Overview The details of operation of the proprietary Timing Tag used in F1 and other motor sports have not been published, but reasonable estimations can be made based on experience in RF tracking systems. What is currently known is that the each Tag operates at a separate frequency; as there are a maximum of 22 cars in a F1 race, there must be at least 22 frequency channels.

The simplest possible implementation is to measure the short range signal strength as the car passes over an antenna buried in the track, so that the time of the received “pulse” can be used for the time estimate. The antenna in the car is typically around 15 cm above the track, and assuming that the signal strength is dominated by the inverse range (near field) effect, the 3 dB pulse width is about twice the height \( H \) , or (say) 30 cm. If the car is travelling at 100 m/s, the pulse width (3 dB points) is about 3 ms. As the stated accuracy is 0.1 ms, the time resolution implies that the position of the pulse must be determined to an accuracy of about 1/30 of the pulse period. This accuracy is possible but difficult, as the absolute magnitude and shape of the pulse is not known, and would vary from car to car. Thus some method of normalisation is desirable. As can be observed in Fig. 10.38 , the actual system utilises two antennas; the following method provides a solution which is simple to implement and can meet the timing accuracy requirements.

Analysis of Timing Detection The timing detection system is assumed to be based on measuring the received signal at the two antennas. These signals are then both summed and subtracted, and finally the ratio computed. Thus this method calculates a function which is independent of the absolute signal strength, and is only weakly dependent on the shape of the received pulse. The Tag antenna is assumed to be a patch antenna with a radiation pattern given by

\( \cos (\theta ) \) . Thus if the height of the antenna above the ground is

\( H \) and the range

\( R \) , the received signal at the antenna in the track is given by

$$ E(\theta ) = \frac{A\cos (\theta )}{R} = \frac{A}{H}\cos^{2} (\theta ) = E_{0} \cos^{2} (\theta ) $$

(10.11)

where

\( A \) is some (unknown) constant. Now consider two antenna

\( D \) apart, with the origin of the x-axis at the mid-point. Clearly the signals will be symmetric about this point, so that this point also represents the position for which the time is to be measured. Thus setting

\( t = 0 \) at this reference point, the received signal strength of the two antennas is given by

$$ E_{1} (\theta ) = E_{0} \frac{{H^{2} }}{{H^{2} + \left( {vt + \frac{D}{2}} \right)^{2} }}\quad \quad \quad E_{2} (\theta ) = E_{0} \frac{{H^{2} }}{{H^{2} + \left( {vt - \frac{D}{2}} \right)^{2} }} $$

(10.12)

where

\( v \) is the car speed, and

\( E_{0} \) is a constant reference signal strength. The sum and difference of these two signals is thus given by

$$ E_{sum} (\theta ) = E_{2} \left( \theta \right) + E_{1} \left( \theta \right)\quad \quad \quad E_{dif} (\theta ) = E_{2} \left( \theta \right) - E_{1} \left( \theta \right) $$

(10.13)

so the ratio of the difference signal to the summed signal is given by

$$ \rho (t) = \frac{{E_{dif} (\theta )}}{{E_{sum} (\theta )}} = \frac{{E_{2} \left( \theta \right) - E_{1} \left( \theta \right)}}{{E_{2} \left( \theta \right) + E_{1} \left( \theta \right)}} = \frac{Dvt}{{H^{2} + \left( {\frac{D}{2}} \right)^{2} + \left( {vt} \right)^{2} }} = \frac{Dvt}{{R_{1}^{2} + \left( {vt} \right)^{2} }} $$

(10.14)

Note that the ratio function is independent of the signal strength, and is zero at

\( t = 0 \) . Thus by detecting the position where

\( \rho \left( t \right) = 0 \) the required timing data has been found. Further, near the origin

\( \left( {vt \ll R_{1} } \right) \) , the function is approximately linear. Alternatively when

\( \left( {vt \gg R_{1} } \right) \) the ratio approaches

\( D/vt \) which approaches zero when the Tag is far from the receiving antennas. The function is shown plotted in Fig.

10.39 . In the ideal case, the time where

\( \rho \left( t \right) = 0 \) can be determined exactly, but measurement errors will result in some computed timing errors. These errors are considered in the next section.

Fig. 10.39 Ratio function with parameters \( H = 15 \) cm, and \( D = 50 \) cm

Effect of Noise The measurement technique described above provides the necessary timing by detecting the time when \( \rho \left( t \right) = 0 \) . However, the actual measurements will have random noise, so that the determination of the time will also have some random errors. The required design accuracy of the Timing Tag system is 0.1 ms, so that this figure can be used to estimate the required signal-to-noise ratio.

