Enhancing Radio Frequency System Performance ... - Semantic Scholar

Thesis for the degree of Licentiate of Engineering

Enhancing Radio Frequency System Performance by Digital Signal Processing

Charles Nader

Signal Processing School of Electrical Engineering KTH (Royal Institute of Technology) Stockholm 2010

Nader, Charles Enhancing Radio Frequency System Performance by Digital Signal Processing c Copyright 2010 Charles Nader except where otherwise stated. All rights reserved.

TRITA-EE 2010:030 ISSN 1653-5146

Signal Processing School of Eletrical Engineering KTH (Royal Institute of Technology) SE-100 44 Stockholm, Sweden Telephone + 46 (0)26-648 850

Abstract In this thesis measurement systems for the purpose of characterization of radio frequency power amplifiers are studied. Methods to increase the speed, accuracy, bandwidth, as well as to reduce the sampling requirements and testing cost are presented. A method intended for signal shaping with respect to peakto-average ratio reduction and its effects-improvements on the radio frequency front-end performance is investigated. A time domain measurement system intended for fast and accurate measurements and characterization of radio frequency power amplifiers is discussed. An automated, fast and accurate technique for power and frequency sweep measurements is presented. Multidimensional representation of measured figure of merits is evaluated for its importance on the production-testing phase of power amplifiers. A technique to extend the digital bandwidth of a measurement system is discussed. It is based on the Zhu-Frank generalized sampling theorem which decreases the requirements on the sampling rate of the measurement system. Its application for power amplifiers behavioral modeling is discussed and evaluated experimentally. A general method for designing multitone for the purpose of out-of-band characterization of nonlinear radio frequency modules using harmonic sampling is presented. It has an application with the validation of power amplifiers behavioral models in their out-of-band frequency spectral support when extracted 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 frequency measurement systems which capture repetitive waveform based on a sampling rate that violates the Nyquist constraint. The method is presented in a compact form, it eliminates ambiguities caused by folded frequency spectra standing outside the Nyquist band, and is relevant for calibration matters. A convex optimization reduction-based method of peaks-to-average ratio of orthogonal frequency division multiplexing signals is presented and experimentally validated for a wireless local area network system. Improvements on the radio frequency power amplifier level are investigated with respect to power added efficiency, output power, in-band and out-of-band errors. The influence of the power distribution in the excitation signal on power amplifier performance was evaluated.

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Sammanfattning I denna avhandling kommer mätsystem för karakterisering av hög frekvens förstärkare att behandlas. Metoder för att öka hastigheten, noggrannheten, bandbredden, minska samplings hastighet och reducerade test kostnader presenteras. En metod för signal formning med avsikt på peak-to-average och dess beteende förbättring för prestandan på högfrekvens front-end är undersökt. Ett tidsdomän mätsystem ämnat för snabba och noggranna mätningar och karakterisering av högfrekvens effektförstärkare diskuteras. En automatiserad, snabb och noggrann mätmetod för effekt och frekvenssvep är presenterad. En fler dimensionell representation av parametrar för effektförstärkare är utvärderad för produktions testning. En metod för att öka mätsystemets bandbred är diskuterad. Den är baserad på Zhu-Frank generaliserad samplings teorem som minska samplingshastigheten för mätsystemet. Dess tillämpning på effektförstärkare och dess beteende modulering är diskuterad och praktiskt utvärderad. En generell metod för att skapa multiton signaler för karakterisering av icke linjariteter utanför kanalen för radio moduler baserat på undersampling är presenterad. Den kan tillämpas vi validering av effektförstärkares beteende modeller för utanför kanal frekvens spektrum vid när undersampling används. En metod för återskapning av frekvensspektrum vid undersampling av bredbandssignaler är presenterad. Detta är viktigt för hög prestanda högfrekvens mätsystem som samplar en repetitiv signal som inte uppfyller Nyquist kriterium. Metoden presenteras i på ett enkelt sätt att arbeta med. Den eliminerar felaktigheter som uppkommer vid vikning av frekvenskomponenter som befinner sig ovanför Nyquist frekvensen. Detta är relevant för kalibrering av mätsystemet. En konvex optimerings metod för att reducera peak-to-average förhållande av orthogonal frequency division multiplexing signaler är presenterad. Den är experimentellt validerad för trådlöst LAN. Förbättringar på högfrekvens effektförstärkare är undersökta med avseende på effektivitet, uteffekt, i kanalen och utanför kanalen fel. Påverkan av effekt fördelningen av insignal till effektförstärkarens prestanda har utvärderats

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Acknowledgements First and foremost, my deepest gratitude goes to my supervisors Professor Peter Händel from the Royal Institute of Technology, and Dr. Niclas Björsell from the University of Gävle, for their relevant guidance, creative ideas and tremendous support. Special thanks also go to Dr. Wendy Van Moer from Vrije Universiteit Brussel and Dr. Olof Bengtsson from Ferdinand Braun Institute, for our inspiring discussions. I would like to thank Dr. Niclas Keskitalo from Ericsson AB for being a supportive project manager, as well as the funders of my research: Ericsson AB, Freescale Semiconductor Nordic AB, Infineon Technologies Nordic AB, Knowledge Foundation, NOTE AB, Rohde & Schwarz AB and Syntronic AB I would like to thank my colleagues at the RF center for measurement technology in Gävle and TB/Electronics department at the University of Gävle for introducing some fun in my last two years of research and making the place as a friendly research environment. Special thanks to Javier Ferrer Coll, Sathyaveer Prasad, Carl Elofsson, Carl Karlsson, Helena Eriksson for their pleasant humor, and Per Landin for our research discussions-argues. I would like to thank also my colleagues in the ”4th floor” at the signal processing Lab-Royal Institute of Technology for making my visits interesting and enjoyable. Special thanks to Samer Medawar for being a good help in Latex. Last but not least, gratitude goes to my family for believing in me, supporting me, and motivating me for a better outcome: my father Tony and my mother Georgette, my sisters Claudia and Roula, and my brother Joseph.

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Contents Abstract

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Sammanfattning

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Acknowledgements

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Contents

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I Introduction

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Introduction 1 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . 2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Concluding remarks and future research . . . . . . . . . . . . . .

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II Included papers

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A Automated Multidimensional Characterization of Power Amplifier for Design and Production A1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1 2 Measurement system . . . . . . . . . . . . . . . . . . . . . . . . A2 2.1 Test-bed . . . . . . . . . . . . . . . . . . . . . . . . . . . A2 2.2 Measurement method . . . . . . . . . . . . . . . . . . . . A3 2.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . A5 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A8 4.1 Measurement Speed and Accuracy . . . . . . . . . . . . . A9 4.2 Faster Approach . . . . . . . . . . . . . . . . . . . . . . A10 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A11 vii

B Wideband Characterization of pling 1 Introduction . . . . . . . . 2 Test Setup . . . . . . . . . 3 Model Identification . . . . 4 Experimental and Results . 5 Model Validation . . . . . 5.1 Multitone . . . . . 5.2 Spectrum Scan . . 6 Conclusions . . . . . . . . References . . . . . . . . . . . .

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C Multi-tone design for out-of-band characterization of nonlinear RF modules using harmonic sampling 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D Unfolding the Frequency Data 1 Introduction . . . . . 2 Main Results . . . . 3 An example . . . . . 4 Conclusions . . . . . References . . . . . . . . .

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E Peak-to-Average Power Reduction of OFDM Signals by Convex Optimization: Experimental Validation and Performance Optimization E1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E1 2 OFDM PAR reduction by convex optimization . . . . . . . . . . . E2 2.1 PAR reduction by convex optimization . . . . . . . . . . E2 2.2 Channel leakage and an extended method . . . . . . . . . E4 2.3 Algorithmic details . . . . . . . . . . . . . . . . . . . . . E4 3 Measurement setup and device under test . . . . . . . . . . . . . E5 3.1 Measurement set-up . . . . . . . . . . . . . . . . . . . . E5 3.2 Device under test . . . . . . . . . . . . . . . . . . . . . . E6 4 Results and Evaluation . . . . . . . . . . . . . . . . . . . . . . . E6 4.1 Power added efficiency . . . . . . . . . . . . . . . . . . . E6 4.2 In-band errors . . . . . . . . . . . . . . . . . . . . . . . . E7 4.3 Spectral mask and out-of-band errors . . . . . . . . . . . E9 4.4 Amplifier saturation . . . . . . . . . . . . . . . . . . . . E12 viii

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E16

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Part I

Introduction

Introduction This thesis is devoted to the subject of radio frequency (RF) measurement systems for testing and characterization of nonlinear microwave products, with a focus on RF power amplifiers (PAs). It also investigates shaping of signals for improving overall RF system performance. RF measurement systems occupy a hot spot in the instrumentation and measurement society, as they are the key player in measuring transmitted signals, extracting information data, and testing/evaluating developed microwave products from a simple cable to PAs by the use of repetitive excitation scenarios. Such microwave products, mainly the PAs, are classified as nonlinear devices, and have a large number of tradeoffs between different parameters that are highly relevant for the overall RF system performance. Characterizing accurately those parameters in the production-testing phase is crucial for an errorless system operation; however it should be performed in a fast and costless way. The first part of this thesis presents a test setup solution for an automated, accurate, fast, and multidimensional characterization of PAs. With wireless communication systems moving towards wideband signals, from 3GPP long term evolution (LTE) with scalable bandwidths up to 20MHz, worldwide interoperability for microwave access (WiMAX) supporting bandwidths up to 28MHz, International Mobile Telecommunications (IMT) advanced using broader band signals, to ultra-wide band (UWB) signals surpassing the 500MHz frequency space, high interest is put on characterizing the nonlinear behavior of the PA. In fact, combining such wideband signals with the input-output nonlinear behavior of PAs, the spectral support of its output does not only cover the spectral support of the input, but also the adjacent channels, spreading over bandwidths in the scale of hundreds of Megahertz. Measuring such wideband output signals and characterizing the input-output relation of the device under test (DUT) is difficult to achieve by normal instrumentation and measurement systems, as large analog baseband bandwidth with high speed analog to digital converters (ADCs) are required. One possible solution to characterize such nonlinear behavior is to use undersampled data by violating the Nyquist-Shannon sampling theorem, as described by Zhu-Frank generalized sampling theorem (ZFGST). In short, ZhuFranks’s work implies that if there is a static invertable function that compresses the 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 the PA with a sampling rate twice its input excitation bandwidth. As ZFGST does only decrease the requirements on sampling rate and not the analogue bandwidth of the system, one cannot rely on the vector signal analyzers on the market today since the intermediate frequency (IF) bandwidth preceding the ADC sampler front-end is limited to increase the dynamic range by avoiding noise folding from the broadband noise. Thus, a specially designed test system that downconverts data from RF to an intermediate frequency (IF) scale followed by a 12-bits pipeline analogue to digital converter (ADC) was used. Wideband characterization of the behavioral model of PAs based on ZFGST and the specially designed test setup is the focus of the second part of this thesis. While extracting behavioral model parameters is of high interest, validating them is of extreme importance, certainly in the out-of-band spectral region where neighbor 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. ACEPR stands for the ratio between the spectral power leaking to adjacent channels, outof-band, compared to the power in-band. A major problem that fronts such evaluation is the frequency spectral overlapping caused by aliasing of frequency tones standing outside the Nyquist band, that is half the sampling rate. Due to that, techniques for extracting original information from their overlapped part are of high interest. Two techniques that can be applied for the out-of-band validation process are presented in the third and fourth part of the thesis respectively. The third part evaluates the generation of a multitone set for characterizing the behavior of nonlinear radio frequency (RF) modules in its out-of-band when harmonic sampling is used as a digitizing technique. It provides the reader with a tool to select proper frequencies and record length for a given application and test-bed. Such technique is highly relevant for improving the performance of previous stat-of-art nonlinear test equipment. In the fourth part, an attention is given for reconstructing the frequency spectrum of an undersampled waveform, which might be useful for validating the outof-band nonlinearities of PAs, and is of relevance to RF measurement systems that capture repetitive waveform based on a sampling rate that violates the Nyquist constraint. The unfolding problem is solved by a frequency based approach based on local oscillator stepping at the RF stage. The problem is presented in a compact form by the inclusion of a complex operator called the CN operator. The ease-of-use problem formulation eliminates the ambiguity caused by folded frequency spectra, in particular those with lines standing on multiples of the Nyquist frequency that are captured with erroneous amplitude and phase values. The presented approach is highlighted by a formalism for handling issues such as calibration. The last part of this thesis is dedicated for shaping signals used in wireless systems, 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 is the 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 in nonlinear distortions presented in the frequency domain on the shape of unwanted intermodulation products, spectral regrowth, and harmonics, which cause spectral interference to neighbor channels. Due to that, the input power of the amplifier has to be reduced; that is, a large number of dBs have to be backed-off to keep the amplifier in linear operation. Such a back-off drastically reduces the power added efficiency (PAE) of the amplifier because a large amount of power (i.e., heat) must be dissipated. Due to that, a strong need rises for reducing the PAR of OFDM signals prior to the conversion to RF. One interesting method/algorithm is based on the merging convex optimization technique. The method reshapes the power spectral density of the signal by minimizing the time domain peak power, subject to some constraints on the error vector magnitude (EVM), and the spectral power of the signal. A study and experimental evaluation of that PAR optimization method, 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 presented in five papers. Paper A: C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keskitalo and H. De La Rosa, "Automated Multidimensional Characterization of Power Amplifier for Design and Production", in Proceedings IEEE International Instrumentation and Measurement Technology conference, I 2 M T C2009, 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 Amplifiers Using Undersampling", in Proceedings IEEE MTT-S International Microwave Symposium, IMS 2009, pp. 1365-1368, Boston, June 2009. Paper C: N. Björsell, C. Nader and P. Händel, "Multi-tone design for out-ofband characterization of nonlinear RF modules using harmonic sampling", to appear in Proceedings IEEE International Instrumentation and Measurement Technology conference, I 2 M T C2010, Austin, Texas, United States of America, May 2010. Paper D: C. Nader, N. Björsell and P. Händel, "Unfolding the Frequency Spectrum for Undersampled Wideband Data", EURASIP Journal on Signal Processing: Fast Communication, submitted, June 2010. Paper E: C. Nader, P. Händel and N. Björsell, "Peak-to-Average Power Reduction of OFDM Signals by Convex Optimization: Experimental Validation

