**2. Frequency measurement based on the pulse coincidence principle**

In the past, pulse coincidence principle has been used for frequency measurement of electrical signals [4-6]. In this measurement method, a desired frequency is measured by comparing it with a standard frequency. The zero crossings of both frequencies are detected, and a narrow pulse is generated at each crossing. Then, two regular independent pulse trains are generated. The desired and standard trains of narrow pulses are compared for coincidence. This is made with an AND-gate, and then a coincidence pulse train is generated. Coincident pulses can be used as triggers to start and stop a pair of digital counters (start and stop events). See Figure 1.

The standard and desired pulse trains are applied to other counters and a measure of the desired frequency is obtained by multiplying the known standard frequency by the ratio between the desired count and the standard count obtained in the two digital counters [5]. And basically, measurement error reduction depends on the ability to detect a pair of *best* coincidences consecutively between a large number of these, for a given measurement time. In other works, measurement error reduction depends on reduction of comparison error of time intervals *n0T0* and *nXTX* by selection of adequate coincidence pulses [6].

Consider *fx* as the desired or unknown frequency and *fo* as the standard frequency, and 1 *T f x x* and 1 *T f o o* are they periods respectively. In Figure 1, *SX(t)* and *S0(t)* are the unknown and standard trains of narrows pulses, is the pulse width on both trains and, *SX(t)* &*S0(t)* is the irregular pulse coincidence train. Then, *n0* and *nX* are integer numbers and represent the amount of whole periods between selected measurement start and stop events (see figure 1).

different frequencies [3], or by electronic detection of two coincident pulses of two regular independent pulse trains [5 -6]. In these methods the quantization error (±1 count error) can be overcome satisfactorily [2-3,6]. But, in [3] a high distinguishability analog circuit for phase coincidence detection is required and in [5] the relative methodical error is pulse width dependent for random selection of stop measurement pulse coincidence and, is experimentally probed that relative methodical error can be reduced by two o three orders

Continuous time stamping principle change the scenario in frequency measurement, because in each measurement has not a defined start (= start trigger event), and a stop (= stop trigger event) plus a dead-time between measurements to read out and clear registers, do interpolation measurements and prepare for next measurement [2]. In this technique Linear regression using the least-squares line fitting is used because for a onesecond frequency measurement in a fast processing counter could contain hundreds o thousands of paced time-stamped events, no just a start event plus a stop event [2]. But fast

However, a fast method for frequency measurement base pulse coincidence principle and rational approximations was proposed and, was shown that under a novel numerical condition for detect the stop trigger event measurement resolution is improved. Instrumental errors are caused only by the reproducibility of the reference frequency and relative measurement error is comparable to the reproducibility of reference oscillator. [7].

In the past, pulse coincidence principle has been used for frequency measurement of electrical signals [4-6]. In this measurement method, a desired frequency is measured by comparing it with a standard frequency. The zero crossings of both frequencies are detected, and a narrow pulse is generated at each crossing. Then, two regular independent pulse trains are generated. The desired and standard trains of narrow pulses are compared for coincidence. This is made with an AND-gate, and then a coincidence pulse train is generated. Coincident pulses can be used as triggers to start and stop a pair of digital

The standard and desired pulse trains are applied to other counters and a measure of the desired frequency is obtained by multiplying the known standard frequency by the ratio between the desired count and the standard count obtained in the two digital counters [5]. And basically, measurement error reduction depends on the ability to detect a pair of *best* coincidences consecutively between a large number of these, for a given measurement time. In other works, measurement error reduction depends on reduction of comparison error of

Consider *fx* as the desired or unknown frequency and *fo* as the standard frequency, and 1 *T f x x* and 1 *T f o o* are they periods respectively. In Figure 1, *SX(t)* and *S0(t)* are the

*SX(t)* &*S0(t)* is the irregular pulse coincidence train. Then, *n0* and *nX* are integer numbers and represent the amount of whole periods between selected measurement start and stop events

is the pulse width on both trains and,

time intervals *n0T0* and *nXTX* by selection of adequate coincidence pulses [6].

