[PPT] - The Magic of Correlation Measurements 29th Ann. Symp & Mini PowerPoint Presentation

SLIDE 1

home page http://rubiola.org

The Magic of Correlation Measurements

Statistics
Spectral measure and estimation
Theory of the cross spectrum
Applications

Enrico Rubiola FEMTO-ST Institute, Besancon, France

Contents MTT-S & AP-S

29th Ann. Symp & Mini Show Hanover Manor, NJ, Oct. 2, 2014

SLIDE 2

Correlation measurements

2 single-channel correlation frequency Sφ(f) 1/√m a(t), b(t) –> instrument noise c(t) –> DUT noise Two separate instruments measure the same DUT. Only the DUT noise is common noise measurements DUT noise,  normal use a, b c instrument noise  DUT noise background,  ideal case a, b c = instrument noise  no DUT background,  real case a, b c ≠ c is the correlated instrument noise Zero DUT noise

Σ

x = a + c c(t) dual-channel FFT analyzer y = b + c a(t)

instr. noise

Σ

b(t) input signal instrument A instrument B DUT

instr. noise

SLIDE 3

Statistics

3

Boring but necessary exercises

SLIDE 4

A random process x(t) is defined through a random experiment e that

associates a function xe(t) with each outcome e.

The set of all the possible xe(t) is called ensemble
The function xe(t) is called realization or sample function.
The ensemble average is called mathematical expectation
A random process is said stationary if its statistical properties are

independent of time.

Often we restrict the attention to some statistical properties.
In physics, this is the concept of repeatability.
A random process x(t) said ergodic if a realization observed in time

has the statistical properties of the ensemble.

Ergodicity makes sense only for stationary processes.
Often we restrict the attention to some statistical properties.
In physics, this is the concept of reproducibility.

Vocabulary of statistics

4

E{ }

Example: thermal noise of a resistor of value R

The experiment e is the random choice of a resistor e
The realization xe(t) is the noise waveform measured across the resistor e
We always measure <x2>=4kTRB, so the process is stationary
After measuring many resistors, we conclude that <x2>=4kTRB always
holds. The process is ergodic.

SLIDE 5

A theorem states that

there is no a-priori relation between PDF1 and spectral measure

For example, white noise can originate from 

Poisson process (emission of a particle at random time)
Random telegraph (random switch between two level)
Thermal noise (Gaussian)

A relevant property of random noise

5 (1) PDF = Probability Density Function

SLIDE 6

Why Gaussian White Noise?

Whenever randomness occurs at microscopic level,

noise tends to be Gaussian (central-limit theorem)

Most environmental effects are not “noise” in strict

sense (often, they are more disturbing than noise)

Colored noise types (1/ƒ, 1/ƒ2, etc) can be

whitened, analyzed, and un-whitened

Of course, GW noise is easy to understand

6

SLIDE 7

Properties of Gaussian White noise with zero mean

7

x(t) <=> X(ıf) = X’(ıf)+ ıX”(ıf)

2N degrees of freedom X' f1 f2 X" statistically independent f0 fN–1/2 statistically independent statistically independent

1. x(t) <=> X(ıf) are Gaussian
2. X(ıf1) and X(ıf2) , f1 ≠ f2
1. are statistically independent,
2. var{X(ıf1)} = var{X(ıf2)}
3. real and imaginary part:
1. X’ and X” are statistically

independent

2. var{X’} = var{X”} = var{X}/2
4. Y = X1 + X2
1. Y is Gaussian
2. var{Y} = var{X1} + var{X2}
5. Y = X1 × X2
1. is Gaussian
2. var{Y} = var{X1} var{X2}

SLIDE 8

Properties of parametric noise

1. Pair x(t) <=> X(ıf)
1. there is no a-priori relation between the

distribution of x(t) and X(ıf) (theorem)

2. Central limit theorem: x(t) and X(ıf) end

up to be Gaussian

2. X(ıf1) and X(ıf2)
1. generally, statistically independent
2. var{X(ıf1)} ≠ var{X(ıf2)} in general
3. Real and imaginary part, same frequency
1. X’ and X” can be correlated
2. var{X’} ≠ var{X”} ≠ var{X}/2
4. Y = X1 + X2, zero-mean independent

Gaussian r.v.  var{Y} = var{X1} + var{X2}

5. If X1 and X2 are zero-mean independent

Gaussian r.v.