4INTRODUCTION

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

1

Contributions of the Thesis

The main contributions of this thesis are presented in five scientific papers. Paper A is devoted for the design of a fast and accurate measurement setup to be used in the production and testing phase of power amplifiers. Due to the high interest in studying the behavioral model of power amplifier with the rising use of broadband signals, mainly in its nonlinear aspect, Paper B evaluates the extraction of the model parameters based on the sub-sampling theory of Zhu-Frank. Paper C and D presents two alternative approach that can be used for the out-of-band performance validation of the model extracted in Paper B. Paper C presents a guide for designing general set of multitone that and their intermodulation products don’t overlap when undersampled. While Paper D presents a frequency based approach for unfolding the frequency spectrum of wideband undersampled waveforms. It introduces a compact formulation of the unfolding problem and solves major ambiguities caused by aliasing. Finally, Paper E is devoted for signal shaping of OFDM based system, e.g. WLAN 802.11a, using a technique based on convex optimization for reducing the signal peak-to-average power ratio, and its impact on the radio frequency power amplifier performance. The papers are summarized in the following sections.

Paper A: Automated Multidimensional Characterization of Power Amplifier for Design and Production Power amplifier (PA) is a key component in a wireless communication chain as it holds the highest power level in the system. Characterizing its behavior and testing its 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 real behavior with a low production-test cost. This paper presents a software-defined measurement setup for fast and cost efficient multidimensional measurements based on highly accurate 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 generated in the baseband excitation waveform. It shows a dramatic reduction in time consumption of the measurements. The measurement system offers the possibility to monitor envelope-tracking dynamic power consumption up to 100 MHz plus the possibility to use high crest factors, which is highly relevant for the new generation of signals.

1 C ONTRIBUTIONS

OF THE

T HESIS 5

Paper B: Wideband Characterization of Power Amplifiers Using Undersampling Wireless communication systems are adopting broad-band signals with bandwidths varying up to 100 MHz. With the nonlinear input-output behavior of the power amplifier (PA), the spectral support of its output spread over hundreds of Megahertz. Such wide spectrum is difficult to characterize with current measurement systems as a sampling frequency ”at least” twice the order of the spectrum bandwidth is required, which is difficult to achieve with the current generation of ADCs based on acceptable dynamic range, resolution, and cost. In this paper the generalized Zhu-Frank sub-sampling theorem is tested for the purpose of PA behavioral modeling. A stat-of-the-art test system has been designed for the purpose. By varying the bandwidths of the excitation signal, behavioral model parameters such as nonlinear order and memory depth are investigated. For the wider signals, the normal cross-correlation based synchronization was found to be no longer sufficient for extracting the optimal linear FIR-filter in the model. Model validation was considered to be a challenging problem within the proposed concept due to aliasing effect. Two different methods for model validation are proposed, using special-designed multitone and frequency spectrum reconstruction based on local oscillator stepping. Paper C: Multi-tone design for out-of-band characterization of nonlinear RF modules using harmonic sampling Due to the interest in validating the performance of power amplifier behavioral model extracted based on sub-sampling theory, as described in the previous paper, a general method for designing multitone signals for that purpose was investigated. By generating a set of tones which and their intermodulation products don’t overlap when undersampled, we allow the validation of the out-of-band behavioral model performance by evaluating the adjacent channel power ratio. The purpose is to provide the reader with a tool to select proper frequencies and record length for a given application and test-bed. The method is based on simulations and the use of Sidon sequences. The proposed method is applicable to sparse discrete frequency multi-tones. Paper D: Unfolding the Frequency Spectrum for Undersampled Wideband Data The problem of unfolding the frequency spectrum for undersampled wideband data is discussed in the paper. The problem is of relevance to state-of-the-art radio frequency measurement systems, which capture repetitive waveform based on a sampling rate that violates the Nyquist constraint. The problem is presented in a compact form by the inclusion of a complex operator called the CN operator. The ease-of-use problem formulation eliminates the ambiguity caused by folded frequency spectra, in particular those with lines standing on multiples of the Nyquist

6INTRODUCTION

frequency that are captured with erroneous amplitude and phase values. The problem formulation is relevant for calibration matters. Paper E: Peak-to-Average Power Reduction of OFDM Signals by Convex Optimization: Experimental Validation and Performance Optimization In this paper the application of convex optimization to peak-to-average power reduction on an orthogonal frequency division multiplexing (OFDM) 802.11a signal is evaluated. A radio frequency power amplifier was excited with an OFDMsignal, and the peak-to-average reduced counterpart and its performance figure of 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 the influence of the power distribution in the excitation signal on power amplifier performance were investigated. Improvements of 6dB in output power and 6.5% in power added efficiency were achieved on average near the operating region. The effect of preserving power-free guard subcarriers was introduced in the optimization algorithm and investigated regarding adjacent channel interference. An improvement of 9dB from that aspect was observed using half of the power-free subcarriers, which reveals the importance of a guard interval.

2 R ELATED

WORK 7

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 Characterization by Undersampling: Zhu-Frank Sampling Theorem,” presented at Radio Frequency Measurement Technology Conference, RFMTC 2009, Gävle, Sweden, October 2009. Paper G: C. Nader, P. Händel and N. Björsell, “OFDM PAPR Reduction by Convex Optimization: A Power Amplifier Point-of-View,” to appear in Proceedings 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 frequency measurement systems. Signal processing was used as an enhancement interface for improving measurement systems merits such as the speed, accuracy, bandwidth, as well as designing special excitation signals for power amplifiers testing purpose, reconstructing undersampled wideband signals, and improving the power amplifiers power performance by a stat-of-art method for peak-to-average signal reduction. • By introducing a new approach for performing power and frequency sweep carried in the baseband signal, testing speed was improved from the scale of hours to minutes/seconds. In addition, measuring the current dissipation envelope through highly accurate current sensor probes, allows the implementation of power added efficiency improvement techniques such as envelope tracking, 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 generalized sampling theorem, requirements on the sampling frequency was reduced in the order of 5-10, alternatively the digital bandwidth of the radio frequency measurement setup increased by the same order. Applying the Zhu-Frank sampling approach as a data extraction tool for the purpose of behavioral modeling of nonlinear power amplifiers resulted in a model characteristics and performance with adequate values. • Measuring the nonlinear behavior of radio frequency modules is of high relevance for the characterization process. Due to the large bandwidth over which the frequency response of the nonlinear behavior can spread, current analogue to digital converters are incapable for collecting such information due to their limitation in sampling speed, and sometime analogue bandwidth, which results in aliasing phenomena. Designing of special set of multitone for which, the fundamental tones and their nonlinear derivations don’t overlap when undersampled, allows the out-of-band characterization of nonlinear radio frequency modules based on harmonic sampling. In the thesis, the method limitation was the large number of samples required for large number of tones and high nonlinear order values. A continuation of this work would be to adjust the presented procedure to reduce the strong demands on the number of samples, and to validate the behavioral models extracted in the second section of this thesis by calculating its adjacent channel error power ratio. • With wireless communication systems adopting more broadband signals and while measurement equipments weakened on the baseband digital part by limited analogue to digital converters, new techniques for collecting data

3 C ONCLUDING

REMARKS AND FUTURE RESEARCH 9

are required. Downconverting the radio frequency signal to an intermediate stage, undersampling the signal, and applying reconstruction algorithms allow the perfect reconstruction of the wideband data. This thesis contributed to a compact formulation of the problem which resolved ambiguities rising from aliased components, especially those standing on multiples of the Nyquist frequency. A continuation of this work would be to validate the reconstruction method on a measurement system. Calibration will be an issue that can be easily handled with the compact formulation. An important application of the work would be to validate the out-of-band performance of models extracted based on section two of this thesis. • OFDM signals are being adopted in most current and future communication systems. Their peak-to-average signal characteristics need to be reduced due to its impact on the power performance of the power amplifier stage. Evaluation of an OFDM signal peak-to-average reduction method based on convex optimization and its experimental validation with respect to power amplifier performance improvements was presented in this thesis. Interesting improvements in power added efficiency, output power was observed. An interesting continuation of this work would be to apply the algorithm for adaptive modulation signals where the sub-carriers allocation changes between symbols, e.g. DVB-T2 based systems. Lot of work have been done for improving the performance of radio frequency measurement systems; lot still remain to be done. Digitally processing data is evolving as a pervasive tool for performance enhancement. Moving the digital part at the radio frequency front-end to the border of the antenna is the focus of future systems. By doing so, all non-idealities generated from mixing effect, downconversion, imbalance effect will be eliminated. More requirement will be on the analogue-to-digital part as higher bandwidth and higher sampling rate will be required. Designing new signal reconstruction algorithms for those ultra wideband direct radio frequency sampling based systems is an interesting future research work to be done. Due to the widenband characteristics of future systems, and the inefficient use of the spectrum, sparsity property can be used to reduce the constraint on the analog-to-information stage. By using the emerging compressive sampling technique, wideband signals requiring ”currently” unachievable sampling rate can be reconstructed from a relatively small amount of information samples. Evaluating the application of compressive sampling for application on radio frequency measurement systems is a highly interesting future research work due to its impact on wideband receivers and application on cognative radios and smart receivers.

Part II

Included papers

Paper A Automated Multidimensional Characterization of Power Amplifier 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 Technology conference, I 2 M T C 2009, pp. 144-148, Singapore, May 2009

c

2009 IEEE

1 INTRODUCTION

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Abstract Designing, optimizing and producing modern power amplifiers (PA) requires new and fast RF (radio frequency) measurement techniques capable of characterizing its real behavior. Power amplifiers are a truly multidimensional device where many desired performance parameters are contradictory to each other. This is especially true for the generation of modern communication PAs that require high efficiency, high linearity as well as high bandwidth. This paper presents a software-defined measurement setup for fast and cost efficient multidimensional measurements based on highly accurate standard instruments and a PC. The test bed as well as the graphical user interface is presented along with a demonstration of its functionality. During tuning of tank networks, drain quiescent current, and bias conditions, 3-dimensional graphs can be selected for the most appropriate axes of trade-off parameters to display a true behavior of the PA under test subjected to real-world or close to real-world signals. The measurement system offers the possibility to monitor envelope-tracking dynamic power consumption up to 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 working with modern communication signals have a large number of tradeoffs between different 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 representation of the PA´s figure of merits is a crucial tool for tuning in on the best performance tradeoffs and selecting the optimum operating point; that is the drain quiescent current, IDQ , the frequency range and the power range near compression. Modern PAs dealing with multiple carrier signals at high bandwidth, high crest-factors as well as high linearity are truly multidimensional devices. Traditionally, these performance parameters are presented in tables and two-dimensional plots. Optimizing the performance in the multidimensional device based on this two-dimensional information is both difficult and time consuming. Thus, multidimensional characterizations offer unique possibilities to make tradeoffs in several dimensions, i.e. superior design and production advantages. To our knowledge, only [1, 2] have considered this need from the test engineers and designers point of view. In addition, [3, 4] characterize multidimensional models of PAs for linearization by using pre-distortion techniques. The models techniques given in [3, 4] are used for existing PAs and not aimed for designing, tuning and production test of PAs. In this paper we suggest a software defined measurement (SDM) method to measure and present critical parameters such as distortion, gain, and efficiency in three-

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

2

Measurement system

Major challenges in designing an automated test system are accuracy, speed and cost. Thus, the used instruments are highly accurate standard instruments, without any special designed equipment. All instruments are connected to a computer that controls the measurements, process the measured data and presents the result. That is, a software-defined measurement setup.

