Last modified on 14 February 2015, at 16:21

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

2nd order and type 2 has become the preferred architecture for high performance CDR (regenerators) as soon as the ever advancing IC technology has allowed to integrate all CDR blocks inside one silicon chip.
All blocks can be integrated ( high gain and low noise amplifiers, A/Ds, D/As and DSPs ) with very low manufacturing costs.
Even the VCO is a block that is integrated, but it can not reach the accuracies possible with a crystal oscillator. This is compensated by the more sophisticated architecture and some additional circuitry (PFD).

This architecture uses the option of a zero (the gain increases at lower frequencies) in the filter stage between phase comparator and VCO.

Single Zero Filter.png
The zero is at angular frequency ωz =1/τz, ωw = 1/τw is the frequency of gain = 1 = 0 dB . τz = Gf τw .
Lower frequencies are amplified more (-20 dB / dec) and the lowest frequency -the d.c. jitter component- gets a theoretical infinite amplification.
The decrease of the amplification continues up to ωz and becomes flat (Gf) at higher frequencies.
In practical applications of type 2 CDRs: Gf << 1 (often referred to as "the filter attenuation" and called β) and ωw << ωz .

The maximum output of the comparator corresponds to much less than the maximum frequency deviation of the VCO, deviation that can be reached gradually thanks to the integration of low frequencies made the loop filter.

The response to an input variation always ends with catching up completely (zero steady-state error in this type 2 loop), even if (if the loop gain is much less than "infinite" or the "latency" is relevant [1] ) every CDR design is tailored to specific applications to avoid that the response overshoots significantly.

It is like using either the accelerator or the brake, increasing pressure indefinitely on the pedal until the response of the system becomes sufficient.
It is easy to understand how this architecture reacts immediately but only to a limited extent, and then increases the correction slowly but progressively if the immediate response has not proven sufficient.This is especially evident during the acquisition phase.

The low gain at high frequencies attenuates more the high frequency input jitter (in many cases it also attenuates the bang-bang tracking jitter) while the integration of the input phase error (assisted by a PFD) provides for acquisition in a much wider range of input frequencies.

Where used and how madeEdit

The 2-2 architecture is well suited for monolithic implementations, which explains its widespread use.

This architecture finds its practical applications where a bang-bang phase detector must be used and where the Es must be reduced to a minimum (possibly zero).

The strong reduction of the low-frequency noise from the VCO is a further bonus.

A bang-bang phase detector is inevitable in monolithic applications at very high line frequencies [1], and monolithic implies also an on-chip oscillator, that is relatively noisy and relatively inaccurate in ωfr.

On the other hand, the high volume applications of today are correspondingly very cost sensitive, and require that a single silicon chip accommodate the whole CDR, and also much more.

In practice a bang-bang phase detector is always used, and the VCO is a linear monolithic one, either a ring oscillator or a low-Q LC.

Consequently all the monolithic:

  • line regenerators,
  • slave clocks in Telecom networks
  • CDRs of portable electronics

are made with this 2 - 2 architecture.

A long run-length can affect the tolerance margin in sampling the received signal, but this is normally mitigated by the use of a ternary phase detector.

This loop type is the most “powerful” of the three, because it incorporates a loop filter that offers a very high gain at low frequencies.
This is the characteristic feature of this architecture, that generates its strong and weak points.
The high gain at low frequencies allows the compression to very low value of any steady state error, unlike the other two loops ( 1 - 1 and 2 - 1 ). The filter gain is maximum at very low frequencies, and decreases up to ωz, then it flattens to the asymptotic value and stays constant in the bandwidth of interest (until parasitic poles, always present, make it drop). Gf is very low, in all cases below 1, so that it is often referred to as "the filter attenuation".
The flat gain of the loop filter at high frequencies allows a good tracking of medium frequency jitter, unlike the other second order loop ( 2 - 1 ).
The very low gain of the loop filter at mid and high frequencies keeps the tracking jitter (i.e. the jitter generated by the bang-bang frequency jumps) adequately low.
The high gain at low frequencies allows the frequency and phase acquisitions even when the VCO free-running frequency is shifted much more than ωz from the frequency of the signal to recover.

This 2 - 2 loop tracks very well, as long as the loop filter and the VCO operate within their range of normal operation.

