ACCURATE MEASUREMENT OF WIDEBAND MODULATED SIGNALS
Phase and/or amplitude modulated signals are often used for information transmission in microwave communications systems. For simulating such a communications system, these types of signals are most naturally represented by their lowpass equivalent (LPE)1 signal. The LPE signal is the carrier envelope with a timevarying amplitude and phase. The microwave signal can be derived from the LPE signal by a linear transformation. In LPE modeling, components such as amplifiers are modeled in terms of their response to the signal envelope. To complement an LPE model, it is desirable to have a time domain measurement of the modulated microwave signal. Such a signal measurement can be used not only for modeling, but also for verifying compliance to systems specifications and for understanding of deviations from ideal performance.2
Two basic methods can be used to measure a modulated microwave signal in the time domain. One method measures the signal directly at the carrier frequency using a high speed digital storage oscilloscope (DSO). The other method downconverts the signal to a lower frequency before measurement. Downconverting before measurement increases complexity, but there are several accuracy advantages in measuring a signal at a lower frequency. These advantages pertain especially if the signal is downconverted all the way to baseband using a local oscillator coherent with the microwave carrier. The measured baseband waveform is the envelope of the microwave signal, which is its LPE representation.
One advantage of coherent downconversion is that it eliminates the phase noise of the carrier from the measurement. Another advantage is that the sample rate can be reduced by the ratio of half the signal bandwidth to the carrier frequency since the carrier need not be sampled. This method allows for a longer time record or higher time resolution of the signal for the same number of samples. (This is the same reason LPE signals are used in modeling.) A third advantage is that DSO accuracy is better at lower frequencies. The higher the carrier frequency, the more inaccurate the direct measurement; the accuracy of the baseband measurement is independent of the carrier frequency. Because of these advantages, major instrument manufacturers have specialized instruments that perform time domain measurements of microwave communications signals at baseband. Signals with bandwidths up to 200 MHz and carrier frequencies up to 110 GHz are supported.2
This article presents a method to accurately measure signals with bandwidths higher than those supported by offtheshelf solutions. How to perform accurate time domain measurements of modulated microwave or millimeterwave signals having gigahertz bandwidths using a calibrated downconverting receiver (DCR) in conjunction with a DSO3 will be presented. By calibrating the DCR the linear distortion of the receiver can be characterized and eliminated from the signal measurement.
Fig. 1 The time domain waveform measurement setup block diagram.
A coherent LO is required for downconversion to baseband. The signal must be repetitive because the LPE representation of a microwave signal has two baseband components: an inphase and a quadrature waveform that are not measured simultaneously but consecutively in this procedure. Waveforms may be measured directly at the output of a modulator or after passing through a linear or nonlinear distorting communications channel. These measurements then may be used to characterize either the modulator or the components in the channel.
MEASUREMENT SETUP
The measurement system consists of a DCR followed by a DSO, as shown in Figure 1. The DCR includes a mixer followed by a baseband amplifier. The mixer LO port is fed by an LO with a phase shifter that is adjusted to enable measurement of the inphase and quadrature baseband waveform components. The DC level, which corresponds to the Fourier component of the signal coherent with the carrier, is measured separately by means of a bias tee and a voltmeter at the output of the downconverting mixer. The DC level must be measured separately because most wideband baseband amplifiers block the DC component. The DCR has two coherent LO outputs: a fixedphase and a variablephase LO. The LO outputs are required for the two test mixers in the described calibration procedure.
CALIBRATION PROCEDURE
The DCR introduces both linear and nonlinear distortion into the signal. The nonlinear distortion caused by the downconverter can be minimized by filtering unwanted mixing products and ensuring that the signal level into the mixer and other components is in its linear range of operation. For baseband downconversion the unwanted mixing products are located at multiples of the LO frequency and can be eliminated by lowpass filtering. The baseband amplifier provides lowpass filtering and, if required, an external lowpass filter can be added. The linear distortion of the downconverter can be minimized by using components with a much wider frequency response than the bandwidth of the signal to be measured and by minimizing SWR interactions, which introduce ripple to the frequency response.
