The magic of correlation in measurements from dc to optics Enrico - - PowerPoint PPT Presentation

SLIDE 1

home page http://rubiola.org

The magic of correlation in measurements from dc to optics

• Short introduction
• Statistics
• Theory
• Applications

Enrico Rubiola

FEMTO-ST Institute, CNRS and UFC, Besancon, France

Contents

SLIDE 2

Correlation measurements

2

single-channel correlation

frequency S(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 = 0 instrument noise no DUT background, real case a, b c ≠ 0 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

Boring exercises before playing a Steinway

3

SLIDE 4

Fourier Statistics

4

SLIDE 5

• 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

5

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 holds
• always. The process is ergodic.

SLIDE 6

Formal definition of the PSD

6 Autocovariance

Improperly referred to as the correlation and denoted with Rxx(τ)

for stationary random process x(t)

For stationary ergodic process, interchange ensemble and time average

process x(t) –> realization x(t)

PSD (two-sided)

In mathematics, called spectral measure

autocorrelation function Rxx(τ) = 1 σ2 E

• [x(t) − µ][x(t − τ) − µ]
• Fourier transform

F

• ξ
• =

∞

−∞

ξ(t) e−ıωtdt 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 C(τ) = E

• [x(t + τ) − µ][x∗(t) − µ]
• S(ω) = F
• C(τ)
• =

∞

−∞

C(τ) e−ıωτdτ C(τ) = lim

T →∞

T/2

−T/2

[x(t + τ) − µ][x∗(t) − µ] dt

µ = E

• x

SLIDE 7

A theorem states that there is no a-priori relation between PDF and spectrum

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

7 PDF = Probability Density Function

SLIDE 8

• 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

8 PDF = Probability Density Function

SLIDE 9

Product of independent zero-mean Gaussian-distributed random variables

9 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

SLIDE 10

• 1. Central limit theorem:

x(t) <=> X(f) are Gaussian distributed

• 2. Energy equipartition (frequency):
• 1. X(f1) and X(f2), f1 ≠ f2, are statistically

independent,

• 2. var{X(f1)} = var{X(f2)}
• 3. Energy equipartition (Re-Im):
• 1. X’ and X” are statistically

independent

• 2. var{X’} = var{X”} = var{X}/2
• 4. Sum: Y = X1 + X2
• 1. Y is Gaussian distributed
• 2. var{Y} = var{X1} + var{X2}
• 5. Product: Y = X1 × X2
• 1. Y has distribution Bessel K0(|y|) / π
• 2. var{Y} = var{X1} × var{X2}

Properties of white Gaussian noise with zero mean

10

x(t) <=> X(f) = X’(f)+ ıX”(f)

a real process has N degrees of freedom X'

f1 f2

X"

statistically independent

f0 fN–1/2

statistically independent statistically independent

SLIDE 11

Properties of flicker noise (Gaussian distributed, with zero mean)

• 1. Central limit theorem: x(t) and X(f) end

up to be Gaussian

• 2. Power distribution (frequency)
• 1. X(f1) and X(f2), f1 ≠ f2, are statistically

independent

• 2. var{X(f2)} < var{X(f1)} for f2 > f1
• 3. Real and imaginary part
• 1. X’ and X” can be correlated
• 2. var{X’} ≠ var{X”} ≠ var{X}/2
• 4. Y = X1 + X2, zero-mean Gaussian r.v.

var{Y} = var{X1} + var{X2}

• 5. If X1 and X2 are zero-mean Gaussian

r.v., the product Y = X1 × X2,

• 1. has distribution Bessel K0(|y|) / π
• 2. has var{Y} = var{X1} var{X2}

11

x(t) <=> X(f) = X’(f)+ ıX”(f)

Central limit theorem: x(t) and X(f) end up to be Gaussian

X'

f1 f2

X"

can be correlated

f0 fN–1/2

statistically independent statistically independent

SLIDE 12

Statistics & finite-duration measurement

12

–T/2 +T/2 1

t f Π(t/T) T sinc(πTf)

1/4T T

File: xsp-truncation-effect

x(t) xT(t) X(f)

product convolution result result

XT(f)

x(t) Π(t/T) X(f) * T sinc(πTf)

• The convolution with sinc( ) scrambles the spectrum, spreading the

power of a single point all around. This introduces correlation

• In the presence of large peaks or sharp roll-off, this is disturbing
• In the measurement of smooth noise, often negligible
• Further consequences in cross-correlation measurements

