Appendix A: Basics of Phase Based Ranging and Multi-Carrier Phase Ranging

Phase Based Ranging (PBR) systems involve measuring the changes to the phase of the radio signal propagating between two entities to determine the distance between them.

Consider a ranging example, wherein Entity A (initiator) is measuring its distance from Entity B (reflector). In an ideal PBR system, Entity A initiates ranging by transmitting an unmodulated continuous wave tone, at a specific frequency f. Entity B receives this RF signal and acts as a true reflector by locking the local oscillator to the incoming RF signal and transmits it back to the initiator. Finally, the initiator measures the difference in the phase of the received signal and its own local oscillator signal and determines the distance based on this phase measurement. Figure 8-1 shows phase based ranging with RF signal at a specific frequency. Entity A is the initiator and Entity B is acting as a true reflector by locking its LO to the incoming RF signal and transmitting it back to the initiator.

Figure 8-1 Phase Based Ranging with RF Signal at a Specific Frequency

If the distance d, between the initiator and the reflector is less than the signal’s wavelength, that is, $\frac{2\times f}{c}$, where f is the frequency of the RF tone and c is the speed of light, then, the measured phase offset or phase difference θ is:

However, in real-world applications, there is a need to measure distances longer than signal’s wavelength and to do this, there is a necessity to keep track of the number of whole cycles that have elapsed when the RF tone is propagating between the two entities. Considering n is the number of whole cycles (integer) that have elapsed, then, distance d is measured as:

To eliminate the need for tracking the number of whole cycles elapsed, multi-carrier phase ranging is considered. In multi-carrier phase ranging systems, the phase measurements (difference in the phase of the received signal and its own local oscillator signal) are taken at multiple RF tone frequencies. RF signals at different frequencies traveling the same distance (and in turn, the same propagation time) can have a different phase offset or shift.

For example, consider that the initiator and reflector devices with distance d between them, perform PBR (as described with Figure 2-1) at two frequencies: f1 and f2. That is, the devices first perform PBR at RF signal frequency f1 and then at RF signal frequency f2. The phase differences measured at f1 and f2 is shown as:

Combining Equation 4 and Equation 5 to remove number of whole cycles (n) ambiguity, the distance d is now represented as:

To get better ranging accuracy and resolution in real-world, the phase offset measurements at more than two frequencies is required. Further, if the phase offset measurements are plotted as a phase vs. frequency curve, then, the slope of the curve represents the distance d between the initiator and the reflector. See Figure 8-2 and Figure 8-3. Equation 6 can be seen as a straight line with the distance proportional to the slope of the line:

Referenced from [2] - Figure 8-2 shows the measured phase differences vs. frequency for two different distances d1 = 10m and d2 = 20m between the initiator and reflector. The phase differences wrap around 2π and can be straightened as shown in Figure 8-3 to calculate the effective phase slope and estimate the distance between the initiator and the reflector.

Figure 8-2 Measured Phase vs. Tone Frequencies (Wrapping Around 2π)

Figure 8-3 Measured Phase vs. Tone Frequencies (Straightened Phase)