1. Y = X1 × X2 is zero-mean Gaussian
2. var{Y} = var{X1} var{X2}

8

x(t) <=> X(ıf) = X’(ıf)+ ıX”(ıf)

The process has N … 2N degrees

f freedom, depending on

correlation between X’ and X” X' f1 f2 X" can be correlated f0 fN–1/2 statistically independent statistically independent

SLIDE 9

Rayleigh  x = √(x12+x22)

Children

f the Gaussian distribution

9

Bessel K0   x = x1 x2 Chi-square  χ2 = ∑i xi2

SLIDE 10

Spectral measure1 and estimation

10

(1) Engineers call it Power Spectral Density (PSD)

SLIDE 11

The Spectral Measure

11 Autocovariance Improperly referred to as the correlation and denoted with Rxx(τ) for stationary random process x(t) For ergodic process, interchange ensemble and time average process x(t) –> realization x(t) Spectral measure (two-sided) autocorrelation function Rxx(τ) = 1 σ2 E n [x(t) − µ][x(t − τ) − µ]

Wiener Khinchin theorem

for stationary ergodic processes In experiments we use the single-sided PSD

SI(f) = 2SII(ω/2π) , f > 0 S(ω) = lim

T →∞

1 T XT (ω) X∗

T (ω) = lim T →∞

1 T |XT (ω)|2

µ = E

x

S(ω) = F

C(τ)

= Z ∞

−∞

C(τ) e−iωτdτ C(τ) = E

[x(t) − µ][x(t − τ) − µ]∗

C(τ) = lim

T →∞

Z T/2

−T/2

[x(t) − µ][x(t − τ) − µ]∗ dt

Fourier transform F

ξ

= Z ∞ −∞ ξ(t) e−iωtdt

SLIDE 12

1. The sum of Gaussian distributed random variables has Gaussian

PDF

2. The central limit theorem states that

For large m, the PDF of the the sum of m statistically   independent processes tends to a Gaussian distribution  Let X = X1+X2+…+Xm be the sum of m processes of mean µ1, µ2, … µm and variance σ12, σ22, … σm2. The process X has Gaussian PDF  expectation E{X} = µ1+µ2+…+µm, and variance σ2 = σ12+σ22+…+σm2

3. Similarly, the average <X>m = (X1+X2+…+Xm)/m has

Gaussian PDF, E{X} = (µ1+µ2+…+µm)/m, and σ2 = (σ12+σ22+…+σm2)/m

4. Since white noise and flicker noise arise from the sum of a large

number of small-scale phenomena, they are Gaussian distributed

Sum of random variables

12 PDF = Probability Density Function

SLIDE 13

Product of independent zero-mean Gaussian-distributed random variables

13 x1 and x2 are normal distributed with zero mean and variance σ12, σ22 x has Bessel K0 distribution  with variance σ = σ12 σ22

f(x) = 1 πσ K0 ✓ −|x| σ ◆ E{f(x)} = 0 E{|f(x) − E{f(x)}|2} = σ2

x = x1 x2

Thanks to the central limit theorem, the average <X>m = (X1+X2+…+Xm)/m

f m products has
Gaussian PDF,
average E{X} = 0
variance V{X} = σ2

quick

SLIDE 14

Spectral Measure Sxx(ƒ)

(Power Spectral Density)

14 Normalization: in 1 Hz bandwidth var{X}= 1, and var{X’}= var{X”}= 1/2 Spectrum white, Gaussian,   avg = 0, var = 1/2 X is white Gaussian noise Take one frequency, S(f) –> S. Same applies to all frequencies white, χ2, with 2m degrees of freedom  avg = 1, var = 1/m the Sxx track on the FFT-SA shrinks as 1/m1/2 dev avg =

