We used the DAC and ADC of the RFSoC to build the VNA measurement circuit, as shown within the dashed blue line in Fig. 1. Fig. 2 shows the internal algorithms of the RFSoC FPGA. A numerically controlled oscillator (NCO) in the FPGA is used to generate a single-frequency cosine signal and a sine signal , where f is the frequency and t is the time. The signal frequency sweeps across the entire observation band, which is 10–100 MHz for our instrument. The cosine signal is converted to analogue via a DAC, filtered with a direct current (DC) block and passed two cascaded directional couplers before being reflected by the device under test (DUT). The signal flowing out from the coupled port of the first directional coupler DC1 is a portion of the cosine signal generated by the NCO, denoted as . The signal flowing out from the coupled port of the second directional coupler DC2 is a portion of the signal reflected by the DUT, denoted as . They will be sampled by two ADCs of RFSoC and then mixed with the I, Q signal, resulting in four signals :

Next, we extract the DC components of these four signals by time-averaging, resulting in four DC signals,

If represent the incident and reflected voltages of the DUT, the reflection coefficient of DUT can be expressed as:

In the practical case with non-ideal directional couplers, and are not exactly the incident and reflected voltages of the DUT, and the raw reflection coefficient is not exactly that of the DUT but needs correction. According to Rytting (1996), a single-port vector measurement network can be modelled by a 3-term error model, including the directivity error , match error , and tracking error . By measuring three known standards, these errors can be calculated to determine the true reflection coefficient of the DUT.

Although, in principle, the errors are eliminated by the VNA calibration process with the open-short-load (OSL) standards, the quantization errors arising from analogue-to-digital conversion could lead to a loss of information. This is particularly noticeable when measuring a 50  load with a very low reflection coefficient or when measuring active components such as LNA with low power. For surface-mounted directional couplers, which is used because of the limited space of the CubeSat, but have higher losses, larger reflection coefficients, and poorer directivity than coaxial directional couplers, this error can be even more significant. In the following subsections, we will investigate the above issues in the RFSoC-based VNA design through simulations and experimentation to assess the measurement accuracy.

We conducted simulation tests using the MATLAB Simulink Platform. A program of VNA measurement simulation built based on the SoC Blockset and SoC Blockset Support Packages for the Xilinx Devices in the MATLAB Simulink library. If the simulation results meet expectations, we can directly generate hardware description language code and deploy it to the RFSoC for laboratory tests, which will be introduced in Section 4.1.3.

The configuration of RFSoC data converter should be the same for both simulation and laboratory tests. The stream clock frequency is 256 MHz. Four NCOs are utilized to generate 16-bit signals with the same frequency and adjacent phases, ensuring that the DAC outputs four samples per clock cycle. Similarly, the ADC get four samples per clock cycle. The interpolation factor for DAC and the decimation factor for ADC are both 4. Based on the configuration above, the sampling rate of ADCs and DACs are 4.096 GSPS. The NCO takes a phase increment as input and accumulates it, using the quantized output of accumulator as an index to the lookup table of the sine wave values and then produce accurate single frequency signal. To implement frequency sweeping functionality, a phase increment value is needed by the NCO to generate signals at varying frequencies. This is given by

where f is the output frequency, N is the length of accumulator word length, S is the sampling rate. Due to the phase increment being an integer, the frequency resolution is constrained by the sampling rate and accumulator word length. In this case, the sampling rate, S is . To achieve a frequency resolution better than 0.1 MHz, the word length is set to 14. Additionally, we configure the dither bit of NCOs to add some dithering noise to help convert quantized noise to white noise, reducing harmonics and spurious signals. The cost of this is a lower signal-to-noise ratio. We tested different values, and chose a trade-off value of 4.

After the ADC samples the RF signal and converts it to a digital signal, the FPGA internally executes the mixing process described in equation (7) and the averaging process described in equation (8). Subsequently, the data are transferred via the AXI bus into the programmable logic module for further computation and storage. In both simulation and laboratory tests, these data will be imported into MATLAB workspace for raw reflection coefficient calculation and calibration.

