A) System concept

While commonly used FMCW-based radar system measure a beat frequency to distinguish the distance to the target, the proposed Six-port-based system evaluates the distance dependent phase of the reflected wave at the target. With this technique, there is no need for a time-consuming fast Fourier transformation (FFT), which leads to a fast responding, low-latency system. The drawback that only a single target can be tracked can often be neglected if there is only a single dominant reflection, as at the addressed applications.

The sketch of the proposed radar system is shown in Fig. 1. A frequency-adjustable CW signal is fed into a coupler, where a small portion is fed as a phase reference into the first input port of the Six-port P ₁. The remaining signal power is coupled to the waveguide structure, reflected by the target and guided to the Six-port's second input port P ₂. A circulator is required to divide the received and transmitted signals, because they have to share the same waveguide feed.

Fig. 1. Measurement setup for waveguide-based distance evaluation.

The core of the system, the Six-port interferometer, can be understood as a homodyne IQ-mixer with differential base band output signals B ₃ to B ₆, forming the base band signal $\underline z $ [Reference Vinci, Lindner, Barbon, Weigel and Koelpin8]:

(1) $$\underline z = \lpar B_5 - B_6 \rpar + j\lpar B_3 - B_4 \rpar .$$

This vector is generated by a cost-efficient and passive structure shown in Fig. 2(a). A Wilkinson power divider and three hybrid couplers superimpose the two input signals P ₁ and P ₂ with discrete phase shift of 0°, 90°, 180°, and 270°. The resulting interfered signals are converted to base band by diode power detectors forming the differential complex vector $\underline z $ . To clarify the phase relations, they are depicted in Fig. 2(b).

Fig. 2. Implemented Six-port structure (a) and phase relations (b).

B) Distance calculation

In the context of a Six-port radar, the distance d of a target can be evaluated if the used transmission frequency f _RF is known and the phase of the complex vector $\underline z $ has been determined:

(2) $$d = \Delta \sigma \cdot \displaystyle{c \over {2 \cdot 2\pi \cdot f_{RF} }}\quad \hbox{with}\; \Delta \sigma = \arg \lcub \underline z \rcub .$$

As in common interferometric approaches, the problem of ambiguity in phase arises, limiting the unambiguous measurement range to half of the wavelength of the used frequency. This ambiguity can be eliminated by measuring the phase difference (Δσ ₁, Δσ ₂) for two tones (f ₁, f ₂) with a spacing of f _B between them as shown in [Reference Lindner, Barbon, Vinci, Mann, Weigel and Koelpin11]. Calculating the difference between the two phase responses, an absolute, coarse distance d _coarse can be evaluated:

(3) $$d_{coarse} = \Delta \sigma _B \cdot \displaystyle{c \over {2 \cdot 2\pi \cdot f_B }}\quad \hbox{with}\; \Delta \sigma _B = \Delta \sigma _1 - \Delta \sigma _2 .$$

After the determination of this coarse distance information, the period of unambiguity is known and an additional high precision distance evaluation based on the two single tones can be done using (2) leading to a micrometer accuracy distance value, which is unambiguous within a range of d _max :

(4) $$d_{max} = \displaystyle{c \over {2 \cdot f_B }} = \displaystyle{c \over {2 \cdot \lpar f_2 - f_1 \rpar }}.$$

C) Waveguide propagation related corrections

A problem arises if the measurement takes place in an enclosed system, e.g. to detect the position of a reflecting target within a waveguide, as the dispersion has to be considered. Due to the field propagation within a waveguide structure, the wavelength λ _w of a transmitted signal within the waveguide depends on the cut-off frequency f _c of the used mode and the free space wavelength λ ₀ = c/f of the transmitted signal:

(5) $$\lambda _w = \lambda _0 \displaystyle{1 \over {\sqrt {1 - \lpar {f_c /f} \rpar ^2 } }}.$$

This equation can be simplified by assuming single mode transmission, e.g. by using a standardized waveguide within its specification. In this case λ _w depends only on the geometrical width a of the used waveguide and λ ₀:

(6) $$\lambda _w = \displaystyle{{\lambda _0 } \over {\sqrt {1 - \lpar {\lambda_0 /2a} \rpar ^2 } }}.$$

