to the high interestin studying the behavioral model of power amplifier with the rising use of broad-band signals, mainly in its nonlinear aspect, Paper B evaluates the extraction of themodel parameters based on the sub-sampling theory of Zhu-Frank. Paper C andD presents two alternative approach that can be used for the out-of-band perfor-mance validation of the model extracted in Paper B. Paper C presents a guide fordesigning general set of multitone that and their intermodulation products don’toverlap when undersampled. While Paper D presents a frequency based approachfor unfolding the frequency spectrum of wideband undersampled waveforms. Itintroduces a compact formulation of the unfolding problem and solves major am-biguities caused by aliasing. Finally, Paper E is devoted for signal shaping ofOFDM based system, e.g. WLAN 802.11a, using a technique based on convexoptimization for reducing the signal peak-to-average power ratio, and its impacton the radio frequency power amplifier performance.

The papers are summarized in the following sections.

Paper A: Automated Multidimensional Characterization of Power Amplifierfor Design and Production

Power amplifier (PA) is a key component in a wireless communication chain as itholds the highest power level in the system. Characterizingits behavior and testingits performance is highly relevant for the overall system performance. Designing,optimizing and producing modern PAs requires new and fast RF(radio frequency)measurement techniques capable of characterizing its realbehavior with a lowproduction-test cost. This paper presents a software-defined measurement setupfor fast and cost efficient multidimensional measurements based on highly accu-rate standard instruments and a PC. A fast approach in measurement is presented,where sweeping both frequency and power in a sequence of series stairs is gener-ated in the baseband excitation waveform. It shows a dramatic reduction in timeconsumption of the measurements. The measurement system offers the possibilityto monitor envelope-tracking dynamic power consumption upto 100 MHz plus thepossibility to use high crest factors, which is highly relevant for the new generationof signals.

1 CONTRIBUTIONS OF THETHESIS5

Paper B: Wideband Characterization of Power Amplifiers Using Undersam-pling

Wireless communication systems are adopting broad-band signals with band-widths varying up to100 MHz. With the nonlinear input-output behavior of thepower amplifier (PA), the spectral support of its output spread over hundreds ofMegahertz. Such wide spectrum is difficult to characterize with current measure-ment systems as a sampling frequency ”at least” twice the order of the spectrumbandwidth is required, which is difficult to achieve with thecurrent generation ofADCs based on acceptable dynamic range, resolution, and cost. In this paper thegeneralized Zhu-Frank sub-sampling theorem is tested for the purpose of PA be-havioral modeling. A stat-of-the-art test system has been designed for the purpose.By varying the bandwidths of the excitation signal, behavioral model parameterssuch as nonlinear order and memory depth are investigated. For the wider signals,the normal cross-correlation based synchronization was found to be no longer suf-ficient for extracting the optimal linear FIR-filter in the model. Model validationwas considered to be a challenging problem within the proposed concept due toaliasing effect. Two different methods for model validation are proposed, usingspecial-designed multitone and frequency spectrum reconstruction based on localoscillator stepping.

Paper C: Multi-tone design for out-of-band characterization of nonlinear RFmodules using harmonic sampling

Due to the interest in validating the performance of power amplifier behavioralmodel extracted based on sub-sampling theory, as describedin the previous paper,a general method for designing multitone signals for that purpose was investi-gated. By generating a set of tones which and their intermodulation products don’toverlap when undersampled, we allow the validation of the out-of-band behavioralmodel performance by evaluating the adjacent channel powerratio. The purpose isto provide the reader with a tool to select proper frequencies and record length for agiven application and test-bed. The method is based on simulations and the use ofSidon sequences. The proposed method is applicable to sparse discrete frequencymulti-tones.

Paper D: Unfolding the Frequency Spectrum for UndersampledWidebandData

The problem of unfolding the frequency spectrum for undersampled widebanddata is discussed in the paper. The problem is of relevance tostate-of-the-art radiofrequency measurement systems, which capture repetitive waveform based on asampling rate that violates the Nyquist constraint. The problem is presented in acompact form by the inclusion of a complex operator called the CN operator. Theease-of-use problem formulation eliminates the ambiguitycaused by folded fre-quency spectra, in particular those with lines standing on multiples of the Nyquist

6INTRODUCTION

frequency that are captured with erroneous amplitude and phase values. The prob-lem formulation is relevant for calibration matters.

Paper E: Peak-to-Average Power Reduction of OFDM Signals byConvex Op-timization: Experimental Validation and Performance Opti mization

In this paper the application of convex optimization to peak-to-average power re-duction on an orthogonal frequency division multiplexing (OFDM) 802.11a sig-nal is evaluated. A radio frequency power amplifier was excited with an OFDM-signal, and the peak-to-average reduced counterpart and its performance figureof merits were measured and compared. Figure of merits such as output power,power added efficiency, in-band and out-of-band errors, spectral emission and theinfluence of the power distribution in the excitation signalon power amplifier per-formance were investigated. Improvements of 6dB in output power and 6.5% inpower added efficiency were achieved on average near the operating region. Theeffect of preserving power-free guard subcarriers was introduced in the optimiza-tion algorithm and investigated regarding adjacent channel interference. An im-provement of 9dB from that aspect was observed using half of the power-freesubcarriers, which reveals the importance of a guard interval.

2 RELATED WORK7

2 Related work

Parts of the enclosed material have been presented at

Paper F: C. Nader, P. Landin, N. Björsell, M. Isaksson, D. Wisell, P. Händel,O. Andersen and N. Keskitalo, “Wideband Power Amplifiers Characteriza-tion by Undersampling: Zhu-Frank Sampling Theorem,” presented atRa-dio Frequency Measurement Technology Conference, RFMTC 2009, Gävle,Sweden, October 2009.

Paper G: C. Nader, P. Händel and N. Björsell, “OFDM PAPR Reduction by Con-vex Optimization: A Power Amplifier Point-of-View,” to appear inProceed-ings IEEE International Microwave Workshop Series, IMWS 2010, Aveiro,Portugal, February 2010.

8INTRODUCTION

3 Concluding remarks and future research

This thesis presented an interdisciplinary research work in the field of radio fre-quency measurement systems. Signal processing was used as an enhancementinterface for improving measurement systems merits such asthe speed, accuracy,bandwidth, as well as designing special excitation signalsfor power amplifierstesting purpose, reconstructing undersampled wideband signals, and improving thepower amplifiers power performance by a stat-of-art method for peak-to-averagesignal reduction.

• By introducing a new approach for performing power and frequency sweepcarried in the baseband signal, testing speed was improved from the scale ofhours to minutes/seconds. In addition, measuring the current dissipation en-velope through highly accurate current sensor probes, allows the implemen-tation of power added efficiency improvement techniques such as envelopetracking, or envelope elimination and restoration.

• Wideband signal are always an issue for measurement and identification,certainly when inputting nonlinear modules. By using the Zhu-Frank gen-eralized sampling theorem, requirements on the sampling frequency wasreduced in the order of 5-10, alternatively the digital bandwidth of the ra-dio frequency measurement setup increased by the same order. Applyingthe Zhu-Frank sampling approach as a data extraction tool for the purposeof behavioral modeling of nonlinear power amplifiers resulted in a modelcharacteristics and performance with adequate values.

• Measuring the nonlinear behavior of radio frequency modules is of high rel-evance for the characterization process. Due to the large bandwidth overwhich the frequency response of the nonlinear behavior can spread, currentanalogue to digital converters are incapable for collecting such informationdue to their limitation in sampling speed, and sometime analogue band-width, which results in aliasing phenomena. Designing of special set ofmultitone for which, the fundamental tones and their nonlinear derivationsdon’t overlap when undersampled, allows the out-of-band characterizationof nonlinear radio frequency modules based on harmonic sampling. In thethesis, the method limitation was the large number of samples required forlarge number of tones and high nonlinear order values. A continuation ofthis work would be to adjust the presented procedure to reduce the strongdemands on the number of samples, and to validate the behavioral mod-els extracted in the second section of this thesis by calculating its adjacentchannel error power ratio.

• With wireless communication systems adopting more broadband signals andwhile measurement equipments weakened on the baseband digital part bylimited analogue to digital converters, new techniques forcollecting data

3 CONCLUDING REMARKS AND FUTURE RESEARCH9

are required. Downconverting the radio frequency signal toan intermediatestage, undersampling the signal, and applying reconstruction algorithms al-low the perfect reconstruction of the wideband data. This thesis contributedto a compact formulation of the problem which resolved ambiguities ris-ing from aliased components, especially those standing on multiples of theNyquist frequency. A continuation of this work would be to validate thereconstruction method on a measurement system. Calibration will be an is-sue that can be easily handled with the compact formulation.An importantapplication of the work would be to validate the out-of-bandperformance ofmodels extracted based on section two of this thesis.

• OFDM signals are being adopted in most current and future communicationsystems. Their peak-to-average signal characteristics need to be reduceddue to its impact on the power performance of the power amplifier stage.Evaluation of an OFDM signal peak-to-average reduction method based onconvex optimization and its experimental validation with respect to poweramplifier performance improvements was presented in this thesis. Interest-ing improvements in power added efficiency, output power wasobserved.An interesting continuation of this work would be to apply the algorithmfor adaptive modulation signals where the sub-carriers allocation changesbetween symbols, e.g. DVB-T2 based systems.

Lot of work have been done for improving the performance of radio frequencymeasurement systems; lot still remain to be done. Digitallyprocessing data isevolving as a pervasive tool for performance enhancement. Moving the digital partat the radio frequency front-end to the border of the antennais the focus of futuresystems. By doing so, all non-idealities generated from mixing effect, downcon-version, imbalance effect will be eliminated. More requirement will be on theanalogue-to-digital part as higher bandwidth and higher sampling rate will be re-quired. Designing new signal reconstruction algorithms for those ultra widebanddirect radio frequency sampling based systems is an interesting future researchwork to be done.

Due to the widenband characteristics of future systems, andthe inefficient useof the spectrum, sparsity property can be used to reduce the constraint on theanalog-to-information stage. By using the emerging compressive sampling tech-nique, wideband signals requiring ”currently” unachievable sampling rate can bereconstructed from a relatively small amount of information samples. Evaluatingthe application of compressive sampling for application onradio frequency mea-surement systems is a highly interesting future research work due to its impact onwideband receivers and application on cognative radios andsmart receivers.

Part II

Included papers

Paper A

Automated Multidimensional Characterization of PowerAmplifier for Design and Production

Charles Nader, Hibah Altahir, Olav Andersen, Niclas Björsell, Edith Condo,Niclas Keskitalo and Hector de la Rosa

in Proceedings IEEE International Instrumentation and Measurement Technologyconference,I2MTC 2009,pp. 144-148, Singapore, May 2009

c©2009 IEEE

1 INTRODUCTION A1

Abstract

Designing, optimizing and producing modern power amplifiers (PA) requiresnew and fast RF (radio frequency) measurement techniques capable of charac-terizing its real behavior. Power amplifiers are a truly multidimensional devicewhere many desired performance parameters are contradictory to each other. Thisis especially true for the generation of modern communication PAs that requirehigh efficiency, high linearity as well as high bandwidth. This paper presents asoftware-defined measurement setup for fast and cost efficient multidimensionalmeasurements based on highly accurate standard instruments and a PC. The testbed as well as the graphical user interface is presented along with a demonstra-tion of its functionality. During tuning of tank networks, drain quiescent current,and bias conditions, 3-dimensional graphs can be selected for the most appropri-ate axes of trade-off parameters to display a true behavior of the PA under testsubjected to real-world or close to real-world signals. Themeasurement systemoffers the possibility to monitor envelope-tracking dynamic power consumption upto 100 MHz plus the possibility to use high crest factors.

1 Introduction

Designers strive to develop a power amplifier (PA) (or a system including a PA)with the best properties for the application. It is well known that a PA workingwith modern communication signals have a large number of tradeoffs betweendifferent parameters. As the demands on high performance PAs continue to grow,novel measurement algorithms are needed as enabler for efficient design, testing,and characterization of future power devices. A multidimensional representationof the PA´s figure of merits is a crucial tool for tuning in on the best performancetradeoffs and selecting the optimum operating point; that is the drain quiescentcurrent,IDQ, the frequency range and the power range near compression. ModernPAs dealing with multiple carrier signals at high bandwidth, high crest-factors aswell as high linearity are truly multidimensional devices.Traditionally, these per-formance parameters are presented in tables and two-dimensional plots. Optimiz-ing the performance in the multidimensional device based onthis two-dimensionalinformation is both difficult and time consuming. Thus, multidimensional charac-terizations offer unique possibilities to make tradeoffs in several dimensions, i.e.superior design and production advantages. To our knowledge, only [1, 2] haveconsidered this need from the test engineers and designers point of view. In addi-tion, [3, 4] characterize multidimensional models of PAs for linearization by usingpre-distortion techniques. The models techniques given in[3, 4] are used for ex-isting PAs and not aimed for designing, tuning and production test of PAs. Inthis paper we suggest a software defined measurement (SDM) method to measureand present critical parameters such as distortion, gain, and efficiency in three-

A2AUTOMATED MULTIDIMENSIONAL CHARACTERIZATION OF POWER AMPLIFIER FOR DESIGN AND

PRODUCTION

AWGDAC

DAC

900

LO

Signal GeneratorI

QRF

DUT

Controllable

Power Supply

ADC

Signal

Analyser

FSQ (R&S)

Hall Sensor

Current ProbeOscilloscope

PC

Matlab

I Q

SMU (R&S)

DPA

Figure 1: The measurement setup.

dimensional plots in order to offer a design view in several dimension and multi-dimensional tuning in production. In fact, presenting the above three merits in onesingle plot will allow designers to optimize the operating point of the PA based onthe best suitable combination for the application. The measurement set up offerslarge possibilities compared to traditional methods.

