Instrumentation and Measurement Technology Conerence IMTC 2007 Warsaw, Poland, May 1-3, 2007 Validation o a crystal detector model or the calibration o the Large Signal Network Analyzer. Liesbeth Gommé, Johan Schoukens, Yves Rolain, Wendy Van Moer Vrije Universiteit Brussel (Dept. ELEC/IW); Pleinlaan 2; B-1050 Brussels (Belgium) Phone: +32.2.629.28.68; Fax: +32.2.629.28.50; Email: lgomme@vub.ac.be Abstract - I the Large Signal Network Analyser [1] is used to measure the waveorms present at the device ports, two additional calibration steps, a power and a phase calibration, are needed besides the classical linear SOLT-calibration. The aim o this paper is to estimate and validate a parametric black-box model or a crystal detector. This detector is used as a reerence element to calibrate the phase distortion o a LSNA operating under narrow band modulated excitation. Index Terms - Squared crystal detector, eedback model, phase calibration. I. INTRODUCTION The absence o the superposition principle or nonlinear systems results in a complexity increase o the characterisation o these systems by at least an order o magnitude. A possible approach to handle this complexity is to use simple models, which describe the behaviour o the device or a certain restricted class o input signals [2]. For such a simple approach to be valid, it is mandatory that the actual signal during the operation o the device and the test signal belong to the same class. In telecom applications, this calls or the use o narrow band (a ew % o the carrier requency) signals with a large number o tones. One o the challenges in modulated measurements resides in the calibration o the instrument s phase distortion or such signals. The calibration o continuous wave (CW) carriers and their harmonics relies on the well-established step recovery diode (SRD) method [1]. This method cannot be used or spectra containing lines that do not lie on the harmonic grid ( 0, 2 0...). As a consequence, it cannot be used to calibrate a narrow band modulated signal. The crystal detector (HP 420C, [3]) can be used as a reerence element in this context, because it translates the envelope o the RF-signal to IF requencies. Assuming a perect detector, this process is independent o the carrier requency and does not inluence the relations between the spectral lines o the modulation signal. The identiication o a model or a realistic crystal detector can then easily be undertaken using dierent RF-signals with a dierent carrier requency and a similar modulation. Unortunately, this approach is not easible in practice. The Digital Sampling Oscilloscope (DSO, [4]) that has to be used to acquire the RF-signals has a restricted memory depth (a ew 1000 samples) and hence is not capable to measure a long modulated signal like the one being used here. To get around this limitation, a two step procedure is proposed: irst, it is conjectured that downconversion o the detector is indeed independent o the RF-carrier requency. This allows one to extract a block oriented baseband model or the nonlinear detector that consists o a linear direct path and a nonlinearity in the eedback path. I the conjecture appears to be valid, the model can be used in the second step to predict the device output when excited with a modulated RF-signal. To veriy the conjecture, a modulated signal that is not exactly known can be used. The envelope o this signal has to be short enough to allow measurements by a DSO. The modulated signal is then translated over the RF-band o the detector at dierent requencies and the relations between the spectral lines are measured. As was previously shown in [5], these relations are indeed independent o c, which proves that the conjecture is indeed valid. In the irst part o the paper the requency converting and phase calibrating properties o the crystal detector are discussed. Subsequently, the model extraction process is described. Section IV discusses the validation measurements, including the experimental setup that incorporates the LSNA. This instrument is used to record input and output signals o the squared detector. Section V describes the model validation applied to RF-signals. The comparison o the measured envelope with the model predicted envelope will be described in section VI. 1-4244-0589-0/07/$20.00 2007 IEEE
