AN279 [SILICON]

ESTIMATING PERIOD JITTER FROM PHASE NOISE; 估算周期抖动相位噪声


型号：	AN279
厂家：	SILICON
描述：	ESTIMATING PERIOD JITTER FROM PHASE NOISE 估算周期抖动相位噪声
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AN279

ESTIMATING PERIOD JITTER FROM PHASE NOISE

1. Introduction

This application note reviews how RMS period jitter may be estimated from phase noise data. This approach is

useful for estimating period jitter when sufficiently accurate time domain instruments, such as jitter measuring

oscilloscopes or Time Interval Analyzers (TIAs), are unavailable.

2. Terminology

In this application note, the following definitions apply:

Cycle-to-cycle jitter—The short-term variation in clock period between adjacent clock cycles. This jitter

measure, abbreviated here as J , may be specified as either an RMS or peak-to-peak quantity.

Jitter—Short-term variations of the significant instants of a digital signal from their ideal positions in time

(Ref: Telcordia GR-499-CORE). In this application note, the digital signal is a clock source or oscillator. Short-

term here means phase noise contributions are restricted to frequencies greater than or equal to 10 Hz

(Ref: Telcordia GR-1244-CORE).

Period jitter—The short-term variation in clock period over all measured clock cycles, compared to the average

clock period. This jitter measure, abbreviated here as J

, may be specified as either an RMS or peak-to-peak

PER

quantity. This application note will concentrate on estimating the RMS value of this jitter parameter.

The illustration in Figure 1 suggests how one might measure the RMS period jitter in the time domain. The first

edge is the reference edge or trigger edge as if we were using an oscilloscope.

Clock Period

Distribution

_PER(RMS) = σ

T =0

T = T_PER

Figure 1. RMS Period Jitter Example

Phase jitter—The integrated jitter (area under the curve) of a phase noise plot over a particular jitter bandwidth.

Phase noise data may be recorded as either SSB phase noise L(f) in dBc/Hz or phase noise spectral density

Sφ(f) in rad /Hz where:

Sϕ(f)

-------------

L(f) ≡

RMS phase jitter may be expressed in units of dBc, radians, time, or Unit Intervals (UI).

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AN279

3. Basic Approach

By definition, period jitter compares two similar instants in time of a clock source such as two successive rising

edges or two successive falling edges. Since the two instants are separated in time by approximately one period, it

is reasonable to expect that higher frequency jitter components will contribute more to period jitter than lower-

frequency jitter components (f<<1/T.)

A number of authors, e.g., Drakhlis (2001), have derived the basic relationship between J

(rms) and phase

PER

noise spectral density Sφ(f). It can be shown that

∞

Δϕ_RMS= 4 Sϕ(f)xsin (πfτ)df

∫

and, therefore:

T₀

Δϕ²_RMS

------

J_PER(rms) =

2π

In short, period jitter is integrated similarly to phase jitter but with a frequency weighting factor of 4sin (πfτ).

See "Appendix A—Derivation" on page 5.

^{There are practical limits to integrating phase noise from f = 0 to}∞. Phase noise is measured from a low-frequency

limit, f , to a high-frequency limit, f . In practice, the minimum and maximum phase noise offset frequencies are

determined by the measurement system bandwidth.

Phase noise integration using the sin weighting factor is not typically sensitive to contributions from offset

frequencies well below the half-carrier frequency. Therefore, the choice of f in theoretical period jitter calculations

is generally not critical. For the purposes of this application note, the minimum offset frequency for phase noise

measurements is 10 Hz, coinciding with the usual dividing line between wander and jitter.

The period jitter sin weighting factor reaches its first maximum at the half-carrier frequency and becomes periodic

thereafter. It is clear that phase noise in the vicinity of this offset will make a significant contribution to the period

jitter.

The maximum offset frequency for the purposes of integration is determined by the bandwidth of interest. One

practical high-frequency limit is to set f equal to the half-carrier frequency. For an example, see Underhill and

Brown (2004). Another, more conservative, approach is to integrate out to the carrier frequency where the sin

weighting factor reaches its first null. This method appears to correlate well with at least one popular Time Interval

Analyzer or TIA, presumably due to aliasing components above f /2. See Smith (2006).