1

m

hSxxim = 1

T hXX⇤im

= 1

T h(X0 + iX00) ⇥ (X0 iX00)im

= 1

T

⌦ (X0)2 + (X00)2↵

m

SLIDE 15

Estimation of |Sxx(ƒ)|

15 Running the measurement, m increases and Sxx shrinks => better confidence level 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=1 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=2 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=4 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=8 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=16 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=32 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=64 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=128 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=256 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=512 frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| m=1024 frequency 50 100 150 200 0.001 0.01 0.1 1 10 File spectraseq1110240316absSxx Sourcexsp.mn E.Rubiola, mar 2010 frequency

SLIDE 16

Cross Spectrum Theory

16

Getting close to the real game

Σ

x = a + c c(t) dual-channel FFT analyzer y = b + c a(t)

instr. noise

Σ

b(t) input signal instrument A instrument B DUT

instr. noise

SLIDE 17

Syx with correlated term (1)

17 Cross-spectrum Expand using A, B = instrument background C = DUT noise channel 1 X = A + C channel 2 Y = B + C A, B, C are independent Gaussian noises  Re{ } and Im{ } are independent Gaussian noises Split Syx into three sets Normalization: in 1 Hz bandwidth var{A} = var{B} = 1, var{C}=κ2  var{A’} = var{A”} = var{B’} = var{B”} = 1/2, and var{C’} = var{C”} = κ2/2

Syx⇥m = Syx⇥m

instr + Syx⇥m
mixed + Syx⇥m
DUT

background

nly

background and DUT noise DUT noise

nly

... and work it out !!! hSyxim =

1 T hY X⇤im

=

1 T h(Y 0 + iY 00) ⇥ (X0 iX00)im X = (A0 + iA00) + (C0 + iC00) and Y = (B0 + iB00) + (C0 + iC00)

SLIDE 18 Real Imaginary

Syx with correlated term κ≠0 (2)

18 Gaussian,   avg = 0, var = 1/2m Gaussian,   avg = 0, var = κ2/2m white, χ2 2m deg. of freedom  avg = κ2, var = κ4/m A, B, C are independent Gaussian noises  Re{ } and Im{ } are independent Gaussian noises Bessel K0,   avg=0, var=κ2/4 Gaussian,   avg = 0, var = κ2/2m white, χ2, 2 DF   avg = κ2, var = κ4 Gaussian,   avg = 0, var = 1/2m Gaussian,   avg = 0, var = κ2/2m Bessel K0,   avg = 0, var = 1/4 Bessel K0,   avg = 0, var = κ2/4 Gaussian,   avg = 0, var = κ2/2m Normalization: in 1 Hz bandwidth var{A} = var{B} = 1, var{C}=κ2  var{A’} = var{A”} = var{B’} = var{B”} = 1/2, and var{C’} = var{C”} = κ2/2 Gaussian, avg = 0, var = (1+2κ2)/2m Gaussian, avg = 0, var = (1+2κ2)/2m var=1/2 var= κ2/2 var=1/2 var= κ2/2 Bessel K0,   avg=0, var=1/4 Note: DF < 2m See vol.XVI p.56 Set A Set C Set B var= κ2/2 var=1/2 All the DUT signal goes in Re{Syx}, Im{Syx} contains only noise <

hSyxim

= 1 T

hB0A0 + B00A00im + hB0C0 + B00C00im + hA0C0 + A00C00im +

⌦ (C0)2 + (C00)2↵ m =

hSyxim

= 1 T {hB00A0 + B0A00im + hB00C0 B0C00im + hA0C00 A00C0im}

SLIDE 19

Estimator Ŝ = |<Syx>m|

19 | hSyxim | = 1 T q [< {hY X∗im}]2 + [= {hY X∗im}]2 = 1 T q [hA im + h ˜ C im]2 + [hBim]2 . κ → 0 Rayleigh distribution hZ im = q [hA im]2 + [hBim]2 . E{hZ im} = r π 4m = 0.886 pm V{hZ im} = 1 m ⇣ 1 π 4 ⌘ = 0.215 m dev{| hSyxim |} E{| hSyxim |} = r 4 π 1 = 0.523 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gauss sigma^2=1/2 Rayleigh sigma^2=1/2 Probability density f(x) figure: xspGaussRayleighpdf sourcealmostallplots