2 Measurement system

Major challenges in designing an automated test system are accuracy, speed andcost. Thus, the used instruments are highly accurate standard instruments, withoutany special designed equipment. All instruments are connected to a computer thatcontrols the measurements, process the measured data and presents the result. Thatis, a software-defined measurement setup.

2.1 Test-bed

The test-bed is based on a vector signal generator (VSG), R&SSMU200A, a vectorsignal analyzer (VSA), R&S FSQ26, an oscilloscope, Agilent54610B, a currentprobe, Agilent N2783A, a driver amplifier, Ericsson 3G, a controllable power sup-ply, Agilent E3631A and a personal computer (PC), see Fig. 1.The instrumentsare connected to the PC via LAN and/or via GPIB interface. Such instruments, orsimilar to, are mainly found in most of RF laboratories whichdoesn’t impose anyadditional hardware cost.

2 MEASUREMENT SYSTEM A3

Figure 2: Envelope of a power sweep sequence.

2.2 Measurement method

A baseband signal is generated in the PC. The signal is a consecutive baseband-sequence of test scenarios, where the amplitude and frequency are increased step-wise. By using coherent sampling, it was assured that the aimed signal power inthe spectrum is contained within the relative tone respectively [5]. The signal isthen downloaded to an arbitrary waveform generator (AWG) inthe VSG, where itis up-converted to radio frequencies (RF). The main advantage of using an AWGis that we can perform several test scenarios in one measurement, and thus we im-prove the speed of the characterization. Fig. 2 shows a sequence of15 power stepsfor a two-Tone signal.

In order to have an accurate and stable power stepping, the total RMS power ofthe carrier should be found. A power formula was derived thatlinks between thetotal RMS power of the carrier,RMST , the highest RMS power step in channel,HSRMS , and the crest factor of the full steps sequence,CFT . Starting from thegeneral relation betweenCFT , RMST and peak power,PeakT , in a logarithmicbasis, as in

RMST = PeakT − CFT , (1)

PeakT can be developed to representHSRMS added to the CF of one single step,


PRODUCTION

CFS , as inPeakT = HSRMS + CFS . (2)

For a multitone system withN tones,HSRMS represents the highest RMS oftotal tones that should be reached in the sweep. For a Wideband Code DivisionMultiple Access (WCDMA) systemHSRMS represents the highest step of totalpower in channel that should be reached in the sweep.

To accurately monitor the current vector of the drain supplya high bandwidthhall sensor current probe is used. Drain supply current to the PA is then measuredby an oscilloscope; by this method, envelope-tracking dynamic power consump-tion can be accurately measured up to about100 MHz. The output power fromthe PA is measured by the VSA, which can be accurately calibrated to an absolutepower sensing power meter to obtain±0.35 dB accuracy, even for modulated timevariant signals.

In order to provide a design tool to select the optimumIDQ, a controllablepower supply is used to sweep the gate bias of the device undertest (DUT) at apercentage of±x%. A look-up table is used to transfer the bias voltage to theIDQ.Stepping bias voltage is also used to study the performance of the PA for differentclasses of operation, identify its sweet spots and their stability with the operatingclass.

The measured data is sent to the PC, where the data sequences are synchronizedin time and frequency domain and each test scenario (amplitude and frequency) isseparated and processed individually [5]. A requirement for that is the use of timerepetitive signals.

One of the main key points in measurement systems is the synchronizationbetween input/output signals, and also between output signal and relative currentmeasurements. The in-phase (I) and quadrature (Q) data on a Nyquist scale arecollected from the VSA. The synchronization is done in two steps. The first stepis done in the time domain, where cross-correlation is used to synchronize datato one-sample accuracy. The second step is done in the frequency domain, whereinterpolations in the phase information are used to identify the exact data sequencerelative to each measurement step.

Synchronizing the current measurements with relative signal sequence is morecomplex due to difference in sampling frequency between theoscilloscope and theVSG and existence of current glitches in the beginning of a sequence. A decou-pler is used on the drain side to eliminate unwanted fluctuations due to suddenshort in the power supply. The current trace is collected from the oscilloscopethrough GPIB cable. Moreover, a fifth order Butterworth low-pass digital filteris used to reduce measurement noise from data. The oscilloscope floor noise issubtracted from the filtered data and to the result a sign function implemented inthe software is applied which gives positive pulses (+1) for positive samples andnegative pulses (−1) for negative ones. The outcome is then differentiated in aseries form which will result in negative pulses (−2) when the power goes fromthe highest step to the lowest one and a positive pulses when the power steps up its

2 MEASUREMENT SYSTEM A5

Figure 3: Current Detection and Synchronization based on digital filtering andsecond order differentiation.

first level. By choosing any two consecutive negative pulses, the period of currentsequence is detected. A demonstration of current synchronization based on slopesidentification is presented in Fig. 3.

Another method for current measurements would be to resample the collecteddata from the oscilloscope in correspondence to the power sequence length, andthen apply the same synchronization procedure used for power measurements. Thedrawback is the loss of speed in the processing due to resampling.

2.3 Visualization

The result is presented in a 3D plot where gain, efficiency anddistortion can bethe three axes. A graphical user interface (GUI) has been designed in order toshow the multidimensional plots, but also for parameter settings, see Fig. 4 andFig. 5. The settings are theIDQ, frequency and input power ranges. A marker canbe used to get numerical data at a specific point in the plot. The marker presentsthe three parameters but also the test conditions (power andfrequency) for thatspecific measurement. Moreover, the GUI can also present data in the traditionaltwo-dimensional plots.

To facilitate the possibility to tune the PA properly and select the most suitable


PRODUCTION

Figure 4: Three-dimensional plot of gain, efficiency and distortion measured ona high power amplifier using a two-tone input signal.

quiescent point, a similar to "live update" of the plots is accessible. The GUI hasslide bars, where theIDQ or the gate voltage (Vg) can be altered and the plotsupdate the performance of the PA simultaneously.

3 Results

The system performance for different power levels and frequencies are shown inFig. 6 and Fig. 7 where dynamic range and power calibration are investigated.Input and output link to the DUT are connected to each other through an adaptorwith a gain of−0.2 dB. Fig. 6 presents the dynamic range of the system (includingthe driver). An average value of−60 dBc can be achieved at an input level of40dBm. The noise created in the system is mainly due to the20 MHz IQ bandwidthand to the53 dB attenuators used in the output link. Fig. 7 shows the variation ofthe system gain regarding frequency and input power. A peak to peak variation of0.4 dB is found up to45 dBm in the band of2.1− 2.2 GHz, which is the flat bandof the driver amplifier used.

To investigate deeper the measurement system, GUI and the new figure ofmerits, a class AB LDMOS WCDMA high power (130 W) high gain amplifier

3 RESULTS A7

Figure 5: Three-dimensional plot of gain, efficiency and distortion measured ona high power amplifier using a WCDMA input signal.

is measured. Fig. 4 shows gain, efficiency versus distortionmeasured using atwo-tone input signal while Fig. 5 shows corresponding results from a WCDMAsignal.

The gain of the PA is the ratio of output power (POUT ) to input power (PIN ),and is measured in decibels. The efficiency is the power addedefficiency (PAE).The distortion parameter depends on the excitation signal.The distortion consid-ered in the case of the two-tone signals is the third order intermodulation distortion(IM3) whereas adjacent channel power ratio (ACPR) is considered in the case ofWCDMA signals [6].

From such representation, designers can choose the operation region of theirproducts with respect to different figure of merits in a fast and reliable way. Inaddition, investigating the operation of an amplifier basedon different bias levelsand the variation of the above figure of merits with respect tothat will give aninsight in the nature of the device and help in characterizing it.

One important and attractive result a PA designer would liketo see is the sweetspot presented as a dip in the intermodulation products. Such sweet spots offeran increase of several dB in the carrier to intermodulation ratio (IMR). A knowproperty is the relation between the existence of those sweet spots, their powerlevel position, and the amplifier class of operation [7]. By changing theIDQ cur-rent slider, one can identify the existence of such sweet spots and find the suitableoperation point relative to it in a similar to ”live update” result.

Finally, by investigating the variation of the IM3 with the tone spacing, asym-


PRODUCTION

Figure 6: Dynamic range versus input power.

metry between intermodulation channels can be identified and modeled for bestlinearization performance [8].

4 Discussion

The measurement system shows strong malleability regarding tuning factors frombias, power and frequency aspect and covers most of the figureof merits a PAdesigner would like to investigate. In this first prototype only two types of signalsare implemented; 2-tone and WCDMA, but the design is generaland the GUI canbe extended to include more type of signals.

A high bandwidth hall sensor current probe is used to accurately monitor thecurrent vector of the drain supply. Supply current to the PA is then measured byan oscilloscope; by this method an envelope tracking dynamic power consumptioncan be accurately measured up to100 MHz. In fact, this property is taken furtherby investigating the maximization of PAE using bias modulation.

More measurement features can be added to the system in orderto improve itsperformance. Implementing coherent averaging can improvethe dynamic range ofthe receiver up to20 dB [5]. Receivers are never ideal, a bandwidth limitation ontheir baseband can vary up to20 − 50 MHz; using frequency stitching can allowmeasurements of signals as wide as hundreds of MHz [5].

4 DISCUSSION A9

Figure 7: Gain variation versus input power and frequency.

Finally, excitation signal purity is a critical issue in a measurement system;applying digital pre-distortion (DPD) can reduce the spurious in the input link tothe DUT [9].

4.1 Measurement Speed and Accuracy

Measurement speed and accuracy is a critical issue nowadaysas RF measurementengineers compete for the best state-of-the-art system fortesting and production.Using stair sequences is a fast method for power sweeping. However some limita-tions in speed and accuracy have been observed and investigated. In fact the usedoscilloscope was the bottleneck in the system due to its slowdata transferring pro-tocol and small number of samples in its trace which limited the number of stepsthat can be swept. To clarify this limitation, time consumption with and withoutcurrent measurement is presented for sequences formed by213 samples/step. Thesystem is capable of running for example10 frequency steps and20 power stepsin 180 seconds (140 seconds without current measurement). While adding10 biassteps will results in a2000 measurements in30 minutes (23 minutes without cur-rent measurement).


PRODUCTION

4.2 Faster Approach

A faster method would be to sweep both frequency and power in baseband byusing a series stair sequences representing both dimensions. By this method thesame number of frequency and power steps is expected to be measured in35− 40seconds (without current measurements it is done in28 seconds). While adding10bias steps and current measurements are expected to be in6.5 minutes. This newsweeping approach by modulating the baseband in both frequency and power do-main will decrease dramatically the time consumption in measurements; howeverit will put more requirements on equipments and processing certainly if a hugenumber of steps are required.

In fact using such long sequences (in the order of millions ofsamples) willinclude limitations regarding processing data e.g. Fourier transform and cross cor-relation for synchronization and hardware limitations regarding processors, mem-ories and current measurements through normal oscilloscope. Solutions for thatare under investigation.

In fact, measuring the current by the use of an external ADC can solve easilythe current measurement limitation. Another method would be to input the currentas I or Q baseband input to the analyzer and sweep between RF and baseband inputin measuring, while synchronization is similar to the powersignal one.

5 Conclusions

In this paper, a software defined measurement for multidimensional power am-plifiers characterization and testing is developed and verified. It is an enabler forefficient design and characterization of PAs.

New figure of merit is presented that shows gain and efficiencyversus distor-tion. A lookup table and a slide bar are used to vary theIDQ. By this method, aninsight study of the PA can be made and the best fitting operation conditions canbe determined. Two type of excitation signal are implemented in the GUI; two-tones and WCDMA signal. However the GUI is malleable and can be extended tosupport further type of signals.

A fast approach in measurement is presented. Sweeping both frequency andpower in a sequence of series stairs is introduced and evaluated. It shows a dra-matic reduction in time consumption of the measurements.

References

[1] T. Driver, “Device Performance Trade-Offs Easily Explored Using New Soft-ware and Measurement Methodology,” inARFTG Conference Digest-Spring,53rd, 1999, pp. 1-9.

5 CONCLUSIONS A11

[2] J. Hu, K.G. Gard, N.B. Carvalho, and M.B. Steer, “Time-frequency charac-terization of long-term memory in nonlinear power amplifiers,” IEEE MTT-SInternational Microwave Symposium Digest, Atlanta, GA, 2008, pp. 269-272.

[3] R.N. Braithwaite, “A self-generating coefficient list for machine learning inRF power amplifiers using adaptive predistortion,”IEEE 36th European Mi-crowave Conference, Manchester, UK, 2006, pp. 1229-1232.

[4] J. Goodman, B. Miller, G. Raz, and M. Herman, “A new approach to achiev-ing high-performance power amplifier linearization,”IEEE Radar Confer-ence 2007, Boston, MA, USA, 2007, pp. 840-845.

[5] D. Wisell, “Measurement Techniques for Characterization of Power Ampli-fiers,” vol. Doctorial Thesis Stockholm, Sweden: Royal Institute of Technol-ogy, 2007.