SLIDE 13

Normal (Gaussian) distribution

13 x is normal distributed with zero mean μ and variance σ2 f(x) = 1 √ 2π σ exp

• −(x − µ)2

2σ2

• E{f(x)} = µ

E{f 2(x)} = µ2 + σ2 E{|f(x) − E{f(x)}|2} = σ2

File: xsp-Gaussian

f(x) = 1 √ 2π σ exp

• −(x − µ)2

2σ2

• µ

σ

PP = P{x > 0} = 1 − 1 2erfc

• µ

√ 2 σ

• PN = P{x < 0} = 1

2erfc

• µ

√ 2 σ

• σ

µ + σ µ − σ µN = µ − 1

1 2erfc

• µ

√ 2 σ

• σ
• 2π exp(µ2/σ2)

µP = µ + 1 1 − 1

2erfc

• µ

√ 2 σ

• σ
• 2π exp(µ2/σ2)

x < 0

x

x > 0

SLIDE 14

One-sided Gaussian distribution

14 x is normal distributed with zero mean and variance σ2

!"! !"# $"! $"# %"! %"# &"! &"# '"! '"# #"! !"! !"$ !"% !"& !"' !"# !"( !") !"* !"+ $"! $"$ $"% $"& $"' !"#$%&'&()*+ !"#$%&'&+ !"#$%&'&+),+

• ./!!"0/0&#%1!!"%.&0"!23"412"5.

,-./012/!3-4/4!56733!4-389-: ;"0<7:-1.6=0>?90%!!*

• ne-sided Gaussian distribution with σ2 = 1/2

quantity value with σ2 = 1/2 [10 log( ), dB] average =

• 1

π 0.564 [−2.49] deviation =

• 1

2 − 1 π 0.426 [−3.70] dev avg = π 2 − 1 0.756 [−1.22] avg + dev avg = 1 +

• 1

2 − 1 π 1.756 [+2.44] avg − dev avg = 1 −

• 1

2 − 1 π 0.244 [−6.12] avg + dev avg − dev = 1 +

• 1/2 − 1/π

1 −

• 1/2 − 1/π

7.18 [8.56]

f(x) = 2 1 √ 2π σ exp

• − x2

2σ2

• E{f(x)} =
• 2

π σ E{f 2(x)} = σ2 E{|f(x) − E{f(x)}|2} =

• 1 − 2

π

• σ2

y = |x|

SLIDE 15

Chi-square distribution

15 is χ2 distributed with r degrees of freedom z! = Γ(z + 1), z ∈ N xi are normal distributed with zero mean and equal variance σ2

! " # $ % &! &" &# &$ &% "! !'! !'& !'" !'( !'# !') !"#"$ !"#"% !"#"& !"#"' !"#"$(

)*+!,-./!0

*+,-./0+!12345-!6+175+8 9'.:38+;,4<.=>5."!!%

Notice that the sum of χ2 is a χ2 distribution f(x) = x

r 2 −1 e− x2 2

Γ 1

2r

• 2

r 2

x ≥ 0 E{f(x)} = σ2r E{[f(x)]2} = σ4r(r + 2) E{|f(x) − E{f(x)}|2} = 2σ4r

χ2 =

r

• i=1

x2

i

χ2 =

m

• j=1

χ2

j ,

r =

m

• j=1

rj

SLIDE 16

Averaging chi-square distributions

16 averaging m variables |X|2, complex X=X’+ıX”, yields a χ2 distribution with r = 2m dev avg = 1 √m relevant case: σ2 = 1/2 avg = 1 dev = 1 √m

!"! !"# !"$ !"% !"& '"! '"# '"$ '"% '"& #"! #"# #"$ #"% !"! !"# !"$ !"% !"& '"! '"# '"$ '"% '"& #"! !"#"$ !"#"% !"#"& !"#"' !"#"$(

)*+",-"!"./0!123)45"*)41

()*+,-.)!/0123+!245!6!78# 9",:17);*2<,=>3,#!!&

!"! !"# !"$ !"% !"& '"! '"# '"$ '"% '"& #"! ! ' # ( $ ) % * & !"#"$ !"#"% !"#"$& !"#"&% !"#"'(&

)*+",-"!"./0!123)45"*)41

+,-./01,!23456.!578!9!:;$ <"/=4:,>-5?/@A6/#!!&

1 m χ2 = 1 m

m

• j=1

(X′

j)2 + (X′′ j )2

E 1 m f(x)

• = σ2r

m = 2σ2 E

• 1

m f(x) − E 1 m f(x)

• 2

= 2σ4r m2 = 4σ4 m

SLIDE 17

Rayleigh distribution

17 x1 and x2 are normal distributed with zero mean and equal variance σ2 x is Rayleigh-distributed

x1 x2 y = (x

1

+ x

2

)

1 / 2

Re Im

!"! !"# $"! $"# %"! %"# &"! &"# '"! '"# #"! !"! !"$ !"% !"& !"' !"# !"( !") !"* !"+ $"! sigma = 0.71 sigma = 1 sigma = 1.41

