Enrico Rubiola – Phase – 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 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

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

ω = 2πν

ω ν = 2π 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- product × − (saturated mixer)

vector s(t) r(t) bridge (interferometric) difference − instruments Enrico Rubiola – Phase Noise – 6 introduction – general problems Why a 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 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 – 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 3 – Variances Enrico Rubiola – Phase Noise – 20 Enrico Rubiola – Phase Noise – 20 Enrico Rubiola – Phase Noise – 20 Variances – classical Classical variance CClaslasssicicala lv varaiancriancee 1 1 normalized reading of a counter that y ! 1 1 $"%t& dt y ! " #$"%t& dt measnormalizuresed ( avreadingerages of) ova cerount a timeer t hatT "0 # # 0 # measures (averages) over a time T

N N 2 1 1 2 classical variance, '2 ! )N y ( )N y '2 ! N1(1 % i( N1 j& fcilelas ofsic Nal c vountariancer re,eadings i)! y j)! y N (1 1 % i N 1 j& file of N counter readings ! ! i 1 j 1average of the N readings average of the N readings

ForFor a g aiv egivn penro cpreocsses, thse, tchelas cslasicals vicaalria vnarceianc e For a given process, the classical variance depdependsends of N of N depends of N EveEnv enwo rwsore, sife, th ife tshepe scpectrumtr iums f-1 is o rf -s1t eore pseteeperr, , -1 thetE hecvlaen scslas iwcaorslic svale,ar ivaifar ntcheiance dsipecvee divrgtreumers ges is f or steeper, the classical variance diverges TheThe filte frilt aerss asocsiaocteiatd tedo th teo mtheea meassure ure taktTheeaks esin f iltt hinere t dheasc csdcoocm ciatpomponentoedne ntot the measure takes in the dc component Enrico Rubiola – Phase Noise – 21 Enrico ERubiolanrico Rubiola – Phase – NoisePhase Nois– 21e – 21 Variances – Allan ZerZerZeroo dead-doe adead-d-timetimtimee twtwo- otw-ssampleo-amsampleple vvara riancviaarnianccee e ((AllanAll(aAllann vvara riancviaarniancce)e) e)

2 1 ! "!2 "y 1%y 2% 2 DDefinitionefinitDionefinition y #$ 2 #$ y1& ' y & ' y2 2 2 1 ((LetLet N( LetN = =2, N 2,and = and2, av ander average)age) average)

m%1 m%1 2 2 1 1 2 EEstimatedstimatEsedtimat Allan edAllan Avllanar variance,ianc vare,ianc e, ! "! " ( $ y(%yy & %y 2 y y2$m%1$& % & 2 $ 12 1& filefile of of m m count counterer readings readings 2 mi"1 i"1 file of m counter readings

TheThe f Theiltfilterer as fassociatediltseroc asiatsedoc iatt oto edt hethe t odif diftheferference encdiffere enc e ofof t wtwooof c contiguoustontwoiguous contiguous meas measures measures isur is esa a band- band-passis a band-pass pass (about(about one oc onetav oce)tave) The estimate converges to the variance The esThetimat esetimat conve erconvgeser toges the t ov artheianc vareiance Enrico Rubiola – Phase Noise – 22 Variances – Allan

The Allan variance is related to the spectrum Sy(f) Enrico Rubiola – Phase Noise – 23 Variances – Allan

Convert Sφ and Sy into Allan variance

noise S (f) S (f) S S σ2(τ) mod σ2(τ) type ϕ y ϕ ↔ y y y

3fH h2 2 white 2 b0 2 τ − 3fH τ0h2 3 b h f h = (2π) τ − PM 0 2 2 ν2 (2π)2 0 2πτf 1 H " 2 flicker b 1 h1 2 0.084 h1 τ − 1 − b 1f − h1f h1 = 2 [1.038+3 ln(2πfH τ)] 2 τ − PM − ν0 (2π) n 1 " white b 2 1 1 1 1 2 − b 2f − h0 h0 = 2 h0 τ − h0 τ − FM − ν0 2 4

flicker 1 b 3 27 3 − b 3f − h 1f − h 1 = 2 2 ln(2) h 1 ln(2) h 1 FM − − − ν0 − 20 −

random 2 2 b 4 (2π) (2π) 4 2 − walk b 4f − h 2f − h 2 = 2 h 2τ 0.824 h 2 τ FM − − − ν0 6 − 6 − 1 1 frequency drift y˙ = D D2 τ 2 D2 τ 2 y 2 y 2 y

fH is the high cutoff frequency, needed for the noise power to be finite. Enrico Rubiola – Phase Noise – 24

