Clock and Data Recovery/Introduction/Definition of (phase) jitter

radians and cycles, radian/sec and Hertz

angles or phases

In the context of periodic phenomena, such as a wave, phase angle is synonymous with phase.

In the context of communication waveforms, the time-variant angle  $\scriptstyle 2\pi ft+\theta ,\,$  or its modulo $\scriptstyle 2\pi$ value, is referred to as instantaneous phase, but often just phase.  

Phases as well as angles are measured in radians or in cycles (or in degrees), and are dimensionless quantities.

When checking the plausibility of an equation by dimensional analysis, it may be useful to include, along with the dimensions of the other involved quantities, also the radian and the cycle (as appropriate) when an angle or a phase is involved.

The correctness check may become easier and the mistake of adding (or of not adding) a 2π may be avoided.

time intervals and frequencies

When checking the correctness of an equation that deals with times and frequencies by dimensional analysis, it is useful to include also the radian and the cycle where appropriate.

It becomes easier to tell time constants from periods, and frequencies from angular frequencies, and to avoid uncertainties or mistakes.

time and frequency quantities

Quantity	Equal to	Expressed in	Physical dimensions	Useful dimensions	Angular and cycle are linked
τ	1/ω	seconds (per radian)	sec	sec/rad	τ = T/2π
ω	1/τ	radians per second	sec-1	rad/sec	ω = f/2π
T	1/f	seconds (per cycle)	sec	sec/cycle	T = 2πτ
f	1/T	cycles per second	sec-1	cycle/sec	f = 2πω

CDRs and parts per million of frequency mismatch

When dealing with frequencies that differ very little from each other, the scale of parts per million is a useful tool.

Parts per million, denoted as ppm, is used to give a relative measure of the frequency difference of two oscillators.

For example if an oscillator runs at 155 MHz and another at 155.13 MHz, they differ by (155.13 10⁶ – 155 10⁶)/(155 10⁶) = 8,38006832978792 10^-4 which is the same as 838 ppm.

The free running frequency of a slave CDR may differ no more than 50 ppm from the frequency of its remote master (not a stringent requirement if assisted by a low cost quartz crystal), or 10000 ppm (easy even with an RC oscillator home made!), or differ less than 1 ppm, still without big cost concerns.

For instance, less than 0.5 ppm accuracy for a local (not locked) oscillator is becoming standard for GPS receivers inside mobile phones nowadays.

Less than 0.01 ppm would mean professional equipment

Definition of (phase) jitter

Although the term jitter, strictly speaking, may refer to several properties of a signal, it refers most often to the phase of the signal.
In this book it always means jitter of the signal phase, only. The phase is defined at discrete instants only, that is at the transitions of the signal through a certain level threshold.

More formally, jitter is here defined as:

The (jittering) time difference between corresponding transitions of two signals

1. In some cases it is convenient to remove the reference to the frequency of the pulses of the transmitted signal (ω_p or f_p where ω_p = 2πf_p).
   Instead of the jitter in seconds it is convenient to represent the related quantity jitter*ω_p (in radian) or jitter*f_p in UI (Unit Interval) i.e. as a fraction of the period corresponding to the transmission of one pulse.
   The true dimension of jitter remains seconds and not the dimension-less related quantities that are used for convenience.
   The reconciliation shall be made with a division by, for instance, ω_p and a phase delay of 0.86 rad is in actuality a jitter (= a time delay) of 0.86/ω_p seconds.
2. The ITU-T defines jitter as the variation of a digital signal’s transitions from their ideal positions in time, for instance in G.810, where jitter and wander are defined accordingly, with separation boundary at 10 Hz:

(timing) jitter: The short-term variations of the significant instants of a timing signal from their ideal positions in time (where short-term implies that these variations are of frequency greater than or equal to 10 Hz).

wander: The long-term variations of the significant instants of a digital signal from their ideal position in time (where long-term implies that these variations are of frequency less than 10 Hz).

In the scope of CDRs however, the jitter is between a clock waveform and (the clock implicit in) a NRZ data stream.
The jitter in such case can only be defined (and measured) when a clock transition and its corresponding data transition are present.
This point, in addition to the reluctance to use an "ideal" reference in a definition, explains the (equivalent) definition given above.

Jitter and phase noise: the same thing?

Jitter and phase noise are descriptions of the same phenomenon from different points of view.

Generally speaking, radio frequency engineers speak of the phase noise of an oscillator, whereas digital system engineers work with the jitter of a clock, as pointed out in the Wikipedia definition of phase noise.

A mountain may look different from different points of view. It may be difficult to imagine the view from a point different from the present one.

Passing from jitter to phase noise may be equally difficult and some caution is necessary. It is important to keep in mind that the jitter in a CDR is the sum of unwanted phase deviations (noise) and of the useful signal (the phase that the CDR wants to acquire and to stay locked into).

Total noise and phase noise (jitter)

Generally speaking, a waveform is affected by both amplitude and phase noise.

