Cyclostationary Characterization of Radiated Emissions Professor, Dr

COST Action IC1407

Cyclostationary Characterization of Radiated Emissions Professor, Dr. Yury Kuznetsov Moscow Aviation Institute (National Research University) Russian Federation

COST Training School: EMC for Future Highly Integrated Systems

MOSCOW AVIATION 01.04.2019 INSTITUTE Outline

1. Motivation 2. Stationary and cyclostationary processes 3. Characteristics of cyclostationary processes 4. Degree of cyclostationarity for stochastic processes 5. Cross-talk characterization 6. Cyclostationary EMI sources separation 7. Conclusion

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 2 INSTITUTE Motivation

 Electromagnetic field (EMF) emitted by the printed circuit board (PCB) contains a stochastic component sensing by the near-field scanning probe and registering by the real-time digital oscilloscope  Statistical signal processing can be used for the characterization of the received stochastic EMF and for the localization and identification of the radiating sources  The measured stochastic signals contain the superposition of different radiating sources located in the environment of the DUT:  emissions from transmission lines transferring the data sequences between distinct blocks of the DUT;  thermal noise and interference from surrounding radiations

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 3 INSTITUTE Outline

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 4 INSTITUTE Classification of random processes

Random process

Time sequence of random The set of random signals variables = , : Ω , = , = 1, is an elementary event 𝒮𝒮 𝑡𝑡 Ω is 𝑆𝑆a sample𝑡𝑡 𝜔𝜔 𝜔𝜔 space∈ 𝑡𝑡 ∈ ℝ = 𝜔𝜔 is 𝒮𝒮a random𝑡𝑡 𝑆𝑆 variable𝑡𝑡𝑖𝑖 𝑖𝑖 at 𝑛𝑛 is a set of real numbers 𝑆𝑆𝑡𝑡𝑖𝑖 𝑡𝑡 𝑡𝑡𝑖𝑖 ℝ  Stochastic phenomena can be mathematically characterized by the model of random process constructed from the measured random signals  The random variable is fully described by the probability density function for each time t = ti  Random signal is a time domain realization of the stochastic phenomena for each event Ω

MOSCOW 𝜔𝜔 ∈ AVIATION COST IC 1407 Training School – Prague, 2019 5 INSTITUTE Cumulative distribution function

 The cumulative distribution function of the stochastic process is:

, = , = : , d = : , Ω 𝐹𝐹𝒮𝒮 𝑠𝑠 𝑡𝑡 𝑃𝑃 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤ 𝑠𝑠 � 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 𝑃𝑃 𝜔𝜔 𝐸𝐸 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 1, : ,  = indicator function : , 0, : , > 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤ 𝑠𝑠 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 �  is the operator of statistical expectation𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 or𝑠𝑠 ensemble averaging1 using Lebesgue integral 𝐸𝐸 �  The expected value of the stochastic process is the statistical mean

= , = , d = d , Ω 𝒮𝒮 𝒮𝒮 𝑚𝑚 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜔𝜔 � 𝑆𝑆 𝑡𝑡 𝜔𝜔 𝑃𝑃 𝜔𝜔 �ℝ 𝑠𝑠 𝐹𝐹 𝑠𝑠 𝑡𝑡 1 A. Napolitano, Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications. John Wiley & Sons Ltd - IEEE Press, 2012.

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 6 INSTITUTE Second-order characterization

 The second-order joint cumulative distribution function of the stochastic process is: , ; , = , , ,

𝐹𝐹𝒮𝒮 𝑠𝑠1 𝑠𝑠2 𝑡𝑡1 𝑡𝑡2 = 𝑃𝑃 𝑆𝑆 𝑡𝑡1: 𝜔𝜔, ≤ 𝑠𝑠1 𝑆𝑆 :𝑡𝑡2 𝜔𝜔, ≤ 𝑠𝑠2 𝐸𝐸 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡1 𝜔𝜔 ≤𝑠𝑠1 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡2 𝜔𝜔 ≤𝑠𝑠2  The autocorrelation function of the stochastic process is: , = , + , = d , ; , +

