Table Of ContentPrimary Calibration of AM and PM Noise
Measurements
Enrico Rubiola
web page http://rubiola.org
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FEMTO-ST Institute
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t CNRS and Universit´e de Franche Comt´e, Besanc¸on, France
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s January 8, 2009
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c Abstract
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s This report describes a method for the primary calibration of phase
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noise and amplitude noise measurement systems.
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Inthefieldofmetrology,thetermprimary referstoastandardwhose
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[ quantity value and measurement uncertainty are established without re-
lation to another measurement standard for a quantity of the same kind;
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or to a procedure used to realize the definition of a measurement unit
v
andobtainthequantityvalueandmeasurementuncertaintyofaprimary
3
measurementstandard. Accordingly,aphaseandanormalizedamplitude
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have to rely on ratio measurements.
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This is a working version, which I expect to update in the following
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. six months adding some crucial experimental results.
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E. Rubiola, AM-PM calibration. January 8, 2009 2
Contents
1 Introduction 3
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Method 7
3 Mixers, scalar products, and Hilbert spaces 8
3.1 Complex signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Real signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 I-Q detector and modulator . . . . . . . . . . . . . . . . . . . . . 10
3.4 Defects of the real I-Q detectors and modulators . . . . . . . . . 11
4 Correction of the I-Q detector 13
4.1 Measurement of the errors ψ (orthogonality) and (cid:15) (symmetry) . 13
4.2 Correction of the orthogonality and symmetry errors . . . . . . . 14
4.3 Correction of the pump phase θ . . . . . . . . . . . . . . . . . . . 15
4.4 Accuracy considerations . . . . . . . . . . . . . . . . . . . . . . . 17
5 Correction of the I-Q modulator 17
5.1 Parameter measurement . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Fixing the defects. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Building the reference AM-PM modulator 19
6.1 Parameter measurement . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 Fixing the defects. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Calibration checklist 22
8 Modulation depth and accuracy 22
8.1 Phase and frequency modulation . . . . . . . . . . . . . . . . . . 23
8.2 Measurement of the modulation depth . . . . . . . . . . . . . . . 24
8.3 Mixer response and linearity. . . . . . . . . . . . . . . . . . . . . 25
8.4 Power levels and signal-to-noise ratio (SNR) . . . . . . . . . . . . 26
8.5 Tentative estimation of the uncertainty. . . . . . . . . . . . . . . 28
9 Application to the bridge (interferometric) instruments 29
10 Where these ideas come from 30
Acknowledgements 31
References 31
E. Rubiola, AM-PM calibration. January 8, 2009 3
1 Introduction
As physics and technology progress, phase noise becomes an increasingly rel-
evant issue in a number of fields. The demand for lower phase noise is seen
in a number of examples, like high speed electronics, and telecommunications
Extreme low phase noise specifications are found in military and space oscilla-
tors. Sometypesofmilitaryradarscanachievethedesiredresolutiononlyifthe
phasenoiseoftheXbandreferencesignaldoesnotexceed10−14 rad2/Hz(−140
dB) at 10 kHz off the carrier. Space research requires a frequency stability in
the lower 10−15 at τ = 1...103 s, which means a phase noise exceed 10−13
rad2/Hz (−130 dB) at 1 Hz off the carrier if the signal is taken at the standard
100MHzoutput. Ofcourse,weexpectthattheneedformeasurementaccuracy
willinevitablyfollowthespecificationsforlowphasenoise,andconsequentlyfor
highmeasurementsensitivity. Besides,primarymetrologyisaroundthecorner.
In fact, in 2010 the definition of the second is expected to switch1 from the
F=3 ↔ F=4, ∆m = 0 hyperfine transition of the 133Cs atom (9.192631770
f
GHz) to an optical transition, to be determined. In this business, we expect
thatthephasenoiseofsomesynthesizersandtransferoscillatorsisapartofthe
metrological chain, for it has to be understood with high accuracy.
This article privileges phase noise vs. amplitude noise because of the needs
and of the culture of our laboratory. Nonetheless, the method proposed here
reliesuponamplitudenoisemeasurements,andprovidesasolutioninwhichAM
and PM calibration have the same accuracy.
In the field of metrology, the term primary [28, 8, 15] refers to a standard
whose quantity value and measurement uncertainty are established without re-
lation to another measurement standard for a quantity of the same kind; or to
a procedure used to realize the definition of a measurement unit and obtain the
quantity value and measurement uncertainty of a primary measurement stan-
dard. By contrast, secondary means that the standard, or to the measurement
procedure, relies upon calibration against, or comparison with, a primary mea-
surement standard for a quantity of the same kind. It is worth mentioning that
an empirical unit can have higher reproducibility than the primary counter-
part, as it happens with the Josephson and with the quantum-Hall electrical
standards, yet inferior absolute accuracy as related to the SI.
