**3.1.2 Periodic Jitter (PJ)**

Periodic jitter denotes periodical timing deviation from the ideal position of a signal that repeat in time, is typically uncorrelated to the data rate or the clock frequency [14]. Electromagnetic interference and crosstalk from some clock line can cause periodic jitter.

The mathematic model of PJ consists of a sum of cosine functions with phase deviation, modulation frequency, and peak amplitude. The model is given by

$$Pf\_T = \sum\_{i=0}^{n} a\_i \cos \left(\alpha\_i t + \theta\_i\right) \tag{32}$$

where *PJT* denotes the total periodic jitter, *n* is the number of cosine components, *ai* is the amplitude in units of time in each tone, *ω<sup>i</sup>* is the angular frequency of the corresponding modulation, *t* is the time, and *θ<sup>i</sup>* is the corresponding phase [15].

Sinusoidal jitter in time domain produces a probability distribution function given by (defining time zero as the center of the distribution)

$$f\_{Pl\_i} \left( \mathbf{x} \right) = \begin{cases} \frac{1}{\pi \sqrt{a\_i^2 - \mathbf{x}^2}} & |\mathbf{x}| \le a\_i \\ 0 & \text{otherwise} \end{cases} \tag{33}$$

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

Fig. 4. Histogram for a) Random jitter (RJ), b) Sinusoidal periodic jitter with added random jitter (PJ+RJ), c) duty cycle distortion with added random jitter (DCD+RJ) y d) data dependent jitter with added random jitter (DDJ+RJ).

#### **3.1.3 Duty Cycle Distortion (DCD)**

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

DCD yields a binomial distribution consisting of two sharp peaks of equal height, unless one separates rising and falling transitions in the measurement. Theoretically those peaks are Dirac delta functions, but in practice random jitter and limited measurement resolution always produce peaks of finite height and finite width. The analytic equation for DCD distribution is the sum of delta functions [14]:

$$J\_{\rm DCD}(\mathbf{x}) = \frac{\delta\left(\mathbf{x} - a\right)}{2} + \frac{\delta\left(\mathbf{x} + a\right)}{2} \tag{34}$$

where *2a* is the peak-to-peak width of the DCD. In Fig. 4c) is shows the histogram for duty cycle distortion with added random jitter.