E. Rubiola, nov 2009

Normalization: in 1 Hz bandwidth var{A} = var{B} = 1, var{C}=κ2  var{A’} = var{A”} = var{B’} = var{B”} = 1/2, and var{C’} = var{C”} = κ2/2

The instrument default

background Im DUT background Re

SLIDE 20

Estimator Ŝ = Re{<Syx>m}

20 0 dB SNR requires that m=1/2κ4. Example κ=0.1 (DUT noise 20 dB lower than single-channel background) averaging on 5x103 spectra is necessary to get SNR = 0 dB.

⌅Z ⇧m = ⌅A ⇧m + ⌅ ˜ C ⇧m

E {⌅Z ⇧m} = κ2 V {⌅Z ⇧m} = 1 + 2κ2 + 2κ4 2m dev {⌅Z ⇧m} =

1 + 2κ2 + 2κ4

2m 1 + κ2 ⌃ 2m dev {⌅Z ⇧m} E {⌅Z ⇧m} = ⌃ 1 + 2κ2 + 2κ4 κ2 ⌃ 2m 1 + κ2 κ2 ⌃ 2m negative values f(x) x κ2 PN PP Normalization: in 1 Hz bandwidth var{A} = var{B} = 1, var{C}=κ2  var{A’} = var{A”} = var{B’} = var{B”} = 1/2, and var{C’} = var{C”} = κ2/2 PN = P{x < 0} = 1 2erfc ✓ κ2 √ 2 σ ◆

Best (unbiased) estimator

background, Re DUT

SLIDE 21

Ergodicity

21 Ergodicity allows to interchange time statistics and ensemble statistics, thus the running index i of the sequence and the frequency f. The average and the deviation calculated on the frequency axis are the same as the average and the deviation of the sequence of spectra. 80 2 y −10 −20 −30 4 dB 3 100 x 60 1 40 20 File: xsp-ergodicity-3d frequency realization |<Syx(f)>32|, dB

Let’s collect a sequence of spectra

SLIDE 22

Example: Measurement of |Syx|

22 20 40 60 80 100 120 140 160 180 200 0.001 0.01 0.1 1 10 file mce!syx!32 E.Rubiola, apr 2008 Sxx Syx m=32 μ = √(π/4m) 5 log(m) – 0.52 dB μ – √[(1-π/4)/m] μ – 3.21 dB μ + √[(1-π/4)/m] μ + 1.83 dB frequency | S y x | C = 0 m, 20 ... 210 frequency |<Syx>m|, dB C ≠ 0 frequency |<Syx>m|, dB m, 20 ... 210

SLIDE 23 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=1 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=2 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=4 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=8 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=16 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=32 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=64 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=128 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=256 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=512 g=0.32 |Scc| frequency 50 100 150 200 0.001 0.01 0.1 1 10 |Sxx| |Syx| m=1024 g=0.32 |Scc| frequency 1 10 100 1000 0.01 0.1 1 a v e r a g e d e v i a t i

n

|Syx| m file spectra!seq!11!1024!0316!absSyx E.Rubiola, apr 2008

Measurement (C≠0), |Syx|

23 Running the measurement, m increases Sxx shrinks => better confidence level Syx decreases => higher single-channel noise rejection

SLIDE 24

Applications

24

The real fun starts here

SLIDE 25

Applications

25

Radio-astronomy (Hanbury-Brown, 1952)
Early implementations
Radiometry (Allred, 1962)
Noise calibration (Spietz, 2003)
Frequency noise (Vessot 1964)
Phase noise (Walls 1976)
Dual delay line system (Lance, 1982)
Phase noise (Rubiola 2000 & 2002)
Effect of amplitude noise (Rubiola, 2007)
Frequency stability of a resonator (Rubiola)
Dual-mixer time-domain instrument (Allan 1975, Stein 1983)
Amplitude noise & laser RIN (Rubiola 2006)
Noise of a power detector (Grop & Rubiola, in progress)
Noise in chemical batteries (Walls 195)
Semiconductors (Sampietro RSI 1999)
Electromigration in thin films (Stoll 1989)
Fundamental definition of temperature
Hanbury Brown - Twiss effect (Hanbury-Brown & Twiss 1956, Glattli 2004)