1.1 Basic definitions
TheSIunitofangle,theradian (rad)isnowconsideredaderivedunit becausean
anglecanalwaysbedefinedintermsoftheratiooftwohomogeneousquantities.
Formerly, itwasconsideredanauxiliaryunit. Inthedomainofelectriccircuits,
√
it is usual to represent the sinusoidal signal v(t) = 2V cos[ω t+ϕ] as the
rms 0
complex number V = V ejϕ obtained by freezing the frequency ω . This
rms 0
complexnumberiscalled‘Fresnelvector’,‘phasevector’,or‘phasor’. WhenV
rms
and ϕ are allowed to vary (slowly), the term ‘analytic signal’ is used instead.
The object of our interest is the primary measurement of the phase ϕ, focusing
on small randomly fluctuating angles.
1ThiswassaidatthepaneldiscussiononthefutureofSIattheCPEMconference,Torino,
Italy, July 9–14 2006. Yet, it seems that this decision is still not in the official documents
(thankstoElioBava,headoftheINRIM,Torino).
E. Rubiola, AM-PM calibration. January 8, 2009 4
A
moVd0uccalaortrsei(eωdr s0itg)nal modulation ωt)ysin(−0
xcos(ω0t)
B
α= x
V
0
amplitude−modulated signal
in−phase
carrier modulation
V0cos(ω0t) xcos(ω0t)
C y
ϕ=
V
d 0
cal-modulate phase−modVu0lcacatoerrdsi( eωsri0gtn)al ϕ ωt)ysin(−0quadraturemodulation
Figure 1: Phasor representation of a randomly modulated signal.
Using the peak amplitudes instead of the rms amplitudes, we introduce the
signal (Fig. 1A)
v (t)=V [1+α(t)]cos[ω t+ϕ(t)] , (1)
i 0 0
whichdefinestherandomfractionalamplitudeα(t)andtherandomphaseϕ(t).
The above can be rewritten as [16, 17]
v (t)=V cosω t+x(t)cosω t−y(t)cosω t , (2)
i 0 0 0 0
where x(t) and y(t) random processes. The two above representations of v (t)
i
are connected by
(cid:114)
(cid:104) x (cid:105)2 (cid:104) y (cid:105)2
α(t)=V 1+ + −1 fractional amplitude (3)
0 V V
0 0
ϕ(t)=arg(y,V +x) . phase, ϕ∈(−π,π] (4)
0
In low-noise conditions, that is, |x/V |(cid:28)1 and |y/V |(cid:28)1, α(t) and ϕ(t) turn
0 0
into
x y
α(t)= and ϕ(t)= . (5)
V V
0 0
For calibration purposes, we need a signal with pure AM (ϕ = 0, thus y = 0),
and one with pure PM (α=0, thus x=0), as in Fig. 1B-C.
Phase noise is usually described in term of S (f), namely, the one-sided
ϕ
power spectral density (PSD) of ϕ(t) as a function of the Fourier frequency f.
The physical dimension of S (f) is rad2Hz−1. Similarly, S (f) is the PSD of
ϕ α
α(t). S (f) has the dimension of Hz−1. The technical quantity L(f) is also
α
E. Rubiola, AM-PM calibration. January 8, 2009 5
used, defined as L(f) = 1S (f) and given in dBc (dB carrier). The technical
2 ϕ
unitdBcisalsousedforAMnoise. Trustingthestationarityandtheergodicity
of the random process, that is, the repeatability and the reproducibility of the
experiment, the PSD is measured as the average square modulus of the one-
sided Fourier transform normalized for the power-type signals. The general
backgroundonphasenoiseandonfrequencystabilityisavailablefromnumerous
references, among which we prefer [5, 10, 27, 4].