2.1

Test-bed

The test-bed is based on a vector signal generator (VSG), R&S SMU200A, a vector signal analyzer (VSA), R&S FSQ26, an oscilloscope, Agilent 54610B, a current probe, Agilent N2783A, a driver amplifier, Ericsson 3G, a controllable power supply, Agilent E3631A and a personal computer (PC), see Fig. 1. The instruments are connected to the PC via LAN and/or via GPIB interface. Such instruments, or similar to, are mainly found in most of RF laboratories which doesn’t impose any additional hardware cost.

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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 basebandsequence of test scenarios, where the amplitude and frequency are increased stepwise. By using coherent sampling, it was assured that the aimed signal power in the spectrum is contained within the relative tone respectively [5]. The signal is then downloaded to an arbitrary waveform generator (AWG) in the VSG, where it is up-converted to radio frequencies (RF). The main advantage of using an AWG is that we can perform several test scenarios in one measurement, and thus we improve the speed of the characterization. Fig. 2 shows a sequence of 15 power steps for a two-Tone signal. In order to have an accurate and stable power stepping, the total RMS power of the carrier should be found. A power formula was derived that links between the total RMS power of the carrier, RM ST , the highest RMS power step in channel, HSRMS , and the crest factor of the full steps sequence, CFT . Starting from the general relation between CFT , RM ST and peak power, P eakT , in a logarithmic basis, as in RM ST = P eakT − CFT , (1) P eakT can be developed to represent HSRMS added to the CF of one single step,

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CFS , as in P eakT = HSRMS + CFS .

(2)

For a multitone system with N tones, HSRMS represents the highest RMS of total tones that should be reached in the sweep. For a Wideband Code Division Multiple Access (WCDMA) system HSRMS represents the highest step of total power in channel that should be reached in the sweep. To accurately monitor the current vector of the drain supply a high bandwidth hall sensor current probe is used. Drain supply current to the PA is then measured by an oscilloscope; by this method, envelope-tracking dynamic power consumption can be accurately measured up to about 100 MHz. The output power from the PA is measured by the VSA, which can be accurately calibrated to an absolute power sensing power meter to obtain ±0.35 dB accuracy, even for modulated time variant signals. In order to provide a design tool to select the optimum IDQ , a controllable power supply is used to sweep the gate bias of the device under test (DUT) at a percentage of ±x%. A look-up table is used to transfer the bias voltage to the IDQ . Stepping bias voltage is also used to study the performance of the PA for different classes of operation, identify its sweet spots and their stability with the operating class. The measured data is sent to the PC, where the data sequences are synchronized in time and frequency domain and each test scenario (amplitude and frequency) is separated and processed individually [5]. A requirement for that is the use of time repetitive signals. One of the main key points in measurement systems is the synchronization between input/output signals, and also between output signal and relative current measurements. The in-phase (I) and quadrature (Q) data on a Nyquist scale are collected from the VSA. The synchronization is done in two steps. The first step is done in the time domain, where cross-correlation is used to synchronize data to one-sample accuracy. The second step is done in the frequency domain, where interpolations in the phase information are used to identify the exact data sequence relative to each measurement step. Synchronizing the current measurements with relative signal sequence is more complex due to difference in sampling frequency between the oscilloscope and the VSG and existence of current glitches in the beginning of a sequence. A decoupler is used on the drain side to eliminate unwanted fluctuations due to sudden short in the power supply. The current trace is collected from the oscilloscope through GPIB cable. Moreover, a fifth order Butterworth low-pass digital filter is used to reduce measurement noise from data. The oscilloscope floor noise is subtracted from the filtered data and to the result a sign function implemented in the software is applied which gives positive pulses (+1) for positive samples and negative pulses (−1) for negative ones. The outcome is then differentiated in a series form which will result in negative pulses (−2) when the power goes from the highest step to the lowest one and a positive pulses when the power steps up its

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

first level. By choosing any two consecutive negative pulses, the period of current sequence is detected. A demonstration of current synchronization based on slopes identification is presented in Fig. 3. Another method for current measurements would be to resample the collected data from the oscilloscope in correspondence to the power sequence length, and then apply the same synchronization procedure used for power measurements. The drawback 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 and distortion can be the three axes. A graphical user interface (GUI) has been designed in order to show the multidimensional plots, but also for parameter settings, see Fig. 4 and Fig. 5. The settings are the IDQ , frequency and input power ranges. A marker can be used to get numerical data at a specific point in the plot. The marker presents the three parameters but also the test conditions (power and frequency) for that specific measurement. Moreover, the GUI can also present data in the traditional two-dimensional plots. To facilitate the possibility to tune the PA properly and select the most suitable

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

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

3

Results

The system performance for different power levels and frequencies are shown in Fig. 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 adaptor with a gain of −0.2 dB. Fig. 6 presents the dynamic range of the system (including the driver). An average value of −60 dBc can be achieved at an input level of 40 dBm. The noise created in the system is mainly due to the 20 MHz IQ bandwidth and to the 53 dB attenuators used in the output link. Fig. 7 shows the variation of the system gain regarding frequency and input power. A peak to peak variation of 0.4 dB is found up to 45 dBm in the band of 2.1 − 2.2 GHz, which is the flat band of the driver amplifier used. To investigate deeper the measurement system, GUI and the new figure of merits, a class AB LDMOS WCDMA high power (130 W) high gain amplifier

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

is measured. Fig. 4 shows gain, efficiency versus distortion measured using a two-tone input signal while Fig. 5 shows corresponding results from a WCDMA signal. 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 added efficiency (PAE). The distortion parameter depends on the excitation signal. The distortion considered 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 of WCDMA signals [6]. From such representation, designers can choose the operation region of their products with respect to different figure of merits in a fast and reliable way. In addition, investigating the operation of an amplifier based on different bias levels and the variation of the above figure of merits with respect to that will give an insight in the nature of the device and help in characterizing it. One important and attractive result a PA designer would like to see is the sweet spot presented as a dip in the intermodulation products. Such sweet spots offer an increase of several dB in the carrier to intermodulation ratio (IMR). A know property is the relation between the existence of those sweet spots, their power level position, and the amplifier class of operation [7]. By changing the IDQ current slider, one can identify the existence of such sweet spots and find the suitable operation point relative to it in a similar to ”live update” result. Finally, by investigating the variation of the IM3 with the tone spacing, asym-

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Figure 6: Dynamic range versus input power.

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

4

Discussion

The measurement system shows strong malleability regarding tuning factors from bias, power and frequency aspect and covers most of the figure of merits a PA designer would like to investigate. In this first prototype only two types of signals are implemented; 2-tone and WCDMA, but the design is general and the GUI can be extended to include more type of signals. A high bandwidth hall sensor current probe is used to accurately monitor the current vector of the drain supply. Supply current to the PA is then measured by an oscilloscope; by this method an envelope tracking dynamic power consumption can be accurately measured up to 100 MHz. In fact, this property is taken further by investigating the maximization of PAE using bias modulation. More measurement features can be added to the system in order to improve its performance. Implementing coherent averaging can improve the dynamic range of the receiver up to 20 dB [5]. Receivers are never ideal, a bandwidth limitation on their baseband can vary up to 20 − 50 MHz; using frequency stitching can allow measurements of signals as wide as hundreds of MHz [5].

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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 to the DUT [9].

4.1 Measurement Speed and Accuracy Measurement speed and accuracy is a critical issue nowadays as RF measurement engineers compete for the best state-of-the-art system for testing and production. Using stair sequences is a fast method for power sweeping. However some limitations in speed and accuracy have been observed and investigated. In fact the used oscilloscope was the bottleneck in the system due to its slow data transferring protocol and small number of samples in its trace which limited the number of steps that can be swept. To clarify this limitation, time consumption with and without current measurement is presented for sequences formed by 21 3 samples/step. The system is capable of running for example 10 frequency steps and 20 power steps in 180 seconds (140 seconds without current measurement). While adding 10 bias steps will results in a 2000 measurements in 30 minutes (23 minutes without current measurement).

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A faster method would be to sweep both frequency and power in baseband by using a series stair sequences representing both dimensions. By this method the same number of frequency and power steps is expected to be measured in 35 − 40 seconds (without current measurements it is done in 28 seconds). While adding 10 bias steps and current measurements are expected to be in 6.5 minutes. This new sweeping approach by modulating the baseband in both frequency and power domain will decrease dramatically the time consumption in measurements; however it will put more requirements on equipments and processing certainly if a huge number of steps are required. In fact using such long sequences (in the order of millions of samples) will include limitations regarding processing data e.g. Fourier transform and cross correlation for synchronization and hardware limitations regarding processors, memories and current measurements through normal oscilloscope. Solutions for that are under investigation. In fact, measuring the current by the use of an external ADC can solve easily the current measurement limitation. Another method would be to input the current as I or Q baseband input to the analyzer and sweep between RF and baseband input in measuring, while synchronization is similar to the power signal one.

5

Conclusions

In this paper, a software defined measurement for multidimensional power amplifiers characterization and testing is developed and verified. It is an enabler for efficient design and characterization of PAs. New figure of merit is presented that shows gain and efficiency versus distortion. A lookup table and a slide bar are used to vary the IDQ . By this method, an insight study of the PA can be made and the best fitting operation conditions can be determined. Two type of excitation signal are implemented in the GUI; twotones and WCDMA signal. However the GUI is malleable and can be extended to support further type of signals. A fast approach in measurement is presented. Sweeping both frequency and power in a sequence of series stairs is introduced and evaluated. It shows a dramatic reduction in time consumption of the measurements.

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

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[2] J. Hu, K.G. Gard, N.B. Carvalho, and M.B. Steer, “Time-frequency characterization of long-term memory in nonlinear power amplifiers,” IEEE MTT-S International Microwave Symposium Digest, Atlanta, GA, 2008, pp. 269272. [3] R.N. Braithwaite, “A self-generating coefficient list for machine learning in RF power amplifiers using adaptive predistortion,” IEEE 36th European Microwave Conference, Manchester, UK, 2006, pp. 1229-1232. [4] J. Goodman, B. Miller, G. Raz, and M. Herman, “A new approach to achieving high-performance power amplifier linearization,” IEEE Radar Conference 2007, Boston, MA, USA, 2007, pp. 840-845. [5] D. Wisell, “Measurement Techniques for Characterization of Power Amplifiers,” vol. Doctorial Thesis Stockholm, Sweden: Royal Institute of Technology, 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 IMD in LDMOS Transistor Amplifiers using a New Large-Signal Model,” IEEE Transactions on Microwave Theory and Techniques, vol. 50, pp. 2834-2842, 2002. [8] N.B. de Carvalho and J.C. Pedro, “Two-tone IMD Asymmetry in Microwave Power Amplifiers,” IEEE MTT-S International Microwave Symposium Digest, Boston, MA, 2000, vol.1, pp. 445-448. [9] C. Luque and N. Björsell, “Improved dynamic range for multi-tone signal using model-based pre-distortion,” 13th Workshop on ADC Modeling and Testing, Florence, Italy, 2008, pp. 1003-1007.

Paper B Wideband Characterization of Power Amplifiers Using Undersampling 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

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1 INTRODUCTION

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Abstract In this paper a radio frequency power amplifier is measured and characterized by the use of undersampling based on the generalized Zhu-Frank sampling theorem. A test system has been designed allowing the bandwidth of the stimuli signal to be 100 MHz in the characterization process. That would not be possible with any vector signal analyzer on the market. One of the more challenging problem within the proposed concept is the model validation process. Here, two different techniques for model validation are proposed, the multitone and the spectrum scan validation methods.

1 Introduction Contemporary wide band code division multiple access (WCDMA) standards utilize bandwidth of 3.84 MHz. The 3GPP long term evolution (LTE) supports scalable bandwidths up to 20 MHz, and its continuation LTE advanced considerably more than that. Moreover, mobile worldwide interoperability for microwave access is supporting up to 28 MHz. It is anticipated that future wireless communication systems will be more and more broad-band. The power amplifier (PA) is a key radio component in any wireless communication system, and behavioral modeling of its input-output characteristics a developed research area. It is also possible that signals from different bands and of different standards may share the same PA. Considering the PA as a nonlinear dynamic device, the spectral support of its outputs does not only cover the spectral support of the input, but also the adjacent channels. Relying on the Nyquist-Shannon sampling theorem, sampling rates of several hundreds of MHz are required to sample the amplifier output according to the classical sampling theorem. For bandwidths at this order of magnitude, there are no available analog-digital converters (ADCs) with the required dynamic range, and thus there is a strong need for alternative radio frequency measurement technologies to circumvent this major obstacle. Alternatives include the use of time-interleaved ADCs, frequency stitching employing repetitive amplifier input [1], or sub-sampling based on the seminal work by Zhu [2]. In short, Zhu’s work implies that if there is a static invertable function that compresses the spectral support of an analog signal, it is sufficient to sample it with a speed corresponding to twice the bandwidth of the compressed signal. After ideal pulse-modulation to obtain the reconstructed signal, the inverse of the compressing function is applied to reconstructed data to obtain a full-band signal. The results of Zhu were later on generalized to nonlinear dynamic systems of certain classes by [3]and [4]. Motivated by the fundamental results on sampling theory for nonlinear system identification, under-sampling (in the Nyquist-Shannon context) has been identified as an emerging radio frequency measurement technology for wireless communication PAs, mostly in terms of theoretical studies. In the paper by Wisell [5],

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it was shown that by using real measurements on a 3G WCDMA PA employing different sampling rates spanning from some 4 MHz up to 40 MHz, the amplifier model 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 increasing the bandwidth of the signal to first 50 and then 96 MHz. This has been possible due to a specially designed test setup described in Sec. 2 [7]. The theory of the model identification is given in Sec. 3 and the results are presented in Sec. 4. Zhu-Frank sampling can utilize the whole Nyquist bandwidth of the ADC for model identification. However, model validation requires special solutions that are discussed in Sec. 5, followed by the conclusions in Sec. 6.