Rayleigh distribution

,-./0123./-45!6-781-9 :"0;<9-=.2>0?@10%!!*

f(x) = x σ2 exp

• − x2

2σ2

• x ≥ 0

E{f(x)} = π 2 σ E{f 2(x)} = 2σ2 E{|f(x) − E{f(x)}|2} = 4 − π 2 σ2

Rayleigh distribution with σ2 = 1/2 quantity value with σ2 = 1/2 [10 log( ), dB] average = π 4 0.886 [−0.525] deviation =

• 1 − π

4 0.463 [−3.34] dev avg =

• 4

π − 1 0.523 [−2.82] avg + dev avg = 1 +

• 4

π − 1 1.523 [+1.83] avg − dev avg = 1 −

• 4

π − 1 0.477 [−3.21] avg + dev avg − dev = 1 +

• 4/π − 1

1 −

• 4/π − 1

3.19 [5.04]

x =

• x2

1 + x2 2

SLIDE 18

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. Though, often they are more disturbing than

noise

• polynomial and other noise types can be whitened,

analyzed, and un-whitened

• … and of course WG noise is easy to understand

18

SLIDE 19

Correlation

• Part 1: Theory

19

SLIDE 20

Power spectral density Sxx

20 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

Sxxm = 1

T XX∗m

= 1

T (X′ + ıX′′) × (X′ − ıX′′)m

= 1

T

• (X′)2 + (X′′)2

m

SLIDE 21

Measurement of |Sxx|

21 Running the measurement, m increases and Sxx shrinks => better confidence level

! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=1

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=2

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=4

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=8

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=16

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=64

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=128

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=256

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=512

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| m=1024

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

&'()*+,)-./0!+)1!##!#!$2!!3#4!05+677 689/-)7+,%:; <%=95'8(0>*:0/*$!#!

frequency

SLIDE 22

Syx with correlated term (1)

22 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 X = (A′ + ıA′′) + (C′ + ıC′′) and Y = (B′ + ıB′′) + (C′ + ıC′′) 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

Syxm = Syxm

• instr + Syxm
• mixed + Syxm
• DUT

background

• nly

background and DUT noise DUT noise

• nly

... and work it out !!! Syxm =

1 T Y X∗m

=

1 T (Y ′ + ıY ′′) × (X′ − ıX′′)m

SLIDE 23

Real Imaginary

Syx with correlated term κ≠0 (2)

23

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

ℜ

• Syxm
• = 1

T

• B′A′ + B′′A′′m + B′C′ + B′′C′′m + C′A′ + C′′A′′m +
• (C′)2 + (C′′)2

m

• ℑ
• Syxm
• = 1

T {B′′A′ + B′A′′m + B′′C′ − B′C′′m + C′′A′ − C′A′′m}

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

SLIDE 24

Expand Syx

24

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

Bessel K0, avg=0, var=κ2/4 white, χ2, 2 DF avg = κ2, var = κ4 Bessel K0, avg=0, var=1/4

Syx = 1

T E {A + ıB + C }

A = B′A′ + B′′A′′ + B′C′ + B′′C′′ + C′A′ + C′′A′′ B = B′′A′ + B′A′′ + B′′C′ − B′C′′ + C′′A′ − C′A′′ C = C′ 2 + C′′ 2 After averaging, the Bessel K0 distribution turns into a Gaussian distribution (central limit theorem) term E V PDF comment A m 1 + 2κ2 2m Gauss average (sum) of zero-mean Bm 1 + 2κ2 2m Gauss Gaussian processes C m κ2 κ4/m χ2 average (sum) of ν = 2m chi-square processes ˜ C

• m

κ2 κ4/m Gauss approximates C m for large m

SLIDE 25

Estimator Ŝ = |<Syx>m|

25 | Syxm | = 1 T

• [ℜ {Y X∗m}]2 + [ℑ {Y X∗m}]2

= 1 T

• [A m + ˜

C m]2 + [Bm]2 . κ → 0 Rayleigh distribution Z m =

• [A m]2 + [Bm]2 .