[6] S.C. Cripps,RF Power Amplifiers for Wireless Communications, Second ed.Norwood, MA: Artech House, 2006.

[7] C. Fager, J.C. Pedro, N.B. de Carvalho, and H. Zirath, “Prediction of IMDin LDMOS Transistor Amplifiers using a New Large-Signal Model,” IEEETransactions on Microwave Theory and Techniques, vol. 50, pp. 2834-2842,2002.

[8] N.B. de Carvalho and J.C. Pedro, “Two-tone IMD Asymmetryin MicrowavePower Amplifiers,” IEEE MTT-S International Microwave Symposium Di-gest, Boston, MA, 2000, vol.1, pp. 445-448.

[9] C. Luque and N. Björsell, “Improved dynamic range for multi-tone signalusing model-based pre-distortion,”13th Workshop on ADC Modeling andTesting, Florence, Italy, 2008, pp. 1003-1007.

Paper B

Wideband Characterization of Power Amplifiers UsingUndersampling

Per Landin, Charles Nader, Niclas Björsell, Magnuss Isaksson, David Wisell,Peter Händel, Olav Andersen and Niclas Keskitalo

in Proceedings IEEE MTT-S International Microwave Symposium, IMS 2009,pp.1365-1368, Boston, June 2009

c©2009 IEEE

1 INTRODUCTION B1

Abstract

In this paper a radio frequency power amplifier is measured and characterizedby the use of undersampling based on the generalized Zhu-Frank sampling theo-rem. A test system has been designed allowing the bandwidth of the stimuli signalto be100 MHz in the characterization process. That would not be possible withany vector signal analyzer on the market. One of the more challenging problemwithin the proposed concept is the model validation process. Here, two differenttechniques for model validation are proposed, the multitone and the spectrum scanvalidation methods.

1 Introduction

Contemporary wide band code division multiple access (WCDMA) standards uti-lize bandwidth of3.84 MHz. The 3GPP long term evolution (LTE) supports scal-able bandwidths up to20 MHz, and its continuation LTE advanced considerablymore than that. Moreover, mobile worldwide interoperability for microwave ac-cess is supporting up to28 MHz. It is anticipated that future wireless communica-tion systems will be more and more broad-band. The power amplifier (PA) is a keyradio component in any wireless communication system, and behavioral modelingof its input-output characteristics a developed research area. It is also possible thatsignals from different bands and of different standards mayshare the same PA.

Considering the PA as a nonlinear dynamic device, the spectral support of itsoutputs does not only cover the spectral support of the input, but also the adjacentchannels. Relying on the Nyquist-Shannon sampling theorem, sampling rates ofseveral hundreds of MHz are required to sample the amplifier output accordingto the classical sampling theorem. For bandwidths at this order of magnitude,there are no available analog-digital converters (ADCs) with the required dynamicrange, and thus there is a strong need for alternative radio frequency measurementtechnologies to circumvent this major obstacle. Alternatives include the use oftime-interleaved ADCs, (frequency stitching employing repetitive amplifier input[1], or sub-sampling based on the seminal work by Zhu [2]. In short, Zhu’s workimplies that if there is a static invertable function that compresses the spectralsupport of an analog signal, it is sufficient to sample it witha speed correspondingto twice the bandwidth of the compressed signal. After idealpulse-modulation toobtain the reconstructed signal, the inverse of the compressing function is appliedto reconstructed data to obtain a full-band signal. The results of Zhu were later ongeneralized to nonlinear dynamic systems of certain classes by [3]and [4].

Motivated by the fundamental results on sampling theory fornonlinear systemidentification, under-sampling (in the Nyquist-Shannon context) has been identi-fied as an emerging radio frequency measurement technology for wireless com-munication PAs, mostly in terms of theoretical studies. In the paper by Wisell [5],

B2 WIDEBAND CHARACTERIZATION OF POWER AMPLIFIERS USING UNDERSAMPLING

it was shown that by using real measurements on a 3G WCDMA PA employingdifferent sampling rates spanning from some4 MHz up to40 MHz, the amplifiermodel kept comparable performance using lower sampling rates. The work in [5]was extended in [6].

In the current work, we take the work of Wisell a step further by increasingthe bandwidth of the signal to first50 and then96 MHz. This has been possibledue to a specially designed test setup described in Sec. 2 [7]. The theory ofthe model identification is given in Sec. 3 and the results arepresented in Sec.4. Zhu-Frank sampling can utilize the whole Nyquist bandwidth of the ADC formodel identification. However, model validation requires special solutions that arediscussed in Sec. 5, followed by the conclusions in Sec. 6.

2 Test Setup

Zhu-Frank generalized sampling theorem (ZFGST) does not decrease the band-width required by the measurement system; only the samplingrate can be de-creased. To utilize ZFGST one cannot rely on the vector signal analyzers on themarket today since the intermediate frequency (IF) bandwidth preceding the ADCsampler front-end is limited to increase the dynamic range by avoiding noise fold-ing from the broadband noise. Thus, a specially designed test system has beendesigned for PA characterization based on ZFGST.

In order to master the wide bandwidth requirements, the test-bed has an ul-tra wideband radio frequency (RF) front-end. The RF input frequency range is500− 2700 MHz and the amplitude range is−10 to+10 dBm for dynamic rangedepending on the signal. The output amplifier has been designed with a frequencyrange of20 − 1000 MHz and14 dB gain. In total this results in a front-end withexceptional properties. It has a1000 MHz bandwidth within±1.5 dB amplitudevariations. It can handle up to30 dBm peak with close to50 dBm third intercep-tion point. That is well enough for the subsequent 12-bit pipelined ADC intendedfor direct IF sampling that operates up to210 MSPS conversion rate with analogbandwidth of700 MHz. A frame grabber interfaces to the ADC and in real-timerecords with a data length of2 MSamples.

3 Model Identification

The model identification procedure has been described in [7]. Sampled input andoutput data records were measured at different time instants with the describedmeasurement system. Synchronization of the acquired time series was neededbefore model identification.

Since the measurement system does not provide any possibilities of precisetriggering (on the order of tenths of a sample interval) and the physical run-time

4 EXPERIMENTAL AND RESULTS B3

through the system is unknown, the synchronization has beendone using cross-correlation and phase-compensation [8] to obtain sub-sample synchronization.

The synchronization was made in two steps. The first step was arough syn-chronization on sample basis using cross-correlation to find the "delay" betweenthe measured input and output signals. The second step was a sub-sample syn-chronization to find the linear phase offset in the frequencydomain.

The models were identified by minimizing the mean square error (MSE) ofthe measured output and the model output. As model structure, the commonlyused parallel Hammerstein (PH) model [9] is chosen. The PH isdefined by itsnonlinear order P and the memory length M. Such a model is henceforth denotedPH(P,M). The models were identified using the measured quantities of the inputand output signals, known at the specific time instant, and formed to a model-specific regression matrixφ . The non-linear model behavior is absorbed byφ. Itwas described with the model predictor

y(n) = ΘTΦ (1)

which is linear in the parametersΘ. The least-squares estimation problem is thenaddressed as an over determined set of equations, linear in the parameters. Power-ful and simple methods can then be used when determining those parameters.

4 Experimental and Results

The tested amplifier is a LDMOS PA intended for being used in base stations inthe 3rd generation of mobile communications. It has52 dB gain and a maximumrated input power of1 dBm. This PA is designed for use in the2110− 2170 MHzband. In small-signalS21−measurements the frequency range with variations lessthan0.5 dB is 2100 − 2220 MHz; therefore the chosen center frequency is2160MHz.

Due to the non-flat gain in the signal bandwidth, memory effects stronger thanusual are expected. To obtain an accurate model these variations must be consid-ered. The linear part of the PH model is simply a FIR-filter. With the variationswithin the signal bandwidth it is not necessarily true that the first coefficient of thisfilter is the largest coefficient. To check for possible "small" initial coefficients inthe linear FIR-filter, a delay in the output signal was introduced with one sampleat a time and identification was done for each such delay. The NMSE was thencomputed for comparison. In no case was more than10 samples delay tested.

For the normal3.84 MHz WCDMA signal the model with lowest normalizedMSE (NMSE) was the PH(9,4) with a NMSE of−40.2 dB and an adjacent channelerror power ratio (ACEPR) of−56.5 dB [10]. No additional delays were requiredto obtain the latter model. These results with the found model order and modelerrors are in line with the results from [8] for this particular PA.

The model order with lowest NMSE for the50 MHz wide signal was thePH(9,7). In Fig. 1 the measured input, output and the model error for the PH(9,7)


-5 -4 -3 -2 -1 0 1 2 3 4 5

x 107

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

No

rmalized

po

wer

[dB

]

Output

Input

Error PH(9,7)

Figure 1: The power spectra of the input, output and the model error of aPH(9,7) for the50 MHz input signal.

are shown. The necessary delay due to the wide bandwidth in this case was onesample yielding an improvement of0.4 dB in NMSE as compared to using no extradelays.

For the96 MHz wide signal the most suitable model based on NMSE wasfound to be a PH(9,9) with a NMSE of−32.8 dB and a delay of 3 samples com-pared to what the synchronization found. Adding this delay improved the NMSEby 0.5 dB compared to no delay and same model order.

5 Model Validation

As shown in [6], [11, 13] undersampling of the output signal of the PA can beused for the purpose of PA modeling with little or no loss in modeling perfor-mance. Common model performance evaluation criteria for PAbehavioral modelsare NMSE, ACEPR and weighted error-to-spectral power ratio(WESPR). How-ever, data sampled according to the ZFGST does only allow direct evaluation ofNMSE. The NMSE has earlier [14, 15] been shown to be an inadequate metric formodel performance evaluation. In fact, ACEPR was in [16] found to be the bestlow-complexity metric to identify nonlinear mismatches.

The different frequency components of the output signal of the PA, due to

5 MODEL VALIDATION B5

-30 -20 -10 0 10 20 30-60

-50

-40

-30

-20

-10

0

10

20

Relative Frequency

Re

lative

Am

plit

ud

e

FundamentalIM productsaliased onetime

IM productsaliased twotimes

IM productsaliased threetimes

Figure 2: Illustration of undersampling.

aliasing, fall in the same frequency bins after the Zhu-Frank sampling making itimpossible to separate linear from nonlinear model errors.In the following twoalternative solutions are discussed.

5.1 Multitone

Here a multitone based approach to the model validation problem is proposed.Multitones have been used extensively for PA modeling purposes and their suit-ability for this task is well established in numerous paperse.g. [17] and the refer-ences therein. First, assume that a PA model has been extracted using Zhu-Franksampling. The model can now be validated by using a multitoneor a set of mul-titones according to [17] and Zhu-Frank sampling under the condition that thesampling frequency is set in such a manner that the intermodulation (IM) productsgenerated in the PA, after the aliasing, fall on frequencies(with some margin) atwhich there is no input signal. This is in practice a mild requirement that still givessufficient freedom to design the input signal and set the sampling frequency. Theprinciple is illustrated in Fig. 2.

A complex four-tone signal at baseband with a normalized bandwidth of60 ispassed through a fifth order nonlinearity. The output signalthen has a normalizedbandwidth of300. The output signal is sampled using a sampling clock of79. This


will cause the IM products to alias multiple times as shown and it is still possibleto determine the amplitude and phase of them and, thus, to compute the frequencydomain evaluation criteria ACEPR and WESPR.

The model validation is done by comparing the amplitudes andphases of thedifferent IM products of the measured output of the PA and of the output fromthe model. Numerous techniques to swiftly determine the amplitude and phase ofmultitone signals of this kind exist. Here a method like the one presented in [18] isrecommended. In this manner, it is possible to calculate theequivalent of adjacentchannel leakage ratio [19] by adding the power of the IM products that fall in theadjacent channels. Further, for each IM product an error vector is calculated. Thepower of these error vectors can then be added and compared tothe channel power,allowing the computation of a WESPR similar to ACEPR.

Using a fine frequency grid in the multitone signal makes the signal closelyresemble a spectrum continuous signal and the calculated validation criteria closeto the ones that would have been obtained with such a signal.

5.2 Spectrum Scan

The spectrum using Zhu-Frank sampling contains information from the true fre-quencies and the frequencies aliased back from higher Nyquist bands.

Ad(w) =

∞∑

k=0

IAc,1(kfs) +QAc,2(fs2

+ kfs) (2)

whereAd = [Ad(0) . . . Ad(π)]T is the sampled spectrum vector,Ac,1(f) is the

IF spectrum for all frequencies within an odd Nyquist band starting with the fre-quencyf , Ac,2(f) is the corresponding spectrum for even Nyquist bands,fs isthe sampling frequency,I is the unity matrix and isQ a matrix with zeros excepton the sub-antidiagonal where it is−1.

In practice, a low pass filter on the ADC input will remove frequencies out-side the interesting frequency band and thus, in the following discussion an ideallow pass filter with a cut-off frequency atK.fs will be considered. The IF spec-trum,Ac, cannot be recovered directly from the sampled spectrum. However, byhaving a sequence of measurements, where the frequency of the local oscillator(LO), fLO, is changed between each measurements the complete spectrum can berecovered. In order to preserve information within a frequency bin the frequencystep of the LO should be an integer,s, multiplied by the distance in frequency be-tween two adjacent bins in the FFT. That isfs divided by the data length,N . Thenotation in (2) will change to include different sets of measurements.