Care must be paid to the very high (closed loop) gain at low jitter frequencies when either:

  • the lack of data transitions temporarily opens the feedback path (this is mitigated by the use of a ternary PD, but presents anyway the risk of a phase error that may increases out of control as long as the loop is open);
  • the (sinusoidal) input jitter has a significant amplitude. The filter gain at high frequencies is kept low to limit the jitter generation resulting from the bang-bang, but this limits the ability of the loop to track a fast and large swing of the input phase. The jitter tolerance depends on the corner frequency of the loop filter and on the length of runs without transitions.
Closed loop gain and corner frequency of the tolerance curve are concepts easily identified if the loop is linear.
When the loop incorporates a bang-bang detector, and the data transitions are random, it is necessary to restrict the study only to conditions of practical interest and to use simulations.

Introductory exampleEdit

This simulation diagram helps understand the operation of the 2 – 2 architecture with bang-bang phase and frequency detector.

The two waveforms of the two phase detectors of the PFD are shifted higher in the diagram, for easier interpretation. The other waveforms are not shifted.
The outputs of the PFD and of the loop filter are scaled differently, with the filter output more amplified in the representation. (The filter output does not reach the clamping level(s), and the VCO drive signal is not different from it, i.e. would be exactly overlapped to it in the diagram. The high frequency parasitic pole of the charge pump is at 200 Mrad/sec, and its effect is barely visible in the shape of the filter bang bang spikes).
To the left the waveforms show that there is no incoming signal. LOS is asserted and the loop is open. The VCO (=the CDR) output simply drifts away with a slope proportional to the difference between the signal frequency and the VCO free-running frequency.
After 1.28 μs (= 150 simulation steps) the signal appears with a phase mismatch equal to -4.0 radian, which happens to reduce the cumulated phase drift that had reached 10.06 rad.
LOS is dis-asserted at that moment and the loop starts catching up.
Initially the output of the PFD is a constant positive level (meaning the VCO is slower than the incoming signal. The filter, that integrates the low frequencies of the comparator output, adds a further positive ramp.
The output phase undershoots the input phase before lock, with even a small overshoot when the filter makes the first negative bang and starts a negative ramp to finally reach a stable continuous bang-bang condition.
During the frequency acquisition three slips have createdted a gap of exactly 3 π between input and output. The gap remains constant from then on, apart from a small additional phase error.
As soon as the loop has caught up with the input, the typical pattern of bang-bang starts: the loop is in lock. The comparator bangs rapidly between is two output states and the filter output maintains a d.c. bias that compensate the ωp - ωfr distance.
After 8.53 μs (= 1000 simulation steps) the input signal phase starts a sinusoidal jittering with a large amplitude (3.20 rad) and a frequency of 0.9 rad/μs.
This jitter brings the loop close to its tolerance limit, which is shown by the detector and by the filter outputs that is alternatively forced out of bang bang, as well as by the error signal that shows deviations from its average of 3π-4 rad in correspondence with those periods of difficult tracking. The error signal deviations are not large in this case, and remain within +0.25 and -0.26 rad.

In this example the transition density is 100%, but the random nature of the input signal must also be taken into account.

Single zero filterEdit

The zero is at angular frequency ωz =1/τz,
ωw = 1/τ is the frequency of gain = 1 = 0 dB . τz = Gf τ .

This loop type is the most “powerful” of the three, because it incorporates a loop filter that offers a very high gain at low frequencies, that is the key feature of the 2 - 2 architecture and that generates its strong and weak points.

The filter gain is maximum at very low frequencies, and decreases up to ωz, then it flattens to the value Gf and stays constant in the bandwidth of interest (until parasitic poles, always present, make it drop).
The high gain at low frequencies allows the frequency and phase acquisitions (a ternary PFD is used in these applications) even when the VCO free-running frequency ωfr is shifted much more than ωz away from the frequency ωp of the signal to recover.
The high gain at low frequencies allows the compression to very low value of any steady state error, unlike the other two loops ( 1 - 1 and 2 - 1 ).

The (almost) infinite gain at low frequencies gives this architecture some very useful properties, but it can also be the origin of unexpected troubles.

As shown in the figure above, Gf is very low, in all cases below 1, so that it is often referred to as "the filter attenuation" and called β. As a consequence, ωz is higher than ωw, the zero-gain frequency.

