Enrico Rubiola – Phase Noise – 1
Phase Noise
Enrico Rubiola
Dept. LPMO, FEMTO­ST Institute – Besançon, France
e­mail rubiola@femto­st.fr or enrico@rubiola.org
Summary
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Spectra
Classical variance and Allan variance
Properties of phase noise
Laboratory practice
Calibration
Bridge (Interferometric) measurements
Advanced methods
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
introduction – noisy sinusoid
Enrico Rubiola – Phase Noise – 3
Chapter 1. Basics
2
Representations
of a sinusoid with noise
Time Domain
V0
v(t)
Phasor Representation
amplitude fluctuation
V0 α(t) [volts]
normalized ampl. fluct.
α(t) [adimensional]
√
V0 / 2
t
V0
v(t)
ampl. fluct.
√
(V0 / 2)α(t)
phase fluctuation
ϕ(t) [rad]
phase time (fluct.)
x(t) [seconds]
√
t
polar coordinates
phase fluctuation
ϕ(t)
V0 / 2
Figure 1.1: .
v(t) = V0 [1 + α(t)] cos [ω0 t + ϕ(t)]
where V0 is the nominal amplitude, and α the normalized amplitude fluctuation,
= V0 cos frequency
ω0 t + nisc (t) cos ω0 t − ns (t) sin ω0 t
Cartesianwhich
coordinates
is adimensional. v(t)
The instantaneous
1 dϕ(t)
ω0
+
holds
2π
2π Itdt
(1.3)
that
nc (t)
ns (t)
of stable
with
α(t)signals
= of the form
and (1.2),
ϕ(t)
=
|nc (t)| ! VThis
and deals
|nswith
(t)| the
! measurement
V0
0 book
0
main focus on phase, thus frequency and time. This V
involves
several topics, V0
under low noise approximation
ν(t) =
Enrico Rubiola – Phase Noise – 4
introduction – noisy sinusoid
Noise broadens the spectrum
v(t) = V0 [1 + α(t)] cos [ω0 t + ϕ(t)]
v(t) = V0 cos ω0 t + nc (t) cos ω0 t − ns (t) sin ω0 t
ω = 2πν
ω
ν=
2π
introduction – general problems
Enrico Rubiola – Phase Noise – 5
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
(low-pass)
time-domain
product
vector
difference
s(t) ∗ hlp (t)
s(t) × r(t − T /4)
s(t) − r(t)
distorsiometer,
audio-frequency instruments
traditional instruments for
phase-noise measurement
(saturated mixer)
bridge (interferometric)
instruments
Enrico Rubiola – Phase Noise – 6
introduction – general problems
Why a spectrum analyzer does not work?
1.
2.
3.
4.
too wide IF bandwidth
noise and instability of the conversion oscillator (VCO)
detects both AM and PM noise
insufficient dynamic range
Some commercial analyzers provide phase noise measurements,
yet limited (at least) by the oscillator stability
Enrico Rubiola – Phase Noise – 4
Enrico Rubiola – Phase Noise – 7
introduction – φ –> vIntroduction
converter – !
The
Schottky-diode
double-balanced
mixer
sa
The Schottky-diode double-balanced mixer saturated at both
at both
inputs
is the
mostdetector
used phase detec
inputs
is the
most used
phase
signal
reference
product
!
s(t) = 2R0 P0 cos [2πν0 t + ϕ(t)]
!
r(t) = 2R0 P0 cos [2πν0 t + π/2]
r(t)s(t) = kϕ ϕ(t) + “2ν0 ” terms
filtered
out
The AM noise is rejected by saturation
Saturation also account for the phase-to-voltage gain kφ
Enrico Rubiola – Phase Noise – 8
2 – Spectra
Spectra – definitions
Enrico Rubiola – Phase Noise – 9
Power spectrum density
In general, the power spectrum density Sv(f) of a random process v(t) is
! "
defined as
#$
Sv (f ) = E F Rv (t1 , t2 )
! "
E ·
! "
F ·
statistical expectation
Fourier transform
Rv (t1 , t2 ) autocorrelation function
In practice, we measure Sv(f) as
! "
# !2
Sv (f ) = !F v(t) !