Ad(w, l) =

K∑

k=0

IAc,1(kfs + lsfsN

) +QAc,2(fs2

+ kfs + lsfsN

) (3)

wherel is an index for the measurement series,l = [0 . . . L− 1].

6 CONCLUSIONS B7

Several measurements will form the set of equations:

Ad(w, 0)...

Ad(w,L− 1)

=

I Q I · · · 0

. . .. . .

. . .

0 · · · I Q I

Ac(f0)...

Ac(fs2+ kfs + (L− 1)s fs

N )

(4)

Under the right conditions, (4) can be used to recover the IF spectrum and tocompute ACEPR and WESPR. The DC components should be excluded and thematrix must be quadratic and full rank. The matrix can be quadratic by excludingthe highest frequency components and full rank can be achieved by using properstep length, s. It can be shown thats = N/2− 1 will fulfil the requirement for fullrank.

6 Conclusions

The ZFGST for the purpose of PA behavioral modeling was tested with differentinput signals of varying bandwidths, going from3.84 MHz through50 MHz to 96MHz. Main difference in the models was the amount of requiredlinear memorydue to gain variations. For the wider signals, the normal cross-correlation basedsynchronization was no longer sufficient to find the optimal linear FIR-filter inthe model. It was shown that introducing additional delays in the output signal ascompared to the input signal improved the model performancewith up to0.5 dBfor the same model order.

Validation of models extracted using the ZFGST was done using the NMSE.As has been shown in [10] and [14, 15], the NMSE is not a well-suited criteriafor PA behavioral model performance evaluation. However, due to the aliasing,criteria using out-of-band frequencies could not be used without special methods.In this paper two different methods are suggested. Multitone signals and frequencyplanning makes it possible to characterize IM products in-band and use criteria likeWESPR. Another method is to achieve full-spectrum coverageby variation of thelocal oscillator frequency in the downconverter and thereby acquire signals forvalidation of ZFGST-extracted models.

References

[1] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidthof a power amplifier test-bed,”IEEE Transactions on Instrumentation andMeasurement, vol. 56, no. 4, pp. 1488-1494, 2007.

[2] Y.M. Zhu, “Generalized sampling theorem,”IEEE Transactions on Circuitsand Systems,vol. 39, pp. 587-588, 1992.

[3] W.A. Frank, “Sampling requirements for Volterra systemidentification,”IEEE Signal Processing Letters, vol. 3, pp. 266-268, 1996.


[4] J. Tsimbinos, and K.V. Lever, “Sampling frequency requirements for identi-fication and compensation of nonlinear systems,” inAcoustics, Speech, andSignal Processing, ICASSP-94, IEEE,vol. 3, pp. 513-516, 1994.

[5] D. Wisell, “Exploring the sample rate limitation for modeling of power am-plifiers,” in IMEKO 2006 Conference Digest, Rio de Janeiro, 2006.

[6] D. Wisell, and P. Händel, “Implementation considerations on the use of Zhu’sgeneral sampling theorem for characterization of power amplifiers,” in In-strumentation and Measurement Technology Conference Proceeding, IMTC2007, IEEE,pp. 1-4, 2007.

[7] O. Andersen, N. Björsell, and N. Keskitalo, “A test-bed designed to utilizeZhu’s general sampling theorem to characterize power amplifiers,” in IEEEInternational Instrumentation and Measurement Technology conference Pro-ceedings,I2MTC 2009,Singapore, pp. 201-204, 2009.

[8] M. Isaksson, D. Wisell, and D. Rönnow, “A comparative analysis of behav-ioral models for RF power amplifiers,”IEEE Transactions on MicrowaveTheory and Techniques,vol. 54, no. 1, pp. 348-359, Jan. 2006.

[9] M.S. Heutmaker, E. Wu, and J.R. Welch, “Envelope distortion models withmemory improve the prediction of spectral regrowth for someRF amplifiers,”in ARFTG Conference Digest-Fall,48th, Volume 30, pp. 10-15.

[10] M. Isaksson, D. Wisell, and D. Rönnow, “Wideband dynamic modeling ofpower amplifiers using radial-basis function neural networks,” IEEE Trans-actions on Microwave Theory and Techniques,vol. 53, pp. 3422-28, 2005.

[11] D. Wisell, “A baseband time domain measurement system for dynamic char-acterization of power amplifiers with high dynamic range over large band-widths,” in Instrumentation and Measurement Technology Conference Pro-ceedings of the 20th IEEE, IMTC 2003,Vol. 2, pp. 1177-1180.

[12] P. Singerl and H. Koeppl, “A low-rate identification method for digital predis-torters based on Volterra kernel interpolation,” in48th Midwest Symposiumon Circuits and Systems,2005, vol. 2, pp.1533-1536.

[13] P. Singerl and H. Koeppl, “Volterra kernel interpolation for system modelingand predistortion purposes,” inInternational Symposium on Signals, Circuitsand Systems,2005, vol. 1, pp. 251-254.

[14] P. Landin, M. Isaksson and P. Händel, “Comparison of evaluation criteriafor power amplifier behavioral modeling,” inIEEE MTT-S International Mi-crowave Symposium Digest,Atlanta, GA, USA, 2008, pp. 1441-1444.

[15] D. Wisell, M. Isaksson and N. Keskitalo, “A general evaluation criteria forbehavioral power amplifier modeling,” inARFTG 69,Honolulu, USA, 2007,pp. 251-255.

6 CONCLUSIONS B9

[16] D. Schreurs, M.O. Broma, A.A. Goacher, and M. Gadringer, “RF Power Am-plifier Behavioral Modeling,” Cambridge University, Press2009, 2008.

[17] N.B. Carvalho, K.A. Remley, D. Schreurs, and K.C. Gard,“Multisine signalsfor wireless system test and design,” inMicrowave Magazine,vol. 9, no. 3,pp. 122-138, June 2008.

[18] D. Wisell, B. Rudlund, and D. Rönnow, “Characterization of memory effectsin RF power amplifiers using digital two-tone measurements,” IEEE Trans-actions on Instrumentation and Measurements,vol. 56, pp. 2757-2766, 2007.

[19] ETSI, “3GPP TS 25.141 V6.3.0.”

Paper C

Multi-tone design for out-of-band characterization of nonlinearRF modules using harmonic sampling

Niclas Björsell, Charles Nader and Peter Händelin Proceedings IEEE International Instrumentation and Measurement Technology

conference,I2MTC 2010,Austin, Texas, USA, May 2010

c©2010 IEEE

1 INTRODUCTION C1

Abstract

In this paper we evaluate the generation of a multi-tone set for characterizingthe behavior of nonlinear radio frequency (RF) modules in its out-of-band whenharmonic sampling is used as digitizer. The purpose is to provide the reader witha tool to select proper frequencies and record length for a given application andtest-bed. The method is based on simulations and the use of Sidon sequences. Theproposed method is applicable to sparse discrete frequencymulti-tones.

1 Introduction

There is a growing interest for measuring nonlinear devices, especially microwavedevices such as power amplifiers (PAs) and mixers. One popular method is to usepolyharmonic distortion model [1], which is used in e.g. large-scale network ana-lyzers (LSNA). The components input-output signal nonlinearity is of importancefor in-band error and out-of-band interference characterization. Considering thePA as a nonlinear device, its output spectrum spread over adjacent channel caus-ing harmful interference to neighboring channels. Characterizing the behavioralmodeling of the PA, with in-band and out-of-band performance validation, is a hotresearch area today.

Communication systems are moving towards wideband signals, from wide-band code division multiple access (WCDMA) with3.84 MHz, 3GPP long termevolution (LTE) with scalable bandwidths up to20 MHz, and worldwide interop-erability for microwave access (WiMAX) supporting bandwidths up to28 MHz. Itis considered that further wireless communication systemswill use even broaderband signals. Consequently, for these systems the nonlinear output of the PA ismasked by a spectrum regrowth covering hundreds of megahertz which is difficultto be characterized by normal measurement systems, as largebaseband bandwidthwith high sampler analog to digital converters (ADCs) are required for such mea-surement. Recently, [2, 3] have introduced a test setup, ZGST, which down con-verts the RF signal to an intermediate frequency band (IF) upto 1000 MHz, andthen try to characterize the signal using the theory of harmonic sampling. As de-scribed in [3] the method showed good performance for in-band validation, whileout-of-band was kept obscure.

In this paper, an attention will be given for validating the out-of-band interfer-ence when harmonic sampling is used as a digitizing method. Multi-tones havebeen used extensively for PA modeling purposes and their suitability for this taskis well established in numerous papers e.g. [4, 5] and the references therein. Inthe current work, we will investigate the design of special sets of multi-tone usedfor validating the out-of-band performance of behavioral modeling using harmonicsampling. Theory about harmonic sampling and the use of multi-tone as excitationof a nonlinear device is presented in Sec. 2. The method of generating the multi-

C2MULTI -TONE DESIGN FOR OUT-OF-BAND CHARACTERIZATION OF NONLINEAR RF MODULES USING

HARMONIC SAMPLING

tone set is described in Sec. 3. Simulation results and discussions are presented inSec. 4, while conclusions are drawn in Sec. 5.

2 Theory

Exciting nonlinear devices with wideband signals, has put demands on samplingthe output signals at much higher rates than the current generation of ADCs allows.The Nyquist theorem states that in order to replicate an analog signal in the digitaldomain, it must be sampled at no less than twice the frequencyof its highest fre-quency component. Using harmonic sampling for sampling of band-pass signals,the received signal is sampled with a frequency less than theanalogue frequencybut at least twice the bandwidth. This technique intentionally aliases input signalfrequencies,fin, to an image frequency,fim, below the Nyquist frequency. Therelation betweenfin andfim is given by

fin = [(n− 1)fs ± fim], (1)

wherefs is the sampling frequency andn is an integer number for thenth Nyquistband.

The model validation is done by comparing the amplitudes andphases of thedifferent intermodulation (IM) products of the measured output of the device andof the output from the model. Numerous techniques to swiftlydetermine the am-plitude and phase of multi-tone signals of this kind exist (see e.g. [4]). In thismanner, it is possible to calculate the equivalent of adjacent channel leakage ratioby adding the power of the IM products that fall in the adjacent channels.

Using a fine frequency grid in the multi-tone signal makes thesignal closelyresemble a spectrum continuous signal and the calculated validation criteria closeto the ones that would have been obtained with such a signal.

The problems with using multi-tones are the peak to average ratio (PAR) andIM distortion. Some methods proposed in the literature to reduce the PAR areclipping, redistributing energy on the free subcarriers inOFDM-signals and to setthe phases for the different tones (see e.g. [6]). The PAR minimization is not thefocus of this work. Instead, the focus will be on the IM product distribution, andin particular on designing a special multi-tone signal whose main and IM productsdoes not overlap when aliased back due to harmonic sampling.

Apply a test signal consisting of the sum of two independent pure sinewaveswith frequencies,f1 andf2, with f2 > f1. IM distortion magnitudes for a two-toneinput signal are found at specified sum and difference frequencies,fimf , notedbelow in (2) and (4) [7]. The difference frequencies are

fimf = |if2 − jf1|, (2)

and the sum frequencies are

fimf = if2 + jf1, (3)

2 THEORY C3

wherei, j = 0, 1, 2, 3, · · · are integers, such that|i|+ |j| > 1.The term "mth-order" is commonly used to describe specific nonlinear systembehavior such as "third-order" intercept points. The "mth-order IM products" arefound for those values ofi and j that satisfym = |i| + |j|, for the sum anddifference frequencies defined by (2) and (4). For example, for m = 3, results inthe frequencies3f1, 3f2, 2f1 + f2, 2f1 − f2, 2f2 + f1, 2f2 − f1. The frequenciesfound fori = 0 or j = 0 correspond to harmonic distortion.

For a multi-tone this will be even worse; for three tones and up to third orderIM (m = 2 andm = 3) the total number of frequencies will be31. The number offrequency components will grow rapidly with the number of tones and the orderof IM products that will be considered.

When using multi-tones stimulus together with nonlinear devices and har-monic sampling the measurements will be somewhat complicated. The frequencycomponents will get scrambled, or in worst case, they can even fall on top of eachother. In Fig. 1, a three tone (upper diagram) is used as stimulus. After it haspassed a third order nonlinear devise the spectrum will include fundamental tones,harmonics and IM frequencies (middle diagram). Here one cansee that that thedifference frequencies from (2) are grouped at low frequencies. The2nd-order sumIM frequencies are around the second harmonics. The3rd-order IM are around thefundamental tones and around the third harmonics. Finally,when using harmonicsampling all tones are folded back to frequencies belowfs/2, the first Nyquistband (lower diagram). In order to be able to recapture the spectrum, it is importantthat all measured frequency components are unique. A solution for the special casewith LSNA is given in [8].

Depending on the application, there are two possible approaches. In the exam-ple given above, the whole spectrum was considered. However, in many applica-tions, especially in wireless communication, it is only interesting to study the IMproducts around the fundamental tones. Consider the tones to be a communicationchannels. The difference IM and frequency components around harmonic distor-tions can be filtered out, but the odd order IM products will interfere with adjacentcommunication channels. These approaches will on this paper be referred to as"full spectrum" and "odd order IM products".

