Clock and Data Recovery/Structures and types of CDRs/Applications of the 2nd order type 1 architecture

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Application of the 2-1 architecture[edit]

This architecture is used in practice for:

  • line regenerators, and for
  • slave clocks in long distance (=Telecom) networks.

The implementation with linear phase comparator and linear VCO is the typical case, as it best fits the target applications.

The linear model in this 2-1 case finds a direct application!

As seen in a previous page, the performance parameters to consider are ωn2 and ζ.

They are related to the design parameters G and τf as follows:

2nd order PLL of type 1: relations between performance and design parameters
 Setting ωn2 and ζ   Setting G and τf 
G = ωn2 / 2ζ ωn22 = G/τf
τf = 1 / 2ζωn2 ζ2 = 1 / 4τfG


τf is the time constant of the added filter and ωf = 1/ τf is its cut off frequency;

G is the open loop DC gain = Gφ * Gf * GVCO.

(It may also be pointed out that the 2 - 1 loop becomes unstable if the natural frequency ωn2 {that normally is about 1/2 ωf or lower} gets close to ωf. As ωn2 grows when the loop gain grows, it is not possible to use a bang-bang detector because its gain varies very much with the phase difference it measures).

The ITU-T Recommendations often indicate as a reference model the 2-1 architecture.
For instance:
“a SEC will generally mimic the behavior of a 2-nd order (type 1) linear analogue phase locked loop. This allows the use of the terms (equivalent) 3 dB bandwidth and (equivalent) damping factor, as they are used in analog PLL theory, irrespective of the fact that in the implementation of a SEC, digital and/or non-linear techniques may be used."[1]
( SEC: a SDH equipment slave clock)


In fact, when requirements are the following, like in telecom networks:

  • continuous transmission mode
  • receiver cost may increase if an increase of the regeneration span offers a larger saving.

they translates into:

  • filtering incoming phase noise is important
  • the cost of a low local noise generation is affordable
  • fast acquisition is not important

then the preferred CDR implementation includes a phase comparator that behaves linearly, a low pass filter and a low noise oscillator whose frequency varies proportionally to the control signal. The PLL behaves as a 2-1 linear loop.

When more CDRs are chained so that their jitter transfer functions combine, then gain peaking must be minimum or absent: ζ >> 0.7.

ωn and ζ in continuous-mode regenerator CDRs[edit]

The ITU-T Recommendations (the G.7XX series and the G.8XX series in particular, see for instance the G.825) describe a lot of the characteristics of clock regeneration recommended for applications in continuous mode.

With a little reverse engineering of the large amount of data available from that source (and making reference to the model of a PLL of the 2nd order, type 1), the preferred values that can be obtained are:

ωn = ω0 / 2,500
ζ ≥ 0.66 (to match the requirement of 0.1 dB of max Jitter amplification).

The damping coefficient ζ must be - in all conditions - larger than 0.66 in order to keep the maximum jitter amplification (maximum value of the jitter transfer function) below 0.1 dB.


Another aspect that suggests values of ζ larger than at least 0.7, is the overshoot when tracking a sinusoidal jitter. Both the diagram of the jitter tolerance and the diagram of the jitter error show that (for ωn2 just above 1) there is an extra deterioration of the tracking performances for low values of ζ (See: 1, 2, 3, 4).


The most important consideration, that forces a substantial margin in the design value chosen for ζ in the CDR design, is however the derating due to the transition density expected in the incoming pulse stream. The transition density generally oscillates around 50%, well below its theoretical maximum of 100%.

It is always necessary to account for a transition density not larger than 50%, and in some cases (depending on the line code and on the product of the maximum run length duration - about 72 bits[2][3]- times the jitter transfer bandwidth) even lower.


Finally the construction tolerances when actually building the physical CDR circuit are to be considered, along with the variations that may come with aging. They can affect the parameters significantly (sometimes as much as +/- 30%, like in the case of monolithic integrated circuits).


Taking into account all these derating factors, in practice the model parameters of CDRs for continuous mode applications fall inside the following ranges:

Ranges of ωn and ζ for the 2 - 1 model of continuous mode CDRs
 Parameter   DT = 100 %   DT = 33 % 
G G = Gφ Gf GVCO 0.33 G
ωn 1/50 to 1/15,000 of ω0 1/90 to 1/26,000 of ω0
ζ 1.0 to 1.3 (typ. 1.1) 1.8 to 2.3 (typ. 1.9)


The extreme cases with ωn < 1/10,000 ω0 can only be found in some long distance links on single mode optical fibre, where a high number of regenerators are located in series along the line, each one with its own CDRs.
In that case the jitter accumulation is the most critical aspect. All the above conditions shall never combine in a worst case that make the jitter amplification of each regenerator exceed +0.1 dB at any frequency, corresponding to ζ equal or lower than 0.652.[4]
Jitter around that particular frequency of peak amplification could be repeatedly amplified along the line and accumulate enough power to become source of error bits.
Another ITU reference goes even further and recommends that any peaking of the transfer function gain is to be avoided in long regenerator chains, which means ζ < 0.7.[5] , which is the same as to recommend ζ never to be smaller than 0.7 .

ωn2 and ωf[edit]

The cut-off frequency of the loop filter block ωf =1/τf is the main parameter that sets the bandwidth of the closed loop, apart from the minor adjustments of ζ as it varies within its allowed range.

The natural frequency ωn2 (which is the jitter cutoff frequency of the CDR) can also be expressed as a function of the loop gain and of ωf :

ωn22 = G ωf

or can be expressed as a function of just ζ and ωf :

ωn21 = ωf /2ζ21

This simple formula tells that (in a 2nd order PLL of type 1 (where ζ21 must remain close to 1) the natural frequency ωn21 remains close to half the cutoff frequency of the loop filter!


It has been shown that the VCO ffr accuracy can set a limit to how narrow the jitter bandwidth can be.

External References[edit]

  1. ITU-T G.813 (03/2003) Timing characteristics of SDH equipment slave clocks (SEC), Appendix II: Considerations on bandwidth requirements, noise accumulation and payload wander accumulation, II.1 Introduction.
  2. The maximum run-length that must be tolerated in SDH systems is 72 CID (Consecutive Identical Digits): G.783-2006 03 Characteristics of synchronous digital hierarchy (SDH) equipment functional blocks; Appendix V Verification of SDH equipment CID immunity, page 275; 72 bits …
  3. G.957-200603 Optical interfaces for equipments and systems relating to the synchronous digital hierarchy; Appendix II, Implementation of the Consecutive Identical Digit (CID) immunity measurement
  4. ITU-T Rec.G.783 (03/2006), in particular Table 15-2 / G.783 - Jitter transfer parameters and 15.1.3 Jitter and wander transfer
  5. ITU-T-REC-G.824 Fascicle III.5 Supplement 36 Page 285- Jitter and wander accumulation in digital networks, ... no peaking in the regenerator jitter/wander transfer characteristic, ...