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
Sv (f0 )
P =
R0
Sv (f0 )
R0
power in 1 Hz bandwidth dissipated by R0
Enrico Rubiola – Phase Noise – 11
Spectra – meaning
Physical meaning of the power spectrum density
The power spectrum density extends the concept of
root-mean-square value to the frequency domain
Spectra – Sφ(f) and ℒ(f)
Enrico Rubiola – Phase Noise – 12
Sφ(f) and ℒ(f) in the presence of (white) noise
SSB
DSB
P0
P0
B
B
B
N
v0
ϕp
v
v0–
f
!
R0 N B
!
R0 P0
ϕrms =
dBc/Hz
v0
+f
N
!
1
ϕp =
2
!
N
L=
2P0
v0
ϕp
!
R0 P0
ϕrms =
3 dB
v
R0 N B/2
!
NB
2P0
v0
+f
!
1
ϕp =
2
!
N
Sϕ =
P0
NB
P0
dBrad2/Hz
Spectra – Sφ(f) and ℒ(f)
Enrico Rubiola – Phase Noise – 13
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)
Spectra – useful quantities
Enrico Rubiola – Phase Noise – 14
Useful quantities
phase
time
1
x(t) =
ϕ(t)
2πν0
x(t) is the phase noise converted into time fluctuation
physical dimension: time (seconds)
fractional
frequency
fluctuation
1
y(t) =
ϕ̇(t) = ẋ(t)
2πν0
y(t) is the fractional frequency fluctuation ν-ν0 normalized
to the nominal frequency ν0
ν − ν0
y(t) =
(dimensionless)
ν0
EnricoRubiola
Rubiola––Phase
PhaseNoise
Noise –– 15
15
Enrico
Spectra –
power –law
Spectra
power-law
Noise
processes
in oscillators
Power-law
and
noise processes
in oscillators
Enrico Rubiola – Phase Noise – 16
Spectra – power law
Relationships between Sφ(f) and Sy(f)
Spectra – jitter
Enrico Rubiola – Phase Noise – 17
Jitter
Jitter
The phase fluctuation can be described in terms of a single
The phase
fluctuation
beordescribed
parameter,
either
phasecan
jitter
time jitterin terms of a single
parameter, either phase jitter or time jitter
phase
noise
mustbebeintegrated
integrated over
over the
the bandwidth
TheThe
phase
noise
must
bandwidthBBofofthe
bebe
difficult
to identify)
thesystem
system(which
(whichmay
may
difficult
to identify)
phase jitter
time jitter
phase jitter
converted into time
&rms !
x rms
S & ' f ( df
%
$
radians
B
1
!
2 " #0
S & ' f ( df
%
$
seconds
B
The jitter is useful in digital circuits because the bandwidth B is known
The jitter
is
useful
in
digital
circuits
because
the
bandwidth
is
known
–lower limit: the inverse propagation time through the system
- lower
limit:
the inverse
propagationdivergent
time through
the system
this
excludes
the low-frequency
processes)
this –upper
excludes
the
low-frequency
divergent
processes)
limit: ~ the inverse switching speed
- upper limit: the inverse witching speed
Victor Reinhardt (invited), A Review of Time Jitter and Digital Systems, Proc. 2005 FCS-PTTI joint meeting
Enrico Rubiola – Phase Noise – 18
white frequency
flicker frequency
random-walk frequency
b−2 f
b−3 f −3
b−4 f −4
Spectra – examples
All these
noise types
are generally
present at
the output of
oscillators,
while
Typical
phase
noise
of some
devices
and
oscillators
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
-40
GH
zD
GH
zS
iel
ec
ap
p
-60
S!(f) (dBrad2 /Hz)
10
10
-80
GH
RO
Hz
-120
es
zS
10
0M
-100
hir
eR
qu
art
micro
wave
a
wi
th
sci
l
cR
ato
r
-160
lat
micro
-180
on
ato
rO
Os
no
sc
lat
de
ge
ne
rat
or
m
plifie
RF am
r
plifie 5 MH
r
z quar
tz osc
.