SLIDE 26

Radio-astronomy

26

R. Hanbury Brown & al., Nature 170(4338) p.1061-1063, 20 Dec 1952 
R. Hanbury Brown, R. Q. Twiss, Phyl. Mag. ser.7 no.366 p.663-682

Measurement of the apparent angular size of stellar radio sources Jodrell Bank, Manchester

The radio link breaks the

hypothesis of symmetry of the two channels, introducing a phase θ

The cross spectrum is complex
The the antenna directivity

results from the phase relationships

The phase of the cross spectrum

indicates the direction of the radio source 500 m2 500 m2 f = 125 MHz B = 200 kHz wave planes Cassiopeia A (or Cygnus A) radio source a few km

X(ıf) X(ıf) Y(ıf) eıθ

Cassiopeia A (Harvard) Cygnus A (Harvard)

DUT

R. Hanbury Brown & al., Nature 170(4338)

SLIDE 27

Thermal noise compensation

27 DUT g g k T0 B k T0 B resistive terminations CP2 interferometer isolation isolation Correlation-and-averaging  rejects the thermal noise E . R u b i

l

a , V . G i

r

d a n

,

R S I 7 3 ( 6 ) , J u n 2 2

SLIDE 28

Radiometry & Johnson thermometry

28

C. M. Allred, A precision noise spectral density comparator, J. Res. NBS 66C no.4 p.323-330, Oct-Dec 1962

correlation and anti-correlation noise comparator

0º 0º 0º 180º T2 A B X = A + B Y = A – B T1 Syx = k (T2 – T1) / 2

C. M. Allred, J. Res. NBS 66C no.4 p.

323-330, Oct-Dec 1962

SLIDE 29

Re-definition of the Kelvin?

29 shot noise thermal noise

S = kT S = 2qIavgR

high accuracy of Iavg with a dc instrument Poisson process μ = σ2 Thermal noise N = kT DC voltmeter Allred noise comparator Josephson effect VDC = hν / 2e null Boltzmann constant Planck constant Electron charge Second (Cesium)

Property of the Poisson process

µ = σ2

SLIDE 30

Noise calibration

30

L. Spietz & al., Primary electronic thermometry

using the shot noise of a tunnel junction, Science 300(20) p. 1929-1932, jun 2003 shot noise thermal noise

S = kT S = 2qIavgR

high accuracy of Iavg with a dc instrument Compare shot and thermal noise with a noise bridge This idea could turn into a re- definition of the temperature

Fig. 1. Theoretical plot of current spectral den-

sity of a tunnel junction (Eq. 3) as a function of dc bias voltage. The diagonal dashed lines indi- cate the shot noise limit, and the horizontal dashed line indicates the Johnson noise limit. The voltage span of the intersection of these limits is 4kBT/e and is indicated by vertical dashed lines. The bottom inset depicts the oc- cupancies of the states in the electrodes in the equilibrium case, and the top inset depicts the

ut-of-equilibrium case where eV

kBT. In a tunnel junction, theory predicts the amount of shot and thermal noise

L. Spietz & al., Science 300(20) p. 1929-1932, jun 2003

SLIDE 31

Early implementations

31 Spectral analysis at the single frequency f0, in the bandwidth B Need a filter pair for each Fourier frequency X–Y X+Y P = X2–2XY+Y2 P = X2+2XY+Y2 ∆P = 4XY thermocouple V ~ 4XY Analog multiplier Analog correlator 1940-1950 technology f0, B f0, B X'(f0)cos(2πf0t) – X"(f0)sin(2πf0t) Y'(f0)cos(2πf0t) – Y"(f0)sin(2πf0t) (Y'X' + Y"X")/2 <Y'X' + Y"X"> / 2 x(t) y(t) Rice representation of noise

