near-field correlation measurement and evaluation of stationary...
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Near-Field Correlation Measurement and Evaluationof Stationary and Cyclostationary Stochastic
Electromagnetic FieldsJohannes A. Russer∗1, Michael Haider∗, Mohd H. Baharuddin†, Christopher Smartt†, Sidina Wane‡,
Damienne Bajon§, Andrey Baev¶, Yury Kuznetsov¶, David Thomas†, and Peter Russer∗∗Institute for Nanoelectronics, Technische Universitat Munchen, Germany
†University of Nottingham, School of Electrical and Electronic Engineering, Nottingham‡NXP-Semiconductors, Caen-France
§ISAE-Universite de Toulouse, Toulouse-France¶Theoretical Radio Engineering Department, Moscow Aviation Institute, Moscow, Russia
Abstract—We present methods for measurement and evalu-ation of stationary and cyclostationary stochastic electromag-netic fields. The radiated electromagnetic interference (EMI)of electronic circuitry is recorded by two-point measurementsof the tangential electric or magnetic field components and byevaluating the field autocorrelation functions and for each pair offield sampling points also the cross correlation functions. In caseof digital circuitry clocked by a single clock pulse, the generatedEMI is a cyclostationary process where the expectation values ofthe EMI are periodically time dependent according to the clockfrequency and which have to be considered in modeling the EMI.
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Cite as: J. A. Russer, M. Haider, M. H. Baharuddin, C. Smartt, S. Wane, D. Bajon, A. Baev, Y. Kuznetsov, D. Thomas, and P. Russer,”Near-Field Correlation Measurement and Evaluation of Stationary and Cyclostationary Stochastic Electromagnetic Fields,”
in Proceedings of the 46th European Microwave Conference (EuMC) 2016,London, U.K., Oct. 3-7, 2016, pp. 481–484, doi: 10.1109/EuMC.2016.7824384
I. INTRODUCTION
Electromagnetic interference (EMI) radiated by electroniccircuitry due to switching operations of the active circuitelements degrades the performance of electronic circuits andsystems. Modeling the EMI radiated by integrated circuitsor circuit boards into a system to be designed, requires theconsideration of the stochastic EM field, usually on the basis ofthe measured radiated EMI of the components, and to establishmodels based on these measurements. Such models alreadywere discussed in literature [1]–[4]. Sampling of stochasticEM fields and source localization have been addressed in [4]–[7]. Numerical simulation methodologies based on the fieldauto- and cross correlation spectra were developed [8]–[13].
In digital circuitry switching operations synchronized by aperiodic clock generate an EMI described by a cyclostationary(CS) process i.e. a process where the expectation valuesof the EMI are periodically time dependent on the clockfrequency [14]–[18]. This has to be considered in modelingthe influence of the EMI. Figure 1 shows an example of a CSrandom signal and of the time dependence of its variance.
Due to the equivalence principle, an equivalent sourcedistribution determined by amplitude and phase scanning ofthe tangential electric or magnetic field on a surface enclosingthe radiating structure is equivalent to the internal sources andallows the modeling of the environmental field. However, due
σ2(t )scsi(t )
0.5 1.0 1.5 2.0t/T0
−1.5−1.0−0.5
0.51.01.5
s(t ), σ2(t )
Fig. 1: Time of three samples of a CS random signal scs1 withperiod T0 and of the time dependence of its variance σ2(t).
to the lack of information about the sources of the radiatedEMI, the near field to be measured has to be treated as astochastic field. The radiated EMI of electronic circuitry isrecorded by two-point measurements of the tangential electricor magnetic field components and by evaluating the fieldautocorrelation functions and for each pair of field samplingpoints also the cross correlation functions. In [19], [20], wehave extended the near-field correlation measurement methodsto CS EM fields. In this work we present experimentalinvestigations of CS noise radiated by electric and electroniccomponents, namely due to the data transfer process of amicrocontroller board, and the recording and evaluation of CSradiated EMI using two-point scanning methods.
II. THE CYCLOSTATIONARY EM FIELD
The time-domain correlation dyadic of two electric fieldvariables E(x1, t1) and E(x2, t2) at points xa and xb, attimes t1 and t2, respectively is given by
ΓE
(xa,xb, t1, t2)= limT→∞
1
2T
⟨ET (xa, t1)E
†T (xb, t2)
⟩, (1)
where the subscript T denotes the amplitude spectrum ofthe field time-windowed by a rectangular window covering
the time interval [−T, T ]. The tilde · denotes time-domainrepresentation whereas field variables without tilde refer tofrequency domain representation.
If the mean value of the field E(x, t) and also the correla-tion dyadic describing the second-order statistics are periodicwith a period T0, i.e.