The following analysis assumes that the receiver output is corrupted by Gaussian noise with the same power in both receivers, but uncorrelated. In this case the ratio function can be expressed as

$$ \rho (t) = \frac{{E_{dif} (t) + n_{2} (t) - n_{1} (t)}}{{E_{sum} (t) + n_{1} (t) + n_{2} (t)}} $$

(10.15)

The main interest is when the signal is much greater than the noise, so that Eq. (

10.15 ) can be approximated by

$$ \begin{aligned} \rho_{n} (t) & = \frac{{\frac{{E_{dif} }}{{E_{sum} }} + \frac{{n_{2} - n_{1} }}{{E_{sum} }}}}{{1 + \frac{{n_{1} + n_{2} }}{{E_{sum} }}}} \approx \left[ {\rho + \frac{{n_{2} - n_{1} }}{{E_{sum} }}} \right]\;\left( {1 - \frac{{n_{1} + n_{2} }}{{E_{sum} }}} \right) \\ & \approx \rho + \frac{{n_{2} - n_{1} }}{{E_{sum} }} - \rho \frac{{n_{1} + n_{2} }}{{E_{sum} }} \\ \end{aligned} $$

(10.16)

As the assumed noise has zero mean, the expected (mean) value of the ratio (

10.16 ) is an unbiased estimate of the ratio. Now consider computing the variance of the ratio function. As the noises are statistically independent, the variance of the ratio is given by

$$ \begin{aligned} \text{var} \left[ {\Delta \rho_{n} (t} \right)] & = \frac{{2\rho^{2} \sigma_{n}^{2} }}{{E_{sum}^{2} }} + \frac{{2\sigma_{n}^{2} }}{{E_{sum}^{2} }} = 2\,\left[ {\frac{{\sigma_{n}^{2} }}{{E_{sum}^{2} }}} \right]\;\left( {1 + \rho^{2} } \right) \\ & = \frac{{2\left( {1 + \rho^{2} } \right)}}{{\gamma_{0} }}\quad \quad \gamma_{0} = \left. {\frac{{E_{1}^{2} }}{{\sigma_{n}^{2} }}} \right|_{t = 0} \\ \end{aligned} $$

(10.17)

where

\( \gamma_{0} \) is the receivers signal-to-noise ratio (SNR) at the origin of the measurements

\( \left( {t = 0} \right) \) . As from Eq. (

10.16 ) the mean error in the ratio function is zero (assuming the noise has zero mean), the standard deviation of the error in the ratio function is given by

$$ \sigma_{\Delta \rho } = \sqrt {\frac{{2\left( {1 + \rho^{2} } \right)}}{{\gamma_{0} }}} \approx \sqrt {\frac{2}{{\gamma_{0} }}} \quad $$

(10.18)

Further, from Eq. (

10.14 ) the standard deviation of the ratio function near the origin can be expressed as

$$ \sigma_{\Delta \rho } = \left( {\frac{Dv}{{R_{1}^{2} }}} \right)\sigma_{t} \quad $$

(10.19)

where

\( \sigma_{t} \) is the standard deviation in the timing measurement, and it is assumed the timing measurement errors

\( \delta t \) are small so that

\( v\delta t \ll R_{1} \) . Thus by equating the two estimates of the error in the ratio function the standard deviation in the timing estimate is given by

$$ \sigma_{t} = \left( {\frac{{R_{1}^{2} }}{Dv}} \right)\sqrt {\frac{{2\left( {1 + \rho^{2} } \right)}}{{\gamma_{0} }}} \approx \left( {\frac{{\sqrt 2 R_{1}^{2} }}{Dv}} \right)\frac{1}{{\sqrt {\gamma_{0} } }}\quad \quad \left( {R_{1} = \sqrt {H^{2} + \left( {D/2} \right)^{2} } } \right) $$

(10.20)

Thus the timing accuracy is a function of the geometry

\( \left( {H,D} \right) \) of the Tag and receiver, the speed

\( \left( v \right) \) , and the SNR at the receiver at the measurement point a distance

\( R_{1} \) from the Tag to

\( t = 0 \) . However, constraints on the geometry and the speed means that the timing accuracy is mainly controlled by the receiver signal-to-noise ratio. The timing accuracy is related to the speed, but the position accuracy is independent of the speed, and is given by

$$ \sigma_{x} = v\sigma_{t} = \sqrt {\frac{2}{{\gamma_{0} }}} \frac{{\left( {\frac{D}{2}} \right)^{2} + H^{2} }}{D} $$

(10.21)

Consider a numerical example, with \( H = 15 \) cm, \( D = 50 \) cm, and peak SNR is \( \gamma_{0} = 30 \) dB. The positional accuracy according to Eq. (10.21 ) is 7.6 mm. Alternatively the SNR required for a timing accuracy of 0.1 ms at 80 m/s is 29.5 dB. Thus the required accuracy can be achieved, but with a fairly high SNR. In practice such a higher SNR should be readily achieved due to the very short range.

The conclusion is that theoretical performance estimates of a simple signal strength measurement system is capable of very accurate timing measurements, meeting the 0.1 ms requirement of F1. However, the actual accuracy based on the testing described in Sect. 10.6 casts some doubts on the actual performance.