4 – Properties of phase noise Enrico Rubiola – Phase Noise – 24 Properties of phase noise – synthesis Enrico Rubiola – Phase Noise – 25 Properties of phase noise – synthesis FrFrequencequencyy s syynthesnthesisis Ideal synthesizer Ideal synthesizer - -noi nose-ise-ffreeee - -zer zeroo deldelaayy ttimimee

titmimee ttrranslanslaattioion:n: outouputtput jjiitteerr == i ninputput jit tjeitrt er phasephase ttiimme xxoo==xxi i linearity of the integral and the linearity of the integral and the derivative operators: derivative operators: φo = (n/d)φi => νo = (n/d)vi !o = (n/d)!i => "o = (n/d)"i spectra spectra 2 n S " f # $ S " f # !o " d # !i Enrico Rubiola – Phase Noise – 26 Properties of phase noise – synthesis Carrier collapse

Simple physical meaning, complex mathematics. Easy to understand in the case of sinusoidal phase modulation

random noise => phase fluctuation Enrico Rubiola – Phase Noise – 26 Properties of phase noise – PLL

EnricoE Rubiolanrico Rubiola – PhaseFilter – P Noisehase Nois ing – e27 – 26<=> Phase LockPropertiesed Loop ofPropert phaseies noise of( phasPLL) – PLLe noise – PLL FilFilterteringing < = <>= P> h Phasase Leo Locckekded Lo Loopop (P (LPLL)L) The signal “2” TheThe trackssignal signal “1”“2” “2” trackstracks “1” “1” k! "V (rad # k! "V (rad # The FFT analyzer (not k "rad (s (V # The FFTTheneeded analyzerFFT analy her (notze)er c(notan be used o " ( ( # neededneeded here) her cane) becan used be used ko rad s V to measure S!(f) to measure Sφ(f) to meastrackurs e“1” S!(f) tracks “1”

2 S " f # " 2 # The PLL low-pass S !2" f # %k%kko kH! H" fc #%f % TheThe PLL P LLlow-pass low-pass !2 $$ o ! c " # 2 2 2 2 2 2 filtersfiltferilter sthe st het hephase phase phase SS!!1" ff# 4&& f''%k k H" #" f #% 1 4 f %ko ko! H!c fc %

2 2 2 SS ""ff## 4&4&f kf !k OutputOutputOutput voltage: vol vtage:oltage: the the the PLL P PLLLL vovo $$ ! S " f # 2 22 2 2 2 isis a a ihigh-passs hia gh-high-paspas sfilter sfi lfiterlter S!1 " f # 4&&f '%'k k! H " f #%" # !1 4 f o%ko k!cH c f % Enrico Rubiola – Phase Noise – 28 Properties of phase noise – discriminator Frequency discriminator

A resonator turns a slow frequency fluctuation Δν into a phase fluctuation

ν

ν ν0 Parameters phase φ ν0 resonant frequency Q merit factor resonator ν0 Q

For slow frequency fluctuations, a delay-line t is equivalent to a resonator of merit factor Enrico Rubiola – Phase Noise – 29 Properties of phase noise – Leeson

Enrico Rubiola – The Leeson Effect – 1 The Leeson effect: phase-to-frequTehnec Lye neosoisne Ecoffnevctersion in oscillators Enrico Rubiola Dept. LPMO, FEMTO-ST Institute – Besançon, France D. B. Leeson, A simplee-ma imodell rubio lfora@ feedfemt oback-st.fr oscillator or enrico noise,@rubi oProc.la.org IEEE 54(2):329 (Feb 1966) D. B. Leeson, A simple model for feed back oscillator noise, Proc. IEEE 54(2):329 (Feb 1966)

out 2 '0 1 S " f # $ 1& S " f # ! 2Q 2 ) resonator % " # f ( oscillator ampli noise noise oscillator noise

S f Leeson !" # effect

noise of electronic circuits

fL = !0/2Q f

E. Rubiola, The Leesonw efwfect,w.r uTutorialbiola. 2A,org Proc. 2005 FCS-PTTI (tutorials) some free documents on noise (amplitude and phase) and on E. Rubiola,p rTheecis iLeesonon elect refonfect,ics we-book,ill be av (ahttp://arxivilable in thi.org/abs/physics/0502143s non-commercial site or rubiola.org) Enrico Rubiola – Phase Noise – 30 5 – Laboratory practice Enrico Rubiola – Phase Noise – 31 Laboratory practice – Practical limitations of the double-balanced mixer