However, in many high-frequency applications, phase noise equipment cannot measure phase noise all the way

out to either the half-carrier or carrier frequencies. Provided that the phase noise data reaches the phase noise

floor at its highest measured offset, the phase noise floor may be extended from f

estimation approach adopted in this application note.

to f . This is the

H, MEASURED

Rev. 0.1

AN279

4. Example

Consider the example in Figure 2, which represents the SSB phase noise in dBc/Hz measured for a 160 MHz

oscillator.

Figure 2. 160 MHz Oscillator SSB Phase Noise

For this particular plot, there are three curves illustrated as follows:

Phase Noise—This is the upper solid curve going from –78.32 dBc/Hz @10 Hz on the left down to

–142 dBc @80 MHz on the right.

Period Jitter Weighting Function—This is the dashed line depicting the 4sin (πfτ) weighting factor in dB. It is

predominately a +20 dB/dec sloped line until reaching a peak at the half-carrier frequency where it turns over to

a null at the carrier frequency.

Resulting Phase Noise—This is the lower, thicker solid curve representing the summation of the phase noise in

dBc/Hz with the Period Jitter Weighting Function in dB.

The particular phase noise instrument that took the original phase noise data had a default maximum offset

frequency of 20 MHz when measuring clocks less than 250 MHz. For the purpose of estimating the period jitter, the

"ultimate noise floor" of the original data set has been extended from the last data point at 20 MHz out to the carrier

frequency of 160 MHz.

In this example, there are no discrete components or spurs. However, if there had been any significant spurs in the

vicinity of the half-carrier frequency, they would need to be weighted similarly as described in "Appendix B—

Calculating with Spurs Included" on page 7.

By numerical integration, we can determine that the integrated phase noise under the entire SSB phase noise

curve from 10 Hz to 160 MHz yields a total phase noise power = –54.46 dBc. This "brick wall" integration is

equivalent to wideband RMS phase jitter of 2.663 ps or 0.00268 radians.

Rev. 0.1

AN279

By contrast, if we weight L(f) using the period jitter weighting function and integrate the resulting curve over 10 Hz

to 160 MHz, we obtain a total phase noise power = –56.84 dBC, equivalent to RMS period jitter of 2.025 ps or

0.00204 radians; so, we can estimate the period jitter as being roughly 2 ps RMS, based solely on the available

phase noise measurement. This estimate can be correlated with measurements from complementary time domain

equipment when available. See Smith (2006) for example.

The estimation process may be simplified further by treating the period jitter weighting function as if it was a single-

pole, high-pass filter function out to fc/2 only. As noted in Figure 2, the J

weighting function, based on the

PER

4sin2(πfτ) weighting factor in dB, looks like a +20 dB/dec sloped line until it gets close to the half-carrier frequency,

where it starts to curve down. If one instead approximates the weighting function as 4(πfτ) in dB, this yields a

straight +20 dB/dec line. The maximum or intercept at the half-carrier frequency is, therefore,

10log(4(π(fc/2)τ) ) = 10log(π ) = 9.94 dB or approximately 10 dB. Integrating the weighted phase noise in this

fashion, one obtains –57.62 dBc, equivalent to RMS period jitter of 1.849 ps or 0.00186 radians. In this particular

example, the simplified “single pole filter” weighting out to the half carrier frequency resulted in a roughly 9% lower

estimate. This is a less conservative approach, which may still yield a sufficiently accurate estimate depending on

the application.

Since this approach does not directly measure period jitter in the time domain, we cannot make definitive

statements about the peak-to-peak values of the period jitter. However, if other evidence suggests that the edge

jitter or period jitter distribution is Gaussian, we can apply the usual statistical estimates.

5. Summary

This application note has reviewed how RMS period jitter may be estimated from phase noise data. The key points

to keep in mind are as follows:

Period jitter is dominated by high-frequency phase noise.

Channel bandwidth plays a determining role in the apparent jitter observed.

Spurs contribute very little to the period jitter unless they are large and located in the vicinity of the half-carrier

frequency.