E{Z m} = π 4m = 0.886 √m V{Z m} = 1 m

• 1 − π

4

• = 0.215

m dev{| Syxm |} E{| Syxm |} =

• 4

π − 1 = 0.523

!!"# !!"$ !%"# !%"$ !$"# $"$ $"# %"$ %"# !"$ !"# &"$ &"# $"$ $"% $"! $"& $"' $"# $"( $") $"* $"+ %"$

,,-./00 0123.4!5%6! 7.89:12; ,0123.4!5%6! <=>?.?191@8,A:B01@8,CDEF

C12/=:G,,E0H!-./00!7.89:12;!HAC 0>/=I:.93>0@!.99!H9>@0 J",7/?1>9.K,B>L,!$$+

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

SLIDE 26

Estimator Ŝ = Re{<Syx>m}

26 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

2 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

• Unbiased estimator

SLIDE 27

Estimator Ŝ = |Re{<Syx>m}|

27

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

• ℜ
• Syxm
• = 1

T |A m + ˜ C m| PN = P{x < 0} = 1 2erfc κ2 √ 2 σ

• f(x)

x

fold the neg values up κ2 PN Pn

File: xsp-estimator-abs-Re

SLIDE 28

Estimator Ŝ = Re{<Syx>m’} averaging on the m’ positive values

28

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

f(x) x

remove the neg values scale f(x) up κ2 PN PN

File: xsp-estimator-Re-discard-neg

PN = P{x < 0} = 1 2erfc κ2 √ 2 σ

SLIDE 29

Estimator Ŝ = <max(Re{ Syx}, 0+)>m

29

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

f(x) x

turn the neg vals to zero κ2 PN PNδ(z)

File: xsp-estimator-Re-make-pos

SLIDE 30

Estimator comparison

30

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

ˆ Syx = ℜ

• max(Syx, 0+)m
• preferred estimator

µ1 > µ2 > µ3 .

contrib to f(x) x

PN

µ1 µ2 µ3=0

|Re{<Syx>m}| Re{<Syx>m}

if > 0, else discard

Re{<Syx>m}

if > 0, else set to 0

PN PN

µN<0

PN

File: xsp-estimator-comparison

SLIDE 31

white, Gaussian, avg = 0, var = 1/2m

Detection, and noise-rejection law (κ=0)

31

Modulus

white, Bessel K0, avg = 0, var = 1/4 white, Gaussian

+ unbiased + fastest convergence

– can’t use log scale (dB!)

Real part

white, Bessel K0, avg = 0, var = 1/4

ℜ

• Syxm
• = Y ′X′ + Y ′′X′′m

white,

• ne-sided Gaussian,

– biased = fair convergence

+ can use log scale (dB!)

Abs Real part

white, Bessel K0, avg = 0, var = 1/4

• ℜ
• Syxm
• = |Y ′X′ + Y ′′X′′m|

| Syx |m =

• [Y ′X′m + Y ′′X′′m]2 + [Y ′′X′m − Y ′X′′m]2

white, Rayleigh

– biased – slowest convergence

+ can use log scale (dB!)

avg =

• 1

πm var = 1 2 − 1 π 1 m avg = 0 var = 1 2m avg = π 4m var =

• 1 − π

4 1 m

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 A, B, C are independent Gaussian noises Re{ } and Im{ } are independent Gaussian noises

SLIDE 32

Noise rejection, |Syx| and |Re{Syx}|

32 E{S} = π 4m Independent X and Y, var{X} = var{Y}= 1/2 |<Syx>m| ~ – 5 log10(m) – 0.53 dB average deviation the dev / avg ratio is independent of m

= 0.886/√m = 0.523

The track thickness on the analyzer logarithmic scale is constant because the dev / avg ratio is independent of m

• E
• |S − E{S}|2

=

• 1 − π

4 1 m

• E{|S − E{S}|2}

E{S} =

• 4

π − 1 |<Re{Syx>m}| ~ – 5 log10(m) – 2.49 dB average deviation

= 0.564/√m = 0.756

E{S} =

• 1

πm

• E
• |S − E{S}|2

= 1 2 − 1 π 1 m

• E{|S − E{S}|2}

E{S} = π 2 − 1

= √(0.215/m) = √(0.182/m)

|Syx| => Rayleigh distribution |Re{Syx}| => one-sided Gaussian distrib. the dev / avg ratio is independent of m

SLIDE 33

The concept of ergodicity

33 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 time series. 80 2 y 10 20 30 4 dB 3 100 x 60 1 40 20

File: xsp-ergodicity-3d

frequency realization | < Syx ( f ) >32 | , d B

SLIDE 34

Example: Measurement of |Syx|

34

! "! #! $! %! &!! &"! &#! &$! &%! "!! !'!!& !'!& !'& & &!

