APPLICATIONS

The above theoretical method for fast frequency count is applicable for resolution improvement of automotive FDS. We can state it for the next reason. Currently, the most of FDS [14, 23, 39] are using DFT (discrete Fourier transform) of the signal spectrum, or even more classic method of events per second count. It is evident that such methods for its proper application requires a full time window of 1 second, and in shorter time intervals has additional source of uncertainty formation. Not necessary to add that jitter or flicker noise just will worsen this situation. Meanwhile, our method introduced in [16-18, 38] has exceptional properties exactly for automotive application. As it easy to proof, for secure control solution in automobile in movement, for example at the speed of 60km/h, we must have a decision during at least in 1 ms. The car will pass in this time a path of 1,67sm. We know that the modern control algorithms are relatively time consuming, so properly the sensory part must to have a response time at least one order less. So, we get value of 10^-4s. It is apriori challenging constrain for mentioned classic frequency measurement methods. Not necessary to add that car speed growth just will lead to measurement time decrease and uncertainty increase. But for our method, as it shown in Section IV and in [16-18, 38], uncertainty of measurement not depends on time of measurement. Moreover, for higher values of frequencies in use, the time sufficient for 5-7 coincidences occurrence is cutting down, but measurement uncertainty is the same. It is worthwhile that in [16] shown that our method has another outstanding property for fast measurement: it is able to use a reference frequency less than unknown, it is not change the way of method functioning. Such property is impossible for classic methods [14, 23]. Finally, in [38] it is shown rigorously that our method application is invariant to timing jitter. In other words, it is reliable to noise in time domain.

All the mentioned above leads us to the one common conclusion: method of rational approximation by mediants based on coincidences principle is the strongest tool for fast and exact measurements in short time intervals. It permits to register fast changes of resonant frequency in short intervals of time, acceptable for automotive application constrains.

Application of this method in automotive FDS we consider in the next way. We can use the sensitive element of the most of commercial sensors without any modification. Just for output signal processing, after conversion of the measuring parameter to proportional frequency changes, we’re applying our original method and algorithm firstly presented in [16]. With the aim to show its efficiency we are carry out the same simulation, similar to [16-18], but with the real frequency domain parameters specific to the one recently known commercial FDS. The unique point, which need additional research and can be objective for our future work, is the option to use our method not just instead of the original method of sensor manufacturer, but use both of them with further averaging. However, such option is not evident and still very questionable.

As the example, we can mention the pressure sensor RPT410 from Druck Incorporate. This device is Resonant Silicon Pressure Transducer (RPT).

One of the recently known automotive applications of this sensor [22] is the pressure sensing in engine test cells. The multi layer sensor structure consists of a resonator and pressure sensitive diaphragm micro-machined from single-crystal silicon, thus achieving the highest level of performance stability. One possible option of this sensor is a frequency output proportional to the barometric pressure range.

The output is a TTL square wave with frequency band since 600 Hz to 1100 Hz, corresponded to operating pressure range of 17.5 to 32.5 inHg (600 to 1100mbar), a relative frequency error of 0.05% [23, p.433] and a response time of 300ms [22]. This device is shown in Fig. 11.

 

 

 

Fig. 11. Resonant Silicon Pressure Transducer RPT410 from Druck Incorporated

 

In particular, this example is relevant because the frequency range of the signal under measurement is considerably low.

In general, the relative error of frequency measurement must be at least five times smaller than sensor error. For the example, frequency must be measured in this case with an error of 0.005%. But the frequency relative error of modern frequency output sensors is 0.01%, then the appropriate and selected limit for relative error in frequency measurements is 0.001% [23, 24].

To illustrate the possibilities of the proposed in Section IV for this application, four frequencies were chosen in the range of operation of the sensor RPT410 (values in first column in Table 1). In addition, three different values for standard frequency (10 kHz, 100 kHz and 1 MHz - values in first row in Table 1) were chosen.

The low frequency value measurement is shown in Table 1. The pulse width in both trains is the same for all simulations (τ₀=τ_x=τ). The duty cycle of the standard signal frequency remained constant, 10%.

Figure 12, shows the behavior of the relative error of the measurement frequency in the time window of 300ms, which is equal to the typical response time of RPT410 under comparison. It is noted that the process is bounded (dotted line) for [24], where is measurement time. However, there is the dynamic of the process, there are values with greater convergence rate than this bound. The mean measurement frequency is calculated using (6) and the perceptual relative error is calculated with

 

% (10)

The results of the calculation of (10) for all the above mentioned cases (combinations of all possible revised values of 3 reference and 4 unknown frequencies) are shown in Table 1. It is shown that using a standard 1 MHz the relative error is less than the limit for the appropriate relative error for the application.