In the 1 - 1 architecture the output of a bang-bang binary PD always makes the VCO jump from one end of its control range to the opposite end. This leaves a strong residual "tracking" jitter in the output phase of the PLL, because only the 1/s slope of the VCO characteristic does filter the sharp and large swings.
The 2 - 2 architecture in principle behaves the same for high frequency jitter as its loop filter does not filter out the high frequency components (higher than ωz) coming from the comparator output. The filter passes the high frequency components of the jitter to the VCO with a flat transfer function. But the value of the filter gain above ωz ( although flat up to where parasitic poles at even higher frequencies make themselves felt ) is much smaller in a 2 - 2 architecture than the equivalent gain in a 1 - 1 architecture and that makes the tracking jitter proportionally smaller.
The use of a ternary phase detector further reduces the peak tracking jitter [2].

This 2 - 2 loop tracks very well, as long as the loop filter and the VCO operate within their range of normal operation.

A unit step input generates an output with an initial step as high as the high frequency gain,
and a following ramp, with a slope equal to the high frequency gain times the cut-off frequency.

Loose-tracking conditionsEdit

When the phase detector outputs a constant request for higher, of for lower, frequency for a significant number of clock cycles ,
a temporary lack of bang-bang around the locking condition occurs, while the VCO lags behind a rising or falling input phase .

As this interval grows, the tracking error increases (see the linear ramp of the step response of the filter), and this might result into a phase error beyond the tolerance limit.

This 2 - 2 loop may drift away from lock more than the other type 1 loops, if the phase information is not refreshed, as indicated also by its linear model.

This may occur for two different causes:

  • the input signal has too few transitions (run-length problem, that can be approximately described using the stability factor ξ) or
  • the phase of the input signal varies too fast for the loop to track (slew-rate problem, that is investigated by simulations with sinusoidal input jitter close to the tolerance limit).

Both causes can reduce the tolerance of the CDR, and the effects of each can be reduced at the expenses of other CDR performances.

Run-length problemEdit

The actual statistic of the transitions in the input signal is so difficult to predict that only a simple worst-case approach can be of use.

The very extreme case when transitions come periodically and separated by a constant number of line-pulse periods may give some insight and explain why large design margins are found in real monolithic CDRs.

In a condition of very rare but periodic transitions, the fundamental parameter is the time that the loop waits before the next update of the input phase, update that comes from the next transition of the input signal.

When a transition comes as soon as possible, the update time is: tupdate = 1/fp.
When a bit of the same sign follows in the input signal, tupdate at least doubles. More precisely:
tupdate = run-length * 1/fp.

It is convenient to define a parameter ξ:

ξ = 2 τz / tupdate

The longer the run length, the more critical the loop response may become . In fact, ξ is called the stability factor[3].

Let's focus now on what happens in the condition of bang-bang tracking.

The last bang corrects the filter output, higher or lower, by a jump of GφGf [v], and a ramp follows, in the same direction, with slope GφGf ωz [volt/sec]. (See the figure above).
The VCO frequency jumps GφGfGVCO = G [rad/sec] and then ramps with a slope of GφGfGVCOωz = Gωz [rad/sec2]
(GφGfGVCO = G, where the phase detector outputs can be -Gφ, +Gφ if the detector is binary and -Gφ, 0, +Gφ if it is ternary)
The quantity G is very closely related to the quantity fbb[4]. Both measure the frequency jump during bang-bang tracking, and are related by the formula G = 2π fbb.
While fbb is easy to interpret as the frequency jump and is fixed by the circuitry of the CDR, G is used as a nominal value for the open loop gain, in the nominal conditions of DT = 1 and maximum phase error (i.e. minimum Gφ). In fact, G is always found multiplied by DT in the formulae that describe the loop behaviour.

The VCO phase, as a function of the time t, is the sum of a linear ramp Gt plus a parabolic ramp \tfrac {1} {2}zt2 .

After tupdate, the VCO phase has increased (or decreased) by a linear part G tupdate [v] plus a parabolic part \tfrac {1} {2} G ωz tupdate2 [rad].
the ratio of the linear part of the phase increase to the parabolic part of the phase increase is exactly the stability factor ξ:
G tupdate / (\tfrac {1} {2} G ωz tupdate2) = 2τz / tupdate = ξ
When the update takes place, the new bang makes the filter output jump in the opposite direction by a step of Gf [v], followed by another ramp, now in this new direction.
In order to stay in tracking and not to sidetrack out of lock, the phase drift during tupdate must not make the VCO phase drift outside the lateral eye opening:
\tfrac {1} {2} G ωz tupdate2 + G tupdate < LEO (0 < LEO ≤ π)
\tfrac {1} {2} G ωz (2τz/ξ)2 + G (2τz/ξ) < LEO (0 < LEO ≤ π)
The equation yields always one positive real root[5]corresponding to ξ ≥ 2.