It is obvious that the wavelength λ _w has a nonlinear relation to λ ₀ for free space. Therefore, the beat frequency f _B has to be calculated from the wavelengths of the single tones λ _w1, λ _w2 within the waveguide:

(7) $$f_{B\comma w} = \displaystyle{{\lambda _{w1} - \lambda _{w2} } \over {\lambda _{w1} \cdot \lambda _{w1} }} \cdot c.$$

Furthermore, the unambiguous distance measurement range inside the waveguide d _max,w , which depends on f _B,w is limited in this case to

(8) $$d_{max\comma w} = \displaystyle{c \over {2 \cdot f_{B\comma w} }}.$$

A few examples of possible unambiguous ranges at free space propagation as well as within the waveguide are provided in Table 1. While at the single tone setup the range is limited by the used RF wavelength, in the case of a dual tone system the unambiguity range can be calculated according (4) and within the waveguide setup according (8), respectively.

Table 1. Unambiguous ranges for different frequency spacings with a center frequency of 24 GHz at free space propagation (no index) and waveguide conditions (index w).

A) Isolation of the radar coupler

Like in all radar-based systems, the isolation between the transmitted and received signal acts as a strong interferer in detection of the correct target position. A similar effect is caused by the often poor matching of the antenna, which also introduces a static reflection in the receive path superimposing the signal from the target. These effects can be reduced by using a bi-static setup with two antennas, but this is not feasible within the shown setup due to the waveguide-transition.

The main paths of transferred power within the system are depicted in Fig. 3. Besides the desired signals P _ref and P _rx at the Six-port there is also a strong static signal P _iso caused by low directivity of the circulator and a second static signal P _feed due to insufficient matching of the feeding structure. Both of these disturbing signals are even higher, if an additional LNA exists between the circulator and the Six-port's input P ₂. The parasitic target reflection P′ _rx can be neglected due to high dampening in this path. Secondary static targets or reflections due to the environment are also neglected for simplicity. Nevertheless, they behave like P _feed , thus they can add to P _feed , if it is necessary to consider them due to their high radar cross section.

Fig. 3. Paths of RF power distribution.

However, as long as the static parasitic reflections do not depend on the target position, they will produce only a static complex offset $\underline z _o $ added to the target's signal $\underline z _T $ at the base band constellation diagram. The measured value will be in this case:

(9) $$\underline z ^{\prime} = \underline z _o +\underline z _T .$$

This behavior is shown in Fig. 4, where an offset effected signal from a moving target is plotted. The target is moving with a constant velocity v, forming a circle in the constellation diagram. To calculate the distance out of the phase, the offset vector $\underline z _o $ can be removed by a simple complex valued subtraction.

Fig. 4. Complex IQ result (a) calculated from base band signals (b) with offsets due to isolation issues observing a moving target.

The offset vector can easily be measured and digitally compensated if the target moves more than half of the used wavelength, but the offset will significantly decrease the dynamic range of the digitization. Alternatively, this can be done analogously by adding a signal with an appropriate power and phase to the received signal at port P ₂, which compensates the sum of P _iso  + P _feed . Such systems can be implemented using a variable attenuator and phase shifter, depicted in Fig. 5, to compensate also a slow changing environmental clutter. But this implementation is difficult, because the regulation of phase and amplitude of the compensation signal has to be very accurate and stable. Therefore, the preferable way is to suppress this offset by a sophisticated RF design, e.g. to consider both the isolation issue and the antenna feed reflection to compensate for each other, or to use the static reflection as a bias signal for the detectors to equalize the gain imbalance within the Six-port junction. Simplified systems, e.g. [Reference Yang, Kim and Hong12] or [Reference Kim, Kim, Lee, Kho and Yook13], based on the same feed-forward idea are already studied and measured but only capable of compensating the isolation P _iso .

Fig. 5. Variable feed-forward compensation structure to reduce offsets.

B) Oscillator influences

For radar-based distance measurements, the phase noise of the used oscillator has to be considered. The resulting root mean square (RMS) error can be calculated, according to Parseval's theorem, by integrating the phase noise density spectrum in the used bandwidth. However, this result is only correct, if the transmitted and received signals are completely uncorrelated. Here, it can be shown [Reference Budge and Burt14] that there is a range correlation effect, which has a high-pass characteristic and suppresses the phase noise density more than 100 dB for low offset frequencies.

