Enrico Rubiola – Phase Noise – 1 Phase Noise Enrico Rubiola Dept. LPMO, FEMTO­ST Institute – Besançon, France e­mail rubiola@femto­st.fr or [email protected] Summary 1. Introduction 2. Spectra 3. Classical variance and Allan variance 4. Properties of phase noise 5. Laboratory practice 6. Calibration 7. Bridge (Interferometric) measurements 8. Advanced methods 9. References www.rubiola.org you can download this presentation, an e-book on the Leeson effect, and some other documents on noise (amplitude and phase) and on precision electronics from my web page Enrico Rubiola – Phase Noise – 2 1 – Introduction Enrico Rubiola – Phase Noise – 3 introduction – noisy sinusoid RCehappterre1.sBeasnicstations of a sinusoid with nois2e Time Domain Phasor Representation amplitude fluctuation v(t) V α(t) [volts] V 0 0 normalized ampl. fluct. α(t) [adimensional] V0/√2 t ampl. fluct. (V0/√2)α(t) phase fluctuation v(t) ϕ(t) [rad] V0 phase time (fluct.) phase fluctuation x(t) [seconds] ϕ(t) t V0/√2 Figure 1.1: . polar coordinates v(t) = V0 [1 + α(t)] cos [ω0t + ϕ(t)] Cartesianw coordinateshere V0 is the nominalvam(tp)lit=udeV, andcosα thωe notrm+alnized(atm)pcoslitudωe flutctuatnion,(t) sin ω t which is adimensional. The instanta0neous freq0uency isc 0 − s 0 ω0 1 dϕ(t) ν(t) = + (1.3) under low noise approximation 2π 2π Itd holdst that nc(t) ns(t) nc(t) VT0his banoodk deanlssw(itt)h the mVe0asurement of staαb(let)si=gnals of the fandorm (1.2ϕ)(, tw)it=h | | ! main focus on| phase|, !thus frequency and time. This Vin0volves several topics, V0 namely: 1. how to describe instability, 2. basic noise mechanisms, 3. high-sensitivity phase-to-voltage and frequency-to-voltage conversion hardware, for measurements, 4. enhanched-sensitivity counter interfaces, for time-domain measurements, 5. accuracy and calibration, 6. the measurement of tiny and elusive instability phenomena, 7. laboratory practice for comfortable low-noise life. We are mainly concerned with short-term measurements in the frequency do- main. Little place is let to long-term and time domain. Nevertheless, problems are quite similar, and the background provided should make long-term and time domain measurement easy to understand. Stability can only be described in terms of the statistical properties of ϕ(t) and α(t) (or of related quantities), which are random signals. A problem arises Enrico Rubiola – Phase Noise – 4 introduction – noisy sinusoid Noise broadens the spectrum ω = 2πν ω ν = 2π v(t) = V0 [1 + α(t)] cos [ω0t + ϕ(t)] v(t) = V cos ω t + n (t) cos ω t n (t) sin ω t 0 0 c 0 − s 0 Enrico Rubiola – Phase Noise – 6 Introduction – general problems How can we measure a low random signal (noise sidebands) close to a strong dazzling carrier? Enrico Rubiola – Phase Noise – 5 introduction – general problems Basic problem: how can we measure a low random signal (noise sidebands) close to a strong dazzling carrier? solution(s): suppress the carrier and measure the noise convolution s(t) h (t) distorsiometer, (low-pass) ∗ lp audio-frequency instruments time-domain traditional instruments for s(t) r(t T/4) phase-noise measurement product × − (saturated mixer) vector s(t) r(t) bridge (interferometric) difference − instruments Enrico Rubiola – Phase Noise – 6 introduction – general problems Why a spectrum analyzer does not work? 1. too wide IF bandwidth 2. noise and instability of the conversion oscillator (VCO) 3. detects both AM and PM noise 4. insufficient dynamic range Some commercial analyzers provide phase noise measurements, yet limited (at least) by the oscillator stability Enrico Rubiola – Phase Noise – 4 Introduction – ! ! v converter Enrico Rubiola – Phase Noise – 7 introduction – φ –> v converter The SchoThettky-d Sciodhottke douyb-lediode-balan cdouble-ed mixebalancr saturateded amixt boerth saturated inatpu tboths is th inputse most uisse thed ph mosase dt eustecedtor phas e detector signal s(t) = 2R0P0 cos [2πν0t + ϕ(t)] ! reference r(t) = 2R0P0 cos [2πν0t + π/2] filtered ! out product r(t)s(t) = kϕϕ(t) + “2ν0” terms The AM noise is rejected by saturation Saturation also account for the phase-to-voltage gain kφ Enrico Rubiola – Phase Noise – 8 2 – Spectra Enrico Rubiola – Phase Noise – 9 Spectra – definitions Power spectrum density In general, the power spectrum density Sv(f) of a random process v(t) is defined as Sv(f) = E v(t1, t2) F R E statistical expectation · ! " #$ Fourier transform F! ·" (t , t ) autocorrelation function R!v "1 2 In practice, we measure Sv(f) as 2 S (f) = v(t) v F This is possible (Wiener-Khinchin theorem)! " #with! ergodic processes ! ! In many real-life cases, processes are ergodic and stationary Ergodicity: ensemble and time-domain statistics can be interchanged. This is the formalization of the reproducibility of an experiment Stationarity: the statistics is independent of the origin of time. This is the formalization of the repeatability of an experiment Enrico Rubiola – Phase Noise – 10 Spectra – meaning Physical meaning of the power spectrum density S (f ) P = v 0 R0 Sv(f0) power in 1 Hz bandwidth dissipated by R0 R0 Enrico Rubiola – Phase Noise – 11 Spectra – meaning Physical meaning of the power spectrum density The power spectrum density extends the concept of root-mean-square value to the frequency domain Enrico Rubiola – Phase Noise – 12 Spectra – Sφ(f) and ℒ(f) Sφ(f) and ℒ(f) in the presence of (white) noise SSB DSB P0 P0 B B B N N v0 v0 v v0– v0 v0 v +f f +f ϕp R0NB R0NB/2 R0P0 ! ! ! 