For sufficiently narrowband signals, the combined frequency response of the DSO and DCR has little variation and may be ignored. However, for wideband signals such as those used in high data rate communication systems, the DCR distortion may be significant. The response variations of the DSO over bandwidths of a few gigahertz are negligible since a 20 GHz bandwidth sampling front end is used.4 In this instance, the bandwidth of the wide band communications signal is 10 percent or less of the bandwidth of the DSO sampling front end. However, since the bandwidth of the DCR used here is approximately 4 GHz, a wideband signal occupies a large portion of the available bandwidth. Thus, linear distortion in the DCR introduces the only significant error in the measured LPE waveforms.
Fig. 2 Block diagrams of the three measurement configurations for DCR calibration.
Since the input to the DCR is located at the signal's carrier frequency and the output of the DCR is located at baseband, its linear distortion cannot be measured using a vector network analyzer (VNA). A twopart procedure has been developed to accurately measure the DCR frequency response. Part 1 measures the DCR response at all frequencies other than the carrier frequency by means of the basebanddoublesideband frequency translating device (FTD) measurement technique.5 Part 2 measures the DCR response at the carrier frequency, which is DC at the baseband output and measured using a voltmeter.
THE BASEBANDDOUBLESIDEBAND FTD PROCEDURE
To calibrate the DCR at frequencies other than the carrier frequency, three test configurations using two test mixers (TM1 and TM2) and the DCR are measured using a VNA, as shown in Figure 2. The three configurations are (A) TM1 to DCR, (B) TM2 to DCR and (C) TM2 to TM1. The VNA sweeps over the baseband frequency range; the first mixer in each configuration serves as an upconverter, and the second serves as a downconverter. Because the input signal is at baseband and there is no filter at the output of the upconverting mixer, an upper and a lower sideband are transmitted to the downconverting mixer.
Two measurements in each test configuration are required to capture the response of both sidebands. The first measurement is performed at a given arbitrary setting of the DCR phase shifter, which is referred to as phase shifter setting 1. In the second measurement, the phase shifter is adjusted by 90°. This setting is referred to as phase shifter setting 2 (phase shifter setting 1 + 90°). The VNA measurements are performed for each of the three configurations A, B and C. The LPE transmission response, PX(f) (expressed as a complex number at each frequency), of each of the backtoback FTD pairs is given by
where
* = the complex conjugate operation
j = square root of –1
MX1(f) = complex S21 response of the backtoback FTD pairs at phase shifter setting 1
MX2(f) = complex S21 response of the backtoback FTD pairs at phase shifting setting 2 (setting 2 – setting 1 = + 90°)
X = A,B, C
The LPE response of the DCR may be derived from the pair responses. First, each pair response is converted from a complex representation to decibels and degrees such that
Then, the frequency response of the DCR (in decibels and degrees) may be expressed as
Note that a VNA cannot measure down to zero signal frequency, so a careful procedure must be followed to interpolate the zero frequency response between the lower and upper sidebands. The overall response must be continuous in phase at zero frequency, and the value of the phase at zero frequency must be correct. Note also that the phase at zero frequency is important even though the signal out of the amplifier has no DC component because it ensures the correct phase at other frequencies.
The DCR phase response at zero baseband frequency can have only one of two possible values, 0° or 180°, depending on whether the signal path through the DCR is noninverting or inverting, respectively. Both the mixer and baseband amplifier can be either inverting or noninverting as the baseband frequency tends to zero. Thus, the sign of the DCR phase response at DC may be either positive or negative depending on the product of the responses of the mixer and the amplifier near DC. This sign is most easily determined if the two components are characterized separately, before they are integrated into the DCR. The mixer can be tested by putting the same CW carrier signal into both the L and R ports simultaneously with zero phase difference and observing whether a positive (noninverting) or a negative (inverting) voltage is obtained. The amplifier can be tested by injecting a low frequency CW signal and determining if the output is in phase (noninverting) or 180° out of phase (inverting).
Part 2 of the DCR frequency response measurement is to locate the response of the DCR at the carrier frequency, which at baseband is DC. The first step in calibrating the DC voltage is to perform a zeroing procedure with no signal applied at the DCR input. The first measurement, Vo1, is made at phase shifter setting 1 used in the DCR response calibration procedure. The next measurement, Vo2, is made at phase shifter setting 2. The Vo3 measurement is made at phase shifter setting 3 (phase shifter setting 1 + 180°), and the Vo4 measurement is made at phase shifter setting 4 (phase shifter setting 1 + 270°). By storing these measured offset voltages at each of these settings and subtracting them from the DC values acquired during a waveform measurement, the offset errors are eliminated.