1 – Power narrow power range: ±5 dB around Pnom = 5-10 dBm r(t) and s(t) should have (about) the same power

2 – due to the mixer internal diodes typical Sφ = –140 dBrad2/Hz at 1 Hz in average-good conditions

3 – Low gain kφ ~ –10 to –14 dBV/rad typical (0.2-0.3 V/rad)

4 – due to the operational amplifier Enrico Rubiola – Phase Noise – 32 Laboratory practice – background noise Typical background noise

RF mixer (5-10) MHz Good operating conditions (10 dBm each input) Low-noise preamplifier (1 nV/√Hz) Enrico Rubiola – Phase Noise – 33 Laboratory practice – background noise The operational amplifier is often misused

Warning: if only one arm of the power supply is disconnected, the LT1028 may delivers a current from the input (I killed a $2k mixer in this way!)

You may duplicate the low-noise amplifier designed at the FEMTO-ST Rubiola, Lardet-Vieudrin, Rev. Scientific Instruments 75(5) pp. 1323-1326, May 2004 Enrico Rubiola – Phase Noise – 34 Laboratory practice – background noise A proper mechanical assembly is vital Enrico Rubiola – Phase Noise – 35 Laboratory practice – useful schemes Enrico Rubiola – Phase Noise – 34 Laboratory practice – useful schemes Two-port device under test (DUT) Enrico Rubiola – Phase Noise – 36 Laboratory practice – useful schemes

Enrico Rubiola – Phase NoisTew –o 35-port device under teLaboratst (oryD pracUTtice) – useful schemes

other configurations are possible Enrico Rubiola – Phase Noise – 37 Laboratory practice – useful schemes A frequency discriminator can be used to measure the phase noise of an oscillator Enrico Rubiola – Phase Noise – 26 Properties of phase noise – PLL Filtering <=> Phase Locked Loop (PLL)

The signal “2”

Enrico Rubiola – Phase Noise – 26 Properties of phasetracks noise – PLL “1” Enrico Rubiola – PhaseFilter Noise ing– 38 <=> Phase LockedLaboratory Loop practice (PLL) – oscillator measurement k! "V (rad # Phase Locked Loop (PLL) The signalThe “2” FFT analyzer (not " ( ( # needed here) can be used ko rad s V tracks “1” to measure S!(f) k! "V (rad # tracks “1” The FFT analyzer (not " ( ( # needed here) can be used ko rad s V to measure S!(f) tracks “1” S " f # %k k H " f #%2 The PLL low-pass !2 $ o ! c " #" # 2 " 2# 2 2 filters the phase SS! !f f %k k!&H f %' " # Phase:The the PLL PLL low-pass 2 1 $ o4 c f %ko k! H c f % is a low-pass filter 2 2 2 filters the phase S! " f # & ' " # 1 4 f %ko k! H c f %

S " f # 4& f 2 k2 OutputOutput voltage: voltage: the P theLL PLL S "vof # 4& f 2 k2 ! Output voltage: the PLL vo $ $ ! " #" # 2 2 2 2 2 2 is a hiisgh- a ihigh-passpass a hisgh- fipaslters fifilterlter S!S!f f 4& f '&k k H'" f # " # 1 1 4% o !f c %k% o k! H c f %

compare an oscillator under test to a reference low-noise oscillator – or – compare two equal oscillators and divide the spectrum by 2 (take away 3 dB) Enrico Rubiola – Phase Noise – 39 Laboratory practice – oscillator measurement Enrico Rubiola – Phase Noise – 38 Laboratory practice – oscillator measurement Phase Locked Loop (PLL) Enrico Rubiola – Phase Noise – 40 Laboratory practice – oscillator measurement

Enrico Rubiola – Phase Noise – 39 Laboratory practice – oscillator measurement A tight PLL shows many advantages

but you have to correct the spectrum for the PLL transfer function Enrico Rubiola – Phase Noise – 41 Laboratory practice – oscillator measurement

Practical measurement of Sφ(f) with a PLL

1. Set the circuit for proper electrical operation

a. power level

b. lock condition (there is no beat note at the mixer out)

c. zero dc error at the mixer output (a small V can be tolerated)

2. Choose the appropriate time constant 3. Measure the oscillator noise 4. At end, measure the background noise Enrico Rubiola – Phase Noise – 42 Laboratory practice – oscillator measurement Enrico Rubiola – PhasWe aNoisrne i–n 41g: a PLL may notLaborat be oryw prachtaicet – iost csillateor emeasmurements Enrico Rubiola – Phase Noise – 43 Laboratory practice – oscillator measurement PLL – two frequencies