1.2 State of the art
Generally,themeasurementofAMandPMnoiseisdonewithasuitabledetector
followed by a fast Fourier transform (FFT) analyzer. In most practical cases,
the phase detector is a double balanced mixer [13, 2, 1] with the two inputs
in quadrature. The amplitude detector [18] is a power detector, that is, a
diode used in the quadratic region, or a double balanced mixer with the two
inputs in phase. A balanced bridge (often referred to as ‘interferometer’) with
amplification and synchronous detection of the noise sidebands is used when
the highest sensitivity is required [24, 11]. The sensitivity is limited by the
equivalent temperature of the instrument [9]. Improved sensitivity is obtained
by correlation and averaging, with two separate—thus independent—systems
that measure the same device under test (DUT) [26, 32]. The dual-bridge
withcorrelationexhibitsthehighestreportedsensitivity,limitedbythethermal
uniformity of the instrument instead of the absolute temperature [22]. The
bridgeworkswellforthemeasurementoftwo-portcomponents,yetitisdifficult
tousefortheoscillatorPMnoise, andvirtuallyunsuitabletotheoscillatorAM
noise.
Calibration is the accurate measurement of the amplitude-to-voltage gain
k and of the phase-to-voltage gain k of the detector. The calibration relies
α ϕ
upon a reference of phase modulation and of amplitude modulation. A de-
terministic modulation is often used because the uncertainty of the base-band
instruments that follow the detector can be made small as compared to that
of the RF/microwave section. Figure 2 shows the calibration schemes generally
used.
FollowingtheschemeA,asidebandoffrequencyν closetoν isaddedtothe
s 0
carrierinonearmofthephasedetector. Thisisequivalenttophasemodulation
(cid:112)
plusamplitudemodulationofrmsvalue P /2P ,andfrequencyν =|ν −ν |.
s 0 b s 0
P and P are the sideband power and the carrier power, respectively. The
s 0
scheme B is similar to the scheme A, but for the use of RF/microwave random
noise instead of the sideband. The close-in noise is S (f) = S (f) = N/P ,
α ϕ 0
where N is the noise PSD.
The schemes A and B suffer from the impossibility to divide AM from PM.
Instead, an equal amount AM and PM is present. Hence, a standard based on
Fig. 2A-B should be referred to as a standard of SNR (signal-to-noise-ratio),
rather than of AM and PM noise. The problem is that the mixer used as
the phase detector is not perfectly balanced, for the dc offset is affected by
the input power. Hence, it turns the AM noise into near-dc noise, which is
mistaken for PM noise. The AM noise rejection can be of 15–35 dB, depending
ontheoperatingpower,frequencyandquadratureerror,onthemixertype,and
on technology [20, 21]. Using the mixer at a sweet point of zero-sensitivity to
AM noise, as suggested in [3, 6]. Consequently, the use of this standard in the
E. Rubiola, AM-PM calibration. January 8, 2009 6
A
synthes. ν P0
s on
output P
s
ν ν
0 s
off
frreefq.. |νs−ν0| <<ν0
P
0
ν ref.
0
ν ν
0 s
B
rf noise P
0
on
output N
ν
0
off
P
0
ν ref.
0
ν
0
C
P
synthes. 0
ν
output
s
ν ν
0 s
freq. |ν −ν | <<ν
s 0 0
ref.
P
0
ν ref.
0
ν ν
0 s
D
P
0
nenaori−sedc αorϕ output N
on modulat. ν
0
s off
hod P0
met ν0 ref.
-
cal ν0
Figure 2: Generally used calibration methods.
calibrationofPMnoisemeasurementsbasedonthesaturatedmixerisincorrect.
IntheschemeofFig.2C,twosinusoidalsignalsaresenttothemixer. These
signals have frequency ν and ν , with a small difference ν =|ν −ν |, and the
0 s b s 0
same power of the final application. The detected signal is a sinusoid of fre-
quency ν crossing the 0 V dc axis with slope dv =ω k , from which k can be
b dt b ϕ ϕ
measured using a digital oscilloscope. Of course, this scheme can only be used
only for the calibration of PM measurements, not for AM. The major prob-
lem with this approach is that it is impossible to assess the mixer’s unwanted
sensitivity to AM noise.
E. Rubiola, AM-PM calibration. January 8, 2009 7
The scheme of Fig. 2D makes use of a true amplitude or phase modulator.
Thisisdefinitelycorrect, providedthemodulatorwascalibrated. AvoidingAM
to PM crosstalk, that is, avoiding residual AM in the phase modulator and
residual PM in the amplitude modulator, is the critical point.
Presently, the state of the art in AM and PM noise standards is the NIST
secondary standard of AM and PM noise [31, 14]. This standard is based on
theschemeofFig.2B,thusitmakestheuseofthemixerincorrectasdiscussed
before. Additionally, however accurate, this standard is a secondary one. Com-
mercial standards of this type also exist, like that of Techtrol Cyclonetics Inc.
(TCI) [25].
A reference modulator (Fig. 2D) is in use at NIST2, calibrated with the
secondary standard. Yet, nothing is said about how to avoid the AM to PM
crosstalk.