In simulation, we assume that the reflection coefficients (i.e. ) of the open, short, load calibration standards are constant across the entire frequency band, with respective values of 1, −1, and 0. We choose two different sources, representing the high and low reflection coefficients. Source 1 has an magnitude of 0.5, while that of source 2 is 0.02, which is close to the LNA used in the receiver. The phase of both varies linearly with frequency. Additionally, we used several power levels to study the effects of quantization error, including −5, −20, −25, and −30 dBm. We implemented two different calibration methodologies: one is to calibrate with −5 dBm using OSL calibration standards, and then measure the DUTs with varing power levels; the other is calibrating and measuring both with the same power level. To speed up the simulation, the frequency sweep interval is set to be 1 MHz.

Fig. 3 shows the measurement errors when calibrating with −5 dBm power level and measuring with different power levels. The VNA works well when measuring a DUT with a high reflection coefficient, but it exhibits significant frequency structural errors when measuring a DUT with low reflection coefficient using low power. Fig. 4 shows the measurement errors when calibrating and measuring with the same power levels. Fig. 5 shows that the measurement error is mainly due to the weak reflected signal of the low reflection coefficient DUT, as the structures are fundamentally similar. Errors during low-power measurement are significantly increased due to the introduction of measurement errors in the raw of the load calibration standard.

In simulation, changes in the output power do not affect the single-port VNA error parameters. However, in practice, different output powers might alter the single-port error parameters, and the parameters of directional couplers are not completely the same as those in the simulation. The most crucial insight from the simulation is the need to be vigilant about the measurement accuracy for low reflection coefficients with low power.

We conducted laboratory tests on the RFSoC-based VNA and compared it with a commercial VNA to evaluate its practical performance. For this comparison, we utilized a N5247A PNA-X Microwave Network Analyser, which is calibrated with the N4433A Electronic Calibration Module. In the following text, it will be referred to as ‘PNA’. For the RFSoC, only a mechanical calibration kit could be used, specifically the Anritsu 3650 calibration kit in this test. To ensure the accuracy of the comparison and eliminate any potential errors from calibration kits, we adopted the following experimental approach:

Fig. 6 shows the test setup. We integrated two ADC-10-4+ directional couplers on a single circuit board and connected the input port and two coupling ports of this board to the DAC and ADC SMA ports on the balun add-on board. A DC blocker was connected to the DAC output on the balun add-on board to prevent the influence of DC signal. A specially designed connection scheme will be used for the satellite, while here it is just for testing purposes.

A variety of test objects were selected as DUTs. Our choices included attenuators with variant resistances, primarily to evaluate the accuracy of the VNA when measuring with flat amplitude and phase. We also chose a narrowband filter with resonance points in their within the band to evaluate the accuracy of VNA in measuring rapidly changing amplitude and phase. Additionally, we selected a cable, as it will be used in noise parameters calculation. Following the aforementioned criteria, we selected a 1 dB attenuator, a 13 dB attenuator, a 22 MHz filter, and a 10-m cable. During measurement, one end of these DUTs is connected to an open calibration standard, and the other end is connected to the output port of the circuit board. We used a 13 dB attenuator because its is approximately −26 dB when terminated with an open calibration standard, which is close to the of the LNA.

First, we examined the raw measurement results of the load calibration standard to see if there are frequency structural errors caused by quantization errors. Fig. 7 compares results using different power levels, with −5 dBm as the standard. A noticeable wiggling pattern appears at measurement with −35 dBm power, and this pattern will become more obvious when using lower power, which is not shown in this figure but has been verified through experiments. Although the measurement result with −25 dBm exhibits higher noise levels, it avoids significant structural errors, and this input power level will not damage the LNA chips. Therefore, in our subsequent tests, we have chosen −25 dBm as the low-power output.

In the accuracy testing of the RFSoC-based VNA, we adopted three different measurement methods, which are listed in Table 1.

In the test, the frequency resolution is 0.0675 MHz and the measurement duration is 0.01 s per frequency point. Fig. 8 shows the measurement results of the RFSoC-based VNA and the PNA, as well as the differences between them. The shaded area represents the measurement uncertainty of PNA, obtained by the VNA uncertainty calculator provided by Keysight Technologies (2020). For sources with low reflection coefficients, the measurement error of RFSoC-based VNA and uncertainty of PNA are both larger, indicating more difficult measurements for sources with low reflection coefficients.