II. THE CRYSTAL DETECTOR AS A PHASE STANDARD Observe an amplitude modulated (AM) signal, x AM (t) consisting o one carrier requency c and two modulation tones, 1 = c - mod and 3 = c + mod, as shown in igure 1. The modulated AM-signal is, x AM (t)=u 1 cos[2π( c - mod )t+φ 1 ]+U 2 cos[2π c t+φ 2 ]+ U 3 cos[2π( c + mod )t+φ 3 ] (II.1) with U 1, U 2, U 3 and φ 1, φ 2, φ 3, the amplitudes and phases at respectively c - mod, c and c + mod. The modulation tones determine the ine requency grid with a spacing in the order o khz. Feeding x AM (t) to an ideal squaring device results in a signal containing spectral lines in the vicinity o DC and 2 c. The lines located around DC are, 1/2(U 1 2 +U 2 2 +U 3 2 )+U 1 U 2 cos[2π mod t-(φ 1 -φ 2 )]+ U 2 U 3 cos[2π mod t+(φ 3 -φ 2 )]+U 1 U 3 cos[2π(2 mod )t+(φ 3 -φ 1 )] (II.2) The phase dierence between the tones can thereore be determined directly. As a result the input signal is known. I the signal, x AM (t) is now ed to both the crystal detector and an uncalibrated instrument, such as an LSNA, the modulated signal will be distorted by the linear dynamics o the measurement channel. (ig.1) This will result in an asymmetry in the upper and lower side band o the measured modulated signal. I one knows the behaviour o the detector without a systematic error, the applied modulated tone can be measured directly, yielding a direct phase calibration or the LSNA. To this end, the crystal detector is modelled and this model will be used to retrieve the modulated signal. o or example 100 khz instead o the standard 600 MHz, which is achievable with LSNAs nowadays. In order to apply this approach we need to have the detector model at our disposal. The way to obtain this model is discussed in the next section. III. THE CRYSTAL DETECTOR MODEL The identiication o nonlinear dynamic systems is a comprehensive issue. No generally valid model ramework exists, as is the case or linear systems. To date, Wiener- Hammerstein and Hammerstein-Wiener systems are commonly used block structured models that replace the nonlinear system by a cascade o linear dynamic and static nonlinear blocks, [6]. However, these models are open loop systems. The observed nonlinear behaviour o the crystal detector cannot be addressed by such models because at RF, eedback is always present. This can be included by adding a eedback path around the nonlinearity [7]. The identiication process o the crystal detector is thoroughly addressed in [8]. The model contains a low-pass ilter in the orward path and a high-pass ilter ollowed by the nonlinearity in the eedback loop o the block oriented structure as is shown in igure 2. u q s G2 constant G LP Filter Non linearity G1 HP Filter y LSNA port 1 r t ADC SyncroCLK Figure 2 : Feedback model o the square law detector Input x c-mod c c+mod b 1 IF ATT a 1 IF ATT SampleCLK IFCLK MΩ Output y mod ADC To interpret this model o the square law detector, we excite the device with an amplitude modulated signal, u(t). The output signal is denoted as y(t). The detector is terminated in a 50 Ohm load. Crystal detector Isolation Ampliier Figure 1 : Fine requency phase calibration setup While the modulation tones determine the ine requency grid, the carrier requency o the amplitude modulated signal is chosen on the coarse requency grid imposed by the SRD-method [1]. Repeating this procedure with AM-signals o varying bandwidth, results in a calibrated phase grid with a spacing Figure 3 : Amplitude modulated input signal
For modulated signals with a low carrier requency, the device acts as a hal wave rectiier as can be seen in igure 4. Figure 4 : Low carrier requency detector output In hal wave rectiication, one hal o the wave is let untouched while the other hal is blocked, depending on the polarity o the rectiier. This operation is valid up to a carrier requency o about 200 MHz. When we increase the carrier requency o the input signal rom 20 MHz to 200 MHz the output signal amplitude decreases which demonstrates a low-pass ilter behaviour. When the requency is increased, the high-pass ilter will attenuate the RF-signal less and high requency contributions will be presented to the nonlinearity. These high requency contributions then undergo the nonlinear squaring operation. This results in contributions at 2 c and at DC that are added to the input o the low-pass ilter, which then transmits the detected low requency envelope. For the model extraction, low requency data has been acquired. This approach is valid, because the