Fast and precise data acquisition for broadband microwave tomography systems

The frequency-dependent scattering parameter S_ij of an N-port network are defined as the ratio between the outcoming wave b_j at port j and the incident wave a_i at port i:

$$\begin{eqnarray}&&{{S}_{ji}}\left(\,\text{j}\,f\right)=\frac{{{b}_{j}}\left(\,\text{j}\,f\right)}{{{a}_{i}}\left(\,\text{j}\,f\right)}{{|}_{{{a}_{m}}=0}},\quad i,j,m=1..N,m\ne i,\end{eqnarray} \tag{ 1 }$$

assuming that the incoming waves at all ports are zero except at port i. In contrast to conventional VNAs based on the continuous wave method, the signal frequency f is varied linearly with time in the frequency range from f₀ to f₁:

$$\begin{eqnarray}\begin{array}{*{35}{l}} f(t) & = & {{f}_{0}}+\sigma \,t, \end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \sigma & = & \frac{{{f}_{1}}-{{f}_{0}}}{{{T}_{\text{s}}}}, \end{array}\end{eqnarray} \tag{ 3 }$$

where σ is the ramp slope and T_s is the ramp sweep time. A single measurement channel of an FMCW VNA can be represented by the mathematical model shown in figure 1 [29]. The input signal x_RF can be expressed by:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{x}_{\text{RF}}}(t) & = & \cos \left({{ \Phi }_{\text{RF}}}(t)\right), \end{array}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{ \Phi }_{\text{RF}}}(t) & = & 2\pi {{f}_{0}}t+\pi \sigma {{t}^{2}}+{{ \Phi }_{\text{RF},0}}, \end{array}\end{eqnarray} \tag{ 5 }$$

where ${{ \Phi }_{\text{RF}}}(t)$ and ${{ \Phi }_{\text{RF},0}}$ are the current and the initial phase of the signal, respectively. The amplitude and phase of the signal are modulated corresponding to the complex-valued transfer function H of the measurement path. The resulting signal y_RF is related to one of the wave quantities (a_i or b_j). It is multiplied with a second input signal y_LO, whose frequency is shifted by a small frequency difference f_d:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{y}_{\text{LO}}}(t) & = & \cos \left({{ \Phi }_{\text{LO}}}(t)\right), \end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{ \Phi }_{\text{LO}}}(t) & = & 2\pi \left(\,{{f}_{0}}+{{f}_{\text{d}}}\right)t+\pi \sigma {{t}^{2}}+{{ \Phi }_{\text{LO},0}}. \end{array}\end{eqnarray} \tag{ 7 }$$

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The output signal of the multiplier contains spectral components at the sum and difference frequency. The output signal of the band-pass filter ${{z}_{\text{IF}}}$ contains only the difference frequency in the intermediate frequency (IF) band, which is constant (${{f}_{\text{IF}}}={{f}_{\text{d}}}$). The filtered output signal is proportional to the real part of the complex-valued transfer function H:

$$\begin{eqnarray}&&{{z}_{\text{IF}}}(t)=0.5\,\Re \left[H\left(\text{j}2\pi {{f}_{\text{RF}}}(t)\right){{\text{e}}^{j2\pi {{f}_{\text{d}}}}}\right].\end{eqnarray} \tag{ 8 }$$

The transfer function of the DUT can be obtained from the analytical signal $z_{\text{IF}}^{+}$, which can be calculated from ${{z}_{\text{IF}}}$ by performing a Hilbert transform:

$$\begin{eqnarray}&&z_{\text{IF}}^{+}(t)=0.5\,H\left(\,\text{j}2\pi {{f}_{\text{RF}}}(t)\right){{\text{e}}^{j2\pi {{f}_{\text{d}}}}}.\end{eqnarray} \tag{ 9 }$$

The analytical signal of one single measurement channel is related to one of 2N wave quantities (a_i or b_j). Therefore, the S-parameters can be calculated by dividing the analytical signals of two measurement channels, which are related to the outcoming and the incident wave:

$$\begin{eqnarray}&&{{S}_{ji}}\left(\,\text{j}\,{{f}_{\text{RF}}}(t)\right)=\frac{z_{\text{IF},\text{b}j}^{+}(t)}{z_{\text{IF},\text{a}i}^{+}(t)}=\frac{{{H}_{\text{b}j}}\left(\text{j}2\pi {{f}_{\text{RF}}}(t)\right)}{{{H}_{\text{a}i}}\left(\,\text{j}2\pi {{f}_{\text{RF}}}(t)\right)},\end{eqnarray} \tag{ 10 }$$

where the time vector t is mapped to an equivalent frequency vector ${{f}_{\text{RF}}}$ using (2).