2

Test Setup

Zhu-Frank generalized sampling theorem (ZFGST) does not decrease the bandwidth required by the measurement system; only the sampling rate can be decreased. To utilize ZFGST one cannot rely on the vector signal analyzers on the market today since the intermediate frequency (IF) bandwidth preceding the ADC sampler front-end is limited to increase the dynamic range by avoiding noise folding from the broadband noise. Thus, a specially designed test system has been designed for PA characterization based on ZFGST. In order to master the wide bandwidth requirements, the test-bed has an ultra wideband radio frequency (RF) front-end. The RF input frequency range is 500 − 2700 MHz and the amplitude range is −10 to +10 dBm for dynamic range depending on the signal. The output amplifier has been designed with a frequency range of 20 − 1000 MHz and 14 dB gain. In total this results in a front-end with exceptional properties. It has a 1000 MHz bandwidth within ±1.5 dB amplitude variations. It can handle up to 30 dBm peak with close to 50 dBm third interception point. That is well enough for the subsequent 12-bit pipelined ADC intended for direct IF sampling that operates up to 210 MSPS conversion rate with analog bandwidth of 700 MHz. A frame grabber interfaces to the ADC and in real-time records with a data length of 2 MSamples.

3

Model Identification

The model identification procedure has been described in [7]. Sampled input and output data records were measured at different time instants with the described measurement system. Synchronization of the acquired time series was needed before model identification. Since the measurement system does not provide any possibilities of precise triggering (on the order of tenths of a sample interval) and the physical run-time

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through the system is unknown, the synchronization has been done using crosscorrelation and phase-compensation [8] to obtain sub-sample synchronization. The synchronization was made in two steps. The first step was a rough synchronization on sample basis using cross-correlation to find the "delay" between the measured input and output signals. The second step was a sub-sample synchronization to find the linear phase offset in the frequency domain. The models were identified by minimizing the mean square error (MSE) of the measured output and the model output. As model structure, the commonly used parallel Hammerstein (PH) model [9] is chosen. The PH is defined by its nonlinear order P and the memory length M. Such a model is henceforth denoted PH(P,M). The models were identified using the measured quantities of the input and output signals, known at the specific time instant, and formed to a modelspecific regression matrix φ . The non-linear model behavior is absorbed by φ. It was described with the model predictor yˆ(n) = ΘT Φ

(1)

which is linear in the parameters Θ. The least-squares estimation problem is then addressed as an over determined set of equations, linear in the parameters. Powerful 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 in the 3rd generation of mobile communications. It has 52 dB gain and a maximum rated input power of 1 dBm. This PA is designed for use in the 2110 − 2170 MHz band. In small-signal S21 −measurements the frequency range with variations less than 0.5 dB is 2100 − 2220 MHz; therefore the chosen center frequency is 2160 MHz. Due to the non-flat gain in the signal bandwidth, memory effects stronger than usual are expected. To obtain an accurate model these variations must be considered. The linear part of the PH model is simply a FIR-filter. With the variations within the signal bandwidth it is not necessarily true that the first coefficient of this filter is the largest coefficient. To check for possible "small" initial coefficients in the linear FIR-filter, a delay in the output signal was introduced with one sample at a time and identification was done for each such delay. The NMSE was then computed for comparison. In no case was more than 10 samples delay tested. For the normal 3.84 MHz WCDMA signal the model with lowest normalized MSE (NMSE) was the PH(9,4) with a NMSE of −40.2 dB and an adjacent channel error power ratio (ACEPR) of −56.5 dB [10]. No additional delays were required to obtain the latter model. These results with the found model order and model errors are in line with the results from [8] for this particular PA. The model order with lowest NMSE for the 50 MHz wide signal was the PH(9,7). In Fig. 1 the measured input, output and the model error for the PH(9,7)

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are shown. The necessary delay due to the wide bandwidth in this case was one sample yielding an improvement of 0.4 dB in NMSE as compared to using no extra delays. For the 96 MHz wide signal the most suitable model based on NMSE was found to be a PH(9,9) with a NMSE of −32.8 dB and a delay of 3 samples compared to what the synchronization found. Adding this delay improved the NMSE by 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 be used for the purpose of PA modeling with little or no loss in modeling performance. Common model performance evaluation criteria for PA behavioral models are NMSE, ACEPR and weighted error-to-spectral power ratio (WESPR). However, data sampled according to the ZFGST does only allow direct evaluation of NMSE. The NMSE has earlier [14, 15] been shown to be an inadequate metric for model performance evaluation. In fact, ACEPR was in [16] found to be the best low-complexity metric to identify nonlinear mismatches. The different frequency components of the output signal of the PA, due to

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IM products aliased one time

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aliasing, fall in the same frequency bins after the Zhu-Frank sampling making it impossible to separate linear from nonlinear model errors. In the following two alternative 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 suitability for this task is well established in numerous papers e.g. [17] and the references therein. First, assume that a PA model has been extracted using Zhu-Frank sampling. The model can now be validated by using a multitone or a set of multitones according to [17] and Zhu-Frank sampling under the condition that the sampling frequency is set in such a manner that the intermodulation (IM) products generated in the PA, after the aliasing, fall on frequencies (with some margin) at which there is no input signal. This is in practice a mild requirement that still gives sufficient freedom to design the input signal and set the sampling frequency. The principle is illustrated in Fig. 2. A complex four-tone signal at baseband with a normalized bandwidth of 60 is passed through a fifth order nonlinearity. The output signal then has a normalized bandwidth of 300. The output signal is sampled using a sampling clock of 79. This

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will cause the IM products to alias multiple times as shown and it is still possible to determine the amplitude and phase of them and, thus, to compute the frequency domain evaluation criteria ACEPR and WESPR. The model validation is done by comparing the amplitudes and phases of the different IM products of the measured output of the PA and of the output from the model. Numerous techniques to swiftly determine the amplitude and phase of multitone signals of this kind exist. Here a method like the one presented in [18] is recommended. In this manner, it is possible to calculate the equivalent of adjacent channel leakage ratio [19] by adding the power of the IM products that fall in the adjacent channels. Further, for each IM product an error vector is calculated. The power of these error vectors can then be added and compared to the channel power, allowing the computation of a WESPR similar to ACEPR. Using a fine frequency grid in the multitone signal makes the signal closely resemble a spectrum continuous signal and the calculated validation criteria close to 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 frequencies and the frequencies aliased back from higher Nyquist bands. Ad (w) =

∞ X

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where Ad = [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 frequency f , Ac,2 (f ) is the corresponding spectrum for even Nyquist bands, fs is the sampling frequency, I is the unity matrix and is Q a matrix with zeros except on the sub-antidiagonal where it is −1. In practice, a low pass filter on the ADC input will remove frequencies outside the interesting frequency band and thus, in the following discussion an ideal low pass filter with a cut-off frequency at K.fs will be considered. The IF spectrum, Ac , cannot be recovered directly from the sampled spectrum. However, by having a sequence of measurements, where the frequency of the local oscillator (LO), fLO , is changed between each measurements the complete spectrum can be recovered. In order to preserve information within a frequency bin the frequency step of the LO should be an integer, s, multiplied by the distance in frequency between two adjacent bins in the FFT. That is fs divided by the data length, N . The notation in (2) will change to include different sets of measurements. Ad (w, l) =

K X

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

(3)

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Several measurements will form the set of equations:     Ac (f0 ) Ad (w, 0) I Q I ··· 0      .. .. .. .. ..  (4)   = . . . . . fs fs 0 ··· I Q I Ac ( 2 + kfs + (L − 1)s N ) Ad (w, L − 1) 

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

6 Conclusions The ZFGST for the purpose of PA behavioral modeling was tested with different input signals of varying bandwidths, going from 3.84 MHz through 50 MHz to 96 MHz. Main difference in the models was the amount of required linear memory due to gain variations. For the wider signals, the normal cross-correlation based synchronization was no longer sufficient to find the optimal linear FIR-filter in the model. It was shown that introducing additional delays in the output signal as compared to the input signal improved the model performance with up to 0.5 dB for 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 criteria for 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 frequency planning makes it possible to characterize IM products in-band and use criteria like WESPR. Another method is to achieve full-spectrum coverage by variation of the local oscillator frequency in the downconverter and thereby acquire signals for validation of ZFGST-extracted models.

References [1] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidth of a power amplifier test-bed,” IEEE Transactions on Instrumentation and Measurement, vol. 56, no. 4, pp. 1488-1494, 2007. [2] Y.M. Zhu, “Generalized sampling theorem,” IEEE Transactions on Circuits and Systems, vol. 39, pp. 587-588, 1992. [3] W.A. Frank, “Sampling requirements for Volterra system identification,” IEEE Signal Processing Letters, vol. 3, pp. 266-268, 1996.

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[4] J. Tsimbinos, and K.V. Lever, “Sampling frequency requirements for identification and compensation of nonlinear systems,” in Acoustics, Speech, and Signal Processing, ICASSP-94, IEEE, vol. 3, pp. 513-516, 1994. [5] D. Wisell, “Exploring the sample rate limitation for modeling of power amplifiers,” in IMEKO 2006 Conference Digest, Rio de Janeiro, 2006. [6] D. Wisell, and P. Händel, “Implementation considerations on the use of Zhu’s general sampling theorem for characterization of power amplifiers,” in Instrumentation and Measurement Technology Conference Proceeding, IMTC 2007, IEEE, pp. 1-4, 2007. [7] O. Andersen, N. Björsell, and N. Keskitalo, “A test-bed designed to utilize Zhu’s general sampling theorem to characterize power amplifiers,” in IEEE International Instrumentation and Measurement Technology conference Proceedings, I 2 M T C 2009, Singapore, pp. 201-204, 2009. [8] M. Isaksson, D. Wisell, and D. Rönnow, “A comparative analysis of behavioral models for RF power amplifiers,” IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 1, pp. 348-359, Jan. 2006. [9] M.S. Heutmaker, E. Wu, and J.R. Welch, “Envelope distortion models with memory improve the prediction of spectral regrowth for some RF amplifiers,” in ARFTG Conference Digest-Fall, 48th, Volume 30, pp. 10-15. [10] M. Isaksson, D. Wisell, and D. Rönnow, “Wideband dynamic modeling of power amplifiers using radial-basis function neural networks,” IEEE Transactions on Microwave Theory and Techniques, vol. 53, pp. 3422-28, 2005. [11] D. Wisell, “A baseband time domain measurement system for dynamic characterization of power amplifiers with high dynamic range over large bandwidths,” in Instrumentation and Measurement Technology Conference Proceedings of the 20th IEEE, IMTC 2003, Vol. 2, pp. 1177-1180. [12] P. Singerl and H. Koeppl, “A low-rate identification method for digital predistorters based on Volterra kernel interpolation,” in 48th Midwest Symposium on Circuits and Systems, 2005, vol. 2, pp.1533-1536. [13] P. Singerl and H. Koeppl, “Volterra kernel interpolation for system modeling and predistortion purposes,” in International Symposium on Signals, Circuits and Systems, 2005, vol. 1, pp. 251-254. [14] P. Landin, M. Isaksson and P. Händel, “Comparison of evaluation criteria for power amplifier behavioral modeling,” in IEEE MTT-S International Microwave Symposium Digest, Atlanta, GA, USA, 2008, pp. 1441-1444. [15] D. Wisell, M. Isaksson and N. Keskitalo, “A general evaluation criteria for behavioral power amplifier modeling,” in ARFTG 69, Honolulu, USA, 2007, pp. 251-255.