experiments perormed in [5] have proven that the phase and amplitude o the detected output modulation do not change signiicantly with respect to the input modulation or carrier requencies above 600 MHz. At this point, it needs to be checked whether the model is still valid or high requency measurements. IV. VALIDATION MEASUREMENTS A square law detector model that matches the actual detector operation or low requencies is now available or validation in the high requency region. To this end, we perormed calibrated measurements o input and output signals with the LSNA. Figure 5 : Increased carrier requency output When the carrier requency is urther increased beyond 200 MHz, the envelope detection starts to show. From 500 MHz on, the low requency envelope is clearly visible, as is shown in igure 5. In the experiments discussed above, all input signals (ig. 3) have the same signal amplitude and modulation envelope: only the carrier requency shits. These initial experiments indicate that the proposed model structure, as shown in igure 2, can be valid. This simple structure is indeed capable o mimicing the measured behaviour. First, assume the eedback loop to be absent. Consider an amplitude modulated signal being ed into the detector. At low requencies, the low-pass ilter does not attenuate the signal, whereas the high-pass device blocks the access o the signal to the nonlinearity. In order to guarantee that the measurement errors are removed, the setup was also calibrated. We perormed a one-port linear calibration and a power calibration. An RFsignal was constructed using an AWG710 generator which was ed into the detector and measured at port 1 o the LSNA. The detector output was measured using an ADC. The setup can be ound in igure 1. Some Remarks: * It sounds peculiar that the detector model is veriied with the LSNA while the latter is in act the measurement instrument we wish to calibrate. However at this point we are not yet calibrating the LSNA and only veriying a model. * One might note that in case o measuring with the LSNA, the phase is calibrated through the SRD-method combined with interpolation on a grid with a requency spacing o 2 MHz. However, the phase dierences at the operating requency vary about a ew tenths o a degree in between two components with a spacing o 2 MHz, which is negligible compared to the luctuations we want to measure.
A. Input Signal Figure 6 shows measurements o the input time domain waveorm and igure 7 shows the input spectrum and its uncertainty. bandwidth has doubled because o the presence o 2 mod contributions o the squaring operation as stated in eq (II.2). Figure 8 : ADC Output Spectrum Figure 6 : LSNA Input Time Domain Waveorm The input signal is a multisine signal composed o 41 lines with a spacing o 20 khz and hence deining a bandwidth o 800 khz with a carrier requency at about 800.4 MHz. This excitation signal is chosen, as the modelling experiments have been perormed using a signal o the same bandwidth. In addition, we keep the power level o the excitation or the validation equal to the one used in the modelling. The LSNA is conigured to measure the 41 components. The variance on the measurements was calculated rom 20 repeated measurements. Inormation on the variance o input and output measurements is necessary because it should not exceed the model variability in order to have meaningul measurement data. V. MODEL VALIDATION In order to determine the model s accuracy we calculate the dierence between the measured envelope and the model predicted envelope. The measured RF-input (ig.6) is applied to the eedback model as depicted in igure 2. For initialisation, we assume the eedback loop to be absent. Signal t (ig.2) is calculated as the output o the multiplication o the transer unction o the low-pass ilter in the eedorward path, and the high-pass ilter in the eedback path. B. Output Signal Figure 7 : LSNA Input Spectrum At the output we ind the downconverted low requency response o the crystal detector with a total bandwidth o 1600 khz. The output signal, measured with the ADC, is shown in igure 8 together with the uncertainty. The The extracted discrete time models are itted with irst order systems in the s-domain by means o the Matlab Toolbox Fdident [9] and extrapolated or RF-requencies. These s- domain models were obtained or the low-pass ilter with transer unction G() and the multiplication o the transer unctions o the low-pass ilter and the high-pass ilter, G tot ()=G() x G 1 (). In igure 9 the solid curves represent the z-domain transer unctions G(z), G 1 (z) and G tot (z). The dashed curves show
