**3.1.4 Data Dependent Jitter (DDJ)**

212 Modern Metrology Concerns

Sinusoidal jitter in time domain produces a probability distribution function given by

1

*i*

(33)


s


s

x 10-12

x 10-12

*x a*

*otra*

0

where *2a* is the peak-to-peak width of periodic jitter [14]. In Fig. 4b) is shows the histogram

2 2

*PJ i*

*J x a x*

 

x 10-12

x 10-12

jitter (PJ+RJ), c) duty cycle distortion with added random jitter (DCD+RJ) y d) data

offsets and differences in the rising and falling edge characteristics [14].

Number of events

c) d)

Fig. 4. Histogram for a) Random jitter (RJ), b) Sinusoidal periodic jitter with added random

Duty Cycle Distortion is often also called pulse width distortion [14], is deviation in the duty cycle value from the ideal value, this equates to a deviation in bit time between a 1 bit (logic 1) and a 0 bit (logic 0). DCD can have several sources. The most ommon are threshold level

a) b)

Number of events

*i*

(defining time zero as the center of the distribution)

for sinusoidal periodic jitter with added random jitter.


s


s

dependent jitter with added random jitter (DDJ+RJ).

**3.1.3 Duty Cycle Distortion (DCD)** 

Number of events

Number of events

Data dependent jitter describes timing errors that depend on the preceding sequence of data bits [14]. DDJ is a predominant form of DJ caused by bandwidth limitations of the system or electromagnetic reflections of the signal [16-17]. Since there is always only a limited number of different possible patterns in a data stream of limited length, data dependent timing errors always produce a discrete timing jitter, theoretically DDJ distribution is the sum of two o more delta functions [14]:

$$J\_{DDJ}\left(\mathbf{x}\right) = \sum\_{i=1}^{N} \left\{ p\_i \mathcal{S}\left(\mathbf{x} - t\_i\right) \right\} \tag{35}$$

where 1 <sup>1</sup> *<sup>N</sup> <sup>i</sup> <sup>i</sup> <sup>p</sup> <sup>N</sup>* is number of distinct patterns, *pi* is the probability of the particular pattern occurring, and *ti* is the timing displacement of the edge following this pattern. In Fig. 4d is shows the histogram for data dependent jitter with added random jitter.

In the simulations, two pulse trains of unitary amplitude are generated. The value of reference frequency was accepted as <sup>7</sup> <sup>0</sup>*f* 1 10 Hz. The hypothetical value of unknown frequency 5878815.277629991 *Xf* Hz is a result of the accepted value of the period <sup>7</sup> 1.701023 10 *TX* s, and value of pulse width in both pulse trains is <sup>9</sup> 1.0 10 s. The RJ model used is Gaussian distributed with RMS value 0.7 ps. The PJ model is a single-tone sinusoidal with frequency 5 MHz and peak-to-peak value of 10 ps, same peak-to-peak value is assumed in DDJ and DCD jitter models. A component of random jitter is added to the last tree models to generate the jitter models PJ+RJ, DDJ+RJ and DCD+RJ. Such jitter values are selected because of the most typical values for such pulse trains according to [15]. Then the jitter components modeled are applied to the time reference points of each narrow pulse on the two pulse trains. In

## **3.2 Jitter simulation results**

In the first computational experiment is assumed the same magnitude of the jitter in the oscillator under measurement and references oscillator on all models. The experimental results when the two pulse trains started in phase are presented in Table 2, to a simulation time of 0.172 s. In the first row of each table are presented the results obtained without jitter models.

In third column is shown the total number of coincidences obtained with each model for the simulation time and, in the fourth column is shown the number of coincidence where appears the stop event associate to the best approximation in the frequency measurement process (the desired count is represented like *10r* [7]). Program code for computational experiment in Matlab is presented in [22].


Table 2. Simulation results with same magnitude of the jitter and pulse trains in phase.

Comparing the results shown in both columns with the values obtained in the simulation that does not include any jitter model, we see that in the presence of jitter the number of coincidence and the number of coincidence pulse associated with the best approach varies depending on model type. The latter implies the position at the time of the coincidence pulse is not fixed since it depends on the jitter component is dominant, in all case *n0=1701023* and *nX=1000000.* 

Fig. 5 is presented the relative error obtained by simulation around best coincidence, obtained under condition (30) with *ε=1×10-12* s. In this plot we can see that due to jitter effect some theoretically expected marginal coincidences may disappear or non-theoretically expected may appear. However, this phenomenon can not be observed in non-marginal coincidences, for instance the expected coincidence under condition (30) and an appropriated pulse width τ.

In third column is shown the total number of coincidences obtained with each model for the simulation time and, in the fourth column is shown the number of coincidence where appears the stop event associate to the best approximation in the frequency measurement process (the desired count is represented like *10r* [7]). Program code for computational

> Total number of coincidences

model 20223 19999

RJ 20222 19998 PJ+RJ 20240 20016 DCD+RJ 20213 19989 DDJ+RJ 20223 19999

RJ 20217 19993 PJ+RJ 20224 20002 DCD+RJ 20218 19995 DDJ+RJ 20218 19994

RJ 20226 20002 PJ+RJ 20218 19995 DCD+RJ 20218 19994 DDJ+RJ 20217 19993

RJ 20227 20003 PJ+RJ 20216 19991 DCD+RJ 20218 19994 DDJ+RJ 20223 20000

Table 2. Simulation results with same magnitude of the jitter and pulse trains in phase.

Comparing the results shown in both columns with the values obtained in the simulation that does not include any jitter model, we see that in the presence of jitter the number of coincidence and the number of coincidence pulse associated with the best approach varies depending on model type. The latter implies the position at the time of the coincidence pulse is not fixed since it depends on the jitter component is dominant, in all case

Fig. 5 is presented the relative error obtained by simulation around best coincidence, obtained under condition (30) with *ε=1×10-12* s. In this plot we can see that due to jitter effect some theoretically expected marginal coincidences may disappear or non-theoretically expected may appear. However, this phenomenon can not be observed in non-marginal coincidences, for instance the expected coincidence under condition (30) and an

Coincidence number for best approximation

Oscillator under test

Without jitter

experiment in Matlab is presented in [22].

Reference Oscillator

Without jitter model

RJ

PJ+RJ

DCD+RJ

DDJ+RJ

*n0=1701023* and *nX=1000000.* 

appropriated pulse width τ.

Fig. 5. Jitter effect on frequency relative error for a series of best coincidences (simulation time 0.2.s)