()*+,-.+!/01!2" 3'456)7*89,8:;,"!!%

Sxx Syx m=32

= (/4m)

5 log(m) – 0.52 dB

– [(1-/4)/m]

– 3.21 dB

+ [(1-/4)/m]

+ 1.83 dB

C = 0 C ≠ 0

frequency | S y x |

m, 20 ... 210 frequency frequency |<Syx>m|, dB |<Syx>m|, dB m, 20 ... 210

SLIDE 35

! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=1 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=2 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=4 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=8 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=16 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=32 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=64 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=128 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=256 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=512 g=0.32 |Scc|

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Syx| m=1024 g=0.32 |Scc|

frequency # #! #!! #!!! !%!# !%# # a v e r a g e d e v i a t i

• n

|Syx| m

&'()*+,)-./0!+)1!##!#!$2!!3#4!05+678 9%:;5'<(0=*0,/*$!!>

Measurement (C≠0), |Syx|

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

SLIDE 36

Measurement (C≠0), |Re{Syx}|

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

! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=1 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=2 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=4 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=8 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=16 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=32 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=64 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=128 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=256 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=512 g=0.32

frequency ! "! #!! #"! $!! !%!!# !%!# !%# # #!

|Sxx| |Scc| |Re{Syx}| m=1024 g=0.32

frequency # #! #!! #!!! !%!# !%# # a v e r a g e d e v i a t i

• n

|Re{Syx}| m

&'()*+,)-./0!+)1!##!#!$2!!3#4!05+6)789 :%6;5'<(0=*0,/*$!!>

SLIDE 37

Linear vs. logarithmic resolution

37

!"# !"$ !"% &'(()*

+ f

,-./0

!1

S 1 + f

,-(2,33./0

#

S"

(4 53333333333,-6./07

N

89:;*)32(6 P

"

2<;3=#38>)&?(2 @3!%A!3,-6 ?B'3(43&C2::A 2:;*)3D:&2*A 2(63!2E3!F

G'D(9)(34()HD):&IE3/0

k T

B

.P @3!!JJA!3,-5(2,337./0

# " "

!"K !!L"AK !!M"AK !!J"AK !!="AK !#""AK

5!!LLA%7 5!!MLA%7 5!!JLA%7 !

!"

5!!%LA%7 5!!KLA%7

plot 471

Linear resolution Logarithmic resolution

Fig.5, G. Cibiel, TUFFC 49(6) jun 2002 Fig.7, E. Rubiola, V. Giordano, RSI 73(6) jun 2002

f0 f1 f2 fn-1 Syx n–1 values M0 Mi MN Joining M values => background reduction of M1/2 because S(fj), S(fk), jk are independent 5 dB Logarithmic resolution: M proportional to f yields a background prop. to M1/2

• ne decade

SLIDE 38

Correlation

• Part 2: Applications

38

SLIDE 39

Applications

39

• 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 40

Radio-astronomy

40

• 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
• f 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

SLIDE 41

Early implementations

41

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(2f0t) – X"(f0)sin(2f0t) Y'(f0)cos(2f0t) – Y"(f0)sin(2f0t)

(Y'X' + Y"X")/2 <Y'X' + Y"X"> / 2 x(t) y(t)

Rice representation of noise

SLIDE 42

Radiometry -- Johnson thermometry

42

• 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 1 8 º T2 A B X = A + B Y = A – B T1

Syx = k (T2 – T1) / 2

SLIDE 43

Re-definition of the Kelvin?

43

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)

SLIDE 44

Noise calibration

44

• 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

SLIDE 45

Measurement of H-maser frequency noise

45

• 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

SLIDE 46

Phase noise measurement

46

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

SLIDE 47

Phase noise

47

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)

Δ Δ

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 48

Effect of amplitude noise

48

• 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)

Δ Δ

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 49

Dual delay line system

49

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)

SLIDE 50

Correlation dual-delay-line method

50

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

• 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

SLIDE 51

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 Δ’

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

51

• 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

!"# !"$ %&''() *+,-)(./'0

1&2'+('.3'(42(,%56.78

9 : f

;<'/;..=78

$

S! : f

;<=78

"9

S

'3 >..........;<0=78?

N

P

"

@A&.'3.%B/,,C /,-)(.2,%/)C /'0.!/6.!D /E-.#F!.*G(%@'/ H.!#CI.;<0 k T

" B

=P

" H.!!IJCI.;<>'/;..?=78 $

!!K"C$ !!J"C$ !!I"C$ !!L"C$ !$""C$

>!!KKCM? >!!JKCM? >!!IKCM? ! >!!FKCM?

!"

>!!MKCM?