𝒮𝒮 2 1 2 𝒮𝒮 1 2 ℛ 𝑡𝑡 𝜏𝜏 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝜔𝜔 �ℝ 𝑠𝑠 𝑠𝑠 𝐹𝐹 𝑠𝑠 𝑠𝑠 𝑡𝑡 𝑡𝑡 𝜏𝜏  The autocovariance of the stochastic process is its autocorrelation function of the zero-mean process: , = , + , +

𝒞𝒞𝒮𝒮 𝑡𝑡 𝜏𝜏 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜔𝜔 − 𝑚𝑚𝒮𝒮 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝜔𝜔 − 𝑚𝑚𝒮𝒮 𝑡𝑡 𝜏𝜏

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 7 INSTITUTE Spectral characterization  The second order stochastic process is harmonizable and can be characterized in the frequency domain if its autocorrelation function can be expressed by the Fourier-Stieltjes integral:

, , = d , 𝑗𝑗2𝜋𝜋 𝑓𝑓1𝑡𝑡1+𝑓𝑓2𝑡𝑡2  , 1 2 2 𝒮𝒮 1 2 where 𝐸𝐸 𝑆𝑆 𝑡𝑡 is𝜔𝜔 a 𝑆𝑆spectral𝑡𝑡 𝜔𝜔 correlation�ℝ 𝑒𝑒 function𝒢𝒢 with𝑓𝑓 𝑓𝑓 bounded variation. It is also known that a stochastic process is harmonizable if 𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2 and only if its covariance function is harmonizable

 The Fourier transform of the realization , of the harmonizable stochastic process can be expressed as: 𝑆𝑆 𝑡𝑡 𝜔𝜔 , = , d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  , ̂ where can contain𝑆𝑆 𝑓𝑓 𝜔𝜔 Dirac�ℝ 𝑆𝑆delta𝑡𝑡 𝜔𝜔 functions𝑒𝑒 𝑡𝑡

MOSCOW 𝑆𝑆̂ 𝑓𝑓 𝜔𝜔 AVIATION COST IC 1407 Training School – Prague, 2019 8 INSTITUTE Time-frequency relation

 The spectral correlation function (Loève bifrequency spectrum) of the harmonizable stochastic process is defined as: , = , ,

𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2 𝐸𝐸 𝑆𝑆̂ 𝑓𝑓1 𝜔𝜔 𝑆𝑆̂ 𝑓𝑓2 𝜔𝜔  The relation between the autocorrelation function and the spectral correlation function is defined by two-dimensional Fourier transform:

= , d d 𝑗𝑗2𝜋𝜋 𝑓𝑓1𝑡𝑡1+𝑓𝑓2𝑡𝑡2 𝐸𝐸 𝑆𝑆 𝑡𝑡1 𝑆𝑆 𝑡𝑡2 � 2 𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2 𝑒𝑒 𝑓𝑓1 𝑓𝑓2 , = ℝ d d −𝑗𝑗2𝜋𝜋 𝑓𝑓1𝑡𝑡1+𝑓𝑓2𝑡𝑡2 𝒮𝒮 1 2 2 1 2 1 2 𝒢𝒢 𝑓𝑓 𝑓𝑓 �ℝ 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝑒𝑒 𝑡𝑡 𝑡𝑡

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 9 INSTITUTE Time-variant spectrum

 The time-variant spectrum of the stochastic process is the Fourier transform of the autocorrelation function with respect to the lag parameter = :

2 1 𝜏𝜏 𝑡𝑡 − 𝑡𝑡 , = , d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝒮𝒮 𝒮𝒮 𝒱𝒱 𝑡𝑡 𝑓𝑓 �ℝ ℛ 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝜏𝜏  By introducing the variables = + /2 and = /2 it can be obtained a time-frequency representation in terms of Wigner-Ville 𝑡𝑡1 𝑡𝑡 𝜏𝜏 𝑡𝑡2 𝑡𝑡 − 𝜏𝜏 spectrum for stochastic processes:

, = + /2 /2 d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝒲𝒲𝒮𝒮 𝑡𝑡 𝑓𝑓 � 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝑆𝑆 𝑡𝑡 − 𝜏𝜏 𝑒𝑒 𝜏𝜏 = ℝ + /2 /2 d 𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 ̂ ̂ �ℝ 𝐸𝐸 𝑆𝑆 𝑓𝑓 𝜈𝜈 𝑆𝑆 𝑓𝑓 − 𝜈𝜈 𝑒𝑒 𝜈𝜈

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 10 INSTITUTE Characteristics of random processes

 Distribution of the probability over random values of the random process  Ensemble averaging of the random process by using expectation operator  Statistical mean and two-dimensional autocorrelation function in time domain  Spectral correlation function in bi-frequency domain  Time-variant and Wigner-Ville spectra in time-frequency domain  Probabilistic approach is more theoretical then practical

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 11 INSTITUTE Stationary process

 Second-order wide-sense stationary process (WSS) can be characterized by an autocorrelation function (ACF) and a power spectral density linked by the Wiener-Khinchin relation

, = = d 𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 ℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 𝑅𝑅𝒮𝒮 𝜏𝜏 � 𝑉𝑉𝒮𝒮 𝑓𝑓 𝑒𝑒 𝑓𝑓 , = = ℝ d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝒮𝒮 𝒮𝒮 𝒮𝒮 𝒱𝒱 𝑡𝑡 𝑓𝑓 𝑉𝑉 𝑓𝑓 �ℝ 𝑅𝑅 𝜏𝜏 𝑒𝑒 𝜏𝜏  Due to the dependency of the ACF for the WSS process only from = , the spectral correlation function , can be non-zero only for = 𝜏𝜏 𝑡𝑡2 − 𝑡𝑡1 = 𝒢𝒢𝒮𝒮 𝑓𝑓1, 𝑓𝑓2 d d 𝑓𝑓1 −𝑓𝑓2 𝑗𝑗2𝜋𝜋 𝑓𝑓1𝑡𝑡1+𝑓𝑓2 𝑡𝑡1−𝜏𝜏 𝐸𝐸 𝑆𝑆 𝑡𝑡1 𝑆𝑆 𝑡𝑡1 − 𝜏𝜏 � 2 𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2 𝑒𝑒 𝑓𝑓1 𝑓𝑓2 = ℝ , d d 𝑗𝑗2𝜋𝜋𝑓𝑓1𝜏𝜏 2 𝒮𝒮 1 1 1 2 �ℝ 𝒢𝒢 𝑓𝑓 −𝑓𝑓 𝑒𝑒 𝑓𝑓 𝑓𝑓

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 12 INSTITUTE Stationary process  Random process realization

0𝑆𝑆 𝑡𝑡

𝑡𝑡  Autocorrelation function

𝑅𝑅𝒮𝒮 𝜏𝜏

1/ 0 1/ 𝜏𝜏  Power− spectral𝛼𝛼 𝛼𝛼 density

𝑉𝑉𝒮𝒮 𝑓𝑓

0 𝑓𝑓

MOSCOW 𝛼𝛼 𝛼𝛼 AVIATION COST IC 1407− Training School – Prague, 2019 13 INSTITUTE 2𝜋𝜋 2𝜋𝜋 Discrete-time random process  Discrete time process can be analyzed as the sampled continuous time realizations of the stochastic processes:

= = = ∞

𝒮𝒮 𝑛𝑛 𝒮𝒮 𝑡𝑡 𝑛𝑛∆ � 𝑆𝑆𝑘𝑘𝛿𝛿 𝑛𝑛 − 𝑘𝑘  where is a sample interval. The Discrete𝑘𝑘=−∞ Time Fourier Transform (DTFT) of the realization of the discrete stochastic process can be expressed∆ as: 𝑆𝑆 𝑛𝑛