Simple digital circuits are needed to practical implementation of this technique.

**2. Frequency measurement based on the pulse coincidence principle** 

of magnitude than frequency meters based on classical method [6].

digital circuits are needed to implement this technique.

counters (start and stop events). See Figure 1.

unknown and standard trains of narrows pulses,

(see figure 1).

Fig. 1. a) Pulse coincidence principle for frequency measurement, b) Practical view of coincidences process (oscilloscope screenshot).

In [6] by possible selection of *partial* coincidences, error of comparison of time intervals *n0T0* and *nXTX* has been reduced to the duration of the coinciding pulses and mathematically expressed by

$$\left| n\_X T\_X - n\_0 T\_0 \right| \le 2\tau. \tag{1}$$

Form (1), relative measurement error for a single measurement can be expressed as follows

$$\beta\_X = \frac{\left| f\_X - \frac{n\_X}{n\_0} f\_0 \right|}{f\_X} \le \frac{2\pi}{n\_0 T\_0} \approx \frac{2\pi}{t\_m} \,\,\,\,\tag{2}$$

where *tm* is the measurement time.

It is experimentally known, that for a standard frequency <sup>6</sup> <sup>0</sup>*f* 1 10 Hz from a thermostatic quartz generator with relative short-term instability does not exceed 10-8 and using a pulses width <sup>9</sup> 7 10 s, root-mean-square (RMS) error <sup>6</sup> 14.3 10 *Xs* Hz for <sup>3</sup> 1 10 *Xf* Hz at 50 observations during total measurement time *tm* 1 s. And <sup>3</sup> 1.79 10 *xs* Hz for <sup>6</sup> 1 10 *Xf* Hz under same measurement conditions [6].

#### **2.1 functioning and uncertainty limitations**

Let us consider two trains of narrow pulses with period *TX* and *T*0 with and pulse width respectively, generated by detection of zero crossings of two sinusoidal signal of frequencies <sup>0</sup>*f* and *Xf* . Suppose that *T*0 is a known parameter and *TX* is unknown and, both pulse trains start in phase, i.e. a time shift is 0.

For an appropriate selection of the pulse width of two regular independent pulse trains, periodic perfect coincidences of these pulses are observed in time axis [13]. Repetition period of perfect coincidences is *TX*<sup>0</sup> , in Fig. 1a). A Practical view of coincidences process is presented in Fig 1 b).

For frequency measurement, the time intervals *n T*0 0 and *n TX X* are compared, where *n*0 is the amount of periods *T*0 in the measurement time and *nX* is the amount of periods *TX* in the same time interval.

Measurement time can be defined by the time interval between the first one pulse of coincidence (start event) after beginning the measurement process, and by any other following pulse of coincidence (stop event). As it were mentioned in the previous section, *n*0 and *nX* are the counts of pulses obtained in two digital counters.

According to [12], pulse coincidence occurs when

$$\left| n\_X T\_X - n\_0 T\_0 \right| \le \varepsilon \tag{3}$$

where is the acceptable tolerance (reasonable error value between time intervals *n T*0 0 and *n TX X* less to pulse with and dependent on the quality of electronic circuits used).

In [6] by possible selection of *partial* coincidences, error of comparison of time intervals *n0T0* and *nXTX* has been reduced to the duration of the coinciding pulses and mathematically

0 0 2 . *n T nT X X*

Form (1), relative measurement error for a single measurement can be expressed as follows

*X*

*<sup>n</sup> f f n*

*X*

*X*

It is experimentally known, that for a standard frequency <sup>6</sup>

<sup>6</sup> 1 10 *Xf* Hz under same measurement conditions [6].

**2.1 functioning and uncertainty limitations** 

trains start in phase, i.e. a time shift is 0.