The input signals x_RF and y_LO can be either generated using two synchronized synthesizers generating two frequency ramps with a small frequency difference (heterodyne principle [29]) or using one synthesizer and a delay line (metadyne principle [27]). The block diagram of one measurement channel of a metadyne FMCW VNA is shown in figure 2. The time delay τ is introduced using one or more non-dispersive (coaxial) lines resulting in a constant frequency difference

$$\begin{eqnarray}&&{{f}_{\text{d}}}=\sigma \tau =\sigma \frac{{{l}_{\text{line}}}}{{{v}_{\text{p}}}},\end{eqnarray} \tag{ 11 }$$

where l_line and v_p are the mechanical length and the wave propagation velocity of the delay line, respectively. The advantages of the metadyne principle are the lower hardware complexity of the system, since only one synthesizer is required, and the phase noise canceling at the mixer, which potentially allows for a higher measurement precision compared to heterodyne VNAs as described in section 2.2.3. However, the flexibility of the system configuration is limited since the ramp slope and length of the delay line must be chosen adequately to achieve a sufficiently high frequency difference.

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Based on the aforementioned considerations, we developed a metadyne 3-channel prototype electronics for fast and precise 2-port reflection and transmission measurements in the frequency range from 0.5 GHz to 5.5 GHz. The major difference between the (metadyne) FMCW method and the (heterodyne) CW method regarding the hardware architecture are the signal generation and distribution. In contrast to CW systems using the stepped frequency method, where the frequency is kept constant during a measurement, a signal synthesizer used in FMCW systems must be able to generate a highly linear analog frequency ramp over the entire frequency range of interest [28]. Furthermore, the synthesizer must provide a low sideband noise (amplitude and phase noise) as well as a sufficiently high and constant output power to achieve a high measurement precision and a large DR. Due to the metadyne principle, only one synthesizer is required, which reduces the hardware complexity compared to heterodyne systems. In contrast to previously published FMCW systems [28, 29], we were able to increase the bandwidth of the signal synthesizer and all other microwave modules and significantly lower the ramp sweep time.

Furthermore, receiving channels with a low noise factor, a high compression point and a good gain flatness are required to achieve a high DR, since the DR is limited by receiver's compression point on the one hand and by noise and spurious signals on the other hand as discussed in section 2.2.3. All channels must be well isolated from each other to avoid a reduction of the spurious free dynamic range (SFDR) of the receiver and therefore the DR.

For N-port measurements the prototype electronics can be either used in conjunction with a $2\times N$ switching matrix as applied in this paper, or it can be extended to a full 2N-channel system due to the module-based structure. A data acquisition unit with a parallel hardware architecture is required to achieve a very low total data acquisition time in conjunction with a high precision and a high DR.

To be able to realize a hardware design at reasonable costs, an architecture with low hardware complexity was developed and all modules are based on passive microstrip circuits as well as commercially available surface mount devices. In contrast to commercial wideband VNAs, which use different microwave modules for different frequency bands [21], all modules of the FMCW prototype systems can be used over the entire frequency range of interest, which reduces the hardware complexity. In this subsection, the hardware architecture and the system's parameters are described as well as a noise analysis of the system is conducted.

2.2.1. Hardware concept.

For FMCW signal generation, we utilize our previously developed synthesizer based on fractional-N phase locked loops (PLL) [30], whose block diagram is shown in figure 3. The linear frequency ramp is synthesized in the frequency range from 8.4 GHz to 13.6 GHz ($\text{VC}{{\text{O}}_{1}}$) and is mixed down to the desired frequency range with a constant frequency of 14 GHz ($\text{VC}{{\text{O}}_{2}}$). Both VCOs are stabilized and synchronized by PLLs with a common stable crystal oscillator (XCO) as reference. The signal at the mixer's output is low-pass filtered and amplified. The frequency-dependent attenuation of the microwave components is compensated by passive microstrip circuits with a high-pass like transfer function. The frequency range of the ramp synthesizer (0.4 GHz to 5.6 GHz) slightly exceeds the desired measurement frequency range to compensate for non-linear effects occurring at the upper and lower band edge, which cause a significant deviation between the actual and measured transfer function of the DUT. The synthesizer delivers a sufficiently high and constant output power around 20 dBm and the phase noise of the output signal is below  −99 dBc Hz⁻¹ at offset frequencies above 10 kHz. A detailed description of the frequency ramp generation is provided in [30].

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The block diagram of the developed prototype electronics is shown in figure 4. The output of the frequency ramp synthesizer is split by a broadband 4-way multi-section Wilkinson power divider (WD). The signal x_RF is coupled via a 2-way multi-section WD into the first receiving channel (R1) as well as towards the device under test (DUT) via a second 2-way multi-section WD. The reflected and transmitted waves are guided into the second (M1) and third (M2) receiving channel. The RF input signals at the mixers ${{y}_{\text{RF},\text{R}1}}$, ${{y}_{\text{RF},\text{M}1}}$, and ${{y}_{\text{RF},\text{M}2}}$ are related to the incident and outcoming waves a₁, b₁, and b₂, respectively.

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To avoid a degradation of the measurement precision and the DR due to poor channel isolation and the low-pass characteristic of the microwave modules, the RF and LO signals are guided towards the mixers using shielded microwave modules with a high reverse isolation ($|{{S}_{12}}{{|}_{\text{dB}}}\leqslant -80$ dB), a low coupling between adjacent channels ($|{{S}_{31}}{{|}_{\text{dB}}}\leqslant -110$ dB) and a good gain flatness ($\Delta |{{S}_{21}}{{|}_{\text{dB}}}\leqslant \pm 1$ dB). The modules consist of broadband low noise amplifiers, attenuators and passive microstrip circuits with a high-pass like transfer function. A detailed analysis of the influence of noise and spurious signals on the performance of the electronics is conducted in section 2.2.3.