These results show that in the actual experiments, the accuracy of method III is higher than that of method II in most cases. For the 13 dB attenuator, although method I deviates the most from the PNA measurements, this is likely due to the stability of cables and connectors for low reflection coefficient measurement (Buber et al. 2019). Regarding method I as a standard, method III still outperforms method II. It is different from the conclusions drawn in the simulation, indicating that the 3-term error parameters of single-port VNA have changed after altering the power level. Fig. 9 shows the 3-term error parameters calculated with two power levels, and slight differences can be observed. Notably, all measurement errors are within the uncertainty of the PNA, except the resonance point of the 22 MHz filter. It may be due to the rapid change of near the resonance point in logarithmic coordinates and is more likely to cause larger measurement errors, which needs to be improved in the future. In Section 4.4, we will explore the impact of VNA measurement errors on the 21-cm sky signal by simulating the entire analogue receiver.

We also conducted tests on the stability and trajectory noise. Fig. 10 illustrates the stability of the RFSoC-based VNA. As expected, when measuring low reflection coefficient devices with low power, the results are the least stable, with the magnitude’s RMS being less than  dB and the phase’s RMS being less than degrees. Fig. 11 illustrates the trace noise of VNA measurement. Here, for better visibility of the noise level, it is assumed that the open calibration standard is ideal, with a reflection coefficient of 0 dB. The noise of a single measurement is about ~ dB across the frequency band, and after 10 times of averaging, the noise can be effectively reduced to less than  dB, much less than the measurement error showed in Fig. 8.

When the isolation of the microwave switch is not perfect, equation (4) is inadequate for its description. More generally, the receiver can be modelled as a multiport microwave network, connecting the antenna, multiple calibrators, an LNA, and two ports of the SP6T2. When measuring a particular source, the PSD function can be expressed as

The terms in the brackets correspond to the contributions of the source temperature, the leakage of ambient temperature, and the noise wave parameters of the LNA, respectively. The receiver gain can be determined by calibration across the entire frequency band, and no additional radiometric noise is added. Here is the source temperature, for most types of calibrators it is the physical temperature, and for the antenna is the sky temperature. The coefficient of the source temperature is calculated using the Python package scikit-rf (Arsenovic et al. 2022). By modelling the source switching subsystem as a multiport microwave network, we can calculate the power received at the LNA port originating from various sources.

is the ambient temperature. The coefficient of the ambient temperature leakage is calculated by the method introduced in Wedge & Rutledge (1991). In this simulation, we have assumed that the switches and the two cables have the same temperature, which is a reasonable assumption given the small size of the satellite and its thermostatic regulation.

are the set of noise wave parameter of the LNA. The corresponding coefficients are calculated by treating the multiport network and all sources as an equivalent antenna, then substituting the reflection coefficient of this antenna into the coefficients of , , in equation (4).

The frequency band of the simulation is 10–100 MHz. The design of the antenna will be detailed in a subsequent paper, here we merely note that it is a foldable matched antenna. In this simulation we scale the reflection coefficient of the REACH dipole antenna (Lera Acedo et al. 2022) into the observation frequency band of this instrument, as shown in Fig. 12. Similarly, the same LNA as REACH is used in this simulation. The impedance of the noise source, hot load, and ambient load are all assumed to be 50 . The reflection coefficients of other calibrators consisting of cables and terminals are obtained from advanced design system simulations, also shown in Fig. 12.

We have selected several commercial microwave switches with relatively good RF performance within the observation frequency band for the simulation. The models and parameters of the microwave switches are shown in Table 3. These parameters are flat in the observation frequency band, so they are set to be frequency-independent in the simulation.

The physical temperature of the hot load is 370 K, while all other components are at 300 K. The noise source has a flat excess noise temperature of 1100 K across the band. The sky temperature is dominated by the Galactic synchrotron radiation, which is a smooth power-law-like spectra. We utilize a five-term polynomial used in Bowman et al. (2018) to model the sky foreground temperature:

Here are the coefficients fitted to the measurement data from EDGES (Bowman et al. 2018), which will be used to extrapolate the foreground temperature to 10–100 MHz as the antenna temperature in the simulation, shown in Fig. 13. In this simulation, we do not consider the 21-cm signal because it is weak compared to the Galactic synchrotron signal, and our main objective is to study the effect of surface-mounted switches and measurement error on the antenna temperature recovery. The noise wave parameters of LNA for the mock data generation are also plotted in Fig. 13, which are derived from the REACH LNA noise parameter measurements in the laboratory and fitted with second-order polynomials. If there are more complex structures in the actual noise wave parameters once the whole receiver being assembled, we will observe it in the analysis, and then use higher order model for the system calibration.