-140
es
cil
ise
zo
on
tri
or
(S
R
ill
ato
r
O)
wave
fe
ion
rrite i
solato
rs
-200
0.1
1
10
100
1k
Fourier frequency (Hz)
10k
100k
Enrico Rubiola – Phase Noise – 19
3 – Variances
Enrico Rubiola – Phase Noise – 20
Enrico Rubiola – Phase Noise – 20
1
y ! 1
y ! "0
"0
1
"%t & dt
$
1
#$
# "%t & dt
##
N
N
%%
Classical variance
Classical
variance
Classical variance
1
1
2
'2 ! 1
yi( 1
' ! N (1 i!1 y i ( N
N (1 i!1
N
)
)
Variances – classical
normalized reading of a counter that
normalized(averages)
reading of over
a counter
measures
a timethat
T
measures (averages) over a time T
&&
2
N
classical variance,
N y 2
classical
variance,
j
file
of
N
counter
readings
j!1 y j
file of N counter readings
j!1average of the N readings
)
)
average of the N readings
For a given process, the classical variance
ForFor
a given
process,
the classical
variance
a
given
process,
the
classical
variance
depends
of
N
depends of N
depends of N
Even
worse,
if the
spectrum
or steeper,
Even
worse,
if the
spectrum
is f-1isorf-1
steeper,
-1 or steeper,
Even
worse,
if
the
spectrum
is
f
the
classical
variance
diverges
the classical variance diverges
the classical variance diverges
filter
associated
to measure
the measure
TheThe
filter
associated
to the
Theinfilter
associated
to the measure
takes
in the
dc component
takes
the
dc
component
takes in the dc component
Enrico
–– Phase
– 21 – 21
Enrico
– Noise
Phase Noise
Enrico Rubiola
RubiolaRubiola
Phase
Noise
–
21
Variances – Allan
Zero
dead-time
two-sample
variance
Zero
dead-time
two-sample
variance
Zero
dead-time
two-sample
variance
(Allan
variance)
(Allan
variance)
(Allan
variance)
#$ #$ & ' & '
1
2
2
! y "! "y 21%yy1 2%y 2
y2
2 2 1
m%1 m%1
Definition
Definition
Definition
(Let
2,N2,
and
(LetN(Let
N= =
average)
= and
2,average)
and
average)
1
2
1
2
! y "! "
y 2 %yy1 2%y 2
y2 $m%1&
1
2 $m%1&
i"1 i"1 2
( $(
$ &
&
Estimated
Allan
variance,
Estimated
Allan
variance,
Estimated
Allan
variance,
fileofof
readings
file
mm
counter
readings
file
of counter
m counter
readings
The
totothe
The
filter
associated
todifference
the difference
Thefilter
filterassociated
associated
the
difference
ofoftwo
measures
isisaaband-pass
two
contiguous
measures
band-pass
ofcontiguous
two contiguous
measures
is a band-pass
(about
one octave)
(about
one octave)
The estimate converges to the variance
The estimate
converges
to thetovariance
The estimate
converges
the variance
Enrico Rubiola – Phase Noise – 22