Coaxial lines are inserted in front of and behind the DUT as well as in front of the first receiving channel to achieve a sufficiently high frequency difference between the input signals of the mixers:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{d},\text{R}1}} & = & \sigma {{\tau}_{\text{ref}}}, \end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{d},\text{M}1}} & = & {{f}_{\text{d},\text{M}2}}=\sigma 2{{\tau}_{1}}. \end{array}\end{eqnarray} \tag{ 13 }$$

The output signals of the mixers are low-pass filtered, amplified and analog-to-digital converted in the IF module, which consists of low noise operational amplifiers, passive filters and three 24-Bit delta-sigma analog-to-digital converters (ADC) with a sampling rate of 3.125 MSPS. The FPGA module is used for the data transfer from the ADCs to a PC. Furthermore, it potentially allows for fast multi-channel digital signal processing. However, in this work, the ADC raw data is sent to the PC, where the signal processing is performed as discussed in section 2.1. A higher order digital band-pass filter is used to suppress spectral components related to unwanted signals and broadband noise.

To achieve maximum reproducibility of the results, the signal synthesizer, the ADCs as well as the FPGA are synchronized using a common crystal oscillator as reference, i.e. all component clocks and trigger signals are derived from or are referred to the reference clock.

2.2.2. System parameter considerations.

2.2.3. Noise analysis.

The precision of the data acquisition unit can be quantified by the relative uncertainty ${{\gamma}_{\text{S}ji}}$ of the measured S-parameter, which can be defined as the ratio of the standard deviation to the mean value of the S-parameter at a certain frequency:

$$\begin{eqnarray}&&{{\gamma}_{\text{S}ji}}(\,f)=\frac{\text{std}\left({{S}_{ji}}\left(\,\text{j}\,f\right)\right)}{|\overline{{{S}_{ji}}\left(\,\text{j}\,f\right)}|}.\end{eqnarray} \tag{ 14 }$$

The relative uncertainty is the inverse of the square root of the signal-to-noise-and-spurious ratio $\Theta$, which is given by:

$$\begin{eqnarray}\begin{array}{*{35}{l}} \Theta & = & \frac{{{P}_{\text{sig}}}}{{{P}_{\text{n}}}+{{P}_{\text{spur}}}}=\frac{1}{\gamma _{\text{S}}^{2}}, \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{P}_{\text{n}}} & = & {\int}_{{{f}_{\text{d}}}-{{B}_{\text{IF}}}/2}^{{{f}_{\text{d}}}+{{B}_{\text{IF}}}/2}{{W}_{\text{n}}}\left(\,\text{j}\,f\right)\text{d}f, \end{array}\end{eqnarray} \tag{ 16 }$$

where ${{P}_{\text{sig}}}$, ${{P}_{\text{n}}}$ and ${{P}_{\text{spur}}}$ are the signal, noise, and spurious power, respectively, and ${{W}_{\text{n}}}$ is the frequency-dependent noise power spectral density referred to the input of the receiver. The total noise power spectral density is the superposition of the sideband noise of the synthesizer and the noise of the receiver:

$$\begin{eqnarray}&&{{W}_{\text{n}}}\left(\,\text{j}\,f\right)={{W}_{\text{n},\text{synth}}}\left(\,\text{j}\,f\right)+{{W}_{\text{n},\text{rec}}}\left(\,\text{j}\,f\right).\end{eqnarray} \tag{ 17 }$$

For a high signal level at the receiver's input, the precision is often limited by the (integrated) sideband noise and the SFDR of the signal synthesizer. In most systems, the phase noise of the synthesizer is dominant and determines the maximum signal-to-noise-and-spurious ratio ${{ \Theta }_{\text{max}}}$:

$$\begin{eqnarray}&&{{ \Theta }_{\text{max}}}=\frac{{{P}_{\text{sig}}}}{{{P}_{\text{n},\phi}}}={{\left[{\int}_{-{{B}_{\text{IF}}}/2}^{{{B}_{\text{IF}}}/2}L_{\phi}^{\text{c}}\left(\,{{f}_{\text{off}}}\right)\,\text{d}{{f}_{\text{off}}}\right]}^{-1}},\end{eqnarray} \tag{ 18 }$$

where ${{P}_{\text{n},\phi}}$ is the integrated phase noise power and $L_{\phi}^{\text{c}}\left(\,{{f}_{\text{off}}}\right)$ is phase noise spectral density referred to the signal power of the carrier as a function of the offset frequency ${{f}_{\text{off}}}$ from the center frequency of the carrier. In metadyne systems the phase noise of the down-converted signal is reduced compared to that of the input signals of the mixer, since the noise sources are correlated and the phases are subtracted (phase noise canceling). Therefore, ${{ \Theta }_{\text{max}}}$ can be significantly higher in metadyne systems compared to heterodyne systems. Neglecting the phase noise suppression factor and assuming a constant phase noise spectral density of $L_{\phi}^{\text{c}}\left(\,{{f}_{\text{off}}}\right)=-98$ dBc Hz⁻¹ as well as an effective IF filter bandwidth of 10 kHz, ${{ \Theta }_{\text{max},\text{cal},10~\text{kHz}}}$ can be determined to 58 dB. For a low signal level, the noise and spurious power of the receiver become dominant and $\Theta$ is reduced.