1 NB 1 NB ϕ = ϕ = ϕ = ϕ = rms 2 p 2P rms 2 p P ! ! 0 ! ! 0 N 3 dB N dBc/Hz = Sϕ = dBrad2/Hz L 2P0 P0 Enrico Rubiola – Phase Noise – 13 Spectra – Sφ(f) and ℒ(f) ℒ(f) (re)defined The first definition of ℒ(f) was ℒ(f) = ( SSB power in 1Hz bandwidth ) / ( carrier power ) The problem with this definition is that it does not divide AM noise from PM noise, which yields to ambiguous results Engineers (manufacturers even more) like ℒ(f) The IEEE Std 1139-1999 redefines ℒ(f) as ℒ(f) = (1/2) × Sφ(f) Enrico Rubiola – Phase Noise – 14 Spectra – useful quantities Useful quantities 1 phase x(t) = ϕ(t) time 2πν0 x(t) is the phase noise converted into time fluctuation physical dimension: time (seconds) fractional 1 frequency y(t) = ϕ˙(t) = x˙(t) 2πν0 fluctuation y(t) is the fractional frequency fluctuation ν-ν0 normalized to the nominal frequency ν0 ν ν0 (dimensionless) y(t) = − ν0 EnricoEnric oRubiola Rubiola – – Phase Phase Noise Noise – – 15 15 Spectra –S pecpowertra –law power-law Power-laNwois aen dpr nocoesises espro inc eosssceillators in osscillators Enrico Rubiola – Phase Noise – 16 Spectra – power law Relationships between Sφ(f) and Sy(f) Enrico Rubiola – Phase Noise – 17 Enrico Rubiola – Phase Noise – 17 Spectra – jitter JJittitteerr The phase fluctuation can be described in terms of a single parTamheet perha,s eie tfherluct uphaseation c jaitnter be or d etismcreib jeitdter in terms of a single parameter, either phase jitter or time jitter TheT hphasee pha snoie nseois me ustmus bet b eint inegrtegatraedted overover thehe bbandwandwididtthh B B o fof th e the ssystysteemm ( w(whihichch m mayay be be dif fdiicfufilcult tot i dtoe nidenttify) ify) & ! % S ' f ( df radians phase jitter rms $ & B time jitter 1 x ! % S ' f ( df seconds phase jitter rms 2"# $ & converted into time 0 B The jitter is useful in digital circuits because the bandwidth B is known The j i t–ter lo wise usefr limiult: tihne diingivetalrs eci prrcuiopatsg abecausetion time thhero ubandwgh the idtsyhst eism known - l ower tlhimis iet:x tchelud ienvers these lo wpr-opagatfrequenicony d tiivmereg ethrnt oughproce sthese ssyst) em t hi s –excl uppudeser lim tihet: ~ l owthe- finrequencyverse swi tdichveringgent spee prd ocesses) - upper limit: the inverse witching speed Victor Reinhardt (invited), A Review of Time Jitter and Digital Systems, Proc. 2005 FCS-PTTI joint meeting Phase Noise Metrology 191 function ϕ(τ). The one-side spectrum is preferred because this is what spectrumRanalyzers display. Complying to the usual terminology, we use the symbol ν for the frequency and f for the Fourier frequency, i.e. the frequency of the detected signal when the sidebands around ν are down converted to baseband. The power-law model is most frequently used for describing phase noise. It assumes that Sϕ(f) is equal to the sum of terms, each of which varies as an integer power of frequency. Thus each term, that corresponds to a noise process, is completely specified by two parameters, namely the exponent and the value at f = 1 Hz. Five power-law processes, listed underneath, are common with electronics. noise type Sϕ(f) 0 white phase b0f 1 flicker phase b 1f − − 2 white frequency b 2f − Enrico Rubiola – Phase Noise – 18 − 3 Spectra – examples flicker frequency b 3f − − 4 random-walk frequency b 4f − − TypAllicathesel phnoiseasteyp nesoareisgenerallye of spresenomet atdtheevoutputices ofaoscillators,nd oscwhileillators two-port devices show white phase and flicker phase noise only. For reference, Fig. 1 reports the typical phase noise of some oscillators and devices. 10 GHz Dielectric Resonator Oscillator -40 10 GHz Sapphire Resonator Oscillator (SRO) -60 10 GHz SRO with noise degeneration -80 100 MHz quartz oscillator 2 -100 -120 microwave amplifier RF amplifier 5 MHz quartz osc. ! -140 S (f) (dBrad /Hz) -160 microwave ferrite isolators -180 -200 0.1 1 10 100 1k 10k 100k Fourier frequency (Hz) Fig. 1. Typical phase noise of some oscillators and devices Phase noise can be measured by means of a phase-to-voltage converter in conjunction with a spectrum analyzer, which can be of the low frequency Enrico Rubiola – Phase Noise – 19