Variances – Allan
The Allan variance is related to the spectrum Sy(f)
Variances – Allan
Enrico Rubiola – Phase Noise – 23
Convert Sφ and Sy into Allan variance
noise
type
white
PM
Sϕ (f )
b0
h2 f
flicker
PM
b−1 f −1
white
FM
b−2 f
−2
b−3 f
−3
flicker
FM
Sy (f )
random
walk
b−4 f −4
FM
2
Sϕ ↔ Sy
b0
h2 = 2
ν0
σy2 (τ )
3fH h2 −2
τ
2
(2π)
2πτ fH "1
mod σy2 (τ )
3fH τ0 h2 −3
τ
2
(2π)
h1 f
b−1
h1 = 2
ν0
h1 −2
[1.038+3 ln(2πfH τ )]
τ
2
(2π)
h0
b−2
h0 = 2
ν0
1
h0 τ −1
2
0.084 h1 τ −2
n"1
1
h0 τ −1
4
h−1
b−3
= 2
ν0
2 ln(2) h−1
27
ln(2) h−1
20
h−2
b−4
= 2
ν0
(2π)2
h−2 τ
6
(2π)2
0.824
h−2 τ
6
1 2 2
Dy τ
2
1 2 2
Dy τ
2
h−1 f
−1
h−2 f −2
frequency drift ẏ = Dy
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
Enrico Rubiola – Phase Noise – 25
Properties of phase noise – synthesis
Properties of phase noise – synthesis
Frequency synthesis
Frequency synthesis
Ideal synthesizer
Ideal synthesizer
- -noise-free
noise-free
- -zero
time
zero delay
delay time
timetranslation:
translation:
time
output jitter
jitter ==input
output
inputjitter
jitter
phase time
time xxoo=x
i i
phase
=x
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
"#
n
S! o " f # $
d
2
S !i " f #
Properties of phase noise – synthesis
Enrico Rubiola – Phase Noise – 26
Carrier collapse
Simple physical meaning, complex mathematics. Easy to understand in the case of
sinusoidal phase modulation
random noise =>
phase fluctuation
Filtering <=> Phase Locked Loop (PLL)
Filtering
Phase
Locked
Loop
(PLL)
Filtering<=>
<=>
Phase
Locked
Loop
(PLL)
Rubiola
– Phase
EnricoEnrico
Rubiola
– Phase
NoiseNoise
– 27 – 26
Properties of
phase noise
– PLLnoise – PLL
Properties
of phase
The signal “2”
The
signal“1”
“2”
The
signal
“2”
tracks
tracks
tracks
“1”“1”
k ! "V (rad #
k ! "V (rad #
The FFT analyzer (n
Theneeded
FFT analyzer
(not b
The FFT
analyzer
(not can
here)
needed
here)
can be
needed
here)
can
be us
to
measure
Sused
!(f)
to measure
Sφ(f)S!(f)
to measure
tracks “1”
k o "rad (s (V #
k o "rad (s (V #
tracks “1”
The
PLL
low-pass
The
PLL
low-pass
The
PLL
low-pass
filters
the
phase
filters
the
phase
filters
the
phase
Output
voltage:
the
PLL
Output
voltage:
the
PLL
Output
voltage:
the
PLL
a high-pass
filter
isisaaishigh-pass
filter
high-pass
filter
SS!!22""ff##
SS !1""ff##
!1
SSvovo""ff# #
SS!1 ""f f# #
!1
$$
$$
2
kko kH! H" fc # #%
%
k
%o ! c %
" 2f
22 2
2
2f
4
&
f
'
k
k
H
"
4 & f ' k o ko! H!c " fc#
%%
2 2
4&
f 2 kf !
4&
k!
