NONLINEAR behavioral modeling and wireless components

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006 2659 A Corrected Microwave Multisine Waveform Generator Nuno Borges Carvalho, Senior Member, IEEE, José Carlos Pedro, Senior Member, IEEE, and João Paulo Martins Abstract This paper proposes a solution for one of the major difficulties associated with arbitrary waveform generation at microwave frequencies: the nonideal RF circuits used for signal generation. It presents a practical implementation of a RF multisine signal generator that corrects for the linear and nonlinear impairments identified with its output RF circuitry. The integration of an adaptive predistortion algorithm with a previously presented methodology for multisine synthesis with prescribed higher-order statistics, proved the feasibility of a simple microwave arbitrary waveform generator that meets the aims of previously developed signal synthesis theory. Index Terms Behavioral science, measurements, nonlinear systems, waveform analysis. I. INTRODUCTION NONLINEAR behavioral modeling and wireless components characterization have been demanding for signals specially designed to fulfill certain statistical characteristics. The multisine excitation is one of the most important available signals. It combines a sum of sine waves of different frequencies, but phase correlated to each other. This correlation is a must, from both the measurement and the data analysis viewpoints, since it enables the identification of each output and input component using the discrete Fourier transform (DFT) or the fast Fourier transform (FFT). Although multisines have, for long, been used in active device characterization and modeling [1] [6], their laboratory generation has not yet received the attention it deserves. For instance, there are several works [4] [7] in which no restrictions are imposed to the phase of each tone in the multisine pattern. Thus, there is no control of even basic signal properties as the amplitude probability density function (pdf). Recently, new theoretical algorithms to obtain multisine signals with desired statistics have been proposed [1], [2]. These algorithms allow the RF design engineer to generate a multisine with some prescribed higher order statistics, at least in the mathematical description and/or in the simulator computer engine. Those statistical properties are achieved for a predetermined phase and amplitude arrangement of the tones. Similar approaches have also been studied and proposed for multisines with minimum crest factors, a goal also satisfied by controlling the phase and amplitude arrangements. Manuscript received October 3, 2005; revised January 12, 2006. This work was supported by the European Union under the Network of Excellence Top Amplifier Research Groups in a European Team Contract IS-1-507893-NoE, and under the MusiLage Project. The authors are with the Instituto de Telecomunicações, Universidade de Aveiro, 3810-193 Aveiro, Portugal (e-mail: nborges@det.ua.pt). Digital Object Identifier 10.1109/TMTT.2006.872947 The authors of this paper have recently established the theoretical framework for the solution of a problem of great engineering significance by proving that the response s power spectral densities of an arbitrary (causal, stable, and of finite memory) nonlinear dynamic system to two different signals are equal if and only if the two signals have equal higher order autocorrelations or power spectral density functions. Thus, they proposed special algorithms to design a standard signal stimulus a multisine capable of mimicking the test of a nonlinear dynamic system under a predetermined excitation [1], [2]. They then also advanced a numerical method for designing a multisine that shows such higher order statistics, which established the methodology necessary to conceive an ideal RF arbitrary waveform generator. Unfortunately, those theoretical results become of limited practical value, unless the laboratory generation schemes allow the designed phase relation between the tones to be unaltered throughout the generator s RF path. If that is not the case, the actually obtained signal statistics and/or crest factors can differ significantly from the desired ones. When these multisine signals are to be created by a real laboratory instrument, several problems appear, which can be traced back to the nonlinear and dynamic path from the digital signal processing (DSP) generator to the output of the RF generator itself (see Fig. 1). Indeed, as the multisine flows through the mixers, amplifiers, and filters, the tones amplitudes and relative phases are modified, necessarily impairing the required signal statistics. If those changes result in mere linear phase shifts and amplitude attenuation, they can be compensated by including that effect in the DSP as a linear equalizer. However, if these effects are nonlinear, the referred phase shifts and attenuation depend on the amplitude of the multisine envelope, and a nonlinear equalizer (some form of dynamic linearizer) should be used [8]. It should be considered that the digital predistortion applied to this case should compensate for amplitude and phase changes. In fact, the authors of [7] also recognize the distortions generated by the generator s front-ends and the need to correct them. However, as their measurement setup can only acquire amplitude information (it is based in a spectrum analyzer), they concentrate on eliminating spurious signals and spectral regrowth. They, therefore, left uncovered the possible change of relative phases experienced by the generated tones (even caused by linear distortions, e.g., in the digital-to-analog converter reconstruction filter), which can be catastrophic for the signal s envelope shape and peak-to-average ratio, and these are known as very important excitation characteristics in the nonlinear test of modern wireless systems and components. 0018-9480/$20.00 2006 IEEE