The value of G however decreases from its nominal value proportionally to the reduction of DT from its maximum of 100 %.

The ability of the CDR to tolerate with minimal phase drift some very long run-lengths, and/or periodic repetitions of them, can be increased increasing the value of τz at the design stage (ξ = 2τz / tupdate).

This reduces the bandwidth of the loop filter ωz and has the adverse effect of reducing the frequency lock-in range and of increasing the lock-in time.

In practice, values of ξ in excess of 1000, even with low transition densities as found in SONET transmissions, are not used.

Slew-rate problemEdit

The 2 - 2 CDR is able to vary rapidly its frequency. If the input signal offers the maximum transition density, the loop can respond to an input phase step with a frequency step equal to G [rad/sec], i.e. with a phase ramp of slope G.

The value G (calculated with DT = 100%) represents the bang bang frequency step but also the reference loop gain ( Gφ is calculated with the largest phase input error, typically ±π , multiplied by DT, Gf is the value of the filter gain in the flat region, GVCO is a linear approximation around the fp working point ).

But if the phase of the input signal varies more rapidly than G rad/sec2, then a phase error appears and it may grow and possibly affect the CDR tolerance.

This problem may be investigated using a sinusoidal input phase jitter. This is also useful as the tolerance curve of the CDR is measured as a function of a sinusoidal input jitter.

A sinusoidal function (of time in the case of jitter) has its maximum rate of variation at the zero crossings, and its maximum rate of variation is the product of its amplitude by its angular frequency, Aω in the figure here below:

Time diagram of a sinusoid of angular frequency ω and amplitude A,
with emphasis on the maximum slope equal to Aω, that relates to the slew-rate concept

Maximum phase slew-rate = maximum frequency deviation.

The value of ωz is kept low in order to have a stability factor ξ large enough.

But ωz can not be too low, or else the slew rate (and the jitter tolerance limit!) at medium/low frequencies become too low as a consequence. (develop this point with comments and a figure)

There always exists the possibility that the VCO is not able to follow the rapidly changing phase of the input, because the rate of change of the VCO phase is insufficient. The VCO is "slew-rate" limited.

This is not normally due to late response of the VCO driven by the signal from the filter output.

The frequency deviation limits of the CDR are often set by the characteristics of the loop filter, rather than by the extremes of the frequency range of the VCO itself.
"The VCO is designed to respond fully in one update time. This is usually very easy to achieve in ring-oscillators and possible with some care using low-Q VCOs." [6]).

The slew-rate is originated by the loop filter that has a very limited gain (= an attenuation) at frequencies higher than ωz.

In fact, the Gf at high frequencies is kept as small as possible to reduce the "tracking" jitter due to the bang-bang jumps, but can not be reduced too much or else the "slew-rate" becomes excessively small.

Jitter Bandwidths in 2nd order type 2 bang-bang CDRsEdit

The bang-bang CDRs are made up by most of the 1-1 (PD) and practically all of the 2-2 (PFD).

These loops are intrinsically non-linear, primarily because of the bang-bang nature of the phase detector: this is the first non-linearity.

The very large (≡ infinite) gain in the detector needs a compensation by a signal level limitation is another loop point.
(The level limitation keeps signal levels within the physical capability of the circuit elements).
The level limitation takes place either:
1. in the limitation of the VCO drive signal made by the circuits driving the VCO, or
2. in the extremes of the VCO voltage-to-frequency characteristic.

It is a frequency limitation. This is the second non-linearity. In both cases of frequency limitation, when the limit is reached, the output phase waveform of the CDR enters a slew-rate condition.

The VCO intrinsic characteristic, at both ends, depends very much (and often unpredictably) on manufacturing and environmental variations.
Therefore, the frequency range is always deliberately limited by a clamp of the output range of the VCO drive stage, +/- Vdr .

Overall, two non-linearities (PD or PFD, and VCO frequency constraints) combine making transfer (e.g. jitter transfer) functions become families of functions.

Each function in a family is associated with a specific input waveform.

In the (transfer functions') case of sinusoidal inputs, each function of ωj is associated with the input amplitude only.

The output is also periodic, and the transfer function is defined as the ratio of the output peak amplitude to the input peak amplitude Aj (that is constant at all frequencies for each function in the family).

Vbb = the absolute value of the voltage step that takes place in these CDRs, at the input of the VCO, when the PD bangs up or down from its intermediate level.