IntheBIPMwebsite3,onlytheFrenchLNEreportsontheaccuracyofphase
noise measurements. The uncertainty is of 2 dB. However disappointing, this
value accounts for all conditions of routine measurements. The NIST4 reports
about typical accuracy of 1 dB in both AM and PM noise measurements. An
old article on the bridge [7]. The declared uncertainty is of 1.9 dB at the NPL,
UK.
2 Method
This article proposes a method for the primary calibration of AM and PM
measurements, based on the following ideas.
• We first understand that the calibration of the detector (amplitude or
phase) is the critical point, while the accuracy of the near-dc complemen-
taryinstruments(FFT,etc.) canbeeasilymadenegligibleforallpractical
purposes.
• A reference modulator is used to calibrate the gain of the detector. Of
course, we need an amplitude modulator free from phase modulation,
and a phase modulator free from amplitude modulation. The conceptual
schemeofthemodulatorisshowninFig.3. Theamplitudemodulationis
obtained by adding to the carrier V cos(ω t) a signal x(t)cos(ω t). Thus,
0 0 0
theamplitudemodulationisα(t)=x(t)/V [Equations(1)and(2)]. Sim-
0
ilarly,thephasemodulationisobtainedbyaddingasignal−y(t)cos(ω t),
0
so that ϕ(t)=y(t)/V
0
• Forourpurpose, itisnecessarytofixthequadratureerrorψ andthegain
asymmetry (cid:15) of the real I-Q modulators and detectors (Fig. 6). This can
be done with null and differential measurements only.
• Realphasedetectorsshowaresidualsensitivitytoamplitudemodulation.
This is due to non-perfect saturation. Conversely, power detectors are
notsensitivetophasemodulation,providedthevideo5 bandwidthislarge
enough to prevent the discriminator effect due to the internal memory.
Consequently, we can use null measurement with a power detector to find
2CraigNelson,PhaseNoiseMeasurements,2006FCStutorial
3http://kcdb.bipm.org/appendixC/...
4http://ts.nist.gov/MeasurementServices/Calibrations/oscillators.cfm
5Theoutputisusuallycalled‘video.’
E. Rubiola, AM-PM calibration. January 8, 2009 8
RF input RF output
V0cos(ω0t) carrier V0cos(ω0t)+xcos(ω0t)−ysin(ω0t)
LO I−Q modulator
I−Q RF modulation sidebands
modul xcos(ω0t)−ysin(ω0t) LO pump
I Q
r 90° RF
ato x y modul. input output
l
u
d
o
m
-
al I Q
c
Figure 3: Reference modulator.
the phase θ of the pump. θ is an arbitrary quantity resulting from the
electrical layout, to be corrected in order to set the appropriate phase
relationship between the modulation sidebands and the carrier.
• Switchingonandoffthemodulationsideband,themodulationismeasured
(cid:112) (cid:112)
as a power ratio. Thus, α = P /P , and ϕ = P /P , where P ,
rms x 0 rms y 0 0
P and P are the power of the carrier, of the in-phase modulation, and
x y
of the quadrature modulation.
It is to be remarked that the entire procedure proposed here relies only on null,
differential,andratiometricmeasurements. Thiswillbediscussedindetailafter
explaining some technical tools.
3 Mixers, scalar products, and Hilbert spaces
It is well known in the classical theory of telecommunications [33, 29, 12] that
with the appropriate definition of the scalar product, the space of the sinusoids
at a given frequency (ω ) has the structure of an Hilbert space. In fact
0
1. the space is a inner-product space, that is, the scalar product (x,y) has
the usual properties of conjugation and linearity, and (x,x) ≥ 0, where
(x,x)=0 only if x=0,
(cid:112)
2. the norm is ||x||= (x,x),
3. the space is complete.
In such space, we can define a orthogonal unitary base, so that all signals can
be expressed as a linear combination of the base.
3.1 Complex signals
The scalar product between two complex signals s(t) and r(t) is defined as
1 (cid:90) T
(s,r)= s(t)r∗(t)dt (6)
T
0
E. Rubiola, AM-PM calibration. January 8, 2009 9
whereT isasuitableintegrationtime,andthesuperscript*denotesthecomplex
conjugate. The base function is the complex sinusoid
r(t)=ejω0t reference (7)
||r(t)||=1 (8)
Mathematically, the integration time should be T =n2π, integer n, so that the
ω and 2ω terms are canceled. In practice, a sufficienωt0ly long T (cid:29) 2π makes
0 0 ω0
the 2ω terms negligible for any practical purposes.