0 0

quartz generator with relative short-term instability does not exceed 10-8 and using a pulses

Let us consider two trains of narrow pulses with period *TX* and *T*0 with and pulse width

respectively, generated by detection of zero crossings of two sinusoidal signal of frequencies <sup>0</sup>*f* and *Xf* . Suppose that *T*0 is a known parameter and *TX* is unknown and, both pulse

For an appropriate selection of the pulse width of two regular independent pulse trains, periodic perfect coincidences of these pulses are observed in time axis [13]. Repetition period of perfect coincidences is *TX*<sup>0</sup> , in Fig. 1a). A Practical view of coincidences process is

For frequency measurement, the time intervals *n T*0 0 and *n TX X* are compared, where *n*0 is the amount of periods *T*0 in the measurement time and *nX* is the amount of periods *TX* in

Measurement time can be defined by the time interval between the first one pulse of coincidence (start event) after beginning the measurement process, and by any other following pulse of coincidence (stop event). As it were mentioned in the previous section,

*n T nT X X* 0 0

and *n TX X* less to pulse with and dependent on the quality of electronic circuits used).

is the acceptable tolerance (reasonable error value between time intervals *n T*0 0

*n*0 and *nX* are the counts of pulses obtained in two digital counters.

According to [12], pulse coincidence occurs when

0 0 2 2 ,

(2)

*X m*

*f nT t* 

 7 10 s, root-mean-square (RMS) error <sup>6</sup> 14.3 10 *Xs* Hz for <sup>3</sup> 1 10 *Xf* Hz at 50 observations during total measurement time *tm* 1 s. And <sup>3</sup> 1.79 10 *xs* Hz for

(1)

<sup>0</sup>*f* 1 10 Hz from a thermostatic

(3)

expressed by

width <sup>9</sup> 

presented in Fig 1 b).

the same time interval.

where

where *tm* is the measurement time.

To find values of *n*0 and *nX* which define the appropriated coincidence it is useful to expand *T T <sup>X</sup>* <sup>0</sup> as simple continued fractions. This is evident rewriting (3)

$$\left| \frac{T\_X}{T\_0} - \frac{n\_0}{n\_X} \right| \le \frac{\varepsilon}{n\_X T\_0} \,. \tag{4}$$

Left side of equation (3) represents the approximation of *T T <sup>X</sup>* <sup>0</sup> using rational numbers and right side is an approximation condition.

For frequency measurement, in a view of 0 0 *f* 1 *T* , 1 *X X f T* , we can write

$$\left| f\_X - \frac{n\_0}{n\_X} f\_0 \right| \le \frac{\varepsilon f\_X f\_0}{n\_0} \,. \tag{5}$$

In (5), *Xf* is the hypothetical true value of the unknown frequency and 0 0 *<sup>X</sup> fn n* is the frequency value obtained by the measurement. Then, dividing both parts of (5) in *Xf* and taking to account 0 0 *f* 1 *T* , relative error of measurement (frequency offset) can been expressed by

$$\mathcal{J} = \left| f\_X - \frac{n\_0}{n\_X} \right| \Big/ f\_X \le \frac{\varepsilon}{n\_0 T\_0} \,. \tag{6}$$

We can see in equation (6) that relative error of measurement is limited by the ratio between the acceptable tolerance of the error of comparison between the time intervals *n T*0 0 and *n TX X* and, the time interval *n T*0 0 . Value of *n T*0 0 is approximately the measurement time.

#### **2.2 Numerical stop condition of measurement**

In frequency measurement, *n*<sup>0</sup> and *nX* are independent counter counts obtained in two digital counters, so they are properly integer numbers. An integer numbers ratio, like the involved in the frequency value obtained by the measurement, is possible to investigate under number theory laws. Let note and briefly explain some of them, especially Euclidean algorithm.