SLIDE 32

Measurement of the frequency noise of a H-maser

32

R. F

. C. Vessot, Proc. Nasa Symp. on Short Term Frequency Stability p.111-118, Greenbelt, MD, 23-24 Nov 1964 H maser correlator H maser common synthesizer

R. F

. C. Vessot, Proc. Nasa Symp. on Short Term Frequency Stability p.111-118, Greenbelt, MD, 23-24 Nov 1964

SLIDE 33

Phase noise measurement

33 F .L. Walls & al, Proc. 30th FCS pp.269-274, 1976  More popular after W. Walls, Proc. 46th FCS pp.257-261, 1992 (relatively) large correlation bandwidth provides low noise floor in a reasonable time F .L. Walls & al, Proc. 30th FCS pp.269-274, 1976 F .L. Walls & al, Proc. 30th FCS pp.269-274, 1976

SLIDE 34 dual integr matrix D R0=50 Ω matrix B matrix G v2 w1 w2 matrix B matrix G w1 w2 FFT analyz. atten atten x t ( ) Q I I−Q modul ’ γ’ atten Q I I−Q detect RF LO Q I I−Q detect RF LO g ~ 40dB g ~ 40dB v1 v2 v1 u1 u2 z2 z1 atten DUT γ ∆’ R R 10−20dB coupl. power splitter pump channel a channel b (optional) rf virtual gnd null Re & Im RF suppression control manual carr. suppr. pump LO diagonaliz. readout readout arbitrary phase

var. att. & phase

automatic carrier arbitrary phase pump I−Q detector/modulator G: Gram Schmidt ortho normalization B: frame rotation inner interferometer CP1 CP2 CP3 CP4 −90° 0° I Q RF LO

Phase noise measurement

34

E. Rubiola, V. Giordano, Rev. Sci. Instrum. 71(8) p.3085-3091, aug 2000 
E. Rubiola, V. Giordano, Rev. Sci. Instrum. 73(6) pp.2445-2457, jun 2002

103 102 correl single arm Fourier frequency, Hz ( ) f dBrad /Hz 2 S! ) f dB/Hz "( S rf [ dBm/Hz] N P two rf chann. angle uncal. arm 1a, 1b avg 351 spectra = 13.8 dBm k T B /P 0 = !187.8 dB[rad ]/Hz 2 !160.2 !170.2 !180.2 !190.2 !200.2 [!166.4] [!176.4] [!186.4] 1 [!156.4] 10 [!146.4] 105 102 104 103 correl !( ) f dBrad /Hz 2 S ) f dB/Hz "( S single arm N [ dBm/Hz] rf P 0= 14.1 dBm avg 32k spectra two rf chann. angle uncal. arm 1a, 1b k T B /P = !188.1 dB[rad ]/Hz 2 [!156.4] [!166.4] [!176.4] [!186.4] [!196.4] !170.5 !180.5 !190.5 !200.5 !210.5 Fourier frequency, Hz background noise noise of a by-step attenuator E . R u b i

l

a , V . G i

r

d a n

,

R S I 7 3 ( 6 ) , J u n 2 2 E . R u b i

l

a , V . G i

r

d a n

,

R S I 7 3 ( 6 ) , J u n 2 2

SLIDE 35

Phase noise

35 dc dc DUT REF REF RF RF LO LO y x arm b arm a FFT analyzer dc dc phase lock phase lock device 2−port Σ Σ FFT analyzer dc dc µw µw FFT analyzer device 2−port phase phase dc dc REF DUT REF RF RF LO LO y x arm a arm b phase and ampl. (ref) ∆ ∆ DUT phase and ampl. bridge b bridge a y x LO LO RF RF meter output (noise only) DUT (ref) (ref) RF RF LO LO x y arm a arm b FFT analyzer