It coincides with the value of Gf(ω) for ω >> ωzω. In fact, the PD output is either +1 or -1 [volt] (or 0 V in the case of a ternary PD).
In 1 -1 loops +/- Vbb is the total drive range of the VCO and coincides with +/- Vdr .
Ed is generated by a deviation of the duty cycle away from 50% in the VCO drive waveform.
In 2 - 2 loops +/- Vbb is much smaller than the drive range of the VCO +/-Vdr (e.g. Vbb / Vdr = 10-3).
The VCO drive waveform shows a tiny bang-bang three-level ripple added to a slower and much larger waveform, that is made by the filter amplification of low jitter frequency components.
Ed is obtained by a variation of the mean level of the bang-bang ripple in the drive waveform.

Jitter transfer functions in bang-bang CDRs of 2nd order and type 2 [7]Edit

This is a type 2 system, which means that -during tracking- the mean values of the input and of the output phase waveforms do not differ .

The difference of mean values is the steady state sampling error Es, and Es = 0 in these systems.
In other words, the average level of the the difference between the input and output waveforms is zero (= the average phase error is zero ).

In 2nd order systems the signal processing between PD and VCO consists of two regions along the frequency axis:

1. a flat attenuation above ωz (that makes the drive signal so small that , in tracking, the VCO frequency bang-bangs within a small range around fp).
2. a region of increasing amplification as jitter frequency decreases from ωz, that slopes at -20 dB/dec.

The two regions of the loop filter define the loop behaviour unless either of the two VCO frequency limits is reached (with an input jitter large enough), where the resulting slew-rate corresponds to either of the VCO limits (+/- Vdr or +/_Vbb).

The two frequency regions of the loop filter, with jitter levels that make Vbb slewing evident in both of them, can be identified in the figure already shown in the jitter tolerance page.

Slew-rate limitations at medium and at high jitter frequencies.
The triangular wave of the output at high frequencies is symmetric: the type 2 loop squeezes to zero any steady state output error.

In steady state with a sinusoidal input, the output of the CDR is just periodic with the same period. The jitter transfer function can be defined as the ratio of the output peak value to the input peak value.

The waveform (in black) of the VCO drive signal bang-bangs when there is no slewing, but does not bang.bang and and slowly ramps away as long as the phase error keeps the same sign (constant slew-rate condition of either sign).
The steep bang transitions, and the slope of the ramps that follow, implement the step response of the loop filter driven by the steps (bang-bangs) of the PD output.
In the PD output, the slow ramp during slewing compensates any possible steady-state error of the loop output.

The figure is drawn with values of jitter amplitudes Aj still within the jitter tolerance boundary, but larger that those used to measure the jitter transfer characteristic of a CDR.

In jitter transfer measurements, at low ωj (= at low jitter frequencies), the slewing that is shown in the left hand side of the figure does not appear.
The amplitude of the jitter input used in those measurements (= Aj) is lower, and the CDR output phase tracks the input phase jitter and the transfer curve is a constant 1, i.e. a flat 0 dB in a Bode plot. A value of Aj = 1.5 UIpp is typical for the measurement of jitter transfer below ωj, and of 0.15 UIpp above ωj.[8]

But for jitter frequencies ωj sufficiently high, even if Aj ( the amplitude of the input sinusoidal jitter ) is relatively low, the tracking is not perfect and the output phase waveform from sinusoidal becomes triangular like in the right hand side of the figure above:

2015 2 14 Acquisition and tracking of a 2nd order type 2 CDR at the very onset of triangular slewing (DT = 1).
There the triangular output has fixed slopes (+/- VbbGVCO) and It takes one quarter of the jitter period, 1/4 * 2π/ωj for the output to vary from 0 to its peak value.
Its peak value is: \tfrac{\pi V_{bb} G_{VCO}}{2 \omega_j}. As the peak varies inversely to ωj, in this region the jitter transfer curve rolls off at- 20 dB/dec.
The ratio of the output peak value to the input peak value (Aj) in the Vbb slew-rate region is therefore :
\tfrac{\pi V_{bb} G_{VCO}}{2 A_j \omega_j}
If the output was measured with the amplitude of its fundamental component instead of with its peak value, the roll off part of the transfer curve would just be translated (downwards) a tiny -0,091 dB.

The overall jitter transfer curve has one horizontal asymptote at 0 dB towards low frequencies, and a - 20 dB/dec asymptote towards high frequencies.