#### **2.2.1 Number theoretic preliminaries**

Let us to assume without loss generality that *T T <sup>X</sup>* <sup>0</sup> , from the division algorithm we can write

$$T\_X = a\_0 T\_0 + \Delta t\_0 \quad T\_0 > \Delta t\_0 \ge 0\tag{7}$$

$$T\_0 = a\_1 \Delta t\_0 + \Delta t\_1 \quad \Delta t\_0 > \Delta t\_1 \ge 0 \tag{8}$$

$$
\Delta t\_0 = a\_2 \Delta t\_1 + \Delta t\_2 \quad \Delta t\_1 > \Delta t\_2 \ge 0 \tag{9}
$$

$$
\vdots \qquad \qquad \qquad \qquad \vdots
$$

$$
\Delta t\_{i-2} = a\_i \Delta t\_{i-1} + \Delta t\_i \quad \Delta t\_{i-1} > \Delta t\_i \ge 0 \tag{10}
$$

$$
\vdots \qquad \qquad \vdots \qquad \qquad \vdots
$$

$$
\Delta t\_{n-2} = a\_n \Delta t\_{n-1} + \Delta t\_n \quad \Delta t\_{n-1} > \Delta t\_n \ge 0 \tag{11}
$$

where the *<sup>i</sup> a* is the *i*th partial quotients for each case and *<sup>i</sup> t* is the *i*th remainder, with *i n* 1,2,3,..., *.* With 1 *<sup>i</sup> a* , *<sup>i</sup> t* is a decreasing sequence for 0. *i*

Each remained obtained in the division step of Euclidean algorithm be could be interpreted as a distance [11], defined by

$$\left| Q\_i T\_X - P\_i T \right|\_0 = \Delta t\_i. \tag{12}$$

where *Pi* and *Qi* are the numerator and denominator of the *i*th convergent of the continued fractions to *T T <sup>X</sup>* <sup>0</sup> defined recursively as [12]

$$P\_i = a\_i P\_{i-1} + P\_{i-2} \tag{13}$$

$$Q\_i = a\_i Q\_{i-1} + Q\_{i-2} \tag{14}$$

for arbitrary 2 *i ,* and

$$\begin{aligned} P\_0 &= a\_{0'} & Q\_0 &= \mathbf{1}\_{\prime} \\ P\_1 &= a\_0 a\_1 + \mathbf{1}\_{\prime} & Q\_1 &= a\_1 \mathbf{1}\_{\prime} \end{aligned}$$

Then, from (12) each remainder *<sup>i</sup> t* is the absolute difference between the time intervals *Q Ti X* and 0. *PTi*

On the other hand, *T*0 can be expressed in terms of two consecutive remainders [11] using the following expression:

$$T\_0 = Q\_i \Delta t\_{i-1} + Q\_{i-1} \Delta t\_i \,. \tag{15}$$

A similar expression can be derived for *TX*

$$T\_X = P\_i \Delta t\_{i-1} + P\_{i-1} \Delta t\_i \,. \tag{16}$$

Supposing that *n* is the number of steps in the Euclidean algorithm to obtain greatest common divisor of *TX* and *T*<sup>0</sup> . Then last remained 0 *<sup>n</sup> t* and time interval *<sup>n</sup>* <sup>1</sup> *t* is the greatest common divisor of both periods *TX* and*T*<sup>0</sup> , in the consecutive division expressed in equations (7) to (11). Because the greatest common divisor is: the last nonzero remainder in this sequence of divisions.

Assuming that *<sup>n</sup>* <sup>1</sup> *t* is greatest common divisor of both periods *TX* and*T*<sup>0</sup> , we can write

$$T\_0 = Q\_n \Delta t\_{n-1} \tag{17}$$

$$T\_X = P\_n \Delta t\_{n-1} \,. \tag{18}$$

Expressing (10) in terms of (15) and (16) it is evident in (17) than step *n* is total equality point for both time intervals

$$\left| Q\_n P\_n \Delta t\_{n-1} - P\_n Q\_n \Delta t\_{n-1} \right| = 0. \tag{19}$$