2660 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006 Fig. 1. Block diagram of the proposed multisine generator. Recently, in [9], a theoretical algorithm was specially conceived to minimize the impact of the generator output RF-frontend. This paper expands upon this groundwork by presenting a practical implementation of such a corrected microwave multisine waveform generator. This paper is organized as follows. First, a revision of the previously advanced theoretical algorithm is presented in Section II. Section III then presents a practical implementation of the proposed generator, and Section IV demonstrates its practical usage via a real laboratory setup. Finally, in Section V, the obtained results are summarized and discussed, and some conclusions are drawn. Expression (1) presents the HB system of equations where and (1) II. ENVELOPE TRACKING BY HARMONIC-BALANCE (HB) ALGORITHM REVISITED As this paper reports on the practical implementation of the algorithm proposed in [9], it is convenient to start by briefly reviewing this theoretical groundwork. As sketched in Fig. 1, the underlying idea to obtain an ideal waveform generator is to compare the actual output multisine with the desired one. The error signal built this way is then used to dynamically adjust the magnitudes and phases of the DSP synthesized multisine up to a point where the error is small enough. The optimization algorithm is based on the HB engine commonly used in the simulation of RF/microwave nonlinear circuits [3]. Similarly to the HB algorithm used for nonlinear circuit simulation, the current algorithm starts from the a priori knowledge of the frequency position of the signal components (the frequency location of the multisine tones), to then balance the amplitude and phase of the positions corresponding to each frequency at all circuit nodes. However, contrary to circuit simulation, in which the circuit and excitation are known and the output is sought, it is now the output signal phase and amplitude of each tone that is perfectly known, while the excitation is unknown. Thus, the algorithm will work on a conceptual port where the error signal between the actually generated and the desired multisine is built. By using an optimization technique, the amplitudes and phases of the DSP synthesized multisine will then be updated to obtain the sought multisine. are the sought digital multisine and multisine obtained at the analog output of the generator itself, respectively. and are the amplitude and phase of the sought multisine, while and are the amplitude and phase of the multisine at the output of the generator. The idea is to minimize the value of the error so that the actually generated multisine approximates the desired one. In order to achieve that goal, we need to measure the output of our generator. As will be seen below, this can be done via a sampling oscilloscope, or a microwave transition analyzer (MTA). III. PRE-DESCRIBED STATISTICS MULTISINE WAVEFORM GENERATOR The first step undertaken in our ideal waveform generator consists of the generation of the intended multisine signal. This was carried out by determining the amplitude and phase of each multisine bin via the MATLAB [10] algorithms previously described in [1] and [2]. The output of these algorithms is the set of amplitudes and phases for each of the multisine bins, resulting in the digital version of the real ( ) and imaginary ( ) components of a periodic low-pass baseband signal sampled at 20 MHz and with 32 768 samples. This digital signal is then downloaded onto the physical memory of an arbitrary RF arbitrary waveform generator, i.e., an HP 4433B ESG-D.

BORGES CARVALHO et al.: CORRECTED MICROWAVE MULTISINE WAVEFORM GENERATOR 2661 Fig. 2. Arbitrary waveform generator. As depicted in the block diagram of the RF arbitrary waveform generator of Fig. 2, this digital baseband signal is then moved to the analog domain via two independent DACs and then up-converted onto the carrier frequency in an modulator. Finally, the bandpass RF signal is then amplified and presented at the output of our arbitrary waveform generator. This step concludes the generation of the uncorrected multisine. Thus, this RF signal prototype must now be acquired and evaluated by comparing it to the desired multisine. In the implemented instrument, the output signal acquisition was made in the time domain via an MTA, i.e., an HP 70820 A, which operates as a fast sub-sampling oscilloscope. This way, the MTA stores at each iteration a time series record of the bandpass signal, which must then be compared with the desired multisine. Asthesetoffrequencypositionsisfixed, themultisinebecomes defined as the amplitude and phase of each tone, which can be acquired via the Fourier transform. In this particular case, and since each of the multisine tones is synchronous to a known reference, wecouldusetheusualdftoritsfastalgorithm,thefft, tocalculate the amplitudes and phases of the recorded time series signal using a multisine amplitude and phase determination algorithm. The amplitudes were determined through the direct application of the FFT because the signal is perfectly correlated to the same phase reference, thus showing no spectral leakage. Unfortunately, the phase determination is significantly more difficult. Since the acquisition instant is random (asynchronous trigger), the raw values of the phases of the desired spectral components are also random. However, if a time alignment is performed on each record, the phase differences between tones become deterministic, enabling the sought comparison between the components of the measured signal and the differences stored for the desired signal. In the optimum case where a perfect