The transition between the two asymptotes can be approximated using the model of a first order linear system, whose corner frequency ωjc is where the sloping asymptote intersects the 0 dB horizontal asymptote. (The approximation given by this model is more than adequate for most engineering purposes) :

ωjc = \tfrac{\pi V_{bb} G_{VCO}}{2 A_j}[9]

The jitter transfer can therefore be modelled as:

JitterTransfer(ω) = \frac{1}{\sqrt{1+\tfrac{\omega_{jc}^2}{\omega^2}}}

The Aj to measure the transfer characteristic may be smaller than Φm.

With large Aj (e.g. 1.5 UIpp ), a sinusoidal jitter is transferred unchanged to the output.
At the onset of triangular slewing, that is a function of Aj, the transfer ratio rolls off at -20dB/dec. This region is typically explored with Aj down to 0.15 UIpp.
The transition from 0 dB flat to -20 dB/dec roll-off takes place at a frequency inversely proportional to Aj.
If Aj is made too small (smaller than Φm), the bandwidth of the transfer curve does not increase any more.

This is a realistic assumption in many practical cases, and may even be used to design the CDR so that a certain mask of jitter transfer is met.

The sinusoidal jitter that is to be applied in the bandwidth from the maximum ωjc and a couple of decades above does not normally exceed 0.15 UIpp [10]
The Φm in a documented 10 Gbps CDR is found to be about 0.26 UIpp, taking into account just the metastability in the PD flip-flops, without other smaller contributions. [11]

The model becomes the linear model of a 2nd order type 2 loop with a linear comparator of gain Gφ = 1/Φm.

The transfer function has a fixed (= independent from Aj in this case) cut-off frequency at

ωn2 = {\sqrt{G\omega_{z}}}
It may be remarked that the linear 2 - 2 loop has an open loop gain G = GφGfGVCO and that in this case Gφ can be computed as Aj/(AjΦm) and Gf can be computed as Vbb.
Substituting inside the formula for ωn2 above, the formula for ωjc is obtained.
As Gφ is in these practical cases very high (higher than 4ωz) , the 2 - 2 model that can be used for small Aj (smaller than Φm) has a damping coefficient ζ larger than 1 and therefore no gain peaking ( ζ = {\sqrt{\tfrac{G}{4\omega_{z}}}}).

Jitter tolerance curves of bang-bang CDRs of 2nd order and type 2Edit

External ReferencesEdit

  1. a b C. Response to Phase Step
  2. Richard C.Walker article, A. Run-length and Latency, pg 10.
  3. Richard C.Walker article, A. Stability Factor, pg 4.
  4. Richard C.Walker article, p. 3.
  5. Roots of \tfrac {1} {2} G ωz (2τz/ξ)2 + G (2τz/ξ) - LEO = 0 are: 1/\xi =\tfrac{-1\pm\sqrt{1+\tfrac{2LEO}{G\tau_z}}} {2}.
  6. Richard C.Walker article, VII. C. VCO Tuning Bandwidth, pg 10.
  7. Jri Lee, Kenneth S. Kundert, Behzad Razavi (SEPTEMBER 2004). "Analysis and Modeling of Bang-Bang Clock and Data Recovery Circuits". IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 39, NO. 9 pages 1571 .. 1579, III., JITTER ANALYSIS, A. Jitter Transfer.;jsessionid=1FD71150D9F00B8D9F8A32252D812190?doi= Retrieved 2015-1-25. 
  8. ITU-T G.8251 The control of jitter and wander within the optical transport network (OTN) (09/2010) A.7 Jitter transfer "The jitter transfer function of a 3R regenerator shall be under the curve given in Figure A.7-1 when input sinusoidal jitter up to the masks of Figures ..., is applied."
  9. Lee, Kundert, Razavi paper of sep. 2004, JITTER ANALYSIS, A. Jitter Transfer ω-3 dB = \tfrac{\pi K_{VCO}I_{p} R_{p} }{2 \phi_{in,p}j} (12)
  10. ITU-T Rec. .8251-201009 The control of jitter and wander within the optical transport network (OTN); A.7 Jitter transfer .
  11. Jri Lee, Kenneth S. Kundert, Behzad Razavi (SEPTEMBER 2004). "Analysis and Modeling of Bang-Bang Clock and Data Recovery Circuits". IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 39, NO. 9 pages 1571 .. 1579, II. BANG-BANG PD MODEL, A. Effect of Metastability.Fig. 3. (b) Simulated characteristic at transistor level..;jsessionid=1FD71150D9F00B8D9F8A32252D812190?doi= Retrieved 2015-1-25.