〈E(x, t+ T0)〉 = 〈E(x, t)〉 , (2a)
ΓE
(xa,xb, t1 + T0, t2 + T0) = ΓE
(xa,xb, t1, t2) , (2b)
the field is said to be second-order cyclostationary in the widesense [14]–[18].
Applying discrete-time measurement systems yields a dis-cretization of the measurement quantities. A discrete-timesecond-order random process X[n]∀n ∈ N is defined to becyclostationary with period N if and only if its mean andautocorrelation are periodic [15], [21], [22]:
µX [n] ≡ 〈X[n]〉 = µX [n+ kN ] ∀k ∈ N , (3)ρX [n,m] ≡ 〈X[n]X∗[n−m]〉
= ρX [n+ kN,m] ∀n,m ∈ N . (4)
III. SPECTRAL REPRESENTATION OF THE CS EM FIELD
Expanding the correlation dyadic, periodic in time with T0,into a Fourier series yields
ΓE
(xa,xb, t, t− τ) =
+∞∑n=−∞
ΓE,n
(xa,xb, τ)enω0t , (5)
with ω0 = 2π/T0. The Fourier coefficients
ΓE,n
(xa,xb, τ) =1
T0
T0/2∫−T0/2
ΓE
(xa,xb, t, t− τ)e−nω0tdt (6)
are referred to as the cyclic time-domain autocorrelationdyadics and the frequencies nf0 = nω0/2π are the so-calledcycle frequencies [15], [19]. The cyclic correlation spectrum(CCS) of the electric field Γ
E,n(xa,xb, ω) is given by
ΓE,n
(xa,xb, ω) =
∫ΓE,n
(xa,xb, τ)e−ωτdω . (7)
The inverse Fourier transform yields the cyclic time-domaincorrelation dyadic
ΓE,n
(xa,xb, τ) =
∫ΓE,n
(xa,xb, ω)eωτdω . (8)
IV. EXCITATION OF A CS EM FIELD
Consider point-like CS stochastic EMI sources Si arrangedas shown in Fig. 2. Describing a deterministic source currentdistribution by the source current density J(x, ω), the excitedelectric field E(x, ω) is given by
E(x, ω) =
∫V
GEJ(x− x′, ω)J(x′, ω)d3x′ , (9)
where GEJ(x − x′, ω) is the Green’s dyadic relating theexcited electric field E(x, ω) to the source current density, andthe integration is extended over the whole volume V where
x
y
z
x1,y1
x2,y2
x3,y3
x4,y4
x5,y5
S1
S2
S3S4
S5
x2‘,y2‘,z‘
P2x
P2y
x1‘,y1‘,z‘
P1x
P1y
z‘0
Source Plane Scan Plane
PEC
Fig. 2: EMI sources sampled at a scan plane [10].
J(x, ω) is non-vanishing [4], [10]. We obtain from (8) and (9)the relation cyclic correlation spectra of the excitation currentdensity and the excited electric field as
ΓE,n
(xa,xb, ω) =
∫∫V
GEJ(xa − x′a, ω)× (10)
ΓJ,n
(xa,xb, ω)G†EJ(xb − x′b, ω − nω0)d3x′ad3x′b .
In the near-field the Green’s dyadic is given by
GEJ(x, ω) =[g1(x, ω)1 + g2(x, ω)xxT
]e−β(ω)|x| (11)
with
g1(x, β) = − ZF0β2
4π
[1
β|x|+
β2|x|2− 1
β3|x|3
], (12a)
g2(x, β) =ZF0β
2
4π
[1
β|x|3+
3
β2|x|4− 3
β3|x|5
], (12b)
where ZF0 =√ε0/µ0 is the free space wave impedance, and
the T denotes the transpose of the vector.
V. CORRELATION OF THE SIGNALS MEASURED BY TWOPROBES IN THE NEAR-FIELD
Following [9], [10] we extend the method applied for themodeling of stationary electromagnetic fields to the investi-gation of CS electromagnetic fields. The field of localizedpoint-like CS stochastic source currents at the locations xµis represented by the CCS dyadic
ΓJ,n
(xa,xb, ω) =
N∑µ=1
N∑ν=1
lµlνCI,n,µν(ω)ΩµΩTν×
δ (xa − xµ) δ (xb − xν) . (13)
The cyclic correlation spectra matrix elements of the dipolecurrents IT,µ(ω) is given by
CI,n,µν(ω) =
+∞∫−∞
cI,n,µν(τ)enω0tdt , (14)
where the cyclic correlation time-domain matrix elements are
cI,n,µν(τ) =1
T0
T0/2∫−T0/2
〈iT,µ(t)iT,ν(t− τ)〉 e−nω0tdt. (15)
Fig. 3: Measurement setup.
0 50 100 150 200 250 300 350 400 450 500
Time (mks)
-10
-5
0
5
10
Level (m
V)
Fig. 4: Realization of the stochastic signal.