In frequency measurement, this term expressed in several forms have a mathematical mean of least common multiple, and practical mean of time interval *TX*0 (see Fig. 1) expressed by:

$$T\_{\chi\_{O}} = \frac{T\_X T\_0}{\Delta t\_{n-1}} = P\_n Q\_n \Delta t\_{n-1} \tag{20}$$

is the condition for periodic perfect coincidences of pulses (see Fig. 1). Assuming to *TX*0 the measurement time in frequency measurement, from (1) and Fig. 1

$$\left| n\_X T\_X - n\_0 T\_0 \right| = 0 \,\,\,\,\,\tag{21}$$

and each time intervals *n T*0 0 and *n TX X* are equal to *TX*<sup>0</sup> . Then,

$$m\_0 T\_0 = P\_n Q\_n \Delta t\_{n-1} \, \prime \tag{22}$$

$$m\_X T\_X = P\_n Q\_n \Delta t\_{n-1} \, . \tag{23}$$

Now, product of two numbers *ab c* can be considered as the sum *aaa a* in which the number of summands is equal to *b* or as the sum *bbb b* in which the number of summands is equal to *a* .

Then, equations (22) and (23) can be rewrited using (17) and (18):

$$n\_0 Q\_n \Delta t\_{n-1} = P\_n Q\_n \Delta t\_{n-1} \prime$$

$$n\_X P\_n \Delta t\_{n-1} = P\_n Q\_n \Delta t\_{n-1} \cdot$$

Expressions (22) and (23) have a reason only when

$$m\_0 = P\_n \tag{24}$$

and

206 Modern Metrology Concerns

where the *<sup>i</sup> a* is the *i*th partial quotients for each case and *<sup>i</sup> t* is the *i*th remainder, with

Each remained obtained in the division step of Euclidean algorithm be could be interpreted

where *Pi* and *Qi* are the numerator and denominator of the *i*th convergent of the continued

00 0 1 01 1 1

Then, from (12) each remainder *<sup>i</sup> t* is the absolute difference between the time intervals

On the other hand, *T*0 can be expressed in terms of two consecutive remainders [11] using

*T Qt Q t* 0 11 *ii i i*

Supposing that *n* is the number of steps in the Euclidean algorithm to obtain greatest common divisor of *TX* and *T*<sup>0</sup> . Then last remained 0 *<sup>n</sup> t* and time interval *<sup>n</sup>* <sup>1</sup> *t* is the greatest common divisor of both periods *TX* and*T*<sup>0</sup> , in the consecutive division expressed in equations (7) to (11). Because the greatest common divisor is: the last nonzero remainder

Assuming that *<sup>n</sup>* <sup>1</sup> *t* is greatest common divisor of both periods *TX* and*T*<sup>0</sup> , we can write

*Pa Q P aa Q a* 

, 1, 1, .

*i n* 1,2,3,..., *.* With 1 *<sup>i</sup> a* , *<sup>i</sup> t* is a decreasing sequence for 0. *i*

as a distance [11], defined by

for arbitrary 2 *i ,* and

the following expression:

in this sequence of divisions.

A similar expression can be derived for *TX*

*Q Ti X* and 0. *PTi*

fractions to *T T <sup>X</sup>* <sup>0</sup> defined recursively as [12]

*i ii i* 2 1 *t at t* 1 0 *i i t t* (10)

*n nn n* 2 1 *t at t* 1 0 *n n t t* (11)

<sup>0</sup> . *QT PT t iX i i* (12)

*P aP P i ii i* 1 2 (13)

*Q aQ Q i ii i* 1 2 (14)

. (15)

*T Pt P t X ii i i* 1 1 . (16)

*T Qt* 0 1 *n n* (17)

*T Pt X nn* <sup>1</sup> . (18)

$$m\_X = Q\_n.\tag{25}$$

#### **2.2.2 Stop condition of measurement**

For small measurement time (less than or equal a 1 s) is evident from equation (4) that: order of magnitude of *<sup>n</sup>* <sup>1</sup> *t* must be of same order of magnitude that the expected relative error of measurement . Then, according to before mentioned, we propose that an acceptable tolerance in (1) is *<sup>n</sup>* <sup>1</sup> *t* .