The dynamic range of the data acquisition unit is the ratio of the maximum signal power at the receiver's input and the minimum detectable signal power:

$$\begin{eqnarray}&&DR=\frac{{{P}_{\text{sig},\text{rec},\text{max}}}}{{{P}_{\text{sig},\text{rec},\text{min}}}}.\end{eqnarray} \tag{ 19 }$$

The maximum signal power depends on the maximum tolerable nonlinearity of the receiving channel and is often chosen according to the 0.1 dB compression point of the microwave amplifiers and the mixer. The minimum detectable signal power is given by the receiver's input referred noise power [31]. Assuming a constant noise power spectral density of the receiver, the dynamic range is given by:

$$\begin{eqnarray}&&DR=\frac{{{P}_{0.1\text{dB},\text{rec}}}}{{{P}_{\text{n},\text{rec}}}}=\frac{{{P}_{0.1\text{dB},\text{rec}}}}{{{W}_{\text{th}}}\centerdot {{F}_{\text{rec}}}\centerdot {{B}_{\text{IF}}}},\end{eqnarray} \tag{ 20 }$$

where ${{W}_{\text{th}}}$ is the thermal noise power density and ${{F}_{\text{rec}}}$ is the noise factor of the receiver. The noise factor of the receiver is dominated by the noise factor of the mixer and the amplitude noise of the LO signal ${{y}_{\text{LO}}}$, which is rectified by the mixer and thus shifted into the IF band. The amplitude noise conversion factor depends on the mixer's hardware architecture and has been determined by measurements to approximately  −40 dB for the utilized double-balanced mixer (Minicircuits SYM-63LH+). The resulting noise factor of the IF channel has been determined by measurements to ${{F}_{\text{IF}}}\approx 300$ (corresponding to a noise figure of $N{{F}_{\text{IF}}}\approx 24.8$ dB). Based on the data sheet specification of the other utilized microwave components, the noise factor of the receiver can be calculated to ${{F}_{\text{rec}}}\approx 1921$ ($N{{F}_{\text{rec}}}\approx 32.8$ dB). This results in a dynamic range of $D{{R}_{\text{cal},10\text{k}\,\text{Hz}\,}}\approx \,91.2$ dB, assuming an effective IF filter bandwidth of 10 kHz.

The accuracy of the data acquisition unit can be quantified determining the relative mean error ${{\delta}_{\text{S}ji}}$ of the measured S-parameter, which can be defined as:

$$\begin{eqnarray}&&{{\delta}_{\text{S}ji}}(\,f)=|\frac{\overline{{{S}_{ji}}\left(\,\text{j}\,f\right)}-{{S}_{ji,\text{true}}}\left(\,\text{j}\,f\right)}{{{S}_{ji,\text{true}}}\left(\,\text{j}\,f\right)}|,\end{eqnarray} \tag{ 21 }$$

where ${{S}_{ji,\text{true}}}$ is the true value. The accuracy is strongly dependent on the static measurement errors introduced by the electronics and the connecting cables, which can be reduced by calibrating the system. The corresponding accuracy enhancement depends on the calibration method and the quality of the calibration standards. Furthermore, the accuracy of FMCW VNAs will be reduced if the IF filter bandwidth is chosen too low (cp. section 2.2.2).

The calibration of the system is used to reduce the static measurement errors and thus shifting the reference planes of the measurement towards the DUT. For MWI systems, the required reference planes are at the boundary or near the imaging domain, which often impedes the system calibration since not all necessary calibration standards are available. Instead of a full N-port calibration, which accounts for all systematic errors, a common approach is the normalization of the measured S-parameters [8], i.e. the measured S-parameters for the DUT are divided by the S-parameters for one calibration standard:

$$\begin{eqnarray}&&{{S}_{ji,\text{norm}}}=\frac{{{S}_{ji,\text{DUT}}}}{{{S}_{ji,\text{CAL}}}}.\end{eqnarray} \tag{ 22 }$$

This reduces the number of required calibration measurements to N², and for a reciprocal network (${{S}_{ji}}={{S}_{ij}}$) to N  +  (N  −  1)N/2 measurements. As calibration standards, highly reflective reference elements ('Open' or 'Short') are used for the normalization of the reflectances S_ii and low-loss standards ('Thru' or 'Line') are used for the normalization of the transmittances S_ji ($j\ne i$).

