AN279

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 JCC, 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. Shortterm 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 JPER, may be specified as either an RMS or peak-to-peak 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. C lock Period Distribution σ J PER(RMS) = σ T =0 T = TPER 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 rad2/Hz where: Sϕ(f ) L ( f ) ≡ ------------2 RMS phase jitter may be expressed in units of dBc, radians, time, or Unit Intervals (UI). Rev. 0.1 7/06 Copyright © 2006 by Silicon Laboratories AN279 A N279 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 lowerfrequency jitter components (f – 2 < ϕ ( t 1 ) > x < ϕ ( t 2 ) > + < ϕ ( t 2 ) > ] [ radians ] 2 2 2 2 Since ϕ ( t ) is stationary: < ϕ ( t1 ) > = < ϕ ( t 2 ) > = < ϕ ( t ) > 2 2 2 Per Parseval’s theorem: ∞ = 2 ∫ Sϕ(f) 0 Further: < ϕ ( t1 ) x ϕ ( t2 ) > = Rϕ ( t2 – t1 ) < ϕ ( t1 ) x ϕ ( t2 ) > = Rϕ ( τ ) Where R ϕ ( τ ) is the autocorrelation of Φ ( f ) , and τ = T 0 The phase noise autocorrelation equals the cosine transform of the phase noise. ∞ < ϕ ( t1 ) x ϕ ( t2 ) > = ∞ 2 Δ ϕ RMS ∫ S ϕ ( f ) cos ( 2 π f τ ) df 0 = 2 ∫ S ϕ ( f ) ( 1 – cos ( 2 π f τ ) ) df 0 ∞ Δ ϕ RMS = 2 ∫ S ϕ ( f ) x 2 sin 2 ( π f τ ) df 2 0 ∞ 2 Δ ϕ RMS = 4 ∫ S ϕ ( f ) x sin 2 ( π f τ ) df 0 Rev. 0.1 5 A N279 ∞ Δ ϕ RMS = 2x 4 ∫ L ( f ) sin 2 ( π f τ ) df 2 0 Now, convert back to time units: 2 Δ t RMS T0 T0 2 = ⎛ ------⎞ Δ ϕ RMS = 2 ----2 ⎝ 2 π⎠ 2 2∞ π T0 2 ∫ L ( f ) sin ( π f τ ) df = 2 -----2 0 2 fU π ∫ L ( f ) sin fL 2 ( π f τ ) df where fL and fU are the practical lower and upper frequency integration limits. J PER RMS = Δ t RMS 2 6 Rev. 0.1 A N279 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 ith spur below: Δ ϕ RMS _ i = 8 L ( f i ) x sin 2 ( π f i τ ) 2 T0 2 2 T0 2 Δτ RMS _ i = ⎛ ------⎞ 8L ( f i ) sin 2 ( π f i τ ) = ----------- L ( f i ) sin 2 ( π f i τ ) 2 ⎝ 2 π⎠ π 2 Each spur's contributions are then added in sum square fashion as follows: Δ t RMS_TOTAL = Δ t RMS_NOISE + Δ t RMS_ 1 + … + Δ t RMS _ i + … + Δ t RMS_N 2 2 2 2 2 or: N 2 Δ t RMS_TOTAL = 2 Δ t RMS_NOISE + ∑ Δ tRMS _ i i=1 2 J PER_TOTAL (rms) = Δ t RMS_TOTAL 2 Rev. 0.1 7 A N279 CONTACT INFORMATION Silicon Laboratories Inc. 4635 Boston Lane Austin, TX 78735 Tel: 1+(512) 416-8500 Fax: 1+(512) 416-9669 Toll Free: 1+(877) 444-3032 Email: VCXOinfo@silabs.com Internet: www.silabs.com The information in this document is believed to be accurate in all respects at the time of publication but is subject to change without notice. Silicon Laboratories assumes no responsibility for errors and omissions, and disclaims responsibility for any consequences resulting from the use of information included herein. Additionally, Silicon Laboratories assumes no responsibility for the functioning of undescribed features or parameters. Silicon Laboratories reserves the right to make changes without further notice. Silicon Laboratories makes no warranty, representation or guarantee regarding the suitability of its products for any particular purpose, nor does Silicon Laboratories assume any liability arising out of the application or use of any product or circuit, and specifically disclaims any and all liability, including without limitation consequential or incidental damages. Silicon Laboratories products are not designed, intended, or authorized for use in applications intended to support or sustain life, or for any other application in which the failure of the Silicon Laboratories product could create a situation where personal injury or death may occur. Should Buyer purchase or use Silicon Laboratories products for any such unintended or unauthorized application, Buyer shall indemnify and hold Silicon Laboratories harmless against all claims and damages. Silicon Laboratories and Silicon Labs are trademarks of Silicon Laboratories Inc. Other products or brandnames mentioned herein are trademarks or registered trademarks of their respective holders. 8 Rev. 0.1