0
Let us consider the scalar product of the signal
s(t)=Vejω0t+ϕ signal (9)
on the reference r(t). Expanding Eq. (6), the scalar product v =(s,r) is
1 (cid:90) T (cid:104) (cid:105)
v = V exp(jω t+ϕ)exp(−jω t) dt
T 0 0
0
1 (cid:90) T (cid:104) (cid:105)(cid:104) (cid:105)
= V cos(ω t+ϕ)+jsin(ω t+ϕ) cos(ω t)−jsin(ω t) dt
T 0 0 0 0
0
1 (cid:90) T (cid:110)(cid:104) (cid:105)
= V cos(ω t)cos(ϕ)−sin(ω t)sin(ϕ) +
T 0 0
0 (cid:104) (cid:105)(cid:111)
+j cos(ω t)sin(ϕ)−sin(ω t)cos(ϕ)
0 0
(cid:104) (cid:105)
× cos(ω t)−jsin(ω t) dt
0 0
and finally
(cid:2) (cid:3)
v =V cos(ϕ)+jsin(ϕ) (after removing the 2ω terms) (10)
0
=Vejϕ (11)
Note that the signal s(t) can be expressed as
s(t)=vr(t) (12)
The important fact is that the scalar product freezes the ω carrier frequency,
0
thus it gives the phasor v =Vejϕ associated to s(t). V is the rms voltage.
3.2 Real signals
Dealing with real signals s(t) and r(t), the scalar product (6) turns into
1 (cid:90) T
(s,r)= s(t)r(t)dt (13)
T
0
because r(t) = r∗(t), thus the complex conjugate is no longer needed. A two-
dimension base is necessary to replace the complex sinusoid
√
r (t)=+ 2cos(ω t) in-phase reference (14)
I 0
√
r (t)=− 2sin(ω t) quadrature reference (15)
Q 0
E. Rubiola, AM-PM calibration. January 8, 2009 10
√
The factor 2 is necessary for ||r || = 1 and ||r || = 1. Let us consider the
I q
scalar product of
√
s(t)=V 2cos(ω t+ϕ) signal (16)
0
on the base functions. Expanding Eq. (13), the scalar product v =(s,r ) is
I I
1 (cid:90) T
v = 2V cos(ω t+ϕ)cos(ω t)dt
I T 0 0
0
1 (cid:90) T (cid:104) (cid:105)
= 2V cos(ω t)cos(ϕ)−sin(ω t)sin(ϕ) cos(ω t)dt
T 0 0 0 0 (17)
1 (cid:90) T (cid:104) (cid:105)
= 2V cos2(ω t)cos(ϕ)−sin(ω t)cos(ω t)cos(ϕ) dt
T 0 0 0
0
=V cos(ϕ) (after removing the 2ω terms)
0
Similarly, the scalar product v =(s,r ) is
Q Q
1 (cid:90) T
v = 2V cos(ω t+ϕ)[−sin(ω t)]dt
Q T 0 0
0
1 (cid:90) T (cid:104) (cid:105)(cid:104) (cid:105)
= 2V cos(ω t)cos(ϕ)−sin(ω t)sin(ϕ) −sin(ω t) dt
T 0 0 0 0 (18)
1 (cid:90) T (cid:104) (cid:105)
= 2V sin(ω t)cos(ω t)cos(ϕ)+sin2(ω t)sin(ϕ) dt
T 0 0 0
0
=V sin(ϕ) (after removing the 2ω terms)
0
Note that the real signal s(t) can be expressed as
s(t)=v r (t)+v r (t) . (19)
I I Q Q
Once again, the signal s(t) is completely defined by the scalar projections on
the base functions. The proof is trivial
√ √
s(t)=V cos(ϕ) 2cos(ω t)−V sin(ϕ) 2sin(ω t) (20)
0 0
√
=V 2cos(ω t+ϕ) . (21)
0
3.3 I-Q detector and modulator
The I-Q detector is a hardware implementation of Eq. (17) and (18). Fig. 4
shows the conceptual scheme of an ideal I-Q detector, based on analog multi-
pliers. The rules of energy management and impedance matching are still not
considered here. In the geometrical analogy, the I-Q detector extracts the real
and imaginary part of the input phasor.
The I-Q modulator (Fig. 4) is similar to the detector, but the signal flow
is reversed. In the geometrical analogy, the I-Q modulator turns v into a real
I
phasorandv intoanimaginaryphasor,andbuildstheoutputphasorbyadding
Q
the real part and imaginary part.