The OSL calibration are performed with respect to a reference plane of the VNA at the ports of SP6T2, while the signal paths used to measure of the antenna and the LNA include additional paths through the DPDT switch, as shown by the solid red lines in Fig. 1. However, when measuring the spectrum, the signal passes through the DPDT switch along the solid blue line. Therefore, the measured needs to be corrected for these differences. In Fig. 1, the reference plane for defining the reflection coefficients of the antenna and the receiver is labelled at the end of the solid blue line near the LNA. To make this translation, we model the difference in signal paths as two-port networks, characterized by the scatter matrix [S], which can be measured in laboratory with high accuracy using the through-reflection-line method (Rytting 2001). The mock measurement data are generated by calculating the ratio of the output and incident voltages at the ports of the SP6T2 switch using the multiport network model described in Section 4.3.1, then corrected as (Pozar 2011),

where and are, respectively, the reflection coefficients before and after embedding the DPDT.

During calibration, the signal path from the calibrators to the S-parameter reference plane can be treated as a two-port network for temperature correction. Fig. 14 illustrates this model for calculating the effective temperature of the hot load, applicable to all other calibrators and antenna. The available power gain for this two-port network is the ratio of the power available from the network to the power available from the source, expressed using the following equation in Pozar (2011):

where and are the forward S-parameters of this two-port network. is the reflection coefficient of the calibrator and is the reflection coefficient measured at the reference plane. Due to the non-negligible insertion loss, the microwave switches also contribute to the noise power. With available gain, we can get the effective temperature of each source as:

where is the physical temperature of calibrator and is the physical temperature of the two-port network. Most components would have nearly the same temperature, there are exceptions, for example the 50  hot load and the antenna. In Section 4.3.1, we assume a common temperature for the switches and cables, and use a multiport network to calculate the ambient temperature leakage.

In simulation, we used different configurations, i.e. different model of switches performance and measurement errors. For each configuration, we generated a corresponding mock data set. Each mock data set contains:

The calibration process is the same for each simulated data set, listed below:

In order to explore the effects of isolation on the calibration results, we generated four different data sets whose differences are listed in Table 4. In RefSet, all microwave switches are ideal – there is no insertion loss, reflection loss, or signal leakage – serving as a reference to validate the calibration approach. In IsoRefSet, the insertion loss and reflection loss follow the parameters specified in Table 3, while isolation remains ideal. In the two remaining cases SwitchOffSet and SwitchOnSet, we consider the impact of non-ideal isolation. When the SP6T1 in Fig. 1 is switched to internal calibrators, due to the non-ideal isolation of the switch, there could still be some leakage caused by antenna through it to the LNA. In SwitchOffSet, we consider the case where the SPDT1 switch is connected to the antenna when not in use, while in SwitchOnSet, we consider the case where it switches to a 50  load. This comparison is intended to demonstrate the impact on the system when the SPDT1 switch enhances the isolation of the antenna signal propagation path. All other aspects of these data sets are identical.

First, we compare the spectrum of these data sets to show the impact of signal leakage. The top plot in Fig. 15 shows the spectrum of various calibrators in IsoRefSet, containing contributions from both the source temperature and LNA noise temperature. The bottom panel illustrates the difference between the spectrum in SwitchOffSet and SwitchOnSet compared to IsoRefSet, showing the effect of non-ideal isolation. If SPDT1 is kept connected to the antenna, signal leakage contributes up to 400 mK to the spectrum. If SPDT1 is switched to the load, the signal leakage is reduced to about 3 mK, indicating that SPDT1 effectively enhance the isolation of the antenna signal propagation path.

Similarly, we study the effect of non-ideal isolation on the measurement of reflection coefficients for various components. Fig. 16 shows the difference between the corrected in SwitchOnSet and IsoRefSet. The greatest impact of isolation on the measurement of S-parameters occurs at the resonance frequency of the antenna, with a magnitude of dB and a phase of deg. It is negligible compared to the potential VNA measurement inaccuracies. Therefore, the effect of isolation on the S-parameters measurement is not significant.