the s-domain transer unctions o the low-pass ilter, G 1 (s), and the product o both, G tot (s). develop a nonlinear solver to calculate a better solution o the nonlinear eedback loop. Figure 9 : Z-domain and s-domain unctions Hence the RF-input is multiplied by the s-domain itted transer unction o the transer unction product, G()xG 1 (), resulting in signal t (ig.2). The result equals a constant gain actor or RF-requencies o about -14dB. This signal is then applied to the static nonlinearity. The output o the static nonlinearity, r, is multiplied with the constant G 2 and applied in negative eedback. Figure 10 : Amplitude o measured and modelled output envelope The phase inormation o the spectral components o the modelled output coincides well with those o the measured output envelope. This signal is then ed to the low-pass ilter in the eed orward path, G(), and evaluated as signal y. The calculated signal is deined as the model predicted output envelope. VI. MODEL ACCURACY Now we can compare the model predicted output envelope to the measured output envelope (ig.8). Figure 10 depicts the magnitude o both signals, while igure 11 shows the phase. Both igures plot the 0-400kHz spectral band o the low requency envelope. The mean deviation between the modelled and measured envelope equals 1.7dB and is relatively small compared to the measurement uncertainty, deined by the 2σ-interval o 0.9dB. The dierence in the lower side o the spectral band o interest equals 1dB, as can be seen rom igure 10. The spectral behaviour o the modelled output coincides less with the measured output envelope or higher requency modulation tones. At this point we have only passed the signal through the eedback loop once, which accounts in part or the deviation that is present. In a next step o the research, we will Figure 11 : Phase o measured and modelled output envelope The mean deviation between the phase o the modelled and measured envelope equals 5.4º. The dierence in the lower side o the spectral band o interest equals 2º. VII. CONCLUSION This work presents a new calibration standard, the crystal detector. The device has been identiied as a nonlinear eedback block structured model. High requency validation measurements were perormed. A irst approach to model the low requency output envelope rom RF-input measurements has been undertaken. The comparison o the measured envelope and modelled envelope shows good results.
ACKNOWLEDGEMENT The research reported here was perormed in the context o the network TARGET and supported by the Inormation Society Technologies Programme o the EU under contract IST-1-507893-NOE, www.target-net.org and supported by the Belgian government (IUAP-V/22) and the Flemish government (GOA-ILiNoS). REFERENCES [1] W. Van Moer, Y. Rolain, A Large-Signal Network Analyzer: Why is it needed? IEEE Microwave Magazine,Vol. 7, No. 6, pp. 46-62, 2006. [2] J. Schoukens, R. Pintelon, T. Dobrowiecki, Y. Rolain, Identiication o linear systems with nonlinear distortions, Automatica, 41, pp. 491-504, 2005. [3] Agilent RF and Microwave Test Equipment, Crystal Detectors. www.agilent.com [4] J. Verspecht. Broadband Sampling Oscilloscope Characterization with the Nose-To-Nose Calibration procedure: a Theoretical and Practical Analysis. IEEE Trans. Instrum. Meas., Vol. 44, No. 6, pp. 991-997, 1995. [5] L. Gommé, A. Barel, Y. Rolain, F. Verbeyst. Fine requency grid phase calibration or the Large Signal Network Analyser. IEEE MTT-S Int. Microwave Symp., pp. 1444-1447, 2006. [6] S. Billings, S. Fakhouri, Identiication o systems containing linear dynamic and static nonlinear elements, Automatica, 18, pp. 15-26, 1982. [7] J. Schoukens, J. Swevers, J. Paduart, D. Vaes, K. Smolders, R. Pintelon, Initial estimates or block structured nonlinear systems with eedback, International Symposium on Nonlinear Theory and its Applications (NOLTA 2005), pp. 622-625, 2005. [8] J. Schoukens, L. Gommé, W. Van Moer, Y. Rolain, Identiication o a crystal detector using a block structured nonlinear eedback model, Accepted or the 2007 IEEE Instrumentation and measurement technology conerence. [9] I. Kollár, Frequency Domain System Identiicaton Toolbox V3.3 or Matlab, Gamax Ltd, Budapest, 2005-2006.