Assuming decimal notation for both periods *TX* and *T*<sup>0</sup> , under the conditions 1 *TX* and <sup>0</sup> *<sup>T</sup>* <sup>1</sup> , and assuming reference period can be expressed as 0 1 10 *<sup>s</sup> <sup>T</sup>* , then the greatest common divisor *<sup>n</sup>* <sup>1</sup> *t* is

$$
\Delta t\_{n-1} = \left(T\_{X'}, T\_0\right) = \frac{1}{10^r} \left(A\_\prime 10^{r-s}\right) \tag{26}
$$

where *A* , *r* , *s* are integer numbers with *r s* , *r* is the exponent associate to expected the order of magnitude of , *r s* is the difference between the expected order of magnitude and the order of magnitude of the time period of the standard.

On the other hand, according with equation (25) the number of time intervals *TX* necessaries to stop the measurement process is *Qn* , and form (15)

$$Q\_n = \frac{T\_0}{\Delta t\_{n-1}}\,. \tag{27}$$

If *A* and 10*r s* in equation (26) are mutually prime then 1 <sup>10</sup>*r s nt* and

$$Q\_n = 10^{r-s} \tag{28}$$

and, if they are not, then 1 <sup>10</sup>*<sup>r</sup> nt a* whit *a* integer number and

$$Q\_n = 10^{r-s} \text{ / } a \text{ .} \tag{29}$$

In both cases 1 10*<sup>r</sup>* is a common divisor of both periods *TX* and *T*<sup>0</sup> .

Then, from equations (16), (24) and (25), the condition that satisfies (19) is

$$m\_X = 10^{r-s}.\tag{30}$$

This is the numeric condition that we propose to stop the measurement process and is easy to implement with basics digital circuits.

A novel fast method to frequency measurement with application in mechatronics and telecommunication is based on this numerical stop condition is presented in [7-9]. Resolution improvement in frequency domain sensors is allowed in automotive applications [10] with this method and is applied in precise optical scanning and structural health monitoring [19-21]

#### **3. Simulation**

In the simulation two pulse trains of unitary amplitude are generated using a computational algorithm sampling independent [7]. The value of reference frequency was accepted as 7 <sup>0</sup>*f* 1 10 Hz. The hypothetical value of unknown frequency is 5878815.277629991 *Xf* Hz, and is a result of the accepted value of the period <sup>7</sup> 1.701023 10 *TX* s. The value of pulse width in both pulse trains is accepted as <sup>9</sup> 1.5 10 *s.* 

In this case, is evident that *TX* and *T*<sup>0</sup> are mutually prime numbers and have the common denominator <sup>13</sup> <sup>1</sup> 1 10 *nt* .

Simulation algorithm provided continuous formation of the segments *n T*0 0 and *n TX X* and compares the magnitude of their difference with parameter 2. When the value of the

where *A* , *r* , *s* are integer numbers with *r s* , *r* is the exponent associate to expected the

On the other hand, according with equation (25) the number of time intervals *TX*

*n*

0 1

*n <sup>T</sup> <sup>Q</sup> <sup>t</sup>*

<sup>10</sup>*r s Qn*

*nt a* whit *a* integer number and

10 . *r s nX*

This is the numeric condition that we propose to stop the measurement process and is easy

A novel fast method to frequency measurement with application in mechatronics and telecommunication is based on this numerical stop condition is presented in [7-9]. Resolution improvement in frequency domain sensors is allowed in automotive applications [10] with this method and is applied in precise optical scanning and structural health

In the simulation two pulse trains of unitary amplitude are generated using a computational algorithm sampling independent [7]. The value of reference frequency was accepted as

<sup>0</sup>*f* 1 10 Hz. The hypothetical value of unknown frequency is 5878815.277629991 *Xf* Hz,

1.5 10 *s.* 

In this case, is evident that *TX* and *T*<sup>0</sup> are mutually prime numbers and have the common

Simulation algorithm provided continuous formation of the segments *n T*0 0 and *n TX X* and

order of magnitude of

and, if they are not, then 1 <sup>10</sup>*<sup>r</sup>*

to implement with basics digital circuits.

monitoring [19-21]