Generally speaking, hybrid crystal oscillators, such as the Si5xx series devices, have excellent period jitter due to

their relatively low noise floor.

6. References

Drakhlis, Boris, "Calculate Oscillator Jitter By Using Phase-Noise Analysis,"

Microwaves & RF, Jan. 2001 pp. 82-90, 157.

Underhill, M. J. and P. J. Brown, "Estimation of Total Jitter and Jitter Probability Density Function from the Signal

Spectrum", Proc. 18th EFTF (European Time and Frequency Forum), University of Surrey, Guildford, UK, 5-7 April

2004.

Smith, Kevin G., “Practical Issues Correlating Theoretical Period Jitter vs. T1A Measured Period Jitter”, Austin

Conference on Integrated Systems & Circuits, 2006.

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AN279

APPENDIX A—DERIVATION

The following calculations are based on the zero crossings derivation of Drakhlis (2001).

Let Δt represent the jitter accumulated in one period.

T₀

------

Δt =

(ϕ(t₁) – ϕ(t₂))[seconds]

2π

Where: T = nominal period.

t = first zero crossing.

t = second zero crossing.

T₀

〈Δt²〉 =

〈(ϕ(t₁) – ϕ(t₂))²〉

⎛

⎝

⎞

⎠

------

2π

We can drop the T /2π factor for simplicity and add it back in later.

<Δϕ²> = [<ϕ(t₁)²> – 2 <ϕ(t₁)> x <ϕ(t₂)> + <ϕ(t₂)²>][radians²]

Since ϕ(t) is stationary:

<ϕ(t₁)²> = <ϕ(t2)²> = <ϕ(t)²>

Per Parseval’s theorem:

∞

<ϕ(t)²> = Sϕ(f)

∫

Further:

<ϕ(t₁) x ϕ(t₂)> = R_ϕ(t₂– t₁)

<ϕ(t₁) x ϕ(t₂)> = R_ϕ(τ)

Where R (τ) is the autocorrelation of Φ(f), and τ = T₀

The phase noise autocorrelation equals the cosine transform of the phase noise.

∞

<ϕ(t₁) x ϕ(t₂)> = Sϕ(f)cos(2πfτ)df

∫

∞

Δϕ²_RMS= 2 Sϕ(f)(1 – cos(2πfτ))df

∫

∞

Δϕ_RMS= 2 Sϕ(f) x 2sin (πfτ)df

∫

∞

Δϕ_RMS= 4 Sϕ(f) x sin (πfτ)df

∫

Rev. 0.1

AN279

∞

Δϕ_RMS= 2x 4 L(f)sin (πfτ)df

∫

Now, convert back to time units:

f_U

∞

T₀

------

2π

T²₀

π²

T²₀

π²

Δt²_RMS

Δϕ_RMS= 2

L(f)sin (πfτ)df = 2

L(f)sin (πfτ)df

⎛

⎝

⎞

⎠

-----

∫

f_L

where f and f are the practical lower and upper frequency integration limits.

_PERRMS = Δt²RMS

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AN279

APPENDIX B—CALCULATING WITH SPURS INCLUDED

Spurs that are small and few in number and do not contribute significantly to phase jitter may typically be omitted

from period jitter calculations. This is especially true for low offset frequency spurs. Each significant spur

contribution should be accounted for separately as for the weighted i spur below:

Δϕ_{RMS_i}= 8L(f_i) x sin (πf_iτ)

T₀

------

2π

2T₀

π²

Δτ²_{RMS_i}

8L(f_i)sin (πf_iτ) =

L(f_i)sin2(πf_iτ)

⎛

⎝

⎞

⎠

------------

Each spur's contributions are then added in sum square fashion as follows:

Δt_R²_{MS_TOTAL}= Δt²_{RMS_NOISE}+ Δt²_{RMS_1}+ … + Δt_R²_{MS_i}+ … + Δt_R²_{MS_N}

or:

Δt²_{RMS_TOTAL}= Δt_R²_{MS_NOISE}

Δt²_{RMS_i}

∑

i = 1

J_{PER_TOTAL}(rms) = Δt_R²_{MS_TOTAL}

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AN279

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Rev. 0.1

AN279 [SILICON]

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