Considering this effect, the RMS phase error σ _err can be calculated depending on the phase noise spectrum of the used oscillator, the distance to the target and the used base band bandwidth f _bw :

(10) $$\sigma _{err} = \sqrt {\int_0^{f_{bw} } 2 \cdot \underbrace {{\cal L}\lpar f\rpar }_{{\rm Phase}\, {\rm noise}} \cdot \underbrace {4\mathop {\sin }\nolimits^2 \left({\displaystyle{{2d\Delta f} \over c}} \right)}_{{\rm Range}\, {\rm correlation}}\hbox{d}f} .$$

Figure 6 shows the evaluation of (10), mapped to a distance error, for different target distances over the used base band bandwidth. The underlying phase noise density originates from a phase-locked loop (PLL) stabilized 24 GHz Silicon Radar voltage controller oscillator measured with a Rohde & Schwarz FSUP26 spectrum analyzer at offset frequencies between 0.1 Hz and 1 MHz.

Fig. 6. Distance error due to oscillator's phase noise depending on the used base band bandwidth.

Although the overall accuracy of the system is limited to the error in the single tones, the phase noise does not influence the performance of the system, as the error is below 1 µm at least for measurement distances of only a few meters.

However, the used frequencies have to be known precisely, otherwise there will be a significant error in the measurement. Nevertheless, the required accuracies can easily be achieved using a PLL stabilized oscillator with a stable quartz reference.

C) Limit of unambiguity solution

The shown equation can lead to the assumption that huge unambiguity ranges can easily be achieved by using two narrow spaced RF signals. However, for narrow frequency spacings between the two tones f _B the slope of the transfer function ∂d/∂σ _B becomes very flat and therefore susceptible for noise.

To clarify this circumstance, simulations were done to investigate the effect of noise at different frequency spacings. Therefore, white Gaussian noise was added to the signals before calculating the coarse distance information between the two signals. The error of this calculation has to be lower than a quarter of the larger single wavelength to distinguish the correct ambiguity range for the fine distance evaluation in a second step.

Figure 7 shows the simulation results of the system using different frequency spacings f _B from 1 MHz to 2.4 GHz at different noise levels which are expressed by the signal-to-noise ratio (SNR). The graph shows the probability for an error lower than it is necessary for a correct determination of the unambiguous region. This probability should be nearly at 100% to develop a usable system. For a bandwidth of 250 MHz, as used in the proposed system achieving an unambiguous range of 50 cm, this means an overall SNR of lower than 40 dB. To achieve reliable measurements in ranges larger than 10 m, an SNR larger than 65 dB has to be reached. The results for the SNR values lower than 10 dB have to be ignored due to mathematical errors within the simulation environment.

Fig. 7. Simulation results of the usability of the dual tone setup with the influence of additive white Gaussian noise.

The shown considerations are only based on noise effects. Additional errors caused for example by multiple target scenarios or non-linearities have to be taken into account separately. While non-linearity errors can be removed using a calibration, the system is not capable of observing multiple moving targets. However, if they are static, they introduce a removable static offset, described in the Section III-A).

A) Measurement setup

An Agilent E8267D signal generator was used as synthesizer to obtain the two accurate frequencies f ₁ and f ₂. Nevertheless, other low cost synthesizer based systems have shown similar accuracies [Reference Mann, Lindner, Lurz, Barbon, Linz, Weigel and Koelpin15], revealing that there is no need for a highly stable source and proving the described phase noise influences. The target is positioned by a PiMicos linear stage, which can move 15 cm in steps of 0.5 µm. Due to its very precise optical encoder, the stage is also used as position reference for all shown measurements.

The Six-port front-end, whose main parameters are summarized in Table 2, is connected by a coaxial-to-waveguide adapter to the 24 GHz capable waveguide type WR-42 shown in Fig. 9. The used waveguide has a length of 1 m and the measurements were performed in the middle of the structure over the entire length of the stage of 15 cm to provide realistic results.

Fig. 9. Measurement setup.

Table 2. Measured parameters of the used front-end.