SLIDE 36

Effect of amplitude noise

36

E. Rubiola, R. Boudot, IEEE Transact. UFFC 54(5) pp.926-932, may 2007

dc DUT (ref) (ref) RF RF LO LO x y arm a arm b A FFT analyzer dc dc DUT REF REF RF RF LO LO y x arm b arm a C FFT analyzer dc dc phase lock phase lock REF DUT REF RF RF LO LO y x arm a arm b B device 2−port Σ Σ FFT analyzer dc dc µw D FFT analyzer device 2−port phase phase dc (noise only) µw phase and ampl. (ref) ∆ ∆ DUT phase and ampl. bridge b bridge a y x LO LO RF RF meter output AM AM delay delay common AM VOS AM VOS AM VOS pink: noise rejected by correlation and averaging Should set both channels at the sweet point, if exists The delay de-correlates the two inputs, so there is no sweet point The effect of the AM noise is strongly reduced by the RF amplification AM VOS VOS Should set both channels at the sweet point of the RF input, if exists, by

ffsetting the PLL or by biasing the IF

SLIDE 37

Dual-delay-line method

37 A.L. Lance, W.D. Seal, F . Labaar ISA Transact.21 (4) p.37-84, Apr 1982 Original idea:

D. Halford’s NBS notebook

F10 p.19-38, apr 1975    First published: A. L. Lance & al, CPEM Digest, 1978 The delay line converts the frequency noise into phase noise The high loss of the coaxial cable limits the maximum delay

Updated version:

The optical fiber provides long delay with low attenuation (0.2 dB/km or 0.04 dB/μs) A . L . L a n c e , W . D . S e a l , F . L a b a a r I S A T r a n s a c t . 2 1 ( 4 ) p . 3 7

8

4 , A p r 1 9 8 2

SLIDE 38

Optical version of the dual-delay-line method

38 splitter F F T

scillator to be

measured semiconductor laser electro-optic modulator microwave amplifier photodetector fiber delay phase shifters dual-channel FFT analyzer

ptical

isolator coupler DC amplifier microwave isolator mixer Φ Φ The only common part of the setup is the power splitter. Two completely separate systems measure the same oscillator under test

E. Salik, N. Yu, L. Maleki, E. Rubiola, Proc. Ultrasonics-FCS Joint Conf., Montreal, Aug 2004 p.303-306

Volyanskiy & al., JOSAB 25(12) 2140-2150, Dec.2008. Also arXiv:0807.3494v1 [physics.optics] July 2008

E. Salik, N. Yu, L. Maleki, E. Rubiola, Proc. Ultrasonics-

FCS Joint Conf., Montreal, Aug 2004 p.303-306

SLIDE 39

Frequency stability of a resonator

39

Bridge in equilibrium
The amplifier cannot flicker around ω0, which it does not know
The fluctuation of the resonator natural frequency is estimated from phase noise
Q matching prevents the master-oscillator noise from being taken in
Correlation removes the noise of the instruments and the reference resonators

Enrico’s weird brain – however, with the cryogenic sapphire oscillators we can do way better – – + – + ψ(t) ϑ(t) φ(t) x=φ–ϑ y=φ–ψ detector FFT analyzer double Wheatstone bridge noise sideband amplification cos(ω0t) DUT

Now obsolete, 3E–16 stability from cryogenic oscillator

SLIDE 40

Amplitude noise & laser RIN

40

E. Rubiola, The measurement of AM noise, dec

2005 arXiv:physics/0512082v1 [physics.ins-det] monitor source under test dual channel FFT analyzer vb va Pb Pa power meter monitor R0 R0 Pa Pb coupler power meter coupler source under test R R va vb dual channel FFT analyzer power meter microwave

ptical

monitor dc dc power meter coupler coupler source under test Pb Pa R R dual channel FFT analyzer va vb monitor

ptical

−123.1 10 102 103 104 105 Fourier frequency, Hz avg 2100 spectra = −10.2 dBm P Wenzel 501−04623E 100 MHz OCXO (f ) Sα dB/Hz −163.1 −153.1 −143.1 −133.1

In PM noise measurements, one can validate the instrument by

feeding the same signal into the phase detector

In AM noise this is not possible without a lower-noise reference
Provided the crosstalk was measured otherwise, correlation

enables to validate the instrument AM noise of RF/microwave sources Laser RIN AM noise of photonic RF/microwave sources 10 1 10 2 10 3 10 4 10 5 −140 −130 −120 −110 −100 −90 −80 Frequency (Hz) SRIN (dB/Hz) CQF935 RIN 20mA 30mA 40mA 60mA 80mA 100mA Kirill Volyanskiy

