Rotating a radio transmitter results in a continuously changing frequency at a stationary observer due to the Doppler effect. The frequency observed by receiver depends on the relative speed of the transmitter with respect to the observer (negative if they move away from each other, positive if they move toward each other):

where is the baseline frequency emitted by the transmitter and is the speed of light.

Since the speed of light is much larger than the velocity of the transmitter, the above formula can be written as

where is the wavelength of the transmitted signal. That is, the doppler shift is

When using radio-interferometry as described in Section 1, that is, a stationary auxiliary transmitter is emitting a sine wave of frequency (where ), it was shown in [15] that the same amount of Doppler shift appears in the low-frequency envelope signal.

Consider Figure 1 where transmitter rotates at a constant angular rate and radius and receiver measures the frequency of the signal. The maximum of the frequency is observed at point where the transmitter moves directly toward the receiver, while the minimum frequency is measured when the transmitter is at point moving exactly away from the receiver. By measuring the time between the two extrema of the frequency shift, the angle can be estimated given the angular speed of the transmitter. The distance between the receiver and the center of rotation of the transmitter is then

Hence, the range between two nodes can be estimated this way.

One of the advantages of the above ranging method is that one does not need the actual magnitude of the frequency shift, only the time of the maximum and minimum frequency values. Figure 2(a) shows the expected Doppler shift observed by the receiver when it is 10 meters away from the rotating transmitter. Unfortunately, any measurement has error. The question is how it affects the accuracy of ranging? Consider (4) again. Reformulating the equation, one can plot the expected value of the measured angle as a function of the distance between the receiver and the center of rotation of the transmitter. Figure 2(b) shows this function with a corresponding rotation radius of 12 cm. One can see that the function gets flat fast. For example, the angle difference between 20 and 21 meters is about 0.03 degrees. That is clearly beyond the expected accuracy of this measurement. In fact, the ranging error beyond only 5 meters would be unacceptably high.

However, introducing a second receiver offers another method for ranging. Consider Figure 3. Both receivers and continuously measure the frequency of the signal. Let denote the velocity vector of the rotating transmitter at time . Let us define and as the (signed) speed of the rotating transmitter with respect to stationary receivers and at time . Formally,

From the velocities at time when observes the maximal frequency, we can compute the angle (the angle at which the segment can be seen from ) as

Given , it can be easily shown that , , and need to be on a circle with a radius of

where is the distance between receivers and . While we obtain an angle, the result is still similar to traditional pairwise ranging in that one "range" estimate constrains the location of the node to a circle. Except the center of the circle in our case is not another node, but a location that can be computed from the locations of the two receivers and the measured angle.

While attractive, this method relies on measuring the Doppler shift at any one receiver accurately. However, in most computers and wireless devices, uncompensated crystal oscillators are used to generate the clock signals. The short-term stability of these oscillators are typically between and for one second. In our case, this corresponds to possibly more than 1 Hz error that cannot be compensated for, because we cannot measure the baseline frequency directly (i.e., when the transmitter is stationary). We need to rely on measuring the difference between the maximum and the minimum frequencies and take their mean. Since the time between these events may not be much less than one second, short term stability can cause a larger error than the phenomenon we are trying to measure. Temperature-compensated crystal oscillators have somewhat better stability, while oven-controlled crystal oscillators are at least an order of magnitude more precise. Unfortunately, their price and power requirements are both significantly higher, and they are not used in everyday devices. The question is then how can we eliminate this significant source of error?

Note that it is only the transmitter instability that is important here, because the radio interferometric technique already eliminates the receiver instability by using the envelope signal. Notice that the transmit frequency instability has the same effect at both receivers because we compare their measurements at the same time. Hence, if we take the difference of the two measured frequencies, the actual transmit frequency is eliminated. This frequency difference relates to the difference of the observed speeds; however, not having the speed measurements available individually, only their difference, makes solving for the location somewhat more complicated. Let denote the difference of the observed frequencies and for receivers and at time . If we assume that and , we can write the measured frequency difference using (2) as

and we now write

Let us define angles and as the angle between the velocity vector of transmitter *T* and its components pointing toward receivers and , respectively. From Figure 4 we see that (5) can be rewritten as and .

To simplify further computation, we assume that the receivers are far from the circle of rotation, that is, and . If the radius of the circle is small compared to the distance between the transmitter and the receiver, the error this assumption introduces is minimal. With this so called *far field* assumption, the angle , denoted as , is fixed. Without the loss of generality, let us assume that . Therefore, and . Substituting these relationships into (9) yields

Using the trigonometric identity for the difference of cosines

we can rewrite (10) as

takes its maximum, where the first sine equals :

From here, using (8), can be expressed as follows:

Therefore, by measuring the maximum difference of the Doppler shifts measured at receivers and , we can estimate . In the presence of noise, however, the maximum of the signal cannot be measured precisely. Obviously, measurement noise can be mitigated by iteratively measuring and averaging the observed values, though such a technique is time consuming.

We observe that not only the maximum measured value, but the magnitude of all the measured values are related to . To make use of this, we can measure the power of the signal instead, because it is more resilient to noise due to the integration. Since the average power of a sine wave is times its amplitude, we get that

where

Therefore, it is sufficient for the two receivers to measure the frequency of the received signal for the duration of merely one rotation in order to compute .