!"# !"$ !"% !"& '())*+

!, -

f

./)0.11234

$

S

• f

./234

",

S

5678+*10)9

N

:1111111111111./9234; )< P

"=1!%>!1./9

0?81&$@15A*'B)0 BC(1)<1'D077> 078+*1E7'0+> 0)91!0F1!G k T

B

2P =1!!HH>!1./:)0.11;234

$ " "

:!!#I>%; :!!II>%; :!!JI>%; :!!HI>%; :!!KI>%;

!!J"># !!H"># !!K"># !$""># !$!">#

L(E)6*)1<)*ME*7'NF134

background noise noise of a by-step attenuator

SLIDE 52

Thermal noise compensation

52

DUT

g g k T0

B

k T0

B resistive terminations CP2

interferometer

isolation isolation

Correlation-and-averaging rejects the thermal noise

SLIDE 53

Thermal noise compensation

53

100 MHz prototype, carrier power Po = 8 dBm injected noise, dBrad2/ Hz mesured noise, dBrad2/ Hz thermal floor

SLIDE 54

Frequency stability of a resonator

54

• 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

– + – +

(t) ϑ(t) (t) x=– y=–

detector FFT analyzer double Wheatstone bridge noise sideband amplification

cos(0t)

DUT Concept, still not engineered

SLIDE 55

Dual-mixer time-domain instrument

55

• S. Stein & al., IEEE Transact. IM 32(1) p.227-230, mar 1983

Original idea:

• D. W. Allan, The measurement of frequency

and frequency stability of precision oscillators, NBS Tech. Note 669, 1975

The average process rejects the mixer noise This scheme is equivalent to the correlation method

SLIDE 56

Amplitude noise & laser RIN

56

• 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

!"

!

!"

#

!"

$

!"

%

!"

&

!!%" !!$" !!#" !!!" !!"" !'" !(" )*+,-+./012345 678912:;<345

=>)'$&1789

20mA 30mA 40mA 60mA 80mA 100mA

Kirill Volyanskiy

SLIDE 57

57

• 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 in progress

SLIDE 58

Noise in chemical batteries

58

• 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 59

Measurement of noise in semiconductors

59

• 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 60

Electromigration in thin films

60

• 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

(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 61

Hanbury Brown - Twiss effect

61

• 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, hν = 1.12×10–24 J, kT/hν = –6.1 dB

SLIDE 62

Conclusions

• Correlation enables the rejection of the instrument noise
• In AM noise, RIN, etc., correlation enables the 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
• |<Re{Syx}>m| and max{<Syx>m,0+} provide easier readout
• Applications in many fields of metrology

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

home page http://rubiola.org

SLIDE 63

home page http://rubiola.org

The FFT Analyzer and the Measurement of Power Spectra

• Basic concepts
• Continuous and discrete Fourier

transform

• Fitting the DFT into a computer memory
• Fourier statistics
• Nice technical issues

Enrico Rubiola

FEMTO-ST Institute, CNRS and UFC, Besancon, France

Contents

SLIDE 64

Nice technical issues

64

SLIDE 65

Magic numbers beyond the display

65

• N/2 available points, from f = 0 to f = fN
• Full span is fspan ≈ 0.8 fN because the anti-aliasing filter rolls off in ≈ 0.2 fN
• Guess fN = 1.25 × fspan, and fS = 2.5 × fspan. Round to a power of 2 if likely
• Example: fspan = 10 kHz, M = 801 displayed points
• Since there are M ≈ 0.8 × N/2 displayed points, guess N = 2048
• Resolution fR = fspan / (M–1), i.e. fR = 10 kHz / 800 = 12.5 Hz
• N-point DFT has fs = N fR , thus fs = 2048 × 12.5 Hz = 25.6 kHz
• The acquisition time is T0 = N/fs = 1/fR, thus T0 = 80 ms

721 values per decade

79 80 800 801 1023 filter roll-off (aliasing)

fN span ≈ 0.8×fN

1024 2047

1024 complex values 801 displayed values (linear-frequency mode) 1024 complex values (alias) full array of 2048 values 0.1 × span fs = 2fN/2 not computed

previous decade

discarded

current decade

(log-frequency mode) fN = Nyquist frequency fs = sampling frequency

• -- Example ---

SLIDE 66

Getting the PSD

66

2047

time series (2048 points)

2046 1 1023 801 useful points, 2048 FFT 1022 1 800

FFT

1 800

|X|2/T

801 point PSD

x(t) X(f) S(f)

2047

time series (2048 points)

2046 1

reversed copy of the originaltime series FFT

2048 2047 1 2047 2046 1 1600 artificially high resolution FFT

Adding a reversed copy of the original time series results in improved estimation at low frequencies

• -- Examples ---

SLIDE 67

Joining decades

67

• -- Example ---

721 values per decade

79 80 800 801 1023

721 values per decade

79 80 800 801 1023

721 values per decade

79 80 800 801 1023 0.1 Hz 1 Hz 10 Hz 100 Hz 1 Hz 10 Hz

0.1 Hz 1 Hz 10 Hz 100 Hz 2.5 mHz resolution 25 mHz resolution 250 mHz resolution 2048 point FFT 2048 point FFT

SLIDE 68

Linear vs. logarithmic resolution

68 Linear resolution

• G. Montress & al, TUFFC 41(5) 1994

Logarithmic resolution (80 pt/dec)

• E. Rubiola, plot 610

Combining Mi values, the no.