Simulation results of the fast frequency count are shown in the Fig. 12), where for all columns the frequency standard in use is the same, and for all rows the value of the hypothetical unknown frequency is the same as well. The analysis of Fig. 12 can draw several important conclusions.

First, as shown earlier in [16], the general character of the desired value approximation is cyclic and converging.

Second, during the typical response time 0.3s of RPT410 the implementation of the method presented in this paper gives, at least, 2 best approximations for the worst case (Fig. 12e,f), and 8-15 best approximations for the best case (Fig. 12l,k) during the presented simulation.

For the rest of combinations of f_x and f_o the average quantity of the best approximations is 5-6. In practice, it means that the implementation of the proposed method gives the possibility to measure the desired value of the unknown parameter converted into frequency in average of 5-6 times faster than original RPT410 can do it.

Third, the averaged uncertainty of such a measurement will be limited only by the instability of reference standard [16], the jitter [21, 38] in this case will not have such effects, because it is well known that short term stability is much better than long one [21, 25].

Fourth, even a quick look at Fig. 12(a-l) shows that there is no universal value of the best reference frequency, which can provide the best quality of approximation in all the considered range of desired frequency variation.

Finally, almost for all variants of the conducted simulation (except for the cases g and h of Fig. 12), the best coincidences appear after 0.05s. So, it is desired for practical measurement to apply the selective time gate under condition

 

where t_m is the measurement time and Tsr is the time of sensor response.

One way to improve the result for low frequency standards is to use an auxiliary time window after the start event of measurement. Table 2 shows the results using a time window of 50ms. It is shown that using standards of 100 kHz and 1 MHz the relative error is less than the limit for the appropriate relative error for the considered application.

Additionally, it is essential to note that the theoretical method of rational approximation based on the theory of mediants and Farey fractions presented in [16], is a strong tool not only for FDS resolution improvement. For example, it was successfully applied for resolution improvement in various metrological tasks related to repetitive pulse cycles buried in noise of different nature. As an example, it is important to mention the optimized positioning of the repetitive pulses of response in a task of circular optical scanning in the application of structural health monitoring [26] or a in similar application of technical vision system for robot navigation [27-34]. In both cases, the method of rational approximation and its mathematical techniques gave the possibility to measure the desired value of unknown parameters several times better.

The proposition to use this method for automotive FDS applications was shown first in [35].

f0 fx	10KHz	100KHz	1MHz
630.40208134 Hz	27.9006x10-3	8.7906x10-4	-2.8357x10-5
910.93749232 Hz	-8.9507x10-3	2.5401x10-4	-1.4123x10-4
1192.9823475 Hz	19.6391x10-3	1.1845x10-3	8.9487x10-5
1509.1255713 Hz	-2.9869x10-3	-8.3079x10-4	1.9556x10-4

 

Table 2. Error relative of frequency measurement using an auxiliary time window.

f0 fx	10KHz	100KHz	1MHz
630.40208134 Hz	3.6155x10-3	-2.2117x10-4	-1.6067x10-5
910.93749232 Hz	5.9505x10-4	2.3534x10-5	-5.5914x10-5
1192.9823475 Hz	6.7321x10-3	4.2967x10-4	4.2389x10-5
1509.1255713 Hz	-3.3408x10-4	-3.7052x10-4	6.8139x10-5

 

VII. CONCLUSIONS

 

Nowadays, the growth of the automotive industry is very fast. Controlling electromechanical processes in cars is a task that requires high processing speeds. The method presented here has been tested under experimental conditions and it has shown that it meets the requirements that are needed to improve processing speed for frequency output sensors in the automotive industry.

The prototype testing with real pulse trains from the independent non-correlated generators shows a proper behavior of the introduced theoretical method of the rational approximations in practice.

Given simulations results shows that implementation of the mentioned theoretical method of fast frequency count on the typical frequency rates of arbitrary selected real industrial automotive FDS gives in average five times faster frequency count. Also it is important that the frequency uncertainty it is caused only by the reference frequency own instability, no more systematic errors.

 

Table 1. Error relative of frequency measurement.

 

a)	b)	c)
d)	e)	f)
g)	h)	i)
j)	k)	l)

 

Fig 12. Behavior of relative error of frequency measurement

 

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