6 C ONCLUSIONS

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[16] D. Schreurs, M.O. Broma, A.A. Goacher, and M. Gadringer, “RF Power Amplifier Behavioral Modeling,” Cambridge University, Press 2009, 2008. [17] N.B. Carvalho, K.A. Remley, D. Schreurs, and K.C. Gard, “Multisine signals for wireless system test and design,” in Microwave Magazine, vol. 9, no. 3, pp. 122-138, June 2008. [18] D. Wisell, B. Rudlund, and D. Rönnow, “Characterization of memory effects in RF power amplifiers using digital two-tone measurements,” IEEE Transactions 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 nonlinear RF modules using harmonic sampling Niclas Björsell, Charles Nader and Peter Händel in Proceedings IEEE International Instrumentation and Measurement Technology conference, I 2 M T C 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 characterizing the behavior of nonlinear radio frequency (RF) modules in its out-of-band when harmonic sampling is used as digitizer. The purpose is to provide the reader with a tool to select proper frequencies and record length for a given application and test-bed. The method is based on simulations and the use of Sidon sequences. The proposed method is applicable to sparse discrete frequency multi-tones.

1 Introduction There is a growing interest for measuring nonlinear devices, especially microwave devices such as power amplifiers (PAs) and mixers. One popular method is to use polyharmonic distortion model [1], which is used in e.g. large-scale network analyzers (LSNA). The components input-output signal nonlinearity is of importance for in-band error and out-of-band interference characterization. Considering the PA as a nonlinear device, its output spectrum spread over adjacent channel causing harmful interference to neighboring channels. Characterizing the behavioral modeling of the PA, with in-band and out-of-band performance validation, is a hot research area today. Communication systems are moving towards wideband signals, from wideband code division multiple access (WCDMA) with 3.84 MHz, 3GPP long term evolution (LTE) with scalable bandwidths up to 20 MHz, and worldwide interoperability for microwave access (WiMAX) supporting bandwidths up to 28 MHz. It is considered that further wireless communication systems will use even broader band signals. Consequently, for these systems the nonlinear output of the PA is masked by a spectrum regrowth covering hundreds of megahertz which is difficult to be characterized by normal measurement systems, as large baseband bandwidth with high sampler analog to digital converters (ADCs) are required for such measurement. Recently, [2, 3] have introduced a test setup, ZGST, which down converts the RF signal to an intermediate frequency band (IF) up to 1000 MHz, and then try to characterize the signal using the theory of harmonic sampling. As described in [3] the method showed good performance for in-band validation, while out-of-band was kept obscure. In this paper, an attention will be given for validating the out-of-band interference when harmonic sampling is used as a digitizing method. Multi-tones have been used extensively for PA modeling purposes and their suitability for this task is well established in numerous papers e.g. [4, 5] and the references therein. In the current work, we will investigate the design of special sets of multi-tone used for validating the out-of-band performance of behavioral modeling using harmonic sampling. Theory about harmonic sampling and the use of multi-tone as excitation of a nonlinear device is presented in Sec. 2. The method of generating the multi-

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

2

Theory

Exciting nonlinear devices with wideband signals, has put demands on sampling the 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 digital domain, it must be sampled at no less than twice the frequency of its highest frequency component. Using harmonic sampling for sampling of band-pass signals, the received signal is sampled with a frequency less than the analogue frequency but at least twice the bandwidth. This technique intentionally aliases input signal frequencies, fin , to an image frequency, fim , below the Nyquist frequency. The relation between fin and fim is given by fin = (n − 1) fs /2 + fim fin = n fs /2 − fim

n odd n even

(1)

where fs is the sampling frequency and n is an integer number for the nth Nyquist band. The model validation is done by comparing the amplitudes and phases of the different intermodulation (IM) products of the measured output of the device and of the output from the model. Numerous techniques to swiftly determine the amplitude and phase of multi-tone signals of this kind exist (see e.g. [4]). In this manner, it is possible to calculate the equivalent of adjacent channel leakage ratio by adding the power of the IM products that fall in the adjacent channels. Using a fine frequency grid in the multi-tone signal makes the signal closely resemble a spectrum continuous signal and the calculated validation criteria close to the ones that would have been obtained with such a signal. The problems with using multi-tones are the peak to average ratio (PAR) and IM distortion. Some methods proposed in the literature to reduce the PAR are clipping, redistributing energy on the free subcarriers in OFDM-signals and to set the phases for the different tones (see e.g. [6]). The PAR minimization is not the focus of this work. Instead, the focus will be on the IM product distribution, and in particular on designing a special multi-tone signal whose main and IM products does not overlap when aliased back due to harmonic sampling. Apply a test signal consisting of the sum of two independent pure sinewaves with frequencies, f1 and f2 , with f2 > f1 . IM distortion magnitudes for a two-tone input signal are found at specified sum and difference frequencies, fimf , noted below in (2) and (3) [7]. The difference frequencies are fimf = |if2 − jf1 |,

(2)

fimf = if2 + jf1 ,

(3)

and the sum frequencies are

2 T HEORY

C3

where i, j = 0, 1, 2, 3, · · · are integers, such that |i| + |j| > 1. The term "mth -order" is commonly used to describe specific nonlinear system behavior such as "third-order" intercept points. The "mth -order IM products" are found for those values of i and j that satisfy m = |i| + |j|, for the sum and difference frequencies defined by (2) and (3). For example, for m = 3, results in the frequencies 3f1 , 3f2 , 2f1 + f2 , 2f1 − f2 , 2f2 + f1 , 2f2 − f1 . The frequencies found for i = 0 or j = 0 correspond to harmonic distortion. For a multi-tone this will be even worse; for three tones and up to third order IM (m = 2 and m = 3) the total number of frequencies will be 31. The number of frequency components will grow rapidly with the number of tones and the order of IM products that will be considered. When using multi-tones stimulus together with nonlinear devices and harmonic sampling the measurements will be somewhat complicated. The frequency components will get scrambled, or in worst case, they can even fall on top of each other. In Fig. 1, a three tone (upper diagram) is used as stimulus. After it has passed a third order nonlinear device the spectrum will include fundamental tones, harmonics and IM frequencies (middle diagram). Here one can see that the difference frequencies from (2) are grouped at low frequencies. The 2nd -order sum IM frequencies are around the second harmonics. The 3rd -order IM are around the fundamental tones and around the third harmonics. Finally, when using harmonic sampling all tones are folded back to frequencies below fs /2, the first Nyquist band (lower diagram). In order to be able to recapture the spectrum, it is important that all measured frequency components are unique. A solution for the special case with LSNA is given in [8]. Depending on the application, there are two possible approaches. In the example given above, the whole spectrum was considered. However, in many applications, especially in wireless communication, it is only interesting to study the IM products around the fundamental tones. Consider the tones to be a communication channels. The difference IM and frequency components around harmonic distortions can be filtered out, but the odd order IM products will interfere with adjacent communication 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 tones as possible in order to cover a wide frequency range. However, the numbers of output tones grows rapidly with the complexity of the nonlinear device and the stimulus. Thus, it is required to be able to measure a lot of output frequency components at unique frequency bins in an FFT spectrum. The number of possible frequencies is proportional to the record length of the measurements. In total the following 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.

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100 80 60 40 20 0

0

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Figure 1: When a three tone (upper) is feed into a (3rd order) nonlinear device it will produce fundamental, harmonic and intermodulation tones at the output (middle). When sampled at relative low frequency, all tones will be aliased back to the first Nyquist band; that is frequencies below fs /2.

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

3

Model Identification

Generating a multi-tone set with non-overlapping IM products when combined with harmonic sampling is, as far as we know, without analytical solutions. Some investigations in number-theory might be useful such as Golomb-ruler and Sidon Sequences [9–11]. A Sidon sequence is a sequence of integers ai < aj , with the property that the sums ai + aj (i < j) are distinct (compare with the sum frequencies in (4)). This theory can be used to ensure the presence of unique frequencies before sampling if we omit IM products of higher order than 2. Higher order IM product include multiplications to the integers a1 , a2 etc. Moreover, the folding (modulo calculations) and mirroring (absolute value) makes the analytic solution even more complex. That would be to find a sequence, S, of integers a1 < a2 . . . < aT such that the following properties will result in unique integer

3 M ODEL IDENTIFICATION

C5

solutions for Y . T X N Y = ( jt at ) mod N − 2 t=1

∀

0≤

T X t=1

|jt | ≤ m, jt ∈ Z

(4)

for the full spectrum approach. Moreover the analog bandwidth must be Bw ≥ aT m.

(5)

If only odd order IM products are considered also the following condition must be fulfilled T X jt = 1, (6) t=1  m−1 m odd aT ( m+1 2 ) − a1 ( 2 ) (7) Bw ≥ m m aT ( 2 ) − a1 ( 2 − 1) m even To our knowledge, there are no theories where Sidon sequences are used together with modulo calculations and polynomial functions similar to the IM products. Thus, we decided to initially tackle the problem by Matlab simulations and the analytic solution might be a future research topic. However, the theory from Sidon 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 and input 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 frequency bin in a FFT spectrum. For coherent sampling the optimum frequencies are given by J (8) fopt = fs N where J is an integer which is relatively prime to N and fs is the sampling frequency. If N is a power of two, which is a natural choice for FFT calculations, then any odd value of J meets this condition. To select suitable frequencies for the stimulus, a set of frequencies is calculated according to   Ji fs : Ji = Jmin , Jmin + 2, . . . , Jmax , i = 1, 2, . . . (9) fi ∈ N Each of the frequencies is calculated corresponding to J being an odd integer. For a given frequency range [fL . . . fH ], the set of useful frequencies is k j fL 1 +1 N − Jmin = 2 2f 2 s (10) l m Jmax = 2

fH 2fs N

−

1 2

+1

M ULTI- TONE C6

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where ⌊·⌋ denotes the nearest integer towards minus infinity and ⌈·⌉ the nearest integer towards infinity. Coherent sample clock and input sine-wave frequencies produce a frequency spectrum that exhibits single-line features also for the nonlinear products, harmonics and IM, generated by the nonlinear device, since the latter have frequencies that are sums and differences at the input frequencies. This is true even if the distortion products are higher than the Nyquist frequency and aliased back. This fact can easily be seen by e.g. using modulo-N calculation. Since the set of suitable frequencies in (9) are a function of the integer, i, the conditions in (4) - (7) 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 calculations are required. For higher order multi-tones, the two-tone is a subset. That is, if two frequencies can not constitute a two-tone they can not be a part of any multi-tone signal.

4

Results

Three different scenarios have been performed in order to illustrate the method. Two scenarios are for the full spectrum and one scenario when only odd order IMproducts are considered. For the two "full spectrum" - scenarios the order of IM products are two (m = 2) and three (m = 3), respectively. For all scenarios the number of tones are stepwise increasing from two (T = 2) to seven (T = 7). The bandwidth was five times the Nyquist frequency. The simulation is done in the following way. A set of possible frequencies for a two-tone are given by the Mian-Chowla sequence [11]. All possible combinations of input frequencies are tested. Those that generate output tones at higher frequencies than the analog bandwidth, BW , and those that generate tones that fall on top of each other are rejected. The minimum record length is the output of the simulations. The simulations are time consuming and it is inconvenient to use trial-and-error in order to find suitable record length. In order to estimate a suitable record length for higher order multi tones a trend line can be applied to set the initial conditions in the simulation. As can be seen in Fig. 2, the record length grows rapidly with the number of tones and order of IM products. To study third order IM product of seven-tone signal the required data length is 2048. That is, seven tones out of 1024. That is not the fine frequency grid that closely resembles a spectrum continuous signal. Thus, the proposed method is mainly appropriate for applications with discrete frequency components. In the simulations a dense Sidon sequence was used. However, several Sidon sequences exist, and it might not be that a dense sequence is the optimal choice.

5 C ONCLUSIONS

C7

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

5 Conclusions A problem in designing a suitable multi-tone stimulus for out-of-band characterization of non-linear components when using harmonic sampling is to select proper frequencies. The non-linear behavior will generate harmonics and IM frequencies in the response signal that, when using harmonic sampling, falls on top of each other. This can be avoided by choosing proper input frequencies. However, the number of possible frequency combinations is highly depending on application. Thus, we have suggested a method to easily find out the requirements on the test set-up, such a record length and analog bandwidth for a specific test scenario and the demands given by the number of tones and the number of IM products that will be studied. Simulations have been used to sort out the fitting combinations. However, the simulations are time consuming. The use of Sidon sequences has been used in order to improve the performance of the simulations. To further improve the performance an analytic solution to the state properties would be desired.