However, the achievable calibration accuracy, especially for the normalized reflectances, is often insufficient for MWI [8]. In FMCW VNAs, it is possible to separate the unwanted signal components, due to the limited isolation, directivity and matching of utilized RF modules, from the desired signals in the IF domain. This requires a sufficiently high difference in electrical length of the signal propagation paths. For example, figure 5 illustrates the different signal propagation paths of the major signal components for a reflection measurement. Due to the finite isolation of the Wilkinson divider, the RF input signal is partly coupled into the receiving channel. Furthermore, the signal guided towards to the DUT, is partly reflected at the input of the switching matrix. The unwanted signal components can be suppressed if the deviations of the frequency differences

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{d},\text{WD}}} & = & 0, \end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{d},\text{SM}}} & = & \sigma \,2\,{{\tau}_{1}}, \end{array}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{f}_{\text{d},\text{DUT}}} & = & \sigma \,2\,\left({{\tau}_{1}}+{{\tau}_{2}}\right), \end{array}\end{eqnarray} \tag{ 25 }$$

are higher than the required IF filter bandwidth, which depends on the modulation bandwidth:

$$\begin{eqnarray}&&{{B}_{\text{mod}}}<{{f}_{\text{d},\text{DUT}}}-{{f}_{\text{d},\text{SM}}}.\end{eqnarray} \tag{ 26 }$$

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Furthermore, the calibration accuracy is limited due to multiple reflections inside the system, which result in signal contributions close to ${{f}_{\text{d},\text{DUT}}}$ and thus cannot be suppressed. The described method is equivalent to the time-gating technique, which is used in pulse-based systems [25], and is also applicable to CW VNAs [32], but requires a high measurement bandwidth in conjunction with a large number of frequency samples resulting in a low measurement rate.

To characterize the performance of the developed prototype electronics as well as to investigate the influence of the system parameters and the applicability of the described calibration method, we conducted measurements for different passive microwave devices and different system parameters. The measurement results are compared to those obtained with a commercial high-end CW VNA ($\text{R} \And\text{S}\circledR$ ZVA40). Based on the results of 100 measurements per configuration, the accuracy, the precision, and the DR of the system are calculated. The ramp sweep time and the maximum power at the receiver's input were chosen to 1 ms and  −10 dBm, respectively. To investigate the influence of the ramp slope on the system's performance, measurements for two different frequency ranges (0.4 GHz to 5.6 GHz and 0.8 GHz to 3 GHz) were conducted. The corresponding frequency differences are ${{f}_{\text{d}1}}\approx 295$ kHz and ${{f}_{\text{d}2}}\approx 125$ kHz. The effective number of frequency samples can be calculated by [30]:

$$\begin{eqnarray}&&{{Z}_{\text{eff}}}=4\centerdot {{f}_{\text{d}}}\centerdot {{T}_{\text{ramp}}},\end{eqnarray} \tag{ 27 }$$

leading to ${{Z}_{\text{eff},1}}\approx 1180$ and ${{Z}_{\text{eff},2}}\approx 500$.

The IF filter bandwidth ${{B}_{\text{IF}}}$ was varied to investigate its influence on the performance parameters and it is chosen in relation to the current frequency difference:

$$\begin{eqnarray}&&{{B}_{\text{IF}}}=k\centerdot {{f}_{\text{d}}}.\end{eqnarray} \tag{ 28 }$$

The ZVA40 was TOSM calibrated and its input power and filter bandwidth were chosen to 0 dBm and 1 kHz, respectively. In the following subsection, the 2-port measurement results of the prototype electronics are presented. Afterwards, the influence of the switching matrix on the system's performance is investigated.

2.4.1. 2-Port measurements.

Figure 6 shows the comparison between the reflection parameter S₁₁ for a coaxial band-pass filter as DUT measured in the frequency range from 0.5 GHz to 5.5 GHz using the prototype electronics (k  =  0.3) and the ZVA40, respectively. The lower and upper cutoff frequencies of the band-pass filter are ${{f}_{\text{c}1,\text{bp}}}\approx 2.2$ GHz and ${{f}_{\text{c}2,\text{bp}}}\approx 2.4$ GHz, respectively. The measurement results are in very good agreement with each other. The relative mean error ${{\delta}_{\text{S}11}}$ of the reflection parameter measured using the prototype electronics and the ZVA40 has been calculated for OSM calibration and normalization of the reflection parameter, respectively, as depicted in figure 7. For OSM calibration, the relative mean error is between $5\centerdot {{10}^{-3}}$ and $2\centerdot {{10}^{-3}}$ in the stop band of the filter and increases to 0.08 and 0.2 in the pass band of the filter. The relative error for the normalization of the reflection parameter is significantly higher compared to the OSM calibration.

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The influence of the IF filter bandwidth ${{B}_{\text{IF}}}$ on the relative error of the reflection parameter S₁₁ is illustrated in figures 8 and 9. The reduction of the filter bandwidth from k  =  0.3 to k  =  0.2 does not degrade the accuracy of the measurement, whereas the further reduction from k  =  0.2 to k  =  0.1 leads to an increase of the relative mean error, especially in the frequency range close to the sharp resonances. The major conclusion is, that the accuracy is independent of the IF filter bandwidth if ${{B}_{\text{IF}}}$ is sufficiently high. However, if ${{B}_{\text{IF}}}$ is chosen too low, i.e. lower than the modulation bandwidth, the accuracy will be significantly reduced as explained in section 2.2.2.