2 22 2
44
&&
f f'%'
k o%kk !kH cH
" f #%"2f #%2
o !
c
2
% #%
Properties of phase noise – discriminator
Enrico Rubiola – Phase Noise – 28
Frequency discriminator
A resonator turns a slow frequency
fluctuation Δν into a phase fluctuation
ν
ν0
phase φ
ν
Parameters
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
Properties of phase noise – Leeson
Enrico Rubiola – Phase Noise – 29
The Leeson effect:
The Leeson
phase-to-frequency
noise Effect
conversion in oscillators
Enrico Rubiola – The Leeson Effect – 1
Enrico Rubiola
Dept. LPMO, FEMTO-ST Institute – Besançon, France
e-mailmodel
rubiola@femto-st.fr
or enrico@rubiola.org
D. B. Leeson, A simple
for feed back oscillator
noise, Proc. 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
%" # (
S ! " f # $ 1&
resonator
oscillator
noise
'0
2Q
2
1
f
2
S) " f #
ampli
noise
oscillator noise
S! " f #
Leeson
effect
noise of electronic circuits
fL = !0/2Q
www.rubiola.org
f
E. Rubiola, The Leeson effect, Tutorial 2A, Proc. 2005 FCS-PTTI (tutorials)
some free documents on noise (amplitude and phase) and on
E. Rubiola,precision
The Leeson
effect,will
e-book,
(http://arxiv.org/abs/physics/0502143
or rubiola.org)
electronics
be available
in this non-commercial site
Enrico Rubiola – Phase Noise – 30
5 – Laboratory practice
Enrico Rubiola – Phase Noise – 31
Laboratory practice – background noise
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 – Flicker noisedue 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 – White noise
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
rico Rubiola – Phase Noise – 34
Laboratory practice – useful schemes
Laboratory practice – useful schemes
Two-port device under test (DUT)
Enrico Rubiola – Phase Noise – 36
Laboratory practice – useful schemes
Two-port device under test (DUT)
nrico Rubiola – Phase Noise – 35
Laboratory practice – 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
Filtering
<=>
Phase
Locked
Loop
(PLL)
k ! "V (rad #
Laboratory practice – oscillator measurement
Enrico Rubiola – Phase Noise – 38
Phase Locked Loop (PLL)
The signalThe
“2” FFT ana
tracks “1” needed here
to measure S
tracks “1”
k o "rad (s (V #
k ! "V (rad #
The FFT analyzer (not
needed here) can be used
to measure S!(f)
tracks “1”
2
k o "rad (s (V #
The PLL low-pass
filters
the
phase
Phase:
PLL
Thethe
PLL
low-pass
is a low-pass
filters thefilter
phase
Output
voltage:
the PLL
Output
voltage:
thePLL
PLL
Output
voltage: the
is a is
high-pass
filter
is a high-pass
filter filter
a high-pass
S! 2 " f #
SS
" f #" f #
! 2!1
S !1 " f #
S vo " f #
$
%k o k ! H c " f #%
$
2 2
2
%k 4k &H "f f #%'%k o k ! H c " f #%
2
o !
2
2
c
4 & f '%k o k ! H c " f #%
2
4 & f 2 k!
2
4 & f 2 k!
$ $
2 2
S !1
S !1
" f #" f # 4 &2 f 24
'%&
k o k !fH c'
"fk
#%2 k H "
o ! c
S vo " f #
2
%
f #%2
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
rico Rubiola – Phase Noise – 38
Laboratory practice – oscillator measurement
Laboratory practice – oscillator measurement
Phase Locked Loop (PLL)
Enrico Rubiola – Phase Noise – 40
o Rubiola – Phase Noise – 39
Laboratory practice – oscillator measurement
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
co Rubiola – Phase Noise – 41
Laboratory practice – oscillator measurement
Laboratory practice – oscillator measurement
Warning: a PLL may not be what it seems
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 injection locking 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
Calibration – measurement of kφ
Enrico Rubiola – Phase Noise – 49
Calibration – measurement of kφ (phase mod.)