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References [1] J. Verspecht and D.E. Root, “Polyharmonic distortion modeling,” Microwave Magazine, IEEE, vol. 7, pp. 44-57, 2006. [2] O. Andersen, N. Björsell, and N. Keskitalo, “A test-bed designed to utilize Zhu’s general sampling theorem to characterize power amplifiers,” in IEEE International Instrumentation and Measurement Technology conference Proceedings, I 2 M T C 2009, Singapore, pp. 201-204, 2009. [3] P.N. Landin, C. Nader, N. Björsell, M. Isaksson, D. Wisell, P. Händel, O. Andersen, and N. Keskitalo, “Wideband Characterization of Power Amplifiers Using Undersampling,” in IEEE MTT-S International Microwave Symposium Proceedings, IMS 2009, pp. 1365-1368, Boston, June 2009 [4] N.B. Carvalho, K.A. Remley, D. Schreurs, and K.C. Gard, “Multisine signals for wireless system test and design,” in Microwave Magazine, IEEE, vol. 9, pp. 122-138, 2008. [5] D. Wisell, B. Rudlund, and D. Rönnow, “Characterization of memory effects in RF power amplifiers using digital two-tone measurements,” IEEE Transactions on Instrumentation and Measurement, vol. 56, pp. 2757-2766, 2007. [6] J. Schoukens and T. Dobrowiecki, “Design of broadband excitation signals with a user imposed power spectrum and amplitude distribution,” in Instrumentation and Measurement Technology Conference Proceeding, IMTC 1998, IEEE, St. Paul, MN, USA, pp. 1002-1005, 1998. [7] IEEE Standard for Digitizing Waveform Recorders, IEEE Standard 10572007 (Revision of IEEE 1057-1994), pp. c1-142, 2008. [8] W.V. Moer and Y. Rolain, “An Improved Broadband Conversion Scheme for the Large-Signal Network Analyzer,” IEEE Transactions on Instrumentation and Measurement, vol. 58, pp. 483-487, 2009. [9] I.Z. Ruzsa, “Sumset of Sidon sets,” Acta Aritmetica, vol. LXXVII.4, pp. 353359, 1996. [10] K. O’Bryan, “A Complete Annotated Bibliography of Work Related to Sidon Sequences,” The electronic journal of combinatorics, vol. DS11, 2004. [11] A.M. Mian and S.D. Chowla, “On the B2 -sequences of Sidon,” in Proceedings National Academy Science India pp. 3-4, 1944.

Paper D Unfolding the Frequency Spectrum for Undersampled Wideband Data Charles Nader, Niclas Björsell and Peter Händel Submitted 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 for undersampled wideband data. The problem is of relevance to state-of-the-art radio frequency measurement systems, which capture repetitive waveform based on a sampling rate that violates the Nyquist constraint. The problem is presented in a compact form by the inclusion of a complex operator called the CN operator. The ease-of-use problem formulation eliminates the ambiguity caused by folded frequency spectra, in particular those with lines standing on multiples of the Nyquist frequency that are captured with erroneous amplitude and phase values.

1 Introduction Digital signal processing has become a pervasive tool for processing measurements that are taken from the real world. Based on the pioneering work by Cooley and Tukey, processing digital data using the fast Fourier transform (FFT) has made a significant impact on the signal processing community [1]. Sampling strategies for the collection of digital data must fulfill the conditions mentioned in Shannon’s sampling theorem, i.e., the sampling must be performed at a rate that is at least twice that of the analog data’s bandwidth [2]. However, sparse signals may relax the sampling speed [3]. For radio frequency (RF) applications, down-conversion to an intermediate frequency (IF) is a standard approach for handling RF signals with limited bandwidth, which is IF band-pass sampling. A test set-up example is presented in Fig. 1. The field of RF measurement systems is a hot spot because it is a key player in the development of wireless systems and RF products. With the increasing use of bandwidth in modern wireless communication systems, requirements on RF measurement systems have become tighter as higher sampling rates and larger analog bandwidths are required to digitally process such wideband signals. The bottleneck in performance is in the analog-to-digital conversion process, where higher sampling rate result in limited resolution, leading to a trade-off between resolution and speed [4]. There is a need for efficient digital signal-processing methods to reconstruct wideband data from undersampled measurements. An important example is the testing of modern power amplifiers for systems such as 3GPP Long-Term Evolution (LTE) and International Mobile Telecommunications (IMT) advanced, with scalable bandwidths above 20 MHz. The nonlinearities of the amplifier result in spectral re-growth leading to signal power spreading in excess of 300 MHz. Characterizing the amplifier requires a minimum sampling frequency of 600 MHz, with a dynamic range of 70-80 dB (that is, 12-bit ADC). These requirements are beyond the capabilities of current ADCs [5], on a reasonable price. Techniques to simultaneously increase bandwidth and dynamic range are of utmost importance, e.g. [6].

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Wideband Downconverter

Amplification

y(t)

Filtration Equalization

FOR

U NDERSAMPLED W IDEBAND DATA

IF

ym

ym (t) ADC

LOm LO Figure 1: Signal processing set-up. The repetitive wideband analog signal is down-converted M times using different local oscillator frequencies, which results in the undersampled data y1 , . . . , yM .

In applications where repetitive measurements are available, the requirements on the speed of the ADC can be reduced by increasing the number of measurement sets in different manners. Due to the violation of the Shannon sampling conditions, aliasing will limit the reproducibility of the undersampled information, which will require the development of state-of-the-art reconstruction methods to overcome the limitations. Nowadays, a main method for such reconstruction is based on time-domain equivalent time sampling (ETS), which is used in many digital oscilloscopes for high frequency acquisition, and standardized in IEEE-STD-1057 [7]. A limitation of ETS is the requirements on the repetition frequency of the waveform [8]. A fixed relationship is required between the sampling frequency and the waveform repetition frequency to rearrange the digitized samples and reconstruct the original waveform. An alternative approach is to work in the frequency domain by estimating the position and complex-valued values of the components of the frequency spectrum. In this letter, we consider the issue of unfolding the frequency spectrum for undersampled wideband data, noted by its discrete Fourier transform (DFT). Although synthetic sampling is straightforward in theory, it introduces a plurality of practical issues such as the calibration of the measurement set-up. From the signal processing point of view, a major issue with FFT-processing is the amplitude/phase ambiguity at the Nyquist frequency due to the violation of the strict inequality in the bandwidth of the analog signal [9, 10]. Such phenomena are exemplified in several engineering textbooks [11, 12]. Such ambiguity is a show-stopper in the error-free reconstruction of undersampled waveforms with bandwidths that surpass multiples of the Nyquist frequency, i.e., the critical frequencies. In this letter, we consider the problem of error-free reconstruction of the DFT corresponding to a Nyquist sampled broadband signal, which is based on a set of

2 M AIN R ESULTS

D3

M measurement sequences that are undersampled by a factor M . The approach utilizes a stepping mechanism in the local oscillator (LO) of the measurement setup shown in Fig. 1. By introducing a complex-notation (CN) operator we present a matrix notation that is suitable for this class of problems. In addition to the compact notation, which has its own right, the results are important for the digital processing 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 higher than half the sampling frequency Fs falls back to the first Nyquist band [i.e., (0, Fs /2)]. Frequency components that are in an odd Nyquist band alias back indistinguishably to the first Nyquist band with the same complex form. Frequency components that are in an even band alias back to the first Nyquist band in a mirrored form relative to the Nyquist frequency Fs /2 with a conjugate complex form. Another issue that must be considered is the amplitude and phase ambiguity caused by the critical frequencies which fall back to their DFT counterparts at DC or Fs /2. Due to the ambiguity phenomenon, DFT frequency bins at DC and Fs /2 are unreliable for the reconstruction of an undersampled signal, and they must be excluded from the calculated DFTs. Although this exclusion violates the inherent structure of the problem, it can be reinforced by combining the CN-operator and the LO stepping mechanism as introduced below. Consider the test set-up shown in Fig. 1 with a repeatable signal applied as the input to the down-converter RF side and one set of measurement data of length N (gathered in the column vector ym ) collected for each setting of the LO. In other words, let the LO span the ordered set in the following equation: LO1 > . . . > LOm > . . . > LOM

[Hz]

(1)

[Hz]

(2)

where, for m = 2, . . . , M , LOm = LOm−1 − Fs



1 1 − 2 N



with initial value LO1 that is determined below. The set of IFs are given by FIF = FRF − LOm , for m = 1, . . . , M . It is assumed that the bandwidth of the RF signal is less than or equal to the IF analog bandwidth of the measurement set-up. For the forthcoming discussion, it is assumed that the analog bandwidth is a multiple M of the LO step, i.e., M Fs (1/2 − 1/N ). Further, the measurement system is assumed to be ideal and distortion-free, and down-conversion is performed in a manner such that no aliasing occurs around the zero frequency. i.e., LO1 is set so that the down-converted RF is properly placed at positive frequencies. The DFT of each set of data y1 , . . . , yM is calculated using N -bin FFTs. If we only consider the bins that correspond to strictly positive frequencies, we obtain

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M DFT column vectors of length N/2 − 1, i.e., z1 . . . . , zM , where the entries zm (k) in each of the vectors zm are calculated using the DFT in the following equation: zm (k) =

N −1 X

y(n)e−

j2πkn N

(3)

n=0

where k goes from 1 to N/2 − 1. Employing a repeatable stimuli, collected and transformed data from each measurement m are synchronized. The aim of this work is to construct a vector u of length M (N/2 − 1) that would been obtained if the IF corresponding to LO1 was sampled by Nyquist rate (1 − 2/N ) M Fs , with 2 M (N/2 − 1) samples collected, followed by a DFT retaining the bins corresponding to strictly positive frequencies. For the sake of a compact notation, divide u into its M sub-vectors, each of length N/2 − 1, according to  u1   u =  ...  . uM 

(4)

With the introduced notation, the relationship between the sought for DFT-vector u and the set of M measurements z1 . . . . , zM can be compactly written as follows:

zk =

M X

Qm+k−1 um .

(5)

m=1

In (5), the matrices {Qi } are squared matrices of dimension N/2 − 1, which link the elements of vectors um to the measured data zk . For i > M , Qi = 0, where 0 is the null matrix representing the zero-effect of vectors um when they are out of the analog bandwidth of the measurement set-up due to the LO step. For i = 1, . . . , M , the matrices {Qi } are given by the following equation: N/2−i

i−2

z

0

    ∗ Qi =  1 |{z}   i odd  0

}|

..

.

1

{

∗

z 0 1 .. .

}|

..

.

{  0      1    0

(6)

3 AN

D5

EXAMPLE



0

    Qi =  1 |{z}   i even  0 |

0

  1       0 } ∗

..

.. .

..

.

.

{z

i−2

1 0 1∗ } |

{z

N/2−i

 (7)

where the CN operator 1∗ has been introduced, with 1∗ c = c∗ for a complexvalued scalar c, and where ∗ denotes a conjugate. The CN-operator is linear, i.e., 1∗ (c1 + c2 ) = c∗1 + c∗2 , and it obeys 1∗ 1∗ c = c. By introducing the operator, we obtain a compact representation of the undersampled data and the constructed full-band DFT. Equations (5)-(7) are key results; basic per se, however a powerful tool for these kinds of problems as demonstrated below. Construction of the unfolded DFT u described in (4) is now straightforward. Stacking the measurements yields,      z1 Q1 . . . QM u1  ..   .   ..  . (8)  .  =  ..  .  .. zM

QM

0

uM

Equation (8) can be solved using back-substitution. Calibration is essential when using sub-Nyquist sampling strategies with nonideal measurement systems characterized by their amplitude and phase distortions. Within the derived framework, filtering of measurements can be expressed by multiplying the data by weighting-matrices Fi that correct the amplitude and phase. The Fi matrices can be found by minimizing the mean square error between the reconstructed u and its ideal counterpart in a least squares sense, which is a convex problem. The mathematical framework described above simplifies the symbolic handling of the calibration phase.

3 An example The proposed scheme is exemplified by the reconstruction of a wideband frequency spectrum that is described by a multisine with random amplitude and phase, spanning DC to 500 MHz, as shown in Fig. 2a. As a reference, the frequency spectrum is Nyquist-sampled at a frequency of 1 GHz with N ′ = 5110 sample points collected. For undersampling, M = 5 sets of data are used, corresponding to a bandwidth of 100 MHz with N = N ′ /5 sample points collected. The performance of the method is evaluated with respect to the normalized mean square error (NMSE) in the difference between the original frequency spectrum and the reconstructed one. As shown in Fig. 2b, perfect reconstruction was

D6

U NFOLDING

THE

FREQUENCY SPECTRUM

FOR

U NDERSAMPLED W IDEBAND DATA

0

Spectrum [dBx]

−50 −100 −150 −200 −250 −300 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(a)

5 8

x 10

0

NMSE [dB]

−50 −100 −150 −200 −250 −300 0

0.5

1

1.5

2

2.5

(b) Frequency [Hz]

3

3.5

4

4.5

5 8

x 10

Figure 2: Frequency spectrum: (a) Original frequency spectrum of the multisine sampled at 1 GHz; (b) Normalized mean square error (dB) of the frequency spectrum reconstruction at 200 MHz, with (star) and without (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 the critical frequencies, which is reflected in the peak errors at these tones.

4

Conclusions

A framework for reconstructing a wideband DFT from undersampled measurements is presented. A stepping mechanism incorporated into the local oscillator in the down-conversion stage allows the measurement of different sets of undersampled digital data, which when set in a matrix form with consideration of the effects of aliasing and stepping on the complex frequency spectrum, lead to perfect reconstruction of the sought for DFT.