compensation is achieved, these differences should be zero. Note that this phase alignment demands for a large amount of data to be acquired, similar to [11]. Fig. 3 presents the block diagram used for this phase-measuring scheme. At this point, we already have a way to evaluate the generated signal and, thus, we can obtain the amplitude and phase of each of the output multisine signal. We can now use the statistical characteristics of the signals, for instance, by evaluating the signal cumulative distribution function (cdf) as a comparison reference. This cdf function is calculated by first quantifying the signal amplitudes according to a set of discrete levels, and then by counting the number of amplitude events that fall in each amplitude interval. This way, we can immediately compare the cdf (or the pdf, which is Fig. 3. Phase-measuring algorithm. nothing but the derivative of the cdf), of the desired multisine and the obtained multisine. Fig. 4 summarizes this stimulus generation and subsequent measurement procedure by showing the system block diagram of a possible multisine test bench. In that setup, we can see the referred arbitrary waveform generator, a digitizer in the current case, the MTA and a computer that will be responsible for the algorithm implementation, controlling the DSP signal that will be programmed at the signal generator. As a final remark, please note that, although a closed box microwave multisine synthesizer solution would naturally include the output signal sampler, the error generator and the whole feedback correction loop inside the same instrument, the availability of an external sampler can be used to guarantee that the device-under-test (DUT) is exactly driven by the desired signal. That may be useful to correct for frequency-dispersive delay and attenuation introduced by the connectors and cables that feeds the DUT, but it will be also useful for correcting any nonlinear distortion present in this signal path. That is of paramount importance in the nonlinear distortion tests of highly linear high power microwave devices and circuits in which a (obviously nonideal) signal booster is normally inserted between the multisine generator and DUT [3]. Moreover, and despite that the operation description has concentrated on the signal fundamental components, the suggested algorithm and setup can also minimize any spectral

2662 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006 Fig. 4. Experimental setup for the ideal waveform generator. regrowth arising in the instrument s RF-front-end and the generator-to-dut signal path since the proposed control is actually implementing some form of adaptive digital predistortion linearization. IV. EXPERIMENTAL RESULTS In order to prove the practical applicability of the described corrected signal generation methodology, the generation and measurement instruments, presented in Section III, were built and the algorithm was implemented into a computer. For demonstrative purposes, we drove the output RF front-end of our generator near its 1-dB compression point so that the nonlinear distortion could be maximized and, thus, more easily handled. The generated signal is a multisine with ten tones, centered at 1.5 GHz, and with a separation of 1 khz. The MTA obtains 5000 samples acquired at a rate of 19.997 MHz, and the algorithm was run through 250 iterations. Due to the nonlinear behavior of the output RF front-end of the used signal generator, the shape of the multisine obtained is significantly different from the one programmed in the instrument. The differences in the output signal are visible, not only in the time waveform, but also in its statistics, as can be seen in Figs. 5 8. In fact, comparing Figs. 5 and 6, the modification faced by the envelope waveform, especially in its amplitude lower end, is evident. The proposed algorithm was then applied to this signal by first acquiring the signal with the MTA. The phase and amplitude of each bin of the multisine was then obtained as previously explained. The total error calculation used for the optimization of the multisine algorithm is the sum of the amplitude error with the phase error. Fig. 5. Signal time waveform of the desired signal. Fig. 6. Signal time waveform at the output of our signal generator.

BORGES CARVALHO et al.: CORRECTED MICROWAVE MULTISINE WAVEFORM GENERATOR 2663 Fig. 7. cdf of the desired and obtained signals. Fig. 10. cdf of the desired and obtained signals after optimization. Fig. 8. pdf of the desired and obtained signals. Fig. 11. pdf of the desired and obtained signals after optimization. obtained from the acquired signal computed after the alignment procedure. This process is done for each iteration, and after running an optimization process, the output mimics the desired signal almost perfectly, as can be seen in the output cdf and pdf presented in Figs. 9 11. As can be seen, the final optimized multisine presents good agreement with the initial proposed one. Fig. 9. Signal time waveform after optimization. The amplitude error is obtained by the sum of the relative differences between the measured components and the desired components. The phase error is thus the sum of the distance between the stored phase differences from the desired signal and the ones V. CONCLUSION In this paper, a corrected waveform generator was proposed to generate a multisine with some prescribed statistics and timedomain waveform. Moreover, a new laboratory setup was also presented in order to implement the proposed algorithm. In this setup, the phase and amplitude determination was also studied. This nonlinear dynamic predistortion overcomes the drawbacks associated to any form of linear or nonlinear distortion impairments. This way, better multisine signals specially designed for RF characterization and/or model extraction are obtained.