• f degrees of freedom

increases by a factor of Mi The confidence interval is reduced by √M, (5 dB left- right in one decade) A weighted average is also possible

720 values/decade

M0 Mi MN 79 80 800 801 1023 previous decade filter roll-off (aliasing polluted)

1024 points, out of the 2048 point FFT 80 points/decade average

Sxx Sxx

SLIDE 69

Insufficient frequency-resolution

69

356

IEEE

TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 43, NO. 3, MAY 1996

C*

• 5v
• Fig. 4.

Simplified schematics of the phase lock of the 100-MHz oscillator to the BVA oscillator.

The phase lock of the 100-MHz oscillator to the BVA can be achieved by multiplying 10 MHz or dividing 100 MHz. It is possible to realize a frequency multiplier that does not degrade the spectral purity of the BVA in the region of interest. The frequency multiplier, especially if the multiplication factor is not a power of two, requires the use of some filter which can be a source of long-term phase instabilities. We investigated the possibility of realizing a digital division scheme, finding discrepancies in the literature. The majority of the authors declare a limit of -150 dB [rad2/Hz] for the white phase noise floor [12], [13], but others show the possibility of reaching lower levels [

• 141. We tested several configurations

and logic families. The configuration which exhibits the best performances uses an emitter-coupled logic (ECL) divided by 10 with an exclusive-or (XOR) gate used as phase comparator. The schematics of the circuit with servo loop amplifier are reported in Fig. 4. From measurements of locked oscillators spectrum reported at 10 MHz, we can infer that the white phase noise floor is better than -140 dB [rad2/Hz] and the flicker is at the level of -135 dB [rad2/Hz] at 1 Hz Fourier frequency. A loop bandwidth four times higher than the crossover frequency of the phase noise spectrum of the two oscillators is chosen to replicate the BVA spectrum up to 30 Hz. This choice is due to the fact that the lowest part of the spectrum dominates in the process of degradation of the clock stability. The output of the 100-MHz quartz is amplified and split by a four-way power splitter. One way is available for test or external frequency comparison; the second is employed in the comparison with the 10 MHz from the BVA, and the third goes to a double-balanced mixer (DBM) which is used to detect the phase difference against the signal coming from a hydrogen

• maser. The output of the DBM can be used to phase lock

the BVA to the H-maser via a second-order loop with 0.5-Hz

• bandwidth. With this configuration the clock can operate with

a fully digital frequency servo loop as described in [7]. In the case that autonomous operation with respect to the maser be required, a switch allows the frequency control of the BVA by an external signal. The fourth way goes to the frequency doubler.

• 60
• 70
• 80
• 90
• 100
• 110
• 120

1o-I

ioo

io1 io2

io3

io4

io5 Frequency [Hz]

Measured phase noise of two sampling mixers (Watkins-Johnson

• Fig. 5.

model 6300-310) at 9.2 GHz in dB [rad2/Hz].

To avoid narrow-band filtering, a sampling mixer [15] is used to phase lock a dielectric resonator oscillator (DRO)

• perating at 9.192 GHz which is the cesium “clock transi-

tion” frequency. The sampling mixer uses a 200-MHz (4~20 MHz) signal at the local oscillator input to down-convert a microwave signal in the 2-18 GHz band to IF’S in the 5-70

MHz range.

To our knowledge, there are no research reports about the phase noise spectrum of a sampling mixer. To measure the noise performances of the sampling mixer, we drive two units with the same 200-MHz signal and DRO, then we compare the two IF outputs in a DBM. This measurement scheme permits a cancellation of the phase noise of the oscillators used. The DRO is phase locked to the 100-MHz quartz to reduce the residual phase noise of the measurement system.

• Fig. 5 reports the measured phase noise of two sampling

mixers; the measurement is performed by driving the mixers with an W level of 0 dBm which is higher than the typical value recommended by the manufacturer. The sampling mixer is driven at the LO port with the 200-MHz signal obtained by doubling and amplifying the 100 MHz coming from the four-way power splitter. The RF input receives 0 dBm of power derived from the DRO’s

• utput signal via a 10-dB directional coupler. The beat note

Sampling mixer Watkins-Johnson 6300-310 Rovera, TUFFC 43, 1996 two units 0 = 9.2 GHz b–1 = –87 dB

Text Resolution OK, the spectral lines are clearly identified Insufficient resolution, the spectral lines cluster in a bump