When designing measurements like this, one would like to have as many tonesas possible in order to cover a wide frequency range. However, the numbers ofoutput tones grows rapidly with the complexity of the nonlinear device and thestimulus. Thus, it is required to be able to measure a lot of output frequencycomponents at unique frequency bins in an FFT spectrum. The number of possiblefrequencies is proportional to the record length of the measurements. In total thefollowing factors will affect the design of the measurement:

• The record length of the measurements,N .

• The order of IM products,m.

• The number of tones,T , in the multi-tone.


HARMONIC SAMPLING

0 1 2 3 4 5 6 7 80

20

40

60

80

100

0 1 2 3 4 5 6 7 80

20

40

60

80

100

0 1 2 3 4 5 6 7 80

20

40

60

80

100

Figure 1: When a three tone (upper) is feed into a (3rd order) nonlinear device it

will produce fundamental, harmonic and intermodulation tones at theoutput (middle). When sampled at relative low frequency, all toneswill be aliased back to the first Nyquist band; that is frequencies be-low fs/2.

• The analog bandwidth,Bw, of the test-bed or the number of Nyquist bandsthat can be used.

3 Model Identification

Generating a multi-tone set with non-overlapping IM products when combinedwith harmonic sampling is, as far as we know, without analytical solutions. Someinvestigations in number-theory might be useful such as Golomb-ruler and SidonSequences [9–11]. A Sidon sequence is a sequence of integersai < aj , withthe property that the sumsai + aj (i < j) are distinct (compare with the sumfrequencies in (4)). This theory can be used to ensure the presence of uniquefrequencies before sampling if we omit IM products of higherorder than2. Higherorder IM product include multiplications to the integersa1, a2 etc. Moreover, thefolding (modulo calculations) and mirroring (absolute value) makes the analyticsolution even more complex. That would be to find a sequence,S, of integersa1 < a2 . . . < aT such that the following properties will result in unique integer

3 MODEL IDENTIFICATION C5

solutions forY .

Y =

∣∣∣∣∣(

T∑

t=1

jtat) mod N − N

2

∣∣∣∣∣

∀ 0 ≤T∑

t=1

|jt| ≤ m, jt ∈ Z (4)

for the full spectrum approach. Moreover the analog bandwidth must be

Bw ≥ aTm. (5)

If only odd order IM products are considered also the following condition mustbe fulfilled ∣

∣∣∣∣

T∑

t=1

jt

∣∣∣∣∣= 1, (6)

Bw ≥{

aT (m+1

2)− a1(

m−1

2) m odd

aT (m2)− a1(

m2− 1) m even

(7)

To our knowledge, there are no theories where Sidon sequences are used to-gether with modulo calculations and polynomial functions similar to the IM prod-ucts. Thus, we decided to initially tackle the problem by Matlab simulations andthe analytic solution might be a future research topic. However, the theory fromSidon sequences is useful in order to optimize the performance of the simulation(i.e. speed).

When capture a record of data, precise selection of the sampling clock andinput sine-wave frequencies, and selection of the record size N , are important.Coherent sampling is used to ensure that each tone is presented in a unique fre-quency bin in a FFT spectrum. For coherent sampling the optimum frequenciesare given by

fopt =J

Nfs (8)

whereJ is an integer which is relatively prime toN andfs is the sampling fre-quency. IfN is a power of two, which is a natural choice for FFT calculations,then any odd value ofJ meets this condition.

To select suitable frequencies for the stimulus, a set of frequencies is calculatedaccording to

fi ∈{JiN

fs : Ji = Jmin, Jmin + 2, . . . , Jmax, i = 1, 2, . . .

}

(9)

Each of the frequencies is calculated corresponding toJ being an odd integer.For a given frequency range[fL . . . fH ], the set of useful frequencies is

Jmin = 2⌊

fL2fs

N − 1

2

⌋

+ 1

Jmax = 2⌈

fH2fs

N − 1

2

⌉

+ 1

(10)


HARMONIC SAMPLING

where⌊·⌋ denotes the nearest integer towards minus infinity and⌈·⌉ the nearestinteger towards infinity. Coherent sample clock and input sine-wave frequenciesproduce a frequency spectrum that exhibits single-line features also for the non-linear products, harmonics and IM, generated by the nonlinear device, since thelatter have frequencies that are sums and differences at theinput frequencies. Thisis true even if the distortion products are higher than the Nyquist frequency andaliased back. This fact can easily be seen by e.g. using modulo-N calculation.

Since the set of suitable frequencies in (7) are a function ofthe integer, i,the conditions in (4) - (5) can be used to design a multi-tone.For a two-tone(m = 2) signal, a Sidon sequences can be used as a starting sequence.However,a Sidon sequence does not consider aliasing. Thus, some subsequent calculationsare required. For higher order multi-tones, the two-tone isa subset. That is, if twofrequencies can not constitute a two-tone they can not be a part of any multi-tonesignal.

4 Results

Three different scenarios have been performed in order to illustrate the method.Two scenarios are for the full spectrum and one scenario whenonly odd order IM-products are considered. For the two "full spectrum" - scenarios the order of IMproducts are two(m = 2) and three(m = 3), respectively. For all scenarios thenumber of tones are stepwise increasing from two(T = 2) to seven(T = 7). Thebandwidth was five times the Nyquist frequency.

The simulation is done in the following way. A set of possiblefrequencies fora two-tone are given by the Mian-Chowla sequence [11]. All possible combina-tions of input frequencies are tested. Those that generate output tones at higherfrequencies than the analog bandwidth,BW , and those that generate tones thatfall on top of each other are rejected.

The minimum record length is the output of the simulations. The simulationsare time consuming and it is inconvenient to use trial-and-error in order to findsuitable record length. In order to estimate a suitable record length for higherorder multi tones a trend line can be applied to set the initial conditions in thesimulation.

As can be seen in Fig. 2, the record length grows rapidly with the number oftones and order of IM products. To study third order IM product of seven-tonesignal the required data length is2048. That is, seven tones out of1024. That isnot the fine frequency grid that closely resembles a spectrumcontinuous signal.Thus, the proposed method is mainly appropriate for applications with discretefrequency components.

In the simulations a dense Sidon sequence was used. However,several Sidonsequences exist, and it might not be that a dense sequence is the optimal choice.

5 CONCLUSIONS C7

Figure 2: The required record length as a function of the number of tones forthree scenarios: Second and third order IM products for the full spec-trum and third order IM products around the fundamental tones.

5 Conclusions

A problem in designing a suitable multi-tone stimulus for out-of-band characteri-zation of non-linear components when using harmonic sampling is to select properfrequencies. The non-linear behavior will generate harmonics and IM frequenciesin the response signal that, when using harmonic sampling, falls on top of eachother. This can be avoided by choosing proper input frequencies. However, thenumber of possible frequency combinations is highly depending on application.Thus, we have suggested a method to easily find out the requirements on the testset-up, such a record length and analog bandwidth for a specific test scenario andthe demands given by the number of tones and the number of IM products that willbe studied.

Simulations have been used to sort out the fitting combinations. However,the simulations are time consuming. The use of Sidon sequences has been usedin order to improve the performance of the simulations. To further improve theperformance an analytic solution to the state properties would be desired.


HARMONIC SAMPLING

References

[1] J. Verspecht and D.E. Root, “Polyharmonic distortion modeling,”MicrowaveMagazine, IEEE,vol. 7, pp. 44-57, 2006.

[2] O. Andersen, N. Björsell, and N. Keskitalo, “A test-bed designed to utilizeZhu’s general sampling theorem to characterize power amplifiers,” in IEEEInternational Instrumentation and Measurement Technology conference Pro-ceedings,I2MTC 2009,Singapore, pp. 201-204, 2009.

[3] P.N. Landin, C. Nader, N. Björsell, M. Isaksson, D. Wisell, P. Händel, O. An-dersen, and N. Keskitalo, “Wideband Characterization of Power AmplifiersUsing Undersampling,” inIEEE MTT-S International Microwave SymposiumProceedings, IMS 2009,pp. 1365-1368, Boston, June 2009

[4] N.B. Carvalho, K.A. Remley, D. Schreurs, and K.C. Gard, “Multisine signalsfor wireless system test and design,” inMicrowave Magazine, IEEE,vol. 9,pp. 122-138, 2008.

[5] D. Wisell, B. Rudlund, and D. Rönnow, “Characterizationof memory effectsin RF power amplifiers using digital two-tone measurements,” IEEE Trans-actions on Instrumentation and Measurement, vol. 56, pp. 2757-2766, 2007.

[6] J. Schoukens and T. Dobrowiecki, “Design of broadband excitation signalswith a user imposed power spectrum and amplitude distribution,” in In-strumentation and Measurement Technology Conference Proceeding, IMTC1998, IEEE,St. Paul, MN, USA, pp. 1002-1005, 1998.

[7] IEEE Standard for Digitizing Waveform Recorders, IEEE Standard 1057-2007 (Revision of IEEE 1057-1994), pp. c1-142, 2008.

[8] W.V. Moer and Y. Rolain, “An Improved Broadband Conversion Scheme forthe Large-Signal Network Analyzer,”IEEE Transactions on Instrumentationand Measurement,vol. 58, pp. 483-487, 2009.

[9] I.Z. Ruzsa, “Sumset of Sidon sets,”Acta Aritmetica,vol. LXXVII.4, pp. 353-359, 1996.

[10] K. O’Bryan, “A Complete Annotated Bibliography of WorkRelated to SidonSequences,”The electronic journal of combinatorics,vol. DS11, 2004.

[11] A.M. Mian and S.D. Chowla, “On theB2-sequences of Sidon,” inProceed-ings National Academy Science Indiapp. 3-4, 1944.

Paper D

Unfolding the Frequency Spectrum for UndersampledWideband Data

Charles Nader, Niclas Björsell and Peter HändelSubmitted to EURASIP Journal on Signal Processing: Fast Communication

c©2010 IEEE

1 INTRODUCTION D1

Abstract

In this letter, we discuss the problem of unfolding the frequency spectrum forundersampled wideband data. The problem is of relevance to state-of-the-art ra-dio frequency measurement systems, which capture repetitive waveform based ona sampling rate that violates the Nyquist constraint. The problem is presented in acompact form by the inclusion of a complex operator called the CN operator. Theease-of-use problem formulation eliminates the ambiguitycaused by folded fre-quency spectra, in particular those with lines standing on multiples of the Nyquistfrequency that are captured with erroneous amplitude and phase values.

1 Introduction

Digital signal processing has become a pervasive tool for processing measurementsthat are taken from the real world. Based on the pioneering work by Cooley andTukey, processing digital data using the fast Fourier transform (FFT) has made asignificant impact on the signal processing community [1]. Sampling strategies forthe collection of digital data must fulfill the conditions mentioned in Shannon’ssampling theorem, i.e., the sampling must be performed at a rate that is at leasttwice that of the analog data’s bandwidth [2]. However, sparse signals may relaxthe sampling speed [3]. For radio frequency (RF) applications, down-conversionto an intermediate frequency (IF) is a standard approach forhandling RF signalswith limited bandwidth, which is IF band-pass sampling. A test set-up example ispresented in Fig. 1.

The field of RF measurement systems is a hot spot because it is akey playerin the development of wireless systems and RF products. Withthe increasinguse of bandwidth in modern wireless communication systems,requirements onRF measurement systems have become tighter as higher sampling rates and largeranalog bandwidths are required to digitally process such wideband signals. Thebottleneck in performance is in the analog-to-digital conversion process, wherehigher sampling rate result in limited resolution, leadingto a trade-off betweenresolution and speed [4]. There is a need for efficient digital signal-processingmethods to reconstruct wideband data from undersampled measurements.

An important example is the testing of modern power amplifiers for systemssuch as 3GPP Long-Term Evolution (LTE) and International Mobile Telecommu-nications (IMT) advanced, with scalable bandwidths above 20 MHz. The nonlin-earities of the amplifier result in spectral re-growth leading to signal power spread-ing in excess of 300 MHz. Characterizing the amplifier requires a minimum sam-pling frequency of 600 MHz, with a dynamic range of 70-80 dB (that is, 12-bitADC). These requirements are beyond the capabilities of current ADCs [5], on areasonable price. Techniques to simultaneously increase bandwidth and dynamicrange are of utmost importance, e.g. [6].

D2 UNFOLDING THE FREQUENCY SPECTRUM FORUNDERSAMPLED WIDEBAND DATA

Wideband Downconverter

ADC

RF IF

LO

LOm

y(t) ym(t) ymAmplification

Filtration

Equalization

Figure 1: Signal processing set-up. The repetitive wideband analog signal isdown-convertedM times using different local oscillator frequencies,which results in theundersampleddatay1, . . . ,yM .

In applications where repetitive measurements are available, the requirementson the speed of the ADC can be reduced by increasing the numberof measurementsets in different manners. Due to the violation of the Shannon sampling conditions,aliasing will limit the reproducibility of the undersampled information, which willrequire the development of state-of-the-art reconstruction methods to overcomethe limitations. Nowadays, a main method for such reconstruction is based ontime-domain equivalent time sampling (ETS), which is used in many digital oscil-loscopes for high frequency acquisition, and standardizedin IEEE-STD-1057 [7].A limitation of ETS is the requirements on the repetition frequency of the wave-form [8]. A fixed relationship is required between the sampling frequency and thewaveform repetition frequency to rearrange the digitized samples and reconstructthe original waveform. An alternative approach is to work inthe frequency do-main by estimating the position and complex-valued values of the components ofthe frequency spectrum.