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The measured transmission parameter for the coaxial band-pass filter is shown in figure 10. The measurement results obtained using the prototype electronics and the commercial VNA, respectively, are in good agreement overall. The amplitude and phase response ripples of the transmission parameter S₂₁ measured using the commercial VNA are caused by static measurement errors, e.g. multiple reflections in the measurement setup, which cannot be fully suppressed due to the limited calibration accuracy. The ripple in the stop band of the filter is higher than that in the pass band, since the desired-signal-to-error-signal-power ratio is larger due to the high attenuation and the high reflection coefficients $|{{S}_{ii}}|$ of the DUT in the stop band. The ripple of the transmission parameter measured using the prototype electronics is significantly lower, because the signal contributions related to multiple reflections have spectral components at higher frequencies than the desired difference frequency and, therefore, are suppressed by the IF filter (cp. section 2.3).

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To determine the precision and the DR of the prototype electronics, transmission measurements for different microwave attenuators with increasing attenuation have been conducted. Figures 11 and 12 show the magnitude of measured transmittance and the relative uncertainty for an effective IF bandwidth ${{B}_{\text{IF}}}\approx 60$ kHz (k  =  0.2), respectively. For low-loss DUTs ($|{{S}_{21}}|\geqslant -20$ dB), the relative uncertainty of approximately $6\centerdot {{10}^{-4}}$ (−65 dB) is determined by the sideband noise of the synthesizer. This corresponds to a measured signal-to-noise-and-spurious ratio for low-loss DUTs of ${{ \Theta }_{\text{max},\text{meas}}}\approx$ 65 dB. It is significantly greater than the theoretical value ${{ \Theta }_{\text{max},\text{cal},60\,\text{kHz}}}=50.1$ dB calculated using (18) and corresponds to an phase noise suppression factor of approximately 15 dB. The increase of the relative uncertainty at frequencies close to 0.6 GHz and 1 GHz results from spurious signals, which are generated by the synthesizer. Since the frequencies of the spurious signals are very close to the signal frequency, the spurious signals cannot be suppressed by the IF filter (in-band spurious). For high-loss DUTs, the relative uncertainty increases as the attenuation increases. The measured dynamic range of the system is approximately 80 dB, which corresponds well to the theoretical value of $D{{R}_{\text{cal},60~\text{kHz}}}=83.3$ dB calculated using (20).

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The relationship between the measurement precision, the DR and the IF filter bandwidth is illustrated in figure 13, comparing the relative uncertainty of the transmittance for low-loss and high-loss DUTs and two different measurement frequency ranges corresponding to different ramp slopes. For the second measurement frequency range (0.8 GHz to 3 GHz), the ramp slope and thus the frequency difference ${{f}_{\text{d}2}}$ is lower than the frequency difference ${{f}_{\text{d}1}}$ for the frequency range from 0.4 GHz to 5.6 GHz (${{f}_{\text{d}2}}\approx 125$ kHz, ${{f}_{\text{d}1}}\approx 295$ kHz). As k is constant (k  =  0.2), the IF filter bandwidth for the second frequency range is lower:

$$\begin{eqnarray}&&\frac{{{B}_{\text{IF}2}}}{{{B}_{\text{IF}1}}}=\frac{{{f}_{\text{d}2}}}{{{f}_{\text{d}1}}}\approx 0.42.\end{eqnarray} \tag{ 29 }$$

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Due to the decreasing integrated noise, this leads to a theoretical increase of ${{ \Theta }_{\text{max}}}$ and the DR of approximately 3.7 dB, which corresponds well to the measured increase of 3–4 $\text{dB}$.

To be able to compare the DR, the maximum signal-to-noise-and-spurious ratio ${{ \Theta }_{\text{max}}}$ (related to the precision), and the temporal resolution of the FMCW prototype electronics to those of the commercial high-end CW VNA, the DR, ${{ \Theta }_{\text{max}}}$, and sweep time ${{T}_{\text{s}}}$ of the ZVA40 have been determined for different effective filter bandwidth values, power levels and number of frequency samples in the frequency range from 0.5 GHz to 5.5 GHz (see tables 1–3). Assuming an effective IF filter bandwidth of 60 kHz and a source power of ${{P}_{\text{in}}}=-10$ dBm, the dynamic range of the prototype system is approximately 5 dB higher than that of the ZVA40 ($D{{R}_{\text{PE},60~\text{kHz}}}\approx 80$ dB and $D{{R}_{\text{ZVA},50~\text{kHz}}}\approx 75$ dB). The maximum signal-to-noise-and-spurious ratio of the prototype electronics is approximately 13 dB higher (${{ \Theta }_{\text{max},\text{PE},60~\text{kHz}}}\approx 65$ dB and ${{ \Theta }_{\text{max},\text{ZVA},50~\text{kHz}}}\approx 52$ dB). The sweep time is significantly lower (${{T}_{\text{s},\text{PE},60~\text{kHz}}}\approx 1$ ms and ${{T}_{\text{s},\text{ZVA},50~\text{kHz}}}\approx 65$ ms) for the same number of frequency sample (Z  =  1180)

Table 1. Approximated average dynamic range of the ZVA40 in the frequency range from 0.5 GHz to 5.5 GHz as a function of IF filter bandwidth and input power.