Vm
The reference signal can be a
tone:
detect with the FFT,
tone:
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
EnricoRubiola
Rubiola –– Phase
PhaseNoise
Noise –– 5352
Enrico
Bridge ––Wheatstone
Interferometer
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
imaginary δZ1 => PM noise
nc(t) cos(ω0t)
–ns(t) sin(ω0t)
EnricoRubiola
Rubiola –– Phase
PhaseNoise
Noise –– 54
Enrico
55
Interferometer
Wheatstone bridge
Bridge ––Wheatstone
Wheatstone bridge – ac version
Enrico Rubiola – Phase Noise – 56
co Rubiola – Phase Noise – 55
Bridge – scheme
Interferometer – scheme
Bridge (interferometric) phase-noise and
amplitude-noise measurement
Enrico
EnricoRubiola
Rubiola– –Phase
PhaseNoise
Noise – – 56
57
– synchronous
detection
InterferometerBridge
– synchronous
detection
Synchronous detection
Enrico Rubiola – Phase Noise – 58
nrico Rubiola – Phase Noise – 57
Bridge – synchronous detection
Interferometer – synchronous detection
Synchronous
in-phase
and
quadrature
detection
Synchronous in-phase and quadrature detection
Bridge – background noise
Interferometer – noise floor
Enrico
EnricoRubiola
Rubiola– –Phase
PhaseNoise
Noise – –5958
White noise floor
Enrico Rubiola – Phase Noise – 60
White noise floor – example
Bridge – background noise
White noise floor – example
Enrico Rubiola – Phase Noise – 61
What really matters (1)
Bridge – summary
Enrico Rubiola – Phase Noise – 62
What really matters (2)
Bridge – summary
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
Advanced
– flicker
reduction
Advanced
– flicker
reduction
Enrico
Rubiola– –Phase
Phase
Noise– –6465
nrico
Rubiola
Noise
Low-flicker scheme
Enrico
Rubiola
– Phase
Noise– –6566
nrico
Rubiola
– Phase
Noise
Advanced – flicker reduction
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
Advanced – matrix
Enrico Rubiola – Phase Noise – 69
Flicker reduction, correlation, and closed-loop
carrier suppression can be combined
channel b (optional)
w1
readout
I v1
RF I−Q
G
B
v
detect Q 2
matrix
matrix w2
g ~ 40dB
LO
pump
rf virtual gnd
null Re & Im
inner interferometer
CP1
x( t )
CP2
R0
R 0 =50 Ω
DUT
power splitter
atten
CP3
Δ’
γ
arbitrary phase
10−20dB
coupl.
’
γ’
arbitrary phase
pump
LO
v1
I−Q
detect Q v2
g ~ 40dB
LO
atten
RF
I u1
I−Q
modul Q u2
z1
dual
integr z 2
channel a
readout
G
matrix
FFT
analyz.
w1
B
matrix w2
G: Gram Schmidt ortho
normalization
B: frame rotation
pump
automatic carrier
suppression control
diagonaliz.
var. att. & phase
atten
I
RF
R0
manual carr. suppr.
atten
CP4
D
matrix
I−Q detector/modulator
I
RF
Q
LO −90°
E. Rubiola, V. Giordano, Rev. Scientific Instruments 73(6) pp.2445-2457, June 2002
0°
Advanced – comparison
Enrico Rubiola – Phase Noise – 70
Comparison of the background noise
Sϕ (f) dBrad2/Hz
real−time
correl. & avg.
−140
sat
−150
mixer, interferometer
ura
int
ted
erf
er
e
r
ble
om
ete
int
e
r
fer r
cor
om
rel
e
. sa
t. m ter
res
ix.
idu
al f
lick
res
idu
er,
res al f
by−
idu lick
ste
al f
res
p in
er,
idu lick fix
ter
al f
er, ed i
fer
lick
om
fix nte
ete
er, ed i rfer
r
nte
fix
o
ed
rfe mete
int
r
erf omet r
ero
e
me r
ter
, ±4
5°
dou
−160
−170
−180
−190
−200
−210
Fourier frequency, Hz
−220
1
10
nested interferometer
mix
10 2
saturated mixer
interferometer
correl. saturated mixer
double interf.
measured floor, m=32k
det
ect
ion
10 3
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.