But a clock waveform (sinusoidal or square) does not vary in amplitude, and just jitters.

Therefore the side-bands of the Power Spectral Density (PSD) of a clock signal are phase-noise side-bands, i.e. jitter side-bands.

More precisely, the upper side-band translated into base-band is nothing but the jitter PSD! (divided by 2)

When the signal is not a clock but a data stream (NRZ or encoded), its PSD explodes into many small replicas of the clock spectrum and may become even a continuous shape if the encoding data are casual.

A dedicated CDR block, the phase comparator, is used to output a meaningful comparison result between an encoded signal and a clock that are presented at its two inputs.

Eye diagram

On the screen of an oscilloscope, triggering the display with the clock signal, it is possible to display such clock and an NRZ data stream associated with it.

Signals on the scope

The subsequent traces of the data waveform trace different patterns, owing to the random nature of the source data.

Note that the physical limitation of slew rate and of signal bandwidth reduce the slope and smooth the corners of the signal transitions.

The presence of noise, of intersymbol interference and of various types of distortions, that affect any real transmission, make the individual traces spread out and differ from each other.

In practice, the pattern of a train of “eyes”, the "eye pattern" will appear on the scope.

Why call it an "eye" diagram?

During the signal transmission, noise, intersymbol interference, channel non linearities and jitter are added to the signal.

The eye diagram at the receiving end (using the regenerated, jittered clock to trigger the scope) shows a closing eye.

The closing eye corresponds in fact to a signal that is less easily detected (= less “visible”).

The receiver in fact "sees" each received pulse only once, stroboscopically at the eye center, every time when the regenerated clock samples the signal.

Although the received signal could be further amplified and clamped before the actual sampling (see below), the eye diagram is very representative of the link conditions and is often used to specify important conditions that the receiver must tolerate.

Errors of regeneration occur when the upper or the lower "lids" of the eye "blink" to the eye mid level.

When the data stream is a coded multilevel signal, the diagram shows a stack of eyes.

3-level eye diagram

Relative phase

The received signal can be strongly amplified and then limited, so that , as a result, it switches rapidly between two opposite levels.
The time position of its abrupt level transitions still betray the analog and imperfect nature of the signal.

In the signal, the transitions through the mid-level amplitude carry the timing information.

The positions of the level transitions move continuously back and forth in an irregular, almost nervous, manner (= they jitter).
If the vibration reaches as far as the middle point before the next transition (= the center of the eye diagram), the bit level in the received signal may be falsely detected (= errored bit).

Jitter (and wander) definition

For a precise mathematical definition of the jitter, the first step is to clearly define the reference.

The instantaneous phase of the signal originated by the transmit clock (= that is the phase of the transmit clock itself) is the reference, and it is represented by $\omega$t.

Any further processing of the signal will add a fixed delay d plus a small, irregularly variable (= jittery) contribution to the signal phase, that will become then: ωt + d ω + j(t).
The phase jitter j(t) is the part additional to the phase of the original signal and to the transit delay.

The reference frequency of the signal is $\omega$₀ (radian/second).

A periodic signal (the shape is not important) is defined as:

p($\omega$₀t)

where $\omega$₀ is a constant. In other words, $\omega$₀t can be viewed as the output phase of an ideal, noise-less and drift-less oscillator of angular frequency $\omega$₀.
A signal of the type:

p($\omega$₀t + x(t))

where x(t) is in radian, represents an angular phase and describes a deviation from the perfectly linear phase increase $\omega$₀t, is a jittered version of p($\omega$₀t), and ${\tfrac {x(t)}{\omega _{0}}}$ is the jitter in seconds.

The jitter added to the (otherwise linear) phase of a constant frequency signal

In some practical cases it is useful to distinguish between the AC part of x(t) – and call it jitter in a restricted sense - from its very low frequency components – and call that wander -.

The wander part of the jitter is made up by the low frequency (or truly DC, which is nothing but a frequency drift) components.

More precisely, the wander components are the low frequency ones that impact in the topic under study only with unidirectional, slow but large, deviations. During the duration of the phenomenon being studied, the drift components last less than one half cycle at their frequency (their period is more than twice the interval of time being considered).

The jitter proper is made by the components relevant to the topic under study as periodic functions of time (or as functions of j$\omega$ in the mathematical model).

A slow, large deviation of the signal phase from $\omega$₀t would be seen on the scope as a drift of the eye diagram to the left (negative time variation) or to the right (positive time variation).

The eye drift of a real signal, although slow, exhibits in practice the same random behavior of the jitter in general. This drifting sometimes to the right, sometimes to the left, is called wander.

To control something, you must first be able to measure it (Engineering principle)

How to measure jitter then?

Oscilloscopes and spectrum analysers are the fundamental tools.

A really excellent tutorial on this fundamental aspect are the 6 lessons from Ransom Stephens in:

Tektronix Jitter 360° Knowledge Series

from http://www.tek.com/learning/ .

A good concise paper with reference to the various standards for jitter is the Agilent article .