${{P}_{\text{in}}}$ (dBm)	$D{{R}_{1~\text{k}}}$ (dB)	$D{{R}_{10~\text{k}}}$ (dB)	$D{{R}_{50~\text{k}}}$ (dB)	$D{{R}_{100~\text{k}}}$ (dB)
0	100	92	85	80
−10	92	80	75	70

Table 2. Approximated average maximum signal-to-noise-and-spurious ratio ${{ \Theta }_{\text{max},\text{ZVA}40}}$ of the ZVA40 in the frequency range from 0.5 GHz to 5.5 GHz as a function of IF filter bandwidth and input power.

${{P}_{\text{in}}}$ (dBm)	${{ \Theta }_{1~\text{k}}}$ (dB)	${{ \Theta }_{10~\text{k}}}$ (dB)	${{ \Theta }_{50~\text{k}}}$ (dB)	${{ \Theta }_{100~\text{k}}}$ (dB)
0	70	63	58	55
−10	68	57	52	48

Table 3. Approximated required sweep time of the ZVA40 without system error correction as a function of IF filter bandwidth and number of frequency samples.

${{Z}_{\text{ZVA}40}}$	${{T}_{\text{s},1~\text{k}}}$ (ms)	${{T}_{\text{s},10~\text{k}}}$ (ms)	${{T}_{\text{s},50~\text{k}}}$ (ms)	${{T}_{\text{s},100~\text{k}}}$ (ms)
1180	1220	159	65	53
1000	1035	135	55	45
100	103	13.5	5.5	4.5
10	10.3	1.35	0.55	0.45

2.4.2. 2-Port measurements with switching matrix.

In this subsection, the influence of the switching matrix on the accuracy, the precision, and the DR is investigated. Figure 14 shows the comparison of the relative error of the reflectance of the coaxial band-pass filter for OSM calibration and normalization of the reflection parameter, respectively. Comparing the results in figures 14 and 7, it can be seen that the relative error for OSM calibration increases in the stop band of the filter, whereas it does not change in the pass band. For the normalized reflection parameter, the accuracy does not decrease due to the switching matrix enabling to utilize this calibration method for multi-port measurements.

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Figure 15 shows the relative uncertainty of the transmittance for different coaxial attenuators. The minimum relative error is slightly increased to approximately $8\centerdot {{10}^{-4}}$ (−62 dB), while the DR is reduced to approximately 70 dB at low frequencies and 60 dB at high frequencies due to the additional insertion loss ($|{{S}_{21,\text{SM}}}|\approx -10$ dB at 1 GHz) of the switching matrix, which increases with increasing frequency.

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Figure 16 shows a photograph of the measurement setup for microwave imaging, which includes the proposed prototype electronics, a $2\times 8$ switching matrix and the 8-port MWT sensor. The utilized MWT sensor consists of a metal pipe with eight wedge-shaped dielectric windows and flush-mounted rectangular waveguides. The sensor is a linear 8-port network and can be described using its scattering matrix. The electric fields of the electromagnetic waves at the interfaces between the dielectric windows and the rectangular waveguides are the input parameters of the reconstruction algorithm and can be calculated based on the measured S-parameters. In previous work [8], we utilized a commercial CW VNA in conjunction with the switching matrix limiting the temporal resolution and leading to a very low accuracy of the normalized reflection parameters S_ii. Therefore, the reconstruction of the permittivity distribution was based on the normalized transmission parameters only.

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The applicability of the calibration method described in section 2.3 for correction of the S-parameters of the 8-port MWT sensor was evaluated in the case that the sensor is homogeneously filled with air or polypropylene (PP), respectively. As one example, figures 17 and 18 show a comparison of the S-parameters S₁₁ and S₅₁ for the air-filled sensor, which were measured using the prototype electronics and the commercial VNA, respectively, in conjunction with the switching matrix, and were simulated using a 3D electromagnetic field simulation software (CST Microwave Studio ®). The measurement results using the prototype electronics and the simulation results are in good agreement, especially in the frequency range of interest from 1.2 GHz to 2.5 GHz, where the mismatch at the interface between the dielectric windows and the waveguides is low. This mismatch is neither included in the 3D simulation nor in the 2D model used for the reconstruction due to the high computational effort.

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In the frequency range of interest, the relative error ${{\delta}_{\text{S}ij}}$ of the measured results obtained using the prototype electronics compared to the simulated results is below 0.11 for the reflection parameter and below 0.09 for the transmission parameter (see figures 19 and 20). Comparing the relative error of the measurement results obtained using the prototype electronics and the commercial VNA, it can be seen that reflection measurement accuracy for the prototype is higher by a factor of 75 (${{\delta}_{\text{S}11,\text{PE}}}=0.11$ to ${{\delta}_{\text{S}11,\text{ZVA}40}}=8.25$), and the transmission measurement accuracy is higher by a factor of 3 (${{\delta}_{\text{S}51,\text{PE}}}=0.09$ to ${{\delta}_{\text{S}51,\text{ZVA}40}}=0.27$). Due to the improved accuracy of the reflection parameters, the permittivity distribution can be reconstructed based on transmission and reflection results.