The output frequency of the two oscillators is not the same. A synthesizer (or two synth.) is necessary to match the frequencies

At low Fourier frequencies, the synthesizer noise is lower than the oscillator noise

At higher Fourier frequencies, the white and flicker of phase of the synthesizer may dominate Enrico Rubiola – Phase Noise – 44 Laboratory practice – oscillator measurement PLL – low noise microwave oscillators With low-noise microwave oscillators (like whispering gallery) the noise of a microwave synthesizer at the oscillator output can not be tolerated.

Due to the lower carrier frequency, the noise of a VHF synthesizer is lower than the noise of a microwave synthesizer. This scheme is useful • with narrow tuning-range oscillator, which can not work at the same freq. • to prevent due to microwave leakage Enrico Rubiola – Phase Noise – 45 Laboratory practice – oscillator measurement Designing your own instrument is simple

Standard commercial parts: • double balanced mixer • low-noise op-amp • standard low-noise dc components in the feedback path • commercial FFT analyzer

Afterwards, you will appreciate more the commercial instruments: – assembly – instruction manual – computer interface and software Enrico Rubiola – Phase Noise – 46 6 – Calibration Enrico Rubiola – Phase Noise – 47 Calibration – general Calibration – general procedure 1 – adjust for proper operation: driving power and quadrature

2 – measure the mixer gain kφ (volts/rad) —> next

3 – measure the residual noise of the instrument Enrico Rubiola – Phase Noise – 48 Calibration – general Calibration – general procedure 4 – measure the rejection of the oscillator noise

Make sure that the power and the quadrature are the same during all the calibration process Enrico Rubiola – Phase Noise – 49 Calibration – measurement of kφ Calibration – measurement of kφ (phase mod.)

Vm

The reference signal can be a tone: tone: detect with the FFT, with a dual-channel FFT, or with a lock-in (pseudo-)random white noise white noise

Some FFTs have a white noise output Dual-channel FFTs calculate the transfer function |H(f)|2=SVm/SVd Enrico Rubiola – Phase Noise – 50 Calibration – measurement of kφ

Calibration – measurement of kφ (rf signal) Enrico Rubiola – Phase Noise – 51 Calibration – measurement of kφ

Calibration – measurement of kφ (rf noise)

A reference rf noise is injected in the DUT path through a directional coupler Enrico Rubiola – Phase Noise – 52

7 – Bridge (interferometric) measurements EnricoEnrico Rubiola Rubiola – – Phase Phase Noise Noise – – 53 52 InterferometBridgeer – – Wheatstone Wheatstone bridge Wheatstone bridge Enrico Rubiola – Phase Noise – 54 Bridge – Wheatstone Wheatstone bridge – ac version

equilibrium: Vd = 0 –> carrier suppression

static error δZ1 –> some residual carrier real δZ1 => in-phase residual carrier Vre cos(ω0t) imaginary δZ1 => quadrature residual carrier Vim sin(ω0t)

fluctuating error δZ1 => noise sidebands real δZ1 => AM noise nc(t) cos(ω0t) imaginary δZ1 => PM noise –ns(t) sin(ω0t) EnricoEnrico Rubiola Rubiola – – Phase Phase Noise Noise – – 55 54 InterferometBridgeer – – Wheatstone Wheatstone bridge Wheatstone bridge – ac version Enrico Rubiola – Phase Noise – 56 Bridge – scheme Enrico Rubiola – Phase Noise – 55 Interferometer – scheme Bridge (interferometric) phase-noise and amplitude-noise measurement EnricEnricoo Rubiola Rubiola – – P hasPhasee Nois Noisee – – 56 57 InterferometerBridge – sync –hronous synchronous detec tdetectionion Synchronous detection Enrico Rubiola – Phase Noise – 58 Bridge – synchronous detection Enrico Rubiola – Phase Noise – 57 Interferometer – synchronous detection Synchronous in-phase and quadrature detection Synchronous in-phase and quadrature detection Enrico Rubiola – Phase Noise – Bridge – background noise Enrico Rubiola – Phase Noise –59 58 Interferometer – White noise floor Enrico Rubiola – Phase Noise – 59 Interferometer – noise floor Enrico Rubiola – Phase Noise – 60 Bridge – background noise