2664 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006 ACKNOWLEDGMENT The authors would like to acknowledge R. Estanqueiro Santos, Instituto de Telecomunicações, Universidade de Aveiro, Aveiro, Portugal, for helping with the measurements. REFERENCES [1] J. C. Pedro and N. B. Carvalho, Designing bandpass multi-sine excitations for microwave behavioral model identification, in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 791 794. [2] J. C. Pedro and N. B. Carvalho, Designing multi-sine excitations for nonlinear model testing, IEEE Trans. Microw. Theory Tech., vol. MTT-53, no. 1, pp. 45 54, Jan. 2005. [3] J. C. Pedro and N. B. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA: Artech House, 2003. [4] S. Boyd, Multitone signals with low crest factor, IEEE Trans. Circuits Syst., vol. CAS-33, no. 10, pp. 1018 1022, Oct. 1986. [5] K. A. Remley, Multi-sine excitation for ACPR measurements, in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 2141 2144. [6] A. R. Varkonyi-Koczy, Synchronized multi-sine measurements via DSP methods, in Proc. IEEE Instrum. Meas. Technol. Conf., 1996, vol. 2, pp. 1056 1060. [7] D. Rabijns, W. Van Moer, and G. Vandersteen, Spectrally pure excitation signals: Only a dream?, IEEE Trans. Instrum. Meas., vol. 53, no. 5, pp. 1433 1440, Oct. 2004. [8] P. B. Kennington, High Linearity RF Design. Norwood, MA: Artech House, 2000. [9] N. B. Carvalho and J. C. Pedro, Laboratory generation of multi-sines with pre-described statistics, in Eur. Microw. Conf., Paris, France, Oct. 2005, pp. 1199 1202. [10] MATLAB 7.0. The Mathworks Inc., Natick, MA, 2004. [11] K. A. Remley, D. F. Williams, D. M. M.-P. Schreurs, G. Loglio, and A. Cidronali, Phase detrending for measured multisine signals, in 61st ARFTG Conf. Dig., Philadelphia, PA, Jun. 13, 2003, pp. 73 83. Nuno Borges Carvalho (S 92 M 00 SM 05), was born in Luanda in 1972. He received the Diploma and Doctoral degrees in electronics and telecommunications engineering from the Universidade de Aveiro, Aveiro, Portugal, in 1995 and 2000, respectively. From 1997 to 2000, he was an Assistant Lecturer with the Universidade de Aveiro, in 2000 was a Professor, and is currently an Associate Professor. He is also a Senior Research Scientist with the Instituto de Telecomunicações, Universidade de Aveiro. He was a Scientist Researcher with the Instituto de Telecomunicações, during which time he was engaged in different projects on nonlinear computer-aided design (CAD) and circuits and RF system integration. He coauthored Intermodulation in Microwave and Wireless Circuits (Artech House, 2003). He has been a reviewer for several magazines. His main research interests include CAD for nonlinear circuits, design of highly linear RF-microwave power amplifiers, and measurement of nonlinear circuits/systems. Dr. Borges Carvalho is a member of the Portuguese Engineering Association. He is a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and an IEEE MTT-11 Technical Committee member. He was the recipient of the 1995 Universidade de Aveiro and the Portuguese Engineering Association Prize for the best 1995 student at the Universidade de Aveiro, the 1998 Student Paper Competition (third place) presented at the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), and the 2000 Institution of Electrical Engineers (IEE) Measurement Prize. José Carlos Pedro (S 90 M 95 SM 99) was born in Espinho, Portugal, in 1962. He received the Diploma and Doctoral degrees in electronics and telecommunications engineering from the Universidade de Aveiro, Aveiro, Portugal, in 1985 and 1993, respectively. From 1985 to 1993 he was an Assistant Lecturer with Universidade de Aveiro, and a Professor since 1993. He is currently a Professor with the Universidade de Aveiro. He is also a Senior Research Scientist with the Instituto de Telecomunicações, Universidade de Aveiro. His main scientific interests include active device modeling and the analysis and design of various nonlinear microwave and opto-electronics circuits, in particular, the design of highly linear multicarrier power amplifiers and mixers. He coauthored Intermodulation Distortion in Microwave and Wireless Circuits (Artech House, 2003). He has authored or coauthored several papers appearing in international journals and symposia. Dr. Pedro has served as a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He was the recipient of the 1993 Marconi Young Scientist Award and the 2000 Institution of Electrical Engineers (IEE) Measurement Prize. João Paulo Martins was born in Sever do Vouga, Portugal, on May 13, 1973. He received the B.Sc. and M.Sc. degrees from the Universidade de Aveiro, Aveiro, Portugal, in 2001 and 2004, respectively. From 2001 to 2003, he was a Researcher with the Instituto de Telecomunicações, Universidade de Aveiro. His main interests are in wireless systems and nonlinear microwave circuit design.