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The measurement and reconstruction procedure is described in detail in [8]. For image reconstruction the linearized one-step Gauss–Newton method [33] combined with the modified perturbation method [34] were utilized, generating a qualitative approximation of the permittivity distribution. To be able to compare the reconstruction results to those published in [8], we reconstructed the permittivity distribution for the same scenarios using air, polypropylene (PP) and tap water (TW) with the relative permittivities of ${{\varepsilon}_{\text{r},\text{air}}}\approx 1$, ${{\varepsilon}_{\text{r},\text{PP}}}\approx 2.2$, and ${{\varepsilon}_{\text{r},\text{TW}}}\approx 78-\text{j}5$, respectively.

In this subsection, the real parts of reconstructed permittivity distributions for two different inhomogeneous scenarios are presented and discussed. The reconstruction results are based on single-frequency measurement data of a broadband sweep from 0.4 GHz to 5.6 GHz of one measurement cycle. The color bar of the presented images is adapted to the reference material, i.e. the color green represents the permittivity of air (${{\varepsilon}_{\text{r},\text{air}}}\approx 1$) or PP (${{\varepsilon}_{\text{r},\text{PP}}}\approx 2.2$). Positive dielectric contrasts are indicated in red and negative contrasts in blue. The color bar is scaled to the maximum dielectric contrast of each reconstructed permittivity distribution.

Figure 21 shows the two measurement scenarios: one PP rod as well as a TW-filled polymethylmethacrylate (PMMA) tube (${{\varepsilon}_{\text{r},\text{PMMA}}}\approx 2.4$) in an air-filled sensor and a PP cylinder with one TW-filled and one air-filled drill hole, respectively. The reconstructed permittivity distribution for the first scenario based on transmission data at 1.5 GHz is shown in figure 22(a). TW appears in dark red with a estimated permittivity of ${{\varepsilon}_{\text{r},\text{TW},\text{est}}}\approx 3.2$ and PP in light orange (${{\varepsilon}_{\text{r},\text{PP},\text{est}}}\approx 1.9$). Figure 22(b) shows the reconstructed permittivity distribution for the second scenario based on transmission data at 1.25 GHz. TW appears in dark red with an estimated permittivity of approximately ${{\varepsilon}_{\text{r},\text{TW},\text{est}}}\approx 6.2$ and air in light blue (${{\varepsilon}_{\text{r},\text{air},\text{est}}}\approx 1$).

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Comparing these reconstruction results to those based on measurement data obtained with a commercial CW VNA (see figure 23), it can been seen, that the image quality in terms of the localization of the objects is higher for the FMCW measurement data due to the higher measurement accuracy. To be able to quantify the accuracy of the reconstructed center of the objects, the reconstructed permittivity along the axis of symmetry (y  =  0) has been determined for both measurement scenarios as depicted in figures 25 and 26. For the first measurement scenario, the relative errors of the reconstructed x-position x₀ of the objects are $\delta _{\text{x}0,\text{TW},\text{FMCW}}^{\text{I}}\approx 0.2$ and $\delta _{\text{x}0,\text{PP},\text{FMCW}}^{\text{I}}\approx 0.2$ for FMCW measurement data as well as $\delta _{\text{x}0,\text{TW},\text{CW}}^{\text{I}}\approx 0.22$ and $\delta _{\text{x}0,\text{PP},\text{CW}}^{\text{I}}\approx 0.29$ for CW measurement data. The deviation in maximum permittivity results from the inaccuracy of the CW measurement data. For the second measurement scenario, the relative errors are $\delta _{\text{x}0,\text{TW},\text{FMCW}}^{\text{II}}\approx 0.16$ and ${{\delta}_{\text{x}0,\text{air},\text{FMCW}}}\approx 0.01$ for the FMCW data, as well as $\delta _{\text{x}0,\text{TW},\text{CW}}^{\text{II}}\approx 0.2$ and $\delta _{\text{x}0,\text{air},\text{CW}}^{\text{II}}\approx 0.17$ for the CW data.

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Figure 24 shows the reconstruction results based on FMCW reflection and transmission data. Compared to the results based on FMCW transmission data only, the centers of the dielectric phantoms are shifted towards the center of the imaging domain, which corresponds to an improved accuracy of the reconstructed distributions. The relative errors are $\delta _{\text{x}0,\text{TW}}^{\text{I}}\approx 0.16$ and $\delta _{\text{x}0,\text{PP}}^{\text{I}}\approx 0.1$ for the first measurement scenario, as well as $\delta _{\text{x}0,\text{TW}}^{\text{II}}\approx 0.1$ and $\delta _{\text{x}0,\text{air}}^{\text{II}}\approx 0$ for the second scenario.

It can be concluded, that the improved accuracy of the scattering parameters measured using the FMCW prototype electronics results in an enhanced accuracy of the images based on transmission data. The additional utilization of reflection data for the reconstruction leads to a further improvement in image accuracy.