4 C ONCLUSIONS

D7

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

References [1] J.W. Cooley, and J.W. Tukey, “An algorithm for the machine calculation of complex 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 Institute 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 Journal 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 extend the bandwidth of a power amplifier test-bed”, IEEE Transactions on Instrumentation and Measurement, vol. 56, no. 4, August 2007, pp. 1488-1494. [7] IEEE standard for Digitizing Waveform Recorders, IEEE Standard 10572007. [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 photodiodes,” 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. 21328, 1939. [10] H.D. Lüke, “The origins of the sampling theorem,” IEEE Communications Magazine, vol. 16, no. 4, pp.106-108, April 1999.

D8

U NFOLDING

THE

FREQUENCY SPECTRUM

FOR

U NDERSAMPLED W IDEBAND DATA

[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, 2rd Edition, 2004.

Paper E Peak-to-Average Power Reduction of OFDM Signals by Convex Optimization: Experimental Validation and Performance Optimization Charles Nader, Peter Händel and Niclas Björsell IEEE 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 power reduction on an orthogonal frequency division multiplexing (OFDM) 802.11a signal. A radio frequency power amplifier was excited with an OFDM-signal, and the peak-to-average reduced counterpart and its performance figure of merits were measured and compared. A state-of-art radio frequency test system with high accuracy was used for this purpose. Improvements due to optimization in output power and power added efficiency and the influence of the power distribution in the excitation signal on power amplifier performance were investigated. Improvements of 6dB in output power and 6.5% in power added efficiency were achieved on average near the operating region. The effect of preserving power-free guard subcarriers was introduced in the optimization algorithm and investigated regarding adjacent channel interference. An improvement of 9dB from that aspect was observed using half of the power-free subcarriers, which reveals the importance of a guard interval.

1 Introduction Orthogonal frequency division multiplexing (OFDM) is a widely used modulation scheme because of its high bandwidth efficiency and robustness against frequency fading due to multipath propagation [1, 2]. The power amplifier is a key component in a wireless communication chain as it holds the highest power level in the system. Its power added efficiency (PAE) directly influences the power consumption of the wireless system [3]. Its input-output signal nonlinearity is important for in-band error and out-of-band interference [3]. A major drawback of OFDM is the generally high peak to average ratio (PAR) of the radio frequency (RF) signal entering the power amplifier, which causes early clipping of the signal due to amplifier saturation and results in nonlinear distortions presented in the frequency domain on the shape of unwanted intermodulation products, spectral regrowth and harmonics [3]. Such nonlinear distortions cause spectral interference to adjacent channels and brake the spectral mask standard for emission [1]. Due that, the input power of the power amplifier has to be reduced; that is, a large number of dBs have to be backed-off to keep the amplifier in linear operation. However, such a back-off drastically reduces the PAE of the amplifier because a large amount of power (i.e., heat) must be dissipated [4]. Several methods have been proposed in the literature to reduce the PAR of OFDM signals prior to the conversion to RF, including clipping, dynamic PAR/bias adapting, or redistributing the nonlinear energy on the free subcarriers [5]- [8]. Recently in [9], PAR minimization was formulated as a convex optimization problem. The power spectral density of the signal to be transmitted was reshaped by minimizing the time domain peak power, subject to some constraints on the error vector

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OFDM SIGNALS BY C ONVEX O PTIMIZATION : E XPERIMENTAL VALIDATION AND PERFORMANCE O PTIMIZATION

magnitude (EVM) and the use of power-free guard subcarriers. By applying the fast Fourier transform (FFT) and its inverse on the OFDM signal, a customized interior point method (IPM) that finds the near-to global minimum PAR by a fast and reliable algorithm was developed [10]. This PAR optimization approach was further developed in [11, 12] by adding a spectral mask constraint and minimizing the EVM while keeping the PAR below a minimum threshold. In this work, the method in [9] is extended to reduce the adjacent channel interference that arises when the guard intervals are excited. Further, there is significant theoretical interest in applying convex optimization to obtain PAR-reduction in OFDM communication systems, although the literature, and in particular [9, 11, 12], present no in-depth experimental validation of the impact of the PAR reduction of the baseband signal on the effect of the PAE of the power amplifier. Such experimental verification is of utmost importance for improving PAE and reducing the overall energy consumption of wireless communication systems. Another important aspect to investigate is the effect of exciting the free-power guard subcarriers on the channel leakage and its induced adjacent channel interference. In this paper, we experimentally evaluate the PAR reduction method introduced in [9] with respect to how the PAR reduced signal influences the power aspects of the power amplifier, which includes the PAE and adjacent channel interference. An extension of the method [9] is developed which preserves a fraction of the power-free subcarriers as a guard interval. The impact of the extended method on the power amplifier performance and reduction of adjacent channel interference is evaluated. The paper is organized as follows. An OFDM signal was generated, and PAR was optimized, as briefly reviewed in Sec. 2. The PAR-optimized signal was then used to excite a commercial power amplifier, using a state-of-art measurement setup as described in Sec. 3. In Sec. 4, a study of the power amplifier figure of merits is presented for both methods, [9] and its extended version, and results are compared to the corresponding measures of the reference signal. Finally, conclusions are drawn in Sec. 5.

2

OFDM PAR reduction by convex optimization

In this section, a review of OFDM PAR reduction using convex optimization, as formulated in [9], is briefly reviewed. To combat the demonstrated effects of exciting guard intervals, such as adjacent channel distortion, an extended method is proposed. The Section ends with some remarks on algorithmic details.

2.1

PAR reduction by convex optimization

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

2 OFDM PAR

REDUCTION BY CONVEX OPTIMIZATION

E3

16-quadrature amplitude modulation (16-QAM), or 64-QAM. A baseband OFDM signal was generated by dividing the information data into multiple data streams. Each data stream was passed to a subcarrier for modulation. The modulated data streams (symbol streams) were sent in parallel on the orthogonal subcarriers. The frequency constellation was then time domain transformed through IFFT. Cyclic prefix (guard interval) as well as windowing (Hamming) was applied for timespreading handling and intersymbol interference elimination (side lobes suppression). The time domain symbols were then serially packed and sent for in-phase and quadrature (IQ) modulation [1]. Consider c = (c1 , . . . , cn )T as a transmitted OFDM frequency constellation, which is a complex-valued vector of length n. Further, let x be the corresponding time domain signal obtained by a ℓ-times oversampling, that is x = IFFTℓ [c]

(1)

where IFFTℓ [·] denotes the inverse discrete Fourier transform of the zero-padded vector c, resulting in the length-nℓ vector x. To learn more about the role of oversampling in this context, see the references in [9]. Now, the PAR can be defined as n2 maxi (|xi |2 ) (2) PAR = β ¯cH c¯ where xi denotes the entries in x = (x1 , . . . , xnℓ )T , β is a real-valued FFT scaling used to lower bound PAR to 1 and c¯ only contains the contribution by the data carriers, that is ¯ c = S c, where S is a diagonal carrier selection matrix with diagonal elements Si,i = 1 for those d carriers i1 , . . . , id that contain data, and zero otherwise – in our context d = 52 and n = 64. For a given use of the subcarriers, minimizing PAR is equivalent to minimizing the peak-power p = maxi (|xi |2 ), where for all i = 1, . . . , nℓ it holds that |xi |2 ≤ p. The minimization of PAR is obtained both by i) adding power to the free carriers, which are given by (I − S)c, and ii) distorting the data/pilot carriers S c. The introduced distortion of the transmitted constellation has to be bounded, given as a constraints imposed on the EVM. Let c0 = (c0,1 , . . . , c0,n )T be a reference constellation, then EVM is defined as [1] v u id u1 X u s |ci − c0,i |2 ud t i=i1 1 ||S(c − c0 )||2 = (3) EVM = P0 d P0

where i1 , . . . , id denotes the location of data/pilot subcarriers, that is determined by the non-zero diagonal elements of S. In the second equality || · || denotes the Euclidian vector norm. Note that c and c0 are scaled to the same average power for evaluation, that is ||c||2 = ||c0 ||2 [9]. The scalar P0 is the average power of the modulation scheme used. A convex formulation of the problem of minimizing the PAR in (2) of an OFDM baseband signal x in (1) by adding power at the free carriers and distorting

PEAK -TO -AVERAGE POWER R EDUCTION E4

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OFDM SIGNALS BY C ONVEX O PTIMIZATION : E XPERIMENTAL VALIDATION AND PERFORMANCE O PTIMIZATION

the transmitted constellation c away from the ideal constellation c0 , subject to a constraint imposed on the EVM (3) is [9] minimize peak power subject to

p = max(|xi |2 ) i

||S(c − c0 )|| ≤ ǫ

ℜ[cH 0

(4) 2

S (c − c0 )] ≥ −ǫ /2

The first constraint in (4) is a bound on the maximum allowed EVM in (3), where ǫ is a real-valued√positive parameter proportional to the allowed EVM and given by ǫ = EVMmax d P0 , where EVMmax is the maximum allowed EVM for a given bit error rate. The second constraint in (4) is a relaxed constraint on the average transmitted data power ||S c||2 , that is a relaxed constraint corresponding to ||S c||2 ≥ ||S c0 ||2 ; see [9] for details.

2.2

Channel leakage and an extended method

Radio frequency receivers-transmitters (Rx-Tx) are an enhanced type of equipments with performance improving drastically with time and technology. Digital processing has been introduced as a tool to achieve perfection and reduce equipment errors caused by the non-ideality of the analog counter-parts [13]. Such errors include IQ-imbalance, analog to digital converter impairments, non-ideal filters and frequency offset, which affect the spectral occupancy characteristics and potentially generate interference [14, 15]. Exciting the free subcarriers of the guard intervals in the minimization process to redistribute the channel power spectrum and reduce PAR has raised questions regarding its applicability because adjacent channel interference can pop-up as a problematic drawback. Using RF channels with equal bandwidth and spacing put strong 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 method introduced in [9] is extended by preserving a fraction of the power-free subcarriers as guards. The modification is achieved by adjusting the matrix I to have zeros on the diagonal elements relative to the preserved subcarriers.

2.3

Algorithmic details

The convex optimization problem (4), and its extension, can be solved by standard methods, yielding a global optimum p∗ , c∗ , x∗ . The reader is referred to [9] for details on general solvers, as well as specific solvers for the problem at hand. In the experimental verifications presented here, the logarithm-barrier-IPM algorithm presented in [9] is employed. The algorithm starts with a strictly feasible point (c, p) and finds a search direction (v, vp ) and a step size α that update the point with a factor αv and αvp , respectively. The updating procedure is designed to reduce the barrier function value based on respecting the feasibility condition of

3 M EASUREMENT

SMU 200A (R&S) AWG

Controllable Power Supply

Signal Generator

I

DAC LO

DAC

RF

Q I

E5

SETUP AND DEVICE UNDER TEST

DPA

900

Signal Analyser

MS2692A (Anritsu)

DUT

ADC

Q

PC Matlab

Oscilloscope

Hall Sensor Current Probe

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 testing are accuracy and the ability to track fast variations in the signal envelope. Because the aim of this study is to validate the improvements in power amplifier performance, 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 vector signal generator, an Anritsu MS2692A signal analyzer, an Agilent 54610B oscilloscope, an Agilent N2783A high bandwidth hall sensor current probe, an Ericsson LDMOS highly linear driver amplifier, an Agilent E3631A controllable power supply, and a personal computer (PC). The instruments were connected to the PC via LAN or GPIB interface. The output power from the amplifier is measured by the MS2692A, which is accurately calibrated in amplitude and phase over the measured bandwidth of interest to obtain ±0.3dB power accuracy, even for modulated time variant signals. To accurately monitor the drain current vector and obtain accurate PAE readings, a high bandwidth hall sensor current probe was used. Measuring the current envelope through an oscilloscope allowed an envelope-tracking dynamic power consumption up to 100MHz. Additional details on the testbed can be found in [16, 17].

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OFDM SIGNALS BY C ONVEX O PTIMIZATION : E XPERIMENTAL VALIDATION AND PERFORMANCE O PTIMIZATION

To study the performance of the test-setup, an evaluation of the EVM as a function of the input power is realized where the power amplifier is replaced by a connector. EVM of the transmitted baseband frequency constellation and the received counterpart was calculated with respect to input power Pin and carrier frequency. An average system error of -45dB was found, showing good performance regarding in-band error.