The second step in the DC calibration process is to calibrate the gain of the DCR at the carrier frequency. Since the DC output of the mixer in the DCR is measured prior to the baseband amplifier, the gain at the carrier frequency must be calibrated independently of other frequencies. This step is performed simply by applying a CW signal of known power P to the input of the DCR, acquiring the DC voltages V1, V2, V3 and V4 at the four LO phase shifter settings, respectively, and calculating a gain factor such that
GDC is a proportionality constant whose value depends on the power and voltage units used. GDC determines the gain of the DCR at the carrier frequency (the DC gain at baseband) and is applied to the measured DC component of the signal. Note that the DC signal path is taken directly from the mixer prior to the amplifier. Thus, GDC will be either positive or negative depending only on whether the mixer response is noninverting or inverting at DC.
MEASUREMENT AND POSTPROCESSING PROCEDURE
Once these calibration procedures are completed, a waveform of interest is applied. At phase shifter setting 1, the waveform w1 is acquired by the DSO and a DC voltage V1 is acquired by the voltmeter. Similarly, w2 and V2 are acquired at phase shifter setting 2. At phase shifter settings 3 and 4, only DC voltage measurements V3 and V4, respectively, are required, not waveform acquisitions. The waveforms acquired at the two settings must be combined and corrected for the receiver response and then combined with the DC voltages to determine a corrected LPE waveform.
As part of the postprocessing procedure, the two waveforms are combined to form an uncorrected LPE waveform:
This waveform must be corrected for the DCR frequency response. One way to accomplish this correction is to transform the LPE waveform to the frequency domain, then divide the frequency domain waveform representation by the DCR response (expressed as a complex number at each frequency). The DCR frequency response must be linearly interpolated and extended in frequency to agree with the discrete frequencies of the waveform. Normally, the DCR response would be calibrated out to frequencies beyond the total span of the signal. In this case, windowing is unnecessary and the amplitude of the DCR response may be set to infinity at frequencies beyond the band of the measured response. This procedure forces the Fourier coefficients of the corrected waveform to zero at these frequencies.
After the LPE waveform is corrected for the receiver response, the corrected LPE waveform is then transformed back to the time domain. The corrected time domain LPE waveform, denoted as wc(n), has no DC component because the amplifier blocks DC. Hence, the DC component measured by the voltmeter must be added to the waveform according to the formula
After the first waveform measurement, the gain factor GAC is adjusted so that the calculated power in wLPE is equal to the signal power as measured with a power meter. GAC is a proportionality constant whose value depends on the power and voltage units used. This gain factor then can be used for all subsequent measurements.
BASEBAND DC VOLTAGE CALIBRATION AND MEASUREMENT
The mixer in the DCR acts as a phase and amplitude detector with regard to the signal component at the carrier frequency. Typically when a mixer is used only as a phase detector (such as in a phase lock loop), the carrier power on both the R and L ports is large enough to saturate the mixer diodes. Here, the signal on the R port must be in the linear range of mixer operation while the CW signal at the L port must be large enough to saturate the mixer diodes. In this case, if the mixer were perfectly balanced, the DC voltage at the I port would be
where
P = power at the carrier frequency at the R port
GDC = a proportionality constant
F = phase difference between the carrier at the L port and the carrier at the R port at phase shifter setting 1
For imperfectly balanced mixers, there is also a temperaturedependent DC offset voltage at the IF output port, even when no signal is applied to the RF port. This DC offset voltage is a function of mixer balance and the carrier power applied at the LO port. Since the phase shifter on the LO port displays some amplitude variation vs. setting, the offset voltage is, in turn, a function of phase setting. Hence, with no signal applied to the R port, four slightly different DC offset voltages (Vo1, Vo2, Vo3 and Vo4 at phase shifter settings 1 through 4, respectively) are measured. When the CW calibration signal of power P is applied to the mixer's R port, the DC output at the I port then consists of the sum of the offset voltage and the DC voltage produced in response to the input signal component at the carrier frequency. At phase setting 1,
At phase setting 2,
Note that settings 3 and 4 are 180° from settings 1 and 2, respectively. Thus,
It is assumed that the proportionality constant GDC is independent of phase shifter setting. Equation 4, the expression for GDC, can be derived by combining Equations 7 through 10. Note that GDC could have been derived equally well from Equations 7 and 8 alone, but in practice using all four phase settings provides more immunity against longterm thermal drift of the DC offsets.