Finally, with (10) we obtain the CCS dyadic of the stochasticelectric field vectors at the points xa and xb as
ΓE,n
(xa,xb, ω) =
N∑µ=1
N∑ν=1
lµlνCI,n,µν(ω)× (16)
GEJ(xa − xµ, ω)ΩµΩTνG†EJ(xb − xν , ω − nω0) .
VI. EXPERIMENTAL INVESTIGATION AND RESULTS
Initial measurements are presented in this paper for thepurpose of investigating the structure of the emission spectraso as to try and design efficient means of characterizing thefield-field correlation spectrum of real devices by measure-ment. Initially, a single probe scan above a PCB is performedto establish the location of major sources of radiation. TwoLanger EMV-Technik RF R50-1 magnetic field probes areconnected to the input ports of an Agilent MSO8104A digitaltime domain oscilloscope. Figure 3 shows the two probe scan-ning measurement in progress for an Arduino Galileo board.Figure 4 shows a typical realization of the stochastic signalmeasured by a near-field probe of the tangential magnetic fieldradiated by the Arduino Galileo board which is programmedfor data transfer between central processor unit (CPU) andinternal memory.
The structure of the measured realization reveals the pres-ence of a regularity in the measured signal and presumescyclostationary properties of the observed stochastic process.
Implementation of the ensemble averaging procedure as-sumes the necessity of a synchronization with the periodicprocess intrinsic to the technology for data transfer in digitaldevices. The synchronized ensemble of the measured realiza-
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Time, [mks]
-10
-5
0
5
10
Le
ve
l, [
mV
]
Mean
Fig. 5: Ensemble of periodic signals.
Fig. 6: Periodic Autocorrelation Function.
tions and its mean, which is a time dependent function, areshown in the Fig. 5.
The information region of the periodic autocorrelation func-tion (4) of the cyclostationary stochastic process is presented inFig. 6. This two-dimensional function shows the dependenceof the autocorrelation from global time and relative time shiftof the stochastic process. The periodic correlation function canbe represented by Fourier series expansion assuming that Nis odd as
ρX [n,m] =
(N−1)/2∑l=−(N−1)/2
RlX [m]e2πlnN , (17)
where RlX [m] are Fourier series coefficients
RlX [m] =1
N
(N−1)/2∑n=−(N−1)/2
ρX [n,m]e−2πlnN . (18)
The Fourier coefficients RlX [m] for the time-discrete pro-cess, compare (6), are called cyclic autocorrelation functions,where l/N, ∀l ∈ N are the cyclic frequencies. This two-dimensional function, visualized in Fig. 7, shows the structureof the Fourier series spectrum for different relative timeshifts of the stochastic process: the correlation interval around40 ns and the spectral width of nearly 1000 MHz of thecyclostationary stochastic process is clearly seen. The Fourier
-2000 -1500 -1000 -500 0 500 1000 1500
Cyclic frequency (MHz)
-50
-40
-30
-20
-10
0
10
20
30
40
Re
lative
tim
e (
ns)
Fig. 7: Cyclic Autocorrelation Function.
-2000 -1500 -1000 -500 0 500 1000 1500
Cyclic frequency (MHz)
-2000
-1500
-1000
-500
0
500
1000
1500
Cyclic
fre
qu
en
cy (
MH
z)
Fig. 8: Spectral Correlation Density Function.
series transform of the cyclic autocorrelation function givesthe spectral correlation density function as
SlX [v] =1
N
(N−1)/2∑m=−(N−1)/2
RlX [m]e−2πvmN , (19)
where v is a discrete frequency. This function, graphed inFig. 8, represents the spectral correlation of global (horizontalaxis) and relative cyclic frequencys content. In particular,spectral correlation reveals a coherent clock frequency of 400MHz and its harmonics (vertical red lines) while cyclic auto-correlation function in Fig. 6 represents only cyclic periodicproperties of the cyclostationary process. The experimentalresults demonstrate the ability of parameter estimation forthe radiated emissions by using the cyclostationary stochasticcharacteristics obtained from the measured random signals inthe near-field region of the digital device.
VII. CONCLUSION AND OUTLOOK
We presented measurements and analysis of correlationdata of CS noisy EM fields. We have extended the near-fieldcorrelation measurement methods to CS EM fields and couldprovide, based on these methods, parameter estimation for CSprocesses on the investigated Adruino microcontroller board.
ACKNOWLEDGMENT
This work was supported in part by the German ResearchFoundation (DFG), under grant no. Ru 314/45-1, the Russian
Foundation for Basic Research Grant 15-08-08951, by anSTSM grant from COST ACTION IC1407 ’ACCREDIT’,and by the European Union’s Horizon 2020 research andinnovation programme under grant no. 664828 (NEMF21).
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