We used polynomial functions of different orders to fit the noise wave parameters, then recover the sky temperature and get the foreground fitting residuals. Basically, we want both the recovery error and the fitting residuals to be as small as possible. Fig. 17 shows the root mean square (RMS) and frequency structure of these two indicators. As expected, in RefSet, the sky temperature can be perfectly recovered by using second-order polynomial functions to fit the noise wave parameters, the same order as used to generate the mock data. For IsoRefSet, a higher order polynomial is needed to fully recover the sky temperature. The reflection coefficient of the noise source measured at the reference plane is no longer zero when considering non-ideal switches, so the equivalent temperature of the noise source will vary with frequency. The graph shows that a fifth-order polynomial fitting results in recovery error less than 1 mK, which is sufficiently small compared to the 21-cm signal. However, in the SwitchOffSet and SwitchOnSet cases, the recovery error cannot be reduced to a very low level. For SwitchOffSet, the recovery error obtained by fitting the noise wave parameters with tenth-order polynomial is above 400 mK at low frequencies, where the sky temperature is particularly high. For SwitchOnSet, the maximum value of the recovery error obtained by fourth-order polynomial fit is less than 60 mK.

In actual experiments, we do not know the true sky temperature, so we are more concerned with foreground fitting residuals as it may contaminate the 21-cm signal. For SwitchOffSet, the fitting residuals are approximately within  mK at the optimal polynomial fitting order, while for SwitchOnSet, they are within  mK. Higher order polynomials lead to higher fitting residuals because they absorb the undesirable leakage of non-ideal isolation of the switches into the noise wave parameters, improving calibrator fits but increasing sky temperature recovery errors. Considering both indicators, SwitchOffSet can achieve low fitting residuals with second-order polynomial but the recovered sky temperature significantly differs from the actual temperature. Overall, the presence of the SPDT1 switch aids in achieving smaller recovery error and fitting residuals.

The previous simulation assumes that all measurements are accurate, however the main sources of actual calibration error are the various measurement errors. From equation (5), calibrating the receiver requires the measurement of three variables: PSDs, the physical temperatures, and reflection coefficients, each with potential errors. We add these measurement errors to SwitchOnSet to evaluate if the resulting foreground fitting residuals meet our requirements.

The PSDs are measured by RFSoC-based digital spectrometer. As demonstrated in Section 2, the dynamic range is sufficient to ensure the linearity of the received PSD. Therefore, PSD measurement errors are not considered in further analyses. For physical temperature measurement, existing commercial probes can achieve an accuracy of less than 0.1 K. Based on Section 4.1.3 tests, we assume for calibrators connected two cables, the errors are 0.01 dB and . For LNA, hot load and ambient load, the error is 0.1 dB and . For other calibrators, the error is 0.01 dB and . The measurement error occurs at the calibration plane of the VNA, so they will be amplified by the imperfect S-parameters of the DPDT switch in the reference plane. Fig. 18 shows that a dB error in the LNA’s reflection coefficient measured at the VNA calibration plane can introduce additional errors of up to 0.07 dB at 100 MHz after shifting the calibration plane.

Since the signs of errors in actual measurements are uncertain, we assign different error signs to the magnitude, phase, and physical temperature measurements, then observe the foreground fitting residuals in different cases. As before, we calculate the RMS for different polynomial fitting orders and select the order with the lowest RMS to show the residual frequency structure. Given the large 10–100 MHz bandwidth, we divide it into two sub-bands to observe residual variations. The results are the black lines in the Fig. 19. In 10–40 MHz range, the foreground fitting residuals are within mK except for the edge of the band. It is relatively large due to high foreground temperature in low frequency, making foreground fitting much more difficult, which needs to be improved in the future. In 40–100 MHz range, which is of more interest for ground-based experiments, the recovery errors are within mK except at the band edges, and up to mK in 50–90 MHz. The frequency structure of the residuals are related to the foreground model we are using. We also try to introduce half of the measurement error, shown as grey lines, resulting in fitting residuals being halved as well.

Fig. 20 illustrates the contribution of different measurement errors to the final fitting residuals in the 40–100 MHz range. The antenna’s measurement error is the primary source of residual, followed by the LNA’s error, while other errors have relatively small contribution to the final fitting residuals.