In this letter, we consider the issue of unfolding the frequency spectrum forundersampled wideband data, noted by its discrete Fourier transform (DFT). Al-though synthetic sampling is straightforward in theory, itintroduces a plurality ofpractical issues such as the calibration of the measurementset-up. From the signalprocessing point of view, a major issue with FFT-processingis the amplitude/phaseambiguity at the Nyquist frequency due to the violation of the strict inequality inthe bandwidth of the analog signal [9, 10]. Such phenomena are exemplified inseveral engineering textbooks [11, 12]. Such ambiguity is ashow-stopper in theerror-free reconstruction of undersampled waveforms withbandwidths that sur-pass multiples of the Nyquist frequency, i.e., the criticalfrequencies.

In this letter, we consider the problem of error-free reconstruction of the DFTcorresponding to a Nyquist sampled broadband signal, whichis based on a set of

2 MAIN RESULTS D3

M measurement sequences that are undersampled by a factorM . The approachutilizes a stepping mechanism in the local oscillator (LO) of the measurement set-up shown in Fig. 1. By introducing a complex-notation (CN) operator we presenta matrix notation that is suitable for this class of problems. In addition to thecompact notation, which has its own right, the results are important for the digitalprocessing of the data, for example, in the calculation of calibration coefficients.

2 Main Results

From the theory of aliasing, we know that any frequency component that is higherthan half the sampling frequencyFs falls back to the first Nyquist band [i.e.,(0, Fs/2)]. Frequency components that are in an odd Nyquist band aliasback in-distinguishably to the first Nyquist band with the same complex form. Frequencycomponents that are in an even band alias back to the first Nyquist band in a mir-rored form relative to the Nyquist frequencyFs/2 with a conjugate complex form.

Another issue that must be considered is the amplitude and phase ambiguitycaused by the critical frequencies which fall back to their DFT counterparts at DCorFs/2. Due to the ambiguity phenomenon, DFT frequency bins at DC andFs/2are unreliable for the reconstruction of an undersampled signal, and they must beexcluded from the calculated DFTs. Although this exclusionviolates the inherentstructure of the problem, it can be reinforced by combining the CN-operator andthe LO stepping mechanism as introduced below.

Consider the test set-up shown in Fig. 1 with a repeatable signal applied as theinput to the down-converter RF side and one set of measurement data of lengthN(gathered in the column vectorym) collected for each setting of the LO. In otherwords, let the LO span the ordered set in the following equation:

LO1 > . . . > LOm > . . . > LOM [Hz] (1)

where, form = 2, . . . ,M ,

LOm = LOm−1 − Fs

(1

2− 1

N

)

[Hz] (2)

with initial value LO1 that is determined below. The set of IFs are given byFIF =FRF− LOm, for m = 1, . . . ,M . It is assumed that the bandwidth of the RF signalis less than or equal to the IF analog bandwidth of the measurement set-up. Forthe forthcoming discussion, it is assumed that the analog bandwidth is a multipleM of the LO step, i.e.,MFs(1/2 − 1/N). Further, the measurement system isassumed to be ideal and distortion-free, and down-conversion is performed in amanner such that no aliasing occurs around the zero frequency. i.e., LO1 is set sothat the down-converted RF is properly placed at positive frequencies.

The DFT of each set of datay1, . . . , yM is calculated usingN -bin FFTs. If weonly consider the bins that correspond to strictly positivefrequencies, we obtain


M DFT column vectors of lengthN/2 − 1, i.e., z1. . . . , zM , where the entrieszm(k) in each of the vectorszm are calculated using the DFT in the followingequation:

zm(k) =

N−1∑

n=0

y(n)e−j2πkn

N (3)

wherek goes from1 to N/2 − 1. Employing a repeatable stimuli, collected andtransformed data from each measurementm are synchronized.

The aim of this work is to construct a vectoru of lengthM(N/2 − 1) thatwould been obtained if the IF corresponding to LO1 was sampled by Nyquist rate(1 − 2/N)M Fs/2, with M(N/2 − 1) samples collected, followed by a DFTretaining the bins corresponding to strictly positive frequencies. For the sake ofa compact notation, divideu into its M sub-vectors, each of lengthN/2 − 1,according to

u =

u1

...uM

. (4)

With the introduced notation, the relationship between thesought for DFT-vectoru and the set ofM measurementsz1. . . . , zM can be compactly written as follows:

zk =

M∑

m=1

Qm+k−1 um. (5)

In (5), the matrices{Qi} are squared matrices of dimensionN/2 − 1, whichlink the elements of vectorsum to the measured datazk. For i > M , Qi = 0,where0 is the null matrix representing the zero-effect of vectorsum when theyare out of the analog bandwidth of the measurement set-up dueto the LO step. Fori = 1, . . . ,M , the matrices{Qi} are given by the following equation:

Qi︸︷︷︸

i odd

=

0 1∗

. ..

1∗

0

0

...

1 0. . .

1

0

(6)

M−2︷︸︸︷

N/2−M︷︸︸︷

3 AN EXAMPLE D5

Qi︸︷︷︸

i even

=

0

1. . .

0 1

...

0

0

1∗

. ..

1∗ 0

(7)

︸︷︷︸

M−2

︸︷︷︸

N/2−M

where the CN operator1∗ has been introduced, with1∗ c = c∗ for a complex-valued scalarc, and where∗ denotes a conjugate. The CN-operator is linear, i.e.,1∗(c1 + c2) = c∗1 + c∗2, and it obeys1∗1∗c = c. By introducing the operator,we obtain a compact representation of the undersampled dataand the constructedfull-band DFT. Equations (5)-(7) are key results; basicper se, however a powerfultool for these kinds of problems as demonstrated below.

Construction of the unfolded DFTu described in (4) is now straightforward.Stacking the measurements yields,

z1...

zM

=

Q1 . . . QM

... . ..

QM 0

u1

...uM

(8)

Equation (8) can be solved using back-substitution.Calibration is essential when using sub-Nyquist sampling strategies with non-

ideal measurement systems characterized by their amplitude and phase distortions.Within the derived framework, filtering of measurements canbe expressed by mul-tiplying the data by weighting-matricesFi that correct the amplitude and phase.TheFi matrices can be found by minimizing the mean square error between the re-constructedu and its ideal counterpart in a least squares sense, which is aconvexproblem. The mathematical framework described above simplifies the symbolichandling of the calibration phase.

3 An example

The proposed scheme is exemplified by the reconstruction of awideband fre-quency spectrum that is described by a multisine with randomamplitude andphase, spanning DC to 500 MHz, as shown in Fig. 2a. As a reference, the fre-quency spectrum is Nyquist-sampled at a frequency of 1 GHz with N ′ = 5110sample points collected. For undersampling,M = 5 sets of data are used, corre-sponding to a bandwidth of 100 MHz withN = N ′/5 sample points collected.

The performance of the method is evaluated with respect to the normalizedmean square error (NMSE) in the difference between the original frequency spec-trum and the reconstructed one. As shown in Fig. 2b, perfect reconstruction was


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 108

−300

−250

−200

−150

−100

−50

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 108

−300

−250

−200

−150

−100

−50

0

Frequency [Hz]

NM

SE

[dB

]S

pect

rum

[dB

x](a)

(b)

Figure 2: Frequency spectrum: (a) Original frequency spectrum of themulti-sine sampled at 1 GHz; (b) Normalized mean square error (dB) ofthe frequency spectrum reconstruction at 200 MHz, with (star) andwithout (line) consideration of the ambiguity at critical frequencies.

achieved with an NMSE of−250 dB, which corresponds to machine precision.Fig. 2b also shows the effect of the lack of consideration of the ambiguities at thecritical frequencies, which is reflected in the peak errors at these tones.

4 Conclusions

A framework for reconstructing a wideband DFT from undersampled measure-ments is presented. A stepping mechanism incorporated intothe local oscillatorin the down-conversion stage allows the measurement of different sets of under-sampled digital data, which when set in a matrix form with consideration of theeffects of aliasing and stepping on the complex frequency spectrum, lead to perfectreconstruction of the sought for DFT.

4 CONCLUSIONS D7

The framework obtained in this study is based on the CN operator 1∗, whichis a key factor in deriving compact matrix formulas for the folded components inthe DFT that are caused by undersampling. The CN operator, combined with localoscillator stepping strategy, enables the exclusion of theambiguity caused by thecritical frequencies with a remained closed form notation.The importance of theapproach is highlighted by the compact form of the reconstruction equations, i.e.(8), and a formalism for handling issues such as calibration. The approach mayalso be a versatile tool for undersampled compressive sampling.

References

[1] J.W. Cooley, and J.W. Tukey, “An algorithm for the machine calculation ofcomplex Fourier series,”Mathematics of Computation, vol. 19, no. 90, pp.1488-1494, April 1965.

[2] C.E. Shannon, “Communication in the presence of noise,”Proceedings Insti-tute of Radio Engineers, vol. 37, no.1, pp. 10-21, January 1949.

[3] E.J. Candes, and M.B. Wakin, “An introduaction to compressive sampling,”IEEE Signal Processing Magazine, vol. 25, no.2, pp. 21-30, 2008.

[4] R.H. Walden, “Analog-to-digital converter survey and analysis,”IEEE Jour-nal on Selected Areas in Communications, vol. 17, no. 4, pp. 539-550, 1999.

[5] B. Murmann, ADC Performance Survey 1997-2010, [Online]. Available:http://www.stanford.edu/ murmann/adcsurvey.html

[6] D. Wisell, D. Rönnow and P. Händel, “A technique to extendthe bandwidthof a power amplifier test-bed”,IEEE Transactions on Instrumentation andMeasurement, vol. 56, no. 4, August 2007, pp. 1488-1494.

[7] IEEE standard for Digitizing Waveform Recorders, IEEE Standard 1057-2007.

[8] T.S. Clement, P.D. Hale, D.F. Williams, C.M. Wang,A. Dienstfrey, and D.A.Keenan, “Calibration of sampling oscilloscopes with high-speed photodi-odes,” IEEE Transactions on Microwave Theory and Techniques, vol. 54,no. 8, pp. 3173-3181, August 2006.

[9] H. Raabe, “Untersuchungen an der wechselzeitigen Mehrfachubertragung(Multiplexubertragung),”Elektrische Nachrichtentechnik, vol. 16, pp. 213-28, 1939.

[10] H.D. Lüke, “The origins of the sampling theorem,”IEEE CommunicationsMagazine, vol. 16, no. 4, pp.106-108, April 1999.


[11] V.K. Ingle, and J.G. Proakis,Digital Signal Processing Using Matlab,Brooks/Cole, California, 2000.

[12] R.G. Lyons,Understanding Digital Signal Processing, Prentice Hall, 2rdEdition, 2004.

Paper E

Peak-to-Average Power Reduction of OFDM Signals by ConvexOptimization: Experimental Validation and Performance

Optimization

Charles Nader, Peter Händel and Niclas BjörsellIEEE Transactions on Instrumentation and Measurements,doi:

10.1109/TIM.2010.2050360, 2010.

c©2010 IEEE

1 INTRODUCTION E1

Abstract

We evaluated the application of convex optimization to peak-to-average powerreduction on an orthogonal frequency division multiplexing (OFDM) 802.11a sig-nal. A radio frequency power amplifier was excited with an OFDM-signal, and thepeak-to-average reduced counterpart and its performance figure of merits weremeasured and compared. A state-of-art radio frequency testsystem with high ac-curacy was used for this purpose. Improvements due to optimization in outputpower and power added efficiency and the influence of the powerdistribution inthe excitation signal on power amplifier performance were investigated. Improve-ments of 6dB in output power and 6.5% in power added efficiencywere achievedon average near the operating region. The effect of preserving power-free guardsubcarriers was introduced in the optimization algorithm and investigated regard-ing adjacent channel interference. An improvement of 9dB from that aspect wasobserved using half of the power-free subcarriers, which reveals the importanceof a guard interval.

1 Introduction

Orthogonal frequency division multiplexing (OFDM) is a widely used modulationscheme because of its high bandwidth efficiency and robustness against frequencyfading due to multipath propagation [1, 2]. The power amplifier is a key compo-nent in a wireless communication chain as it holds the highest power level in thesystem. Its power added efficiency (PAE) directly influencesthe power consump-tion of the wireless system [3]. Its input-output signal nonlinearity is important forin-band error and out-of-band interference [3].

A major drawback of OFDM is the generally high peak to averageratio (PAR)of the radio frequency (RF) signal entering the power amplifier, which causes earlyclipping of the signal due to amplifier saturation and results in nonlinear distor-tions presented in the frequency domain on the shape of unwanted intermodulationproducts, spectral regrowth and harmonics [3]. Such nonlinear distortions causespectral interference to adjacent channels and brake the spectral mask standard foremission [1]. Due that, the input power of the power amplifierhas to be reduced;that is, a large number of dBs have to be backed-off to keep theamplifier in linearoperation. However, such a back-off drastically reduces the PAE of the amplifierbecause a large amount of power (i.e., heat) must be dissipated [4].