The data were acquired at a rate of 400 kSa/s after a 8th order Butterworth low pass filter with a cut-off frequency of 50 kHz. The digitized data are transferred to a PC, where the data processing is done with Matlab to simplify the development of the algorithms. In future, the whole calculations will be implemented within a micro-controller to achieve a low-cost, low-power and low-latency signal processing.

B) Measurement results

Figure 10 shows the linearized single phase and corresponding beat phase responses at 24 GHz with a spacing of 2.4 GHz. The shortened wavelength of the beat frequency, compared to the free space setup, is clearly visible. Two beat frequency periods are recognizable with a spacing of 5.1 cm corresponding to the value in Table 1.

Fig. 10. Linearized phases for waveguide measurements at 24 GHz and a frequency spacing of f ₁ − f ₂ = 2.4 GHz.

From these phase responses, it is possible to detect the ambiguity period of the phase measurement for a single frequency and thus to give an accurate distance information. The results are plotted in Fig. 11 for the first ambiguity period, i.e. the interval from about 15 to 65 mm. Due to the high accuracy of the system, the error calculated with respect to the reference system is directly shown. The first plot (a) shows the coarse distance evaluation results for the beat frequency without any compensation or offset correction. The small offset and gain error is compensated in the second plot (b), while in (c) the fine distance evaluation error of one of the single tones is displayed. This is calculated by searching for the correct period of ambiguity in the coarse distance and adding an appropriate offset to the distance evaluation based on a single tone as shown in [Reference Barbon, Vinci, Lindner, Weigel and Koelpin16]. The error is below ±25 µm exceeding the accuracy for FMCW, e.g. shown in [Reference Ayhan, Pauli, Kayser, Scherr and Zwick17], where a bandwidth of 1 GHz is used achieving a maximum error of 150 µm.

Fig. 11. Waveguide-based distance measurements at a frequency spacing of 2.4 GHz within the first ambiguity period of the beat frequency.

To provide an ISM conform and long range capable measurement, a second evaluation with a frequency spacing of only 250 MHz was done. According to Table 1 this leads to an unambiguous range of approximately 50 cm, although the measurement is limited by the stage to only 15 cm. The results are shown in Fig. 12. Caused by the smaller bandwidth, this measurement has significantly higher errors at the coarse distance detection. In (a) the raw coarse distance evaluation for the beat frequency can be observed. The offset error here is caused by the start position of the measurement, while the gain error can have different influences, e.g. an erroneous calculation of λ _w .

Fig. 12. Waveguide-based distance measurement with ISM conform bandwidth of 250 MHz.

Nevertheless, these linear errors can easily be removed and the resulting compensated coarse distance error in (b) is below a quarter of the value of the single tone wavelengths. Therefore, a fine distance evaluation based on the single phases is possible. The results of this evaluation are shown in Fig. 12(c). Here, the error is higher than for the first measurement, but still below  ±40 µm over the whole distance range. The high noise, especially in the mid range, is probably caused by mechanical friction between the slider in the waveguide used as target and the wall of the waveguide, as the error is significantly lower at the beginning of the waveguide within the first distance values.

C) Statistical analysis

For a better comparison with other systems, Figs 13 and 14 show a statistical analysis of the coarse distance error and the overall error, respectively. In both plots (a) depicts a histogram of the achieved error, while (b) shows the cumulative histogram of the absolute value. To distinguish the region for the fine distance evaluation, an error lower than a quarter of the used RF wavelength has to be ensured. Here, this was achieved for more than 99.9% of all measurements. The slight mean error in Fig. 13(a) is due to an imperfect gain correction (see Fig. 12(b)), but it has no influence to the overall system accuracy as long as the error is below a quarter of the used RF wavelength, because the coarse distance is only used to detect the period of unambiguity.

Fig. 13. Histogram of coarse distance detection (f _B  = 250 MHz).

Fig. 14. Histogram of fine distance detection (f _B  = 250 MHz).

The statistical analysis of the fine distance evaluation is shown in Fig. 14. The cumulative frequency of measurements yielding an error lower then  ±35 µm is more than 99.73%, i.e. within the 3σ range. More than 75% of the measurement errors are even lower than  ±10 µm.