White noise floor – example White noise floor – example Enrico Rubiola – Phase Noise – 61 Bridge – summary What really matters (1) Enrico Rubiola – Phase Noise – 62 Bridge – summary What really matters (2) Enrico Rubiola – Phase Noise – 63 Bridge – commercial A bridge (interferometric) instrument can be built around a commercial instrument

You will appreciate the computer interface and the software ready for use Enrico Rubiola – Phase Noise – 64 8 – Advanced Techniques EnricEnricoo Rubiola Rubiola – –P hasPhasee Nois Noisee – – 64 65 AdvAdvancedanced – f–lic flickerker reduc reductiontion Low-flicker scheme Enrico Rubiola – Phase Noise – Advanced – flicker reduction Enrico Rubiola – Phase Noise – 6566 Advanced – flicker reduction Interpolation is necessary Enrico Rubiola – Phase Noise – 67 Advanced – correlation

Correlation can be used to reject the mixer noise Enrico Rubiola – Phase Noise – 68 Advanced – correlation

Correlation – how it works Enrico Rubiola – Phase Noise – 69 Advanced – matrix Flicker reduction, correlation, and closed-loop carrier suppression can be combined

channel b (optional) I v1 readout w1 rf virtual gnd RF I−Q G B null Re & Im detect Q v2 matrix matrix w2 g ~ 40dB LO inner interferometer pump FFT CP1 CP2 x( t ) CP4 R0 analyz. DUT channel a I v1 readout w1 CP3

Ω Δ’ RF I−Q G B γ detect Q v2 matrix matrix w2 =50

0 10−20dB g ~ 40dB LO R R coupl. 0 arbitrary phase pump G: Gram Schmidt ortho atten normalization B: frame rotation manual carr. suppr. automatic carrier atten ’ γ’ suppression control I−Q detector/modulator

power splitter var. att. & phase atten diagonaliz. I RF RF I u1 z1 Q arbitrary phase pump LO I−Q dual D atten u z modul Q 2 integr 2 matrix LO −90° 0°

E. Rubiola, V. Giordano, Rev. Scientific Instruments 73(6) pp.2445-2457, June 2002 Enrico Rubiola – Phase Noise – 70 Advanced – comparison Comparison of the background noise

2 Sϕ (f) dBrad /Hz real−time −140 correl. & avg. mixer, interferometer −150 saturated mixer interferometer nested interferometer double interferometer −160 correl. sat. mix. saturated mixer

−170 residual flicker, by−step interferometer

residual flicker, fixed interferometer interferometer −180 residual flicker, fixed interferometer residual flicker, fixed interferometer, ± correl. saturated mixer −190 double interf.

−200 measured floor, m=32k

45° detection −210 Fourier frequency, Hz −220 1 10 102 103 104 105 Enrico Rubiola – Phase Noise – 71 9 – References STANDARDS J. R. Vig, IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology--Random Instabilities, IEEE Standard 1139-1999

ARTICLES J. Rutman, Characterization of Phase and Frequency Instabilities in Precision Frequency Sources: Fifteen Years of Progress, Proc. IEEE vol.66 no.9 pp.1048-1075, Sept. 1978. Recommanded E. Rubiola, V. Giordano, Advanced Interferometric Phase and Amplitude Noise Measurements, Rev. of Scientific Instruments vol.73 no.6 pp.2445-2457, June 2002. Interferometers, low-flicker methods, correlation, coordinate transformation, calibration strategies, advanced experimental techniques BOOKS Chronos, Frequency Measurement and Control, Chapman and Hall, London 1994. Good and simple reference, although dated W. P. Robins, Phase Noise in Signal Sources, Peter Peregrinus,1984. Specific on phase noise, but dated. Unusual notation, sometimes difficult to read. Oran E. Brigham, The Fast Fourier Transform and its Applications, Prentice-Hall 1988. A must on the subject, most PM noise measurements make use of the FFT W. D. Davenport, Jr., W. L."Root, An Introduction to Random Signals and Noise, McGraw Hill 1958. Reprinted by the IEEE Press, 1987. One of the best references on electrical noise in general and on its mathematical properties. E. Rubiola, The Leeson effect (e-book, 117 pages, 50 figures) arxiv.org, document arXiv:physics/0502143 E. Rubiola, Phase Noise Metrology, book in preparation

ACKNOWLEDGEMENTS I wish to express my gratitude to the colleagues of the FEMTO-ST (formerly LPMO), Besancon, France, firsts of which Vincent Giordano and Jacques Groslambert for a long lasting collaboration that helped me to develop these ideas and to put them in the present form.