Several methods have been proposed in the literature to reduce the PAR ofOFDM signals prior to the conversion to RF, including clipping, dynamic PAR/biasadapting, or redistributing the nonlinear energy on the free subcarriers [5]- [8]. Re-cently in [9], PAR minimization was formulated as a convex optimization problem.The power spectral density of the signal to be transmitted was reshaped by mini-mizing the time domain peak power, subject to some constraints on the error vector

E2PEAK-TO-AVERAGE POWER REDUCTION OFOFDM SIGNALS BY CONVEX OPTIMIZATION : EXPERIMENTAL

VALIDATION AND PERFORMANCE OPTIMIZATION

magnitude (EVM) and the use of power-free guard subcarriers. By applying thefast Fourier transform (FFT) and its inverse on the OFDM signal, a customizedinterior point method (IPM) that finds the near-to global minimum PAR by a fastand reliable algorithm was developed [10]. This PAR optimization approach wasfurther developed in [11, 12] by adding a spectral mask constraint and minimizingthe EVM while keeping the PAR below a minimum threshold. In this work, themethod in [9] is extended to reduce the adjacent channel interference that ariseswhen the guard intervals are excited. Further, there is significant theoretical inter-est in applying convex optimization to obtain PAR-reduction in OFDM commu-nication systems, although the literature, and in particular [9, 11, 12], present noin-depth experimental validation of the impact of the PAR reduction of the base-band signal on the effect of the PAE of the power amplifier. Such experimentalverification is of utmost importance for improving PAE and reducing the overallenergy consumption of wireless communication systems. Another important as-pect to investigate is the effect of exciting the free-powerguard subcarriers on thechannel leakage and its induced adjacent channel interference.

In this paper, we experimentally evaluate the PAR reductionmethod introducedin [9] with respect to how the PAR reduced signal influences the power aspects ofthe power amplifier, which includes the PAE and adjacent channel interference.An extension of the method [9] is developed which preserves afraction of thepower-free subcarriers as a guard interval. The impact of the extended method onthe power amplifier performance and reduction of adjacent channel interference isevaluated.

The paper is organized as follows. An OFDM signal was generated, and PARwas optimized, as briefly reviewed in Sec. 2. The PAR-optimized signal was thenused to excite a commercial power amplifier, using a state-of-art measurementsetup as described in Sec. 3. In Sec. 4, a study of the power amplifier figure ofmerits is presented for both methods, [9] and its extended version, and results arecompared to the corresponding measures of the reference signal. Finally, conclu-sions are drawn in Sec. 5.

2 OFDM PAR reduction by convex optimization

In this section, a review of OFDM PAR reduction using convex optimization, asformulated in [9], is briefly reviewed. To combat the demonstrated effects of ex-citing guard intervals, such as adjacent channel distortion, an extended method isproposed. The Section ends with some remarks on algorithmicdetails.

2.1 PAR reduction by convex optimization

According to the standards, WLAN 802.11a used an OFDM signalcomprising 64subcarriers, distributed into 48 data, 4 pilot, and 12 free subcarriers. The 52 mod-ulated subcarriers used binary or quadrature phase shift keying (BPSK/QPSK),

2 OFDM PARREDUCTION BY CONVEX OPTIMIZATION E3

16-quadrature amplitude modulation (16-QAM), or 64-QAM. Abaseband OFDMsignal was generated by dividing the information data into multiple data streams.Each data stream was passed to a subcarrier for modulation. The modulated datastreams (symbol streams) were sent in parallel on the orthogonal subcarriers. Thefrequency constellation was then time domain transformed through IFFT. Cyclicprefix (guard interval) as well as windowing (Hamming) was applied for time-spreading handling and intersymbol interference elimination (side lobes suppres-sion). The time domain symbols were then serially packed andsent for in-phaseand quadrature (IQ) modulation [1].

Considerc = (c1, . . . , cn)T as a transmitted OFDM frequency constellation,

which is a complex-valued vector of lengthn. Further, letx be the correspondingtime domain signal obtained by aℓ-times oversampling, that is

x = IFFTℓ[c] (1)

where IFFTℓ[·] denotes the inverse discrete Fourier transform of the zero-paddedvectorc, resulting in the length-nℓ vectorx. To learn more about the role ofoversampling in this context, see the references in [9]. Now, the PAR can bedefined as

PAR=n2 maxi(|xi|2)

β cH c(2)

wherexi denotes the entries inx = (x1, . . . , xnℓ)T , β is a real-valued FFT scaling

used to lower bound PAR to 1 andc only contains the contribution by the data car-riers, that isc = Sc, whereS is a diagonal carrier selection matrix with diagonalelementsSi,i = 1 for thosed carriersi1, . . . , id that contain data, and zero other-wise – in our contextd = 52 andn = 64. For a given use of the subcarriers, min-imizing PAR is equivalent to minimizing the peak-powerp = maxi(|xi|2), wherefor all i = 1, . . . , nℓ it holds that|xi|2 ≤ p. The minimization of PAR is obtainedboth byi) adding power to the free carriers, which are given by(I − S)c, andii)distorting the data/pilot carriersSc. The introduced distortion of the transmittedconstellation has to be bounded, given as a constraints imposed on the EVM. Letc0 = (c0,1, . . . , c0,n)

T be a reference constellation, then EVM is defined as [1]

EVM =

√√√√√√

1

d

id∑

i=i1

|ci − c0,i|2

P0

=

√

1

d

||S(c − c0)||2P0

(3)

wherei1, . . . , id denotes the location of data/pilot subcarriers, that is determinedby the non-zero diagonal elements ofS. In the second equality|| · || denotes theEuclidian vector norm. Note thatc andc0 are scaled to the same average powerfor evaluation, that is||c||2 = ||c0||2 [9]. The scalarP0 is the average power ofthe modulation scheme used.

A convex formulation of the problem of minimizing the PAR in (2) of anOFDM baseband signalx in (1) by adding power at the free carriers and distorting



the transmitted constellationc away from the ideal constellationc0, subject to aconstraint imposed on the EVM (3) is [9]

minimize peak power p = maxi

(|xi|2)

subject to ||S(c − c0)|| ≤ ǫ (4)

ℜ[cH0 S (c − c0)] ≥ −ǫ2/2

The first constraint in (4) is a bound on the maximum allowed EVM in (3), whereǫ is a real-valued positive parameter proportional to the allowed EVM and givenby ǫ = EVMmax

√dP0, where EVMmax is the maximum allowed EVM for a given

bit error rate. The second constraint in (4) is a relaxed constraint on the aver-age transmitted data power||Sc||2, that is a relaxed constraint corresponding to||Sc||2 ≥ ||Sc0||2; see [9] for details.

2.2 Channel leakage and an extended method

Radio frequency receivers-transmitters (Rx-Tx) are an enhanced type of equip-ments with performance improving drastically with time andtechnology. Digitalprocessing has been introduced as a tool to achieve perfection and reduce equip-ment errors caused by the non-ideality of the analog counter-parts [13]. Sucherrors include IQ-imbalance, analog to digital converter impairments, non-idealfilters and frequency offset, which affect the spectral occupancy characteristicsand potentially generate interference [14, 15].

Exciting the free subcarriers of the guard intervals in the minimization processto redistribute the channel power spectrum and reduce PAR has raised questionsregarding its applicability because adjacent channel interference can pop-up as aproblematic drawback. Using RF channels with equal bandwidth and spacing putstrong requirements on the Rx-Tx RF equipments if a 100% bandwidth is excited,without any spectrum guard margin. To combat the aspect of leakage, the methodintroduced in [9] is extended by preserving a fraction of thepower-free subcarriersas guards. The modification is achieved by adjusting the matrix I to have zeros onthe diagonal elements relative to the preserved subcarriers.

2.3 Algorithmic details

The convex optimization problem (4), and its extension, canbe solved by standardmethods, yielding a global optimump∗, c∗, x∗. The reader is referred to [9] fordetails on general solvers, as well as specific solvers for the problem at hand. Inthe experimental verifications presented here, the logarithm-barrier-IPM algorithmpresented in [9] is employed. The algorithm starts with a strictly feasible point(c, p) and finds a search direction(v, vp) and a step sizeα that update the pointwith a factorαv andαvp, respectively. The updating procedure is designed toreduce the barrier function value based on respecting the feasibility condition of

3 MEASUREMENT SETUP AND DEVICE UNDER TEST E5

AWGDAC

DAC

900

LO

DPA

Signal GeneratorI

QRF

DUT

Controllable

Power Supply

ADC

Signal Analyser

MS2692A

(Anritsu)

Hall Sensor

Current ProbeOscilloscope

PC

Matlab

I Q

SMU 200A

(R&S)

Figure 1: The measurement setup.

the point [10]. The procedure is iterated until a global optimum point is reached,or almost reached, which solves the PAR minimization problem [9].

3 Measurement setup and device under test

Two major characteristics of RF measurement systems for power amplifier testingare accuracy and the ability to track fast variations in the signal envelope. Becausethe aim of this study is to validate the improvements in poweramplifier perfor-mance, a state-of-art measurement system was needed.

3.1 Measurement set-up

The test-setup presented in Fig. 1 was mainly based on an R&S SMU200A vectorsignal generator, an Anritsu MS2692A signal analyzer, an Agilent 54610B oscillo-scope, an Agilent N2783A high bandwidth hall sensor currentprobe, an EricssonLDMOS highly linear driver amplifier, an Agilent E3631A controllable power sup-ply, and a personal computer (PC). The instruments were connected to the PC viaLAN or GPIB interface. The output power from the amplifier is measured by theMS2692A, which is accurately calibrated in amplitude and phase over the mea-sured bandwidth of interest to obtain±0.3dB power accuracy, even for modulatedtime variant signals. To accurately monitor the drain current vector and obtain ac-curate PAE readings, a high bandwidth hall sensor current probe was used. Mea-suring the current envelope through an oscilloscope allowed an envelope-trackingdynamic power consumption up to 100MHz. Additional detailson the testbed canbe found in [16, 17].



To study the performance of the test-setup, an evaluation ofthe EVM as afunction of the input power is realized where the power amplifier is replaced bya connector. EVM of the transmitted baseband frequency constellation and thereceived counterpart was calculated with respect to input powerPin and carrierfrequency. An average system error of -45dB was found, showing good perfor-mance regarding in-band error.

3.2 Device under test

A class AB LDMOS high power (47dBm) amplifier was used for the validationprocess. It has the capability to handle high PAR up to 15dB. WLAN 802.11a usesa bandwidth of 20MHz. In order to ensure inband flatness, the power amplifierwas operated at 2GHz with a gain variation of 0.4dB over an 80MHz bandwidth.Such characteristics simulate the behavior of a typical WLAN power amplifier.

4 Results and Evaluation

The reference WLAN OFDM 20MHz signal was generated based on 802.11a stan-dards. It had 64 subcarriers with 128 OFDM symbols, a cyclic prefix of 1/4, anoversampling rate of 4 and 14dB PAR after being Hamming windowed. The PAR-optimized counterpart, based on full use of the guard subcarriers, reached a PAR of9.5dB after three Newton iterations in the employed algorithm. Optimizing OFDMsignals allocates power in the free subcarriers that resideat the channel sides. Suchallocation extends the signal bandwidth from 16.6MHz for the reference signal to20.0MHz for the optimized one. To combat this effect, the reference OFDM signalis optimized based on the method in Sec. 2.3, where half of thepower-free subcar-riers are preserved as a guard interval, which results in an effective bandwidth of18.0MHz and a PAR of 9.75dB. This guard margin is sufficient for reducing chan-nel leakage, but requires increasing the number of Newton iterations because twoadditional iterations were needed to reach the optimal solution in the algorithm.

4.1 Power added efficiency

The main goals of reducing PAR are extending the input power level at the 1 dBcompression point of the amplifier, reducing the back-off margin, and allowingan efficient use of the available power. Fig. 2 shows the PAE ofthe amplifier asfunction ofPin for both signals, reference and PAR-optimized with six reservedguard subcarriers, where the 1dB compression points have been specified.

As shown in Fig. 2, the 1dB compression point of the amplifier was extendedby 1.5dB which improved the PAE by 4.2% near the compression region. Thepower amplifier gain was found to have similar shaping with a 1.5dB extendedcompression region. Considering the operating region of the amplifier in a realapplication, which is the compression region backed-off bythe respective PAR, an

4 RESULTS AND EVALUATION E7

14 16 18 20 22 24 26 28 30 32 340

5

10

15

20

25

30

35

40

X: 29Y: 28.85

Input power [dBm]

PA

E [%

]

X: 15Y: 3.197

X: 21Y: 9.651

X: 30.5Y: 33.04

ReferenceOptimized with guard margin

Backed−off Region

1dB Compression

Figure 2: Power added efficiency of the device under test versus input powerfor the reference signal (dashed line) and PAR-optimized signal with6 guard subcarriers (solid line).

improvement of 6.5% in PAE can be seen between the reference and optimizedsignal. In fact, reducing the PAR by 4.5dB and extending the compression regionby 1.5dB leads to a total output power improvement of 6dB. Such improvement(that is, 6.5%) varies with the amplifier technology and design.

Measurements based on PAR-optimized signal without guard subcarriers re-sulted in similar performance regarding PAE and output power compared to thatof a PAR-optimized signal with guard subcarriers.