In the case of a modulated signal input, Equations 7 through 10 still apply if P is interpreted as the power in the Fourier component of the signal at the carrier frequency. The voltage measurement at setting 1 provides the real part of the Fourier coefficient and the measurement at setting 2 provides the imaginary part in Equation 6.
Fig. 3 Nonlinear amplifier block models; the (a) onebox and (b) twobox models.
APPLICATION TO POWER AMPLIFIER MODELING
An application of the previously described technique to the measurement and modeling of a 20 GHz wideband solidstate amplifier will now be presented. Measurements of the input and output waveforms using binary phase shiftkeying (BPSK) modulation were performed at several data rates. By using different data rates, the measurement and modeling techniques were evaluated over a range of bandwidths. The performance of two nonlinear block models for the 20 GHz solidstate amplifier is compared to measured results. The first model, the onebox model, shown in Figure 3, consists of a memoryless nonlinearity only. In this model, the nonlinear device is characterized by the bandcenter nonlinear amplitude (AMAM) and phase (AMPM) conversion functions, which operate on the instantaneous envelope of the input signal. The second model, the twobox model, includes an input filter prior to a memoryless nonlinearity. Here, the first box is the smallsignal frequency response of the nonlinear device. The second box is the same AMAM, AMPM nonlinearity used in the onebox model. Both of these models are constructed strictly from singletone VNA measurements. In all cases the measurements and simulations were performed at an amplifier operating point near saturation.
QUANTIFYING WAVEFORM FIDELITY
A means of quantifying the agreement between the modeled and measured time domain waveforms is necessary. A convenient metric to use for this purpose is the normalized mean square error (NMSE):
where the measured and modeled inphase (yI) and quadrature (yQ) waveforms have M sample points. To calculate the NMSE it is first necessary to line up the two waveforms in both time and phase. This lineup can be conveniently achieved by maximizing the cross correlation between the two waveforms. The metric is the total error vector magnitude power between the measured and modeled waveforms normalized to the measured signal power. It is assumed that the true waveform is much closer to the measured waveform than to the modeled waveform, so the NMSE is a model fidelity metric. The NMSE metric can be useful for assessing model predictive fidelity for many communications applications.
TABLE I  

 Data Rate (Mbps)  
 150  600  1200  2400 
Onebox model NMSE (dB)  27.92  22.64  19.84  18.72 
Twobox model NMSE (dB)  33.69  28.34  26.24  24.31 
Twobox model improvement (dB)  5.77  5.70  6.4  5.59 
Fig. 4 Modeled vs. measured outputs for a 2.4 Gbps BPSK modulated input.
RESULTS
The BPSK signals applied vary in bit rate from 150 Mbps to 2.4 Gbps. A 5 GHz bandlimiting filter was inserted before the amplifier for these signals. The operating point was chosen at the 3 dB gain compression point. Table 1 lists the NMSE results for the various BPSK signals as well as the difference between the two models. Note that the fidelity of both models decreases as the bandwidth of the signal increases. However, the twobox model provides more than 5 dB improvement over the onebox model for all bandwidths. To illustrate one of these cases, Figure 4 shows the amplitude envelope of the 2.4 Gbps BPSK output for the two models vs. the measured waveform. A small portion of the time record is shown to enable the differences between the measured and modeled waveforms to be seen.
CONCLUSION
A novel technique for accurately measuring wideband communications waveforms has been presented. The waveform measurements were made at baseband using a calibrated downconverting receiver to take advantage of the inherent accuracy of available DSOs. A correction technique removed any errors in the waveforms caused by the frequency response of the downconverting receiver. The measured input and output communications signal waveforms have been applied to develop a simulation model of a 20 GHz solidstate amplifier over a bandwidth of 4.8 GHz.
References
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