SLIDE 70

Spectrum of the quantization noise

70

Ergodicity suggests that the quantization noise can be calculated statistically The Parseval theorem states that energy and power can be evaluated by integrating the spectrum

NB = V 2

q

12

σ2 = V 2

q

12

Changing B in geometric progression (decades) yields naturally 1/B (flicker) noise

Vq

sampling

x

error

v(t) t Vq 1/Vq p(x) σ2 = V 2

q

12 Sv(f) f N B σ2 = NB

N = V 2

q

12B

The analog-to-digital converter introduces a quantization error x, –Vq/2 ≤ x ≤ +Vq/2

Sv(f) f

B1 B2 B4 N1 N2 N3 N4

B3

log-log

1/12 is –10.8 dB

SLIDE 71

Noise of the real FFT analyzer

71

The quantization noise scales with the frequency span, the front-end noise is constant The energy is equally spread in the full FFT bandwidth, including the upper region not displayed because of aliasing

ADC

Sv(f)

log-log FFT algorithm input

Nquant B3 B2 B1 B4 f N1 N2 N3 N4 B5 N5 Nampli Nampli Ntot Nquant

anti aliasing sampling

720 values/decade

79 80 800 801 1023 previous decade filter roll-off (aliasing)

1024 points, out of the 2048 point FFT

Sxx

SLIDE 72

Example of FFT analyzer noise

72

HP-3562A

(E.Rubiola notebook v.5 p.177)

Theoretical evaluation

DAC 12 bit resolution, including sign range 10 mVpeak Vfsr = 20 mV (±10 mV) resolution Vq = Vfsr / 212 = 4.88 μV total noise σ2 = (4.88 μV)2 / 12 = 2×10–12 V2 (–117 dB) quantization noise PSD Sv = σ2 / B = –117 dBV2/Hz with B = 1 Hz (etc.)

Front-end noise, evaluated from the plot

Sv = 2×10–15 V2 (–150 dB), at 10–100 kHz

• r 45 nV/Hz1/2

use Sv = 4kTR R = 125 kΩ

• r R = 100 kΩ and F = 1 dB (noise figure)

Experimental observation

SLIDE 73

Oscillator noise measurement

73

1 10 100 1000 10000 1e+05 !180 !160 !140 !120 !100 !80 !60 !40 !20 Sphi(f), dBrad^2/Hz frequency, Hz

480 MHz SAW oscillator

file oscillator!noise!with!jump

• E. Rubiola, may 2008

A tight loop is preferred because: – reduces the required dynamic range – overrides (parasitic) injection locking

under test reference FFT analyzer control

VCO in

log-log

• scillator noise

PLL out P L L r e s p

• n

s e frequency

Steps are sometimes observed, due to the FFT quantization noise

SLIDE 74

FFT noise in oscillator measurements

74

! !" !"" !""" !"""" !#$"% !!&" !!'" !!(" !!)" !!"" !&" !'" !(" !)" *+,-./012345637)89: /5#;<#=>?129:

Quantization noise

+656@#A#5B ;<6=A2=C-B#2D2)E"#!!F2824 GGH2/ICC52D2FE"#!!J @2D2K!'12F)12'(12!)&12)%'L MNN2/>2D2F)""" CB>-II2O"22D2)E"#!!& CB>-II2O!!2D2!E"#!!F CB>-II2O!)2D2!E"#!P CB>-II2O!F2D2&E"#!%

/-I#2CB>-II6AC5!>6I>!Q-A,!R<@+ SE2T<O-CI612@6?2)""&

• bserv. PLL out

ideal PLL out true oscill. noise FFT noise

• bserv. oscill. noise

step error ! !" !"" !""" !"""" !#$"% !!&" !!'" !!(" !!)" !!"" !&" !'" !(" !)" *+,-./012345637)89: /5#;<#=>?129:

Quantization noise

+656@#A#5B ;<6=A2=C-B#2D2)E"#!!F2824 GGH2/ICC52D2FE"#!!J @2D2K!'12F)12'(12!)&12)%'L MNN2/>2D2F)""" CB>-II2O"22D2)E"#!!& CB>-II2O!!2D2!E"#!!F CB>-II2O!)2D2!E"#!P CB>-II2O!F2D2&E"#!%

/-I#2CB>-II6AC5!B-@<I!Q-A,!R<@+ SE2T<O-CI612@6?2)""&

• bserv. PLL out

true oscill. noise FFT noise

• bserv. oscill. noise

step error

calculated simulated

The steps are due to the FFT quantization noise The problem shows up when the dynamic range is insufficient, often in the presence of large stray signals Systematic errors are also possible at high Fourier frequencies Explanation: the steps occurring at the transition between decades are due the quantization noise, when the resolution is insufficient