7

denominator <sup>13</sup>

**3. Simulation** 

10 / *r s Q a <sup>n</sup>*

and the order of magnitude of the time period of the standard.

necessaries to stop the measurement process is *Qn* , and form (15)

If *A* and 10*r s* in equation (26) are mutually prime then 1 <sup>10</sup>*r s*

In both cases 1 10*<sup>r</sup>* is a common divisor of both periods *TX* and *T*<sup>0</sup> .

and is a result of the accepted value of the period <sup>7</sup> 1.701023 10 *TX*

compares the magnitude of their difference with parameter 2

width in both pulse trains is accepted as <sup>9</sup>

<sup>1</sup> 1 10 *nt* .

Then, from equations (16), (24) and (25), the condition that satisfies (19) is

1 0 <sup>1</sup> , ,10 10

*r s n X <sup>r</sup> t TT A* (26)

. (27)

(28)

. (29)

(30)

s. The value of pulse

. When the value of the

*nt* and

, *r s* is the difference between the expected order of magnitude

specified difference was less than 2 on corresponding steps of simulation, it was identified like a coincidence of pulses and the integer numbers *n*0 and *nX* are stored.

The unknown frequency is calculated using *Xm x* 0 0 *f nf n* and frequency relative error is obtained using *ff f X Xm x* , both results are stored also.

Simulation results are partially presented in the Table 1 and, frequency relative error calculated (non absolute value) is presented in Fig. 2 for a simulation time of 0.2 s.

The simulation process star in 0 *n* 0 and 0 *nX* , and the best approximation is selected (in this case) using the condition, <sup>6</sup> 1 10 *nX* .

Table 1 represents an interesting fact. For thousands of data we have the same uncertainty range <sup>13</sup> <sup>10</sup> , as for first and third rows. And only when *nX* takes a form of 1 with six zeros (in this case, second row of Table 1) we are getting up to <sup>17</sup> 10 .

In Fig. 2, we can see a global convergence to zero of frequency relative error. An alternated convergence and a non monotone decreasing characteristic are evident. However, we can identify in the graphic a point where is minimum for an approximated time of 0.17 s.

Fig. 2. Frequency offset from the simulation process

Frequency offset from computational selection of best coincidences, obtained under condition (1) with <sup>12</sup> 1 10 is presented in Fig. 3. In this graphic, we can observe a convergence by the left and a divergence by the right around 0.17 s (first point in the Fig. 3). This condition is repeated with time, and we can see five points where absolute value of is minimum, for a simulation time of 1 s. In each mentioned points the condition expressed in (30) is fulfilled.


Table 1. Simulation results of frequency measurement process.

Fig. 3. Frequency offset from the simulation process for 1 s.

#### **3.1 Jitter effect in frequency measurement**

In order to evaluated the jitter effect on the non-electrically detectable stop event for frequency measurement method based on the direct comparison of two regular independent trains of narrow pulses and rational approximations. Deterministic and random components of jitter are modeled and, are added in both pulse trains one deterministic jitter component and random jitter in each case. Simulation results are presented when both pulse trains start in phase and when start with a phase shift.

Timing jitter (henceforth referred to as jitter) is defined as short-term non-cumulative variations of the significant instants of a signal from their ideal positions in time [18].

For modeling, Total Jitter (TJ) consists of two components: Deterministic Jitter (DJ) and Random Jitter (RJ) [14]. In time domain, TJ is the sum of the RJ and DJ components [15]. RJ is characterized by a Gaussian distribution. It has been shown that it is theoretically unbounded in amplitude.

DJ consists of several components caused by different and mostly physically-based phenomena, such electronic interference, cross-talk and bandwidth limitation. All DJ subcomponents have a bounded peak-to-peak value that does not increase when more measurement samples are taken [15].

Deterministic jitter has four components: duty cycle distortion (DCD), intersymbol interference (ISI), periodic jitter (PJ) and bounded uncorrelated jitter (BUJ).

DCD and ISI are referred as data correlated jitter, while PJ and BUJ are referred as data uncorrelated jitter. RJ is unbounded and uncorrelated [15].