3.2

Device under test

A class AB LDMOS high power (47dBm) amplifier was used for the validation process. It has the capability to handle high PAR up to 15dB. WLAN 802.11a uses a bandwidth of 20MHz. In order to ensure inband flatness, the power amplifier was 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 standards. It had 64 subcarriers with 128 OFDM symbols, a cyclic prefix of 1/4, an oversampling rate of 4 and 14dB PAR after being Hamming windowed. The PARoptimized counterpart, based on full use of the guard subcarriers, reached a PAR of 9.5dB after three Newton iterations in the employed algorithm. Optimizing OFDM signals allocates power in the free subcarriers that reside at the channel sides. Such allocation extends the signal bandwidth from 16.6MHz for the reference signal to 20.0MHz for the optimized one. To combat this effect, the reference OFDM signal is optimized based on the method in Sec. 2.3, where half of the power-free subcarriers are preserved as a guard interval, which results in an effective bandwidth of 18.0MHz and a PAR of 9.75dB. This guard margin is sufficient for reducing channel leakage, but requires increasing the number of Newton iterations because two additional 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 dB compression point of the amplifier, reducing the back-off margin, and allowing an efficient use of the available power. Fig. 2 shows the PAE of the amplifier as function of Pin for both signals, reference and PAR-optimized with six reserved guard subcarriers, where the 1dB compression points have been specified. As shown in Fig. 2, the 1dB compression point of the amplifier was extended by 1.5dB which improved the PAE by 4.2% near the compression region. The power amplifier gain was found to have similar shaping with a 1.5dB extended compression region. Considering the operating region of the amplifier in a real application, which is the compression region backed-off by the respective PAR, an

4 R ESULTS

AND

E VALUATION

E7

40 Reference Optimized with guard margin X: 30.5 Y: 33.04

35 1dB Compression

30 X: 29 Y: 28.85

PAE [%]

25

20 Backed−off Region 15

10 X: 21 Y: 9.651

5 X: 15 Y: 3.197

0 14

16

18

20

22 24 26 Input power [dBm]

28

30

32

34

Figure 2: Power added efficiency of the device under test versus input power for the reference signal (dashed line) and PAR-optimized signal with 6 guard subcarriers (solid line).

improvement of 6.5% in PAE can be seen between the reference and optimized signal. In fact, reducing the PAR by 4.5dB and extending the compression region by 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 resulted in similar performance regarding PAE and output power compared to that of a PAR-optimized signal with guard subcarriers.

4.2 In-band errors Adjusting the frequency constellation power might raise questions regarding inband errors in the system as well as out-of-band errors due to spectrum regrowth. An evaluation of the output signal EVM versus input power, before and after optimization, is presented in Fig. 3. Despite the 12.6dB EVM difference between the signals in the backed-off region, the EVM of the optimized signal (with guard subcarriers) follows the standard limit value (-19dB for the used signal [1]) with a margin error of -0.3dB. Similar results were obtained when exciting with the

PEAK -TO -AVERAGE POWER R EDUCTION E8

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OFDM SIGNALS BY C ONVEX O PTIMIZATION : E XPERIMENTAL VALIDATION AND PERFORMANCE O PTIMIZATION

−10 Reference Optimized with guard margin EVM Standards for 16QAM

X: 30.5 Y: −14.19

−15 X: 21 Y: −18.66

X: 29 Y: −17.1

EVM [dB]

−20

−25

−30

−35 14

X: 15 Y: −31.31

16

18

20

22 24 26 Input power [dBm]

28

30

32

34

Figure 3: Error vector magnitude of the device under test versus input power for the reference signal (dashed line) and PAR-optimized signal with 6 guard subcarriers (pointed line).

PAR-optimized signal without guard intervals. Such behavior is expected as the EVM constraint in the optimization algorithm was set to the standard limit in order to study the maximum improvements in PAE and ACPR. One should mention that the state of the art coding/decoding techniques are successful in correcting for in-band errors as long as the standards limits for EVM are fulfilled. By that, the out-of-band emissions are the most problems tackled nowadays as their effects (intermodulation products and spectrum regrowth) interfere with the adjacent channels and violate the regulations for spectral emission. In summary, despite the changes made to the constellation diagram after optimization, the EVM achieved at the output of the power amplifier in the operation region (backed-off) is still sufficient for allowing decoding algorithms to correct caused errors and restore original information. It complies with the IEEE 802.11a error standard [1] in that region of interest.

4 R ESULTS

AND

E VALUATION

E9

10 Reference Optimized guard−free Spectral Mask 802.11a

20.0MHz 16.6MHz 0

30KHz Interference Spectrum [dBX]

−10

−20

−30

−40

−50 −4

−3

−2

−1

0 Frequency [Hz]

1

2

3

4 7

x 10

Figure 4: Power spectrum of both the reference signal (pointed line) and the PAR-optimized signal (dashed line) at back-off region, compared to 802.11a spectral mask standard for emission (solid line).

4.3 Spectral mask and out-of-band errors Adding power to the sideband free subcarriers increases the bandwidth of the optimized signal. Fig. 4 presents the power spectrums of both reference and optimized signals without guard interval at back-off region, compared to the 802.11a spectral mask standard for emission [1]. As shown, a 16.6MHz bandwidth is found for the reference signal, while exciting the free subcarriers leads to a 20.0MHz bandwidth. Such an increase in the spectrum breaks the standard constraint mask for emission on the lower side of the channel by 30KHz and raises questions about the applicability of such a method. A worse behavior is found at compression region where the spectral mask is violated at both the upper and lower channels. Violating the spectral mask for emission by 30KHz will generate strong interference to neighboring channels which marks the importance of preserving a fraction of the power-free subcarriers as guard interval. Fig. 5 presents the power spectrum of both signals, reference and optimized with 6 guard subcarriers, at back-off region. It shows complete agreement with the IEEE 802.11a standards spectral mask emission.

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10 Reference Optimized with 6 guard sub−carriers Spectral Mask 802.11a 0

18.0MHz −10 Spectrum [dBx]

16.6MHz

−20

−30

−40

−50 −4

−3

−2

−1

0 Frequency [Hz]

1

2

3

4 7

x 10

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

Regarding the out-of-band errors, measurements of the adjacent channel power ratios (ACPRs) of the three signals at 20.0MHz channel spacing and bandwidth, at different input power levels, as presented in Fig. 6, show a large variation between the reference signal and the guard-free optimized one, which is clearly revealed in the lower channel side. Such an increase in the ACPR is due to the 30kHz leakage/interference from the main channel, which biases the value of ACPR. Comparing the above result with the ACPR of the optimized signal with a guard interval shows an improvement of 9dB in the lower channel near the back-off region with respect to not using guard margin, while a 1dB improvement was found in the transition region with respect to the reference signal. Considering the upper adjacent channel, an average ACPR improvement of 1.5dB in the transition region between the backed-off and compression regions was found between the guard-free optimized signal and the reference one; while preserving a guard interval in the optimization process showed an ACPR improvement of 2dB in the back-off region compared to not using a guard margin.

4 R ESULTS

AND

E VALUATION

E11

−18

−20

Lower Channel Optimized with 6 Guard sub−carriers Upper Channel Optimized with 6 Guard sub−carriers Lower Channel Optimized without Guard sub−carriers Upper Channel Optimized without Guard sub−carriers Lower Channel Reference Upper Channel Reference

−22

ACPR [dBc]

−24

−26

−28

−30

−32 14

16

18

20

22 24 Input power [dBm]

26

28

30

32

Figure 6: Comparison of adjacent channel power ratio for both optimized signals, with and without power-free guard subcarriers, and the reference signal.

The achieved improvements between both optimized signals points the necessity of having a frequency guard interval in the signal for adjacent channel interference reduction. From an application point-of-view, the method introduced in [9], and its derivations, need to consider this aspect. Exciting part of the powerfree guard subcarriers is sufficient to achieve similar-to-better power performance compared to that with the full use of the guard interval, with just two extra Newton iterations in the optimization algorithm. In summary, optimizing the signal while preserving a fraction of the powerfree subcarriers as a guard interval improves the ACPR by an average of 2dB. This is caused by the absence of the clipped high peaks that usually cause early bird spectrum regrowth. Such small improvements can still reduce the adjacent channel interference and the associated errors, but more importantly, this improvement is advantageous compared to the clipping-based algorithms that usually generate undesired regrowth in the output spectrum of the power amplifier and require sophisticated methods to filter the regrowth without regenerating the peaks.

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2

10

Reference Optimized

1

10

0

CCDF [%]

10

−1

10

−2

10

−3

10

0

5

10

15

PAR [dB]

Figure 7: Complementary cumulative density function of the signal peaks to average ratio for the reference signal (dashed line) and the PARoptimized signal with guard margin (solid line).

4.4

Amplifier saturation

Reducing the PAR should generally increase the average power at the output of the amplifier by a couple of dBs, which will lead to a compression point at a higher input power level, because fewer peaks excite the amplifier’s nonlinearities. However, contrary to what was expected, an average increase of 1.5dB was found near compression, and requires further investigation to explain this behavior. A study of the complementary cumulative density function (CCDF) of the peaks distribution in the measured signals, reference and optimized with guard interval, would justify such a result. Fig. 7 presents the CCDF of all signal peaks normalized to the signal average, for both measured signals while Fig. 8 shows the CCDF of the normalized 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 at the cost of increasing the lower peaks density distribution, which in turn limited the improvement near saturation. In fact, despite the reduction of the high peaks, the optimization algorithm allowed the “smaller” peaks to increase. Ultimately the

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2

10

Reference Optimized Reference Average Optimized Average 1

10

X: 0.652 Y: 5.593

0

CCDF [%]

10

X: 0.6519 Y: 0.3247 −1

10

−2

10

−3

10

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Envelope normalized to relative Peak

0.8

0.9

1

Figure 8: Complementary cumulative density function of the normalized envelope for both the reference signal (dashed line) and the PARoptimized signal with guard margin (solid line).

total energy, due to nonlinearity, retained a comparable value to the non-optimized case, sufficient to push the power amplifier into compression. Such aspect in power improvement near saturation represents the tradeoff that must be considered when choosing between advanced optimization method like convex optimization and the low-complexity clipping based methods. In fact, clipping the signal only reduce the dominant peaks, resulting in an extra power margin for saturation to be reached. However, due to the distortion created in the excitation signal, higher side band levels are expected to arise which cause higher out-of-band errors.

5 Conclusion Optimizing the PAR of OFDM signals based on convex optimization algorithms was performed, and before/after figure of merits of a contemporary RF power amplifier were evaluated. The employed PAR optimization algorithm based on [9] resulted in limited im-

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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, in the 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 such limited gain improvement near compression is the increase in lower peak density distribution in the excitation signal. Spectral emission was considered to be a drawback in the method because power leakage was seen by the high ACPR. The main cause of the leakage is the absence of guard subcarriers. An extended version of the method in [9] was implemented in which half of the guard subcarriers were used to prevent such interference. The extended method showed the necessity of preserving part of the guard interval with a low iterative cost in optimization. An improvement up to 9dB was found in ACPR for the lower channel side, while the power performance maintained its merits values. Even though PAR reduction by advanced methods, such as convex optimization, costs more in terms of additional digital signal processing, it is commonly considered to be a worthwhile technology because the digital processing is “free”; according to Moore’s law, it will be cheaper and cheaper with time. We have experimentally shown an extra 6dB in output power and 6.5% in power added efficiency due to digital processing of the transmitted signal, without any additional requirements from the hardware. This result is believed to be of significance in a world “where every dB is worth a Billion” [18].

References [1] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface 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 Standard 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,” in Workshop on RF Circuits for 2.5G and 3G Wireless Systems, Feb 2001. [5] H. Ochiai, and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” 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, “Performance improvement of power amplifiers with digital linearization technol-

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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 amplifier adaptation for power efficient operation of WiMAX OFDM transmitters,” IEEE, Radio and Wireless Symposium, pp. 21-24, Jan 2008. [8] J. Tellado, “Multicarrier Modulation with Low PAR: Applications to DSL and Wireless, Norwell, MA: Kluwer, Sept 2000. [9] A. Aggarwal, and T.H. Meng, “Minimizing the peak-to-average power ratio of OFDM signals using convex optimization,” IEEE Transactions on Signal Processing, vol. 54, pp. 3099-3110, Aug 2006. [10] S. Boyd, and L. Vandenberghe, Convex Optimization, Cambridge, UK: Cambridge Univ. Press, Mar 2004. [11] Q. Liu, R.J. Baxley, and G.T. Zhou, “Free subcarrier optimization for peakto-average power ratio minimization in OFDM systems,” in Proceedings IEEE ICASSP, Las Vegas, NV, Mar 2008. [12] Q. Liu, R.J. Baxley, X. Ma and G.T. Zhou, “Error vector magnitude optimization for OFDM systems with a deterministic peak-to-average power ratio 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 Information Networks, vol. 14, no. 2, pp. 133-148, June 2007. [14] A. Behzad, Wireless LAN Radios, IEEE Press on Digital and Mobile Communication, 2007. [15] T.C.W. Schenk, and E.R. Fledderus, “RF impairments in high-rate wireless systems - understanding the impact of TX/RX-asymmetry,” in 3rd International Symposium on Communications, Control and Signal Processing, ISCCSP 2008, pp. 117-122, Mar 2008. [16] C. Nader, H. Altahir, O. Andersen, N. Björsell, E. Condo, N. Keskitalo, and H. de la Rosa, “Automated multidimensional characterization of power amplifier for design and production,” in International Instrumentation and Measurement Technology Conference Proceedings, I 2 M T C 2009, pp. 144-148, May 2009. [17] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend the bandwidth of an RF power amplifier test bed,” IEEE Transactions on Instrumentation and Measurement, vol. 56, pp. 1488-1494, 2007.

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