SLIDE 41 41

Remove the noise of the source by balancing C–A and C–B
Use a lock-in amplifier to get a sharp null measurement
Channels A and B are independent –> noise is averaged out
Two separate JFET amplifiers are needed in the C channel
JFETs have virtually no bias-current noise
Only the noise of the detector C remains

Basic ideas

B Pc Rc Pa va Ra Pb vb Rb

diff. ampli

dual channel FFT analyzer g(Pc−P a) g(Pc−P b) dual channel FFT analyzer

diff. ampli

power meter source low noise input lock−in amplifier Im Re

ut
sc. out

input lock−in amplifier Im Re

ut
sc. out

Re output to be zero adjust the gain for the AM input vc monitor

adj. gain
adj. gain

JFET input A C In all previous experiments, the amplifier noise was higher than the detector noise

Measurement of the detector noise

Grop & Rubiola

SLIDE 42

Noise in chemical batteries

42

C. K. Boggs, A. D. Doak, F

. L. Walls, Proc. IFCS p.367-373 1995 noise sideband amplification

Do not waste DAC bits for a constant

DC, V = VB2–VB1 has (almost) zero mean

Two separate amplifiers measure the

same quantity V

Correlation rejects the amplifier nose,

and the FFT noise as well

SLIDE 43

Noise in semiconductors

43

M. Sampietro & al, Rev. Sci. Instrum 70(5) p.2520-2525, may 1999
FIG. 2. Schematics of the building blocks of our correlation spectrum ana-

lyzer performing the suppression of the uncorrelated input noises by a digi- tal processing of sampled data.

FIG. 9. Experimental frequency spectrum of the current noise from DUT

resistances of 100 k and 500 M ⇥continuous line⇤ compared with the limits ⇥dashed line⇤ given by the instrument and set by residual correlated noise components.

FIG. 3. Schematics of the active test fixture for current noise measurements.

SLIDE 44

Electro-migration in thin films

44

A. Seeger, H. Stoll, 1/f noise and defects in thin metal films, proc. ICNF p.162-167, Hong Kong 23-26 aug 1999

RF/microwave version: E. Rubiola, V. Giordano, H. Stoll, IEEE Transact. IM 52(1) pp.182-188, feb 2003 Re Im Up Dn v(t)/√2 v(t)/√2 v(t) null fluct – + +45º –45º FFT u(t) d(t) pump bridge error ampli Δ(t) DUT DUT

Random noise: X’ and X” (real and imag part) of a

signal are statistically independent

The detection on two orthogonal axes eliminates

the amplifier noise.  This work with a single amplifier!

The DUT noise is detected

Sud(f) = 1 2

Sα(f) − Sϕ(f)

⇥

SLIDE 45

Hanbury Brown - Twiss effect

45

R. Hanbury Brown, R. Q. Twiss, Correlation between photons in two coherent beams of light, Nature 177 (1956) 27-29

1/2 Source 1/2

in single-photon regime, anti-correlation shows up Also observed at microwave frequencies

C. Glattli & al. (2004), PRL 93(5) 056801, Jul 2004

4.2 K 300 K c) 1 K source a 3dB splitter source b 20 mK 28dB 1-2GHz 50dB 1.6-1.8GHz 0-1MHz x1000 spectrum analyser kT = 2.7×10–25 J at 20 mK, hν = 1.12×10–24 J at 1.7 GHz, kT/hν = –6.1 dB

SLIDE 46

Conclusions

Rejection of the instrument noise
AM noise, RIN, etc. –> validation of the instrument

without a reference low-noise source

Display quantities

<Re{Syx}>m is faster and more accurate <Im{Syx}>m gives the background noise max{<Syx>m,0+} provide easier readout

Applications in many fields of metrology

46

The cross spectrum method is magic Correlated noise sometimes makes magic difficult

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