Time Transfer
4.3 Range Uncertainty
There are many real-life realities that make the ideal system described earlier unattainable although approachable. These are factors that should be taken into account for time transfer ranging over relatively short distances:
• Difference of time between the counter event and the time the deliminator instance leaves or enters the transceiver antenna (Section 4.2);
• Absolute and relative drifts of time bases of terminals;
• Pulse time resolution due to finite bandwidth and noise;
• Multipath interference.
4.3.1 Clock Drift and Measurement Time
Time base drifts reduce the accuracy of half duplex two way ranging [3]. Assume that the nominal rate of the clocks of Figure 4.2 is R0ticks per second. The actual clock rates are
RA= R0(1+ ⌬) (4.11)
RB= R0(1+ ⌬ + ␦) (4.12)
where⌬ is the absolute drift of the clock at A from perfect time and␦is the relative drift of B’s clock compared to that of A. ⌬ and ␦can be positive or negative. In this discussion we do not consider the influence of the granularity of the clock counters.
Now the elapsed time readings of the A and B clocks are estimates that are expressed as follows:
TA′ = TA(1+ ⌬) (4.13)
TB′= TB(1+ ⌬ + ␦) (4.14)
where TAand TBare the true elapsed times of (4.7) and (4.8) and shown on Figure 4.2.
Referring to (4.9), the estimated propagation delay can then be expressed as
Tp′=TA′ − T ′B
2
Tp′= Tp+ ⌬ ⭈ Tp−␦⭈ TB
2 (4.15)
Clock accuracy in a communication system will generally not be worse than around 20 ppm= 2 × 10−5. At this or better accuracy the second term on the right of (4.15) can be ignored, leaving
Tp′= Tp−␦⭈ TB
2 (4.16)
Equation (4.16) indicates that the propagation time estimation error, ␦TB/2, is proportional to the difference in deviation of the two terminal clocks, that is, the clock drift between them, and the delay time of the responder’s range acknowledgement. For a worst-case relative drift of 40 ppm, range errors for three values of TB are shown in Table 4.1.
Table 4.1 Range Error Versus Responder Elapsed Time for Interrogator-Responder Drift Equal to 40 ppm
TB(microseconds) Range Error (meters)
10 0.12
100 1.2
1,000 12
There are three ways to reduce the effect of relative time base drift between interrogator and responder. A short ranging packet length is necessary for minimum error, as shown in Table 4.1. While message length can usually be kept short for ranging, the preamble length will be fixed by the protocol and will probably be the limiting factor determining the length of the ranging packet. Synchronizing the responder clock rate to the incoming packet during the preamble will also reduce the error.
A third way is to essentially cancel out the error term by performing back-to-back ranging, where range is measured with A being the interrogator and B the responder, then measured again with reversed roles: B is the interrogator and A the responder. Figure 4.4 shows the exchange of packets. Averaging the two measurements gives the final range estimate. The reduction of the error term is shown as follows. The result of the first range estimation, with A the interrogator and B the responder, adapted from (4.15), is
Tp, 1′ = Tp+ ⌬ ⭈ Tp−␦ ⭈ TB, 1
2 (4.17)
For the second estimation B is the interrogator and A the responder, giving
Tp, 2′ = Tp+ ⌬ ⭈ Tp+␦ ⭈ TB, 2
2 (4.18)
We now calculate a new estimate for the propagation time, Tp″, by averaging Tp, 1′ and Tp, 2′ , getting:
TA,1
TB,1 TB,2
TA,2
Tp
Tp Tp
Tp
A
B
Figure 4.4 Back-to-back ranging timing diagram.
Tp″ =Tp, 1′ + T ′p, 2
2
Tp″ = Tp+ ⌬ ⭈ Tp+␦
4 ⭈(TB, 2− TB, 1) (4.19) Comparing (4.19) and (4.15), only the last term has changed. When packet lengths and receive to transmit delay are the same for the two propagation time estimation trials, it can be seen from Figure 4.4 that (TB, 2− TB, 1) approximately equals 2Tp. Substituting in (4.19) gives:
Tp″ ≈ Tp+ ⌬ ⭈ Tp+␦ 2 ⭈Tp
Tp″ ≈ Tp冉1+ ⌬ +␦2冊 (4.20)
The factor in parenthesis is very close to unity, so the result of the ‘‘back to back’’ propagation time estimation is to reduce the clock rate drift.
4.3.2 Noise
As explained in Chapter 2, noise and limited bandwidth reduce time instant resolu-tion. In the time transfer method of distance measuring, noise can prevent an accurate determination of the reference point in the received signal, and thus an erroneous reading of the clock (Figures 4.1 and 4.2). The method requires at least two clock readings of the received reference point, and an error in one or both of them will affect the propagation time, and hence the distance, estimation. A high noise or interference level may disrupt the measurement all together, and the system should be designed to determine that a clock reading is not plausible and the measurement must be abandoned or tried again.
The degree to which noise affects the clock readings and the accuracy of the distance measurement is a function of signal-to-noise ratio, bandwidth, and the clock resolution. Noise causes jitter in the clock reading. While the clock itself is stable, the noise in the input signal produces a relative jitter of the sampling point.
The relationship between the rms jitter , signal bandwidth f0, and the signal-to-noise ratio S /N can be approximated by [4]:
= 1
2⭈ ⭈ f0⭈ (S/N) (4.21)
Averaging of multiple measurements can improve accuracy at the expense of measuring time. By making N independent measurements, the improvement of the rms jitter is:
av=
√
N (4.22)4.3.3 Multipath
Multipath interference creates the possibility that the propagation delay on a path other than the direct path between transmitter and receiver will be measured erroneously as time of flight for the purpose of finding the range. For example, if the deliminator is detected by a matched filter, multipath signals will result in multiple output pulses that indicate the timing epoch. The receiver should be capable of examining the pulse times and deciding which is the desired one. The earliest pulse of those that arrive within the expected range of multipath reflections should represent the direct path, but it may very well not be the strongest pulse.
The direct path may be blocked and not detected at all, in which case the estimated range will be too high. When a correlator is used to detect the deliminator, a chip period that is short compared with the multipath span facilitates rejection of multipath reflections because of the autocorrelation properties of the deliminator.
For indoor systems particularly, path lengths are short and a relatively high chip rate must be implemented. This makes UWB (ultra-wideband) systems inherently advantageous for distance measuring (see Chapter 11).
4.3.4 Relative Motion
Relative motion between the interrogator and the responder could affect the dis-tance measurements. In a single one-way time transfer measurement, relative motion will effect the propagation time estimate if the distance between the terminals changes more than the wavelength of the measuring clock frequency during the measurement. The bound on the relative velocity that will not affect the distance measurement is [6]:
v<TC⭈ c
TM (4.23)
where v is the relative velocity, TCis the period of the clock frequency, TM is the total measurement time, and c is the speed of light. An example shows the degree to which target velocity is apt to affect the distance measurement in a short-range environment.
Example 4.1
In an ultra-wideband distance measuring system we assume the following parame-ters:
Clock rate, fC= 528 MHz
Measurement time, TM = 50 microseconds c = 3 × 108m/s
vmax=(1/ fC) ⭈ c
TM (4.24)
vmax = 11.364 km/s
It is clear that during a single measurement, the relative velocity is not a factor.
However, if many measurements are taken over a much longer period of time in order to improve accuracy with noisy signals, vmax is reduced proportionally to the increase in the measurement time.