4.2 In-band errors

Adjusting the frequency constellation power might raise questions regarding in-band errors in the system as well as out-of-band errors due tospectrum regrowth.An evaluation of the output signal EVM versus input power, before and after op-timization, is presented in Fig. 3. Despite the 12.6dB EVM difference betweenthe signals in the backed-off region, the EVM of the optimized signal (with guardsubcarriers) follows the standard limit value (-19dB for the used signal [1]) witha margin error of -0.3dB. Similar results were obtained whenexciting with the



14 16 18 20 22 24 26 28 30 32 34−35

−30

−25

−20

−15

−10

X: 29Y: −17.1

Input power [dBm]

EV

M [d

B]

X: 30.5Y: −14.19

X: 15Y: −31.31

X: 21Y: −18.66

ReferenceOptimized with guard marginEVM Standards for 16QAM

Figure 3: Error vector magnitude of the device under test versus inputpowerfor the reference signal (dashed line) and PAR-optimized signal with6 guard subcarriers (pointed line).

PAR-optimized signal without guard intervals. Such behavior is expected as theEVM constraint in the optimization algorithm was set to the standard limit in or-der to study the maximum improvements in PAE and ACPR. One should mentionthat the state of the art coding/decoding techniques are successful in correctingfor in-band errors as long as the standards limits for EVM arefulfilled. By that,the out-of-band emissions are the most problems tackled nowadays as their ef-fects (intermodulation products and spectrum regrowth) interfere with the adjacentchannels and violate the regulations for spectral emission.

In summary, despite the changes made to the constellation diagram after opti-mization, the EVM achieved at the output of the power amplifier in the operationregion (backed-off) is still sufficient for allowing decoding algorithms to correctcaused errors and restore original information. It complies with the IEEE 802.11aerror standard [1] in that region of interest.


−4 −3 −2 −1 0 1 2 3 4

x 107

−50

−40

−30

−20

−10

0

10

Frequency [Hz]

Spe

ctru

m [d

BX

]

ReferenceOptimized guard−freeSpectral Mask 802.11a16.6MHz

20.0MHz

30KHz Interference

Figure 4: Power spectrum of both the reference signal (pointed line) and thePAR-optimized signal (dashed line) at back-off region, compared to802.11a spectral mask standard for emission (solid line).

4.3 Spectral mask and out-of-band errors

Adding power to the sideband free subcarriers increases thebandwidth of the opti-mized signal. Fig. 4 presents the power spectrums of both reference and optimizedsignals without guard interval at back-off region, compared to the 802.11a spectralmask standard for emission [1]. As shown, a 16.6MHz bandwidth is found forthe reference signal, while exciting the free subcarriers leads to a 20.0MHz band-width. Such an increase in the spectrum breaks the standard constraint mask foremission on the lower side of the channel by 30KHz and raises questions about theapplicability of such a method. A worse behavior is found at compression regionwhere the spectral mask is violated at both the upper and lower channels.

Violating the spectral mask for emission by 30KHz will generate strong in-terference to neighboring channels which marks the importance of preserving afraction of the power-free subcarriers as guard interval. Fig. 5 presents the powerspectrum of both signals, reference and optimized with 6 guard subcarriers, atback-off region. It shows complete agreement with the IEEE 802.11a standardsspectral mask emission.



−4 −3 −2 −1 0 1 2 3 4

x 107

−50

−40

−30

−20

−10

0

10

Frequency [Hz]

Spe

ctru

m [d

Bx]

ReferenceOptimized with 6 guard sub−carriersSpectral Mask 802.11a

16.6MHz

18.0MHz

Figure 5: Power spectrum of both the reference signal (pointed line) and thePAR-optimized signal with 6 guard subcarriers (dashed line) at back-off region, compared to 802.11a spectral mask standard for emission(solid line).

Regarding the out-of-band errors, measurements of the adjacent channel powerratios (ACPRs) of the three signals at 20.0MHz channel spacing and bandwidth, atdifferent input power levels, as presented in Fig. 6, show a large variation betweenthe reference signal and the guard-free optimized one, which is clearly revealed inthe lower channel side. Such an increase in the ACPR is due to the 30kHz leak-age/interference from the main channel, which biases the value of ACPR. Compar-ing the above result with the ACPR of the optimized signal with a guard intervalshows an improvement of 9dB in the lower channel near the back-off region withrespect to not using guard margin, while a 1dB improvement was found in thetransition region with respect to the reference signal.

Considering the upper adjacent channel, an average ACPR improvement of1.5dB in the transition region between the backed-off and compression regionswas found between the guard-free optimized signal and the reference one; whilepreserving a guard interval in the optimization process showed an ACPR improve-ment of 2dB in the back-off region compared to not using a guard margin.


14 16 18 20 22 24 26 28 30 32

−32

−30

−28

−26

−24

−22

−20

−18

Input power [dBm]

AC

PR

[dB

c]

Lower Channel Optimized with 6 Guard sub−carriersUpper Channel Optimized with 6 Guard sub−carriersLower Channel Optimized without Guard sub−carriersUpper Channel Optimized without Guard sub−carriersLower Channel ReferenceUpper Channel Reference

Figure 6: Comparison of adjacent channel power ratio for both optimized sig-nals, with and without power-free guard subcarriers, and the referencesignal.

The achieved improvements between both optimized signals points the neces-sity of having a frequency guard interval in the signal for adjacent channel in-terference reduction. From an application point-of-view,the method introducedin [9], and its derivations, need to consider this aspect. Exciting part of the power-free guard subcarriers is sufficient to achieve similar-to-better power performancecompared to that with the full use of the guard interval, withjust two extra Newtoniterations in the optimization algorithm.

In summary, optimizing the signal while preserving a fraction of the power-free subcarriers as a guard interval improves the ACPR by an average of 2dB. Thisis caused by the absence of the clipped high peaks that usually cause early birdspectrum regrowth. Such small improvements can still reduce the adjacent chan-nel interference and the associated errors, but more importantly, this improvementis advantageous compared to the clipping-based algorithmsthat usually generateundesired regrowth in the output spectrum of the power amplifier and require so-phisticated methods to filter the regrowth without regenerating the peaks.



0 5 10 1510

−3

10−2

10−1

100

101

102

PAR [dB]

CC

DF

[%]

ReferenceOptimized

Figure 7: Complementary cumulative density function of the signal peaks toaverage ratio for the reference signal (dashed line) and thePAR-optimized signal with guard margin (solid line).

4.4 Amplifier saturation

Reducing the PAR should generally increase the average power at the output of theamplifier by a couple of dBs, which will lead to a compression point at a higherinput power level, because fewer peaks excite the amplifier’s nonlinearities. How-ever, contrary to what was expected, an average increase of 1.5dB was found nearcompression, and requires further investigation to explain this behavior. A studyof the complementary cumulative density function (CCDF) ofthe peaks distribu-tion in the measured signals, reference and optimized with guard interval, wouldjustify such a result. Fig. 7 presents the CCDF of all signal peaks normalized tothe signal average, for both measured signals while Fig. 8 shows the CCDF of thenormalized envelope signals with respect to their highest peak value, respectively.

As shown in Fig. 7, a 4.5dB PAR reduction was observed after optimization.However, by analyzing Fig. 8, one realizes that such reduction was achieved atthe cost of increasing the lower peaks density distribution, which in turn limitedthe improvement near saturation. In fact, despite the reduction of the high peaks,the optimization algorithm allowed the “smaller” peaks to increase. Ultimately the

5 CONCLUSION E13

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−3

10−2

10−1

100

101

102

X: 0.652Y: 5.593

Envelope normalized to relative Peak

CC

DF

[%]

X: 0.6519Y: 0.3247

ReferenceOptimizedReference AverageOptimized Average

Figure 8: Complementary cumulative density function of the normalized en-velope for both the reference signal (dashed line) and the PAR-optimized signal with guard margin (solid line).

total energy, due to nonlinearity, retained a comparable value to the non-optimizedcase, sufficient to push the power amplifier into compression.

Such aspect in power improvement near saturation represents the tradeoff thatmust be considered when choosing between advanced optimization method likeconvex optimization and the low-complexity clipping basedmethods. In fact, clip-ping the signal only reduce the dominant peaks, resulting inan extra power marginfor saturation to be reached. However, due to the distortioncreated in the ex-citation signal, higher side band levels are expected to arise which cause higherout-of-band errors.

5 Conclusion

Optimizing the PAR of OFDM signals based on convex optimization algorithmswas performed, and before/after figure of merits of a contemporary RF power am-plifier were evaluated.

The employed PAR optimization algorithm based on [9] resulted in limited im-



provement in power performance near the saturation region of the power amplifier:an extra 1.5 dB in output power and 4.2% in power added efficiency. However, inthe backed-off region where the amplifier normally will operates, it had a 6.5%PAE improvement with a gain of 6dB in the output power. One reason for suchlimited gain improvement near compression is the increase in lower peak densitydistribution in the excitation signal.

Spectral emission was considered to be a drawback in the method becausepower leakage was seen by the high ACPR. The main cause of the leakage isthe absence of guard subcarriers. An extended version of themethod in [9] wasimplemented in which half of the guard subcarriers were usedto prevent suchinterference. The extended method showed the necessity of preserving part of theguard interval with a low iterative cost in optimization. Animprovement up to9dB was found in ACPR for the lower channel side, while the power performancemaintained its merits values.

Even though PAR reduction by advanced methods, such as convex optimiza-tion, costs more in terms of additional digital signal processing, it is commonlyconsidered to be a worthwhile technology because the digital processing is “free”;according to Moore’s law, it will be cheaper and cheaper withtime. We haveexperimentally shown an extra 6dB in output power and 6.5% inpower added ef-ficiency due to digital processing of the transmitted signal, without any additionalrequirements from the hardware. This result is believed to be of significance in aworld “where every dB is worth a Billion” [18].

References

[1] IEEE Standard for Local and Metropolitan Area Networks Part16: Air Inter-face for Fixed Broadband Wireless Access Systems, IEEE Standard 802.11a,Sept 1999.

[2] Wireless LAN Medium Access Control (MAC) and Physical Layer(PHY)Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Stan-dard 802.16-2004 (Revision of IEEE Std. 802.16-2001), 2004.

[3] S. Cripps,RF Power Amplifiers for Wireless Communications, Second ed.Norwood, MA: Artech House, 2006.

[4] J.L. Dawson, “Power Amplifier Linearization Techniques: An Overview,” inWorkshop on RF Circuits for 2.5G and 3G Wireless Systems, Feb2001.

[5] H. Ochiai, and H. Imai, “Performance analysis of deliberately clipped OFDMsignals,”IEEE Transactions on Communications, vol. 50, no. 1, pp. 89-101,Jan 2002.

[6] W. Jiang, B. Xing, J. Wang, Y. Ni, C. Peng, X. Zhu, and W. Hang, “Perfor-mance improvement of power amplifiers with digital linearization technol-

5 CONCLUSION E15

ogy,” in Asia-Pacific Microwave Conference Proceedings, APMC 2007, pp.1-4, Dec 2007.

[7] S. Sen, R. Senguttuvan, and A. Chatterjee, “Concurrent PAR and power am-plifier adaptation for power efficient operation of WiMAX OFDM transmit-ters,” IEEE, Radio and Wireless Symposium, pp. 21-24, Jan 2008.

[8] J. Tellado,“Multicarrier Modulation with Low PAR: Applications to DSLand Wireless, Norwell, MA: Kluwer, Sept 2000.

[9] A. Aggarwal, and T.H. Meng, “Minimizing the peak-to-average power ratioof OFDM signals using convex optimization,”IEEE Transactions on SignalProcessing, vol. 54, pp. 3099-3110, Aug 2006.

[10] S. Boyd, and L. Vandenberghe,Convex Optimization, Cambridge, UK: Cam-bridge Univ. Press, Mar 2004.

[11] Q. Liu, R.J. Baxley, and G.T. Zhou, “Free subcarrier optimization for peak-to-average power ratio minimization in OFDM systems,” inProceedingsIEEE ICASSP, Las Vegas, NV, Mar 2008.

[12] Q. Liu, R.J. Baxley, X. Ma and G.T. Zhou, “Error vector magnitude opti-mization for OFDM systems with a deterministic peak-to-average power ra-tio constraint,”Information Sciences and Systems, 42nd Annual Conference,CISS 2008, pp. 101-104, Mar 2008.

[13] G. Fettweis, M. Lohning, D. Petrovic, M. Windisch, P. Zillmann, and W.Rave, “Dirty RF: a new paradigm,”International Journal of Wireless Infor-mation Networks, vol. 14, no. 2, pp. 133-148, June 2007.

[14] A. Behzad,Wireless LAN Radios, IEEE Press on Digital and Mobile Com-munication, 2007.

[15] T.C.W. Schenk, and E.R. Fledderus, “RF impairments in high-rate wirelesssystems - understanding the impact of TX/RX-asymmetry,” in3rd Interna-tional Symposium on Communications, Control and Signal Processing, IS-CCSP 2008, pp. 117-122, Mar 2008.

[16] C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keskitalo, andH. de la Rosa, “Automated multidimensional characterization of power am-plifier for design and production,” inInternational Instrumentation and Mea-surement Technology Conference Proceedings,I2MTC 2009, pp. 144-148,May 2009.

[17] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidthof an RF power amplifier test bed,”IEEE Transactions on Instrumentationand Measurement, vol. 56, pp. 1488-1494, 2007.



[18] C. Beckman, L. Eklund, B. Karlsson, B. Lindmark, D. Ribbenfjärd, and P.Wirdemark, “Verifying 3G license requirements when every dB is worth abillion,” in First European Conference on Antennas and Propagation, Eu-CAP 2006, France, Nov 2006.

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