= ∞ −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝑆𝑆̂ 𝜑𝜑 � 𝑆𝑆𝑛𝑛𝑒𝑒  where = is a normalizes frequency.𝑛𝑛=−∞ The inverse DTFT gives the initial realization of the discretized stochastic process: 𝜑𝜑 𝑓𝑓∆ / = 1 2 / 𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝑆𝑆 𝑛𝑛 � 𝑆𝑆̂ 𝜑𝜑 𝑒𝑒 MOSCOW −1 2 AVIATION COST IC 1407 Training School – Prague, 2019 14 INSTITUTE Discrete-time stationary process  Discrete time process

𝒮𝒮 𝑛𝑛

𝑛𝑛  Discrete autocorrelation function [ ]

𝑅𝑅𝒮𝒮 𝑚𝑚

1 0 1 𝑚𝑚  Periodical −power spectral density

𝑉𝑉𝒮𝒮 𝑓𝑓

2 1 0 1 2 MOSCOW 𝜑𝜑 AVIATION −COST IC 1407− Training School – Prague, 2019 15 INSTITUTE Characteristics of discrete random process  The second order discretized stochastic process is harmonizable and can be characterized in the frequency domain if its autocorrelation function can be expressed by the Fourier-Stieltjes integral: = d , / , / 𝑗𝑗2𝜋𝜋 𝜑𝜑1𝑛𝑛1+𝜑𝜑2𝑛𝑛2 1 2 2 𝒮𝒮 1 2  The spectral𝐸𝐸 𝑆𝑆 𝑛𝑛 correlation𝑆𝑆 𝑛𝑛 � function−1 2 1 2 𝑒𝑒(Loève bifrequency𝒢𝒢 𝜑𝜑 spectrum)𝜑𝜑 of the harmonizable discretized stochastic process is defined as: , =

 The relation between𝒢𝒢 the𝒮𝒮 𝜑𝜑 autocorrelation1 𝜑𝜑2 𝐸𝐸 𝑆𝑆̂ 𝜑𝜑1 function𝑆𝑆̂ 𝜑𝜑2 and the spectral correlation function is defined by two-dimensional DTFT: = , d d / , / 𝑗𝑗2𝜋𝜋 𝜑𝜑1𝑛𝑛1+𝜑𝜑2𝑛𝑛2 1 2 2 𝒮𝒮 1 2 1 2 𝐸𝐸 𝑆𝑆 𝑛𝑛 𝑆𝑆 𝑛𝑛 �−1 2 1 2 𝒢𝒢 𝜑𝜑 𝜑𝜑 𝑒𝑒 𝜑𝜑 𝜑𝜑 , = ∞ ∞ −𝑗𝑗2𝜋𝜋 𝜑𝜑1𝑛𝑛1+𝜑𝜑2𝑛𝑛2 𝒢𝒢𝒮𝒮 𝜑𝜑1 𝜑𝜑2 � � 𝐸𝐸 𝑆𝑆 𝑛𝑛1 𝑆𝑆 𝑛𝑛2 𝑒𝑒 MOSCOW 𝑛𝑛1=−∞ 𝑛𝑛2=−∞ AVIATION COST IC 1407 Training School – Prague, 2019 16 INSTITUTE Characteristics of stationary random processes

 Distribution of the probability over random values of the random process doesn’t depend of time  Ensemble averaging of the random process by using expectation operator  Statistical mean and autocorrelation function doesn’t depend of time  Spectral correlation function in a Fourier transform of the ACF  Discretization of the stationary random processes gives the periodic power spectrum density and discrete ACF

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 17 INSTITUTE Outline

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 18 INSTITUTE Cyclostationary random process

 The cyclostationary random process is a non-stationary stochastic process whose statistical properties are periodically vary 𝒮𝒮 𝑡𝑡 with respect to time. The period is called a cycle, and its inverse = 1/ is a cyclic frequency. 𝑇𝑇0 𝛼𝛼 𝑇𝑇0  The process is almost-cyclostationary (ACS) if its statistical properties can be represented by a superposition of periodic functions with distinct cyclic frequencies .

𝛼𝛼 ∈ 𝒜𝒜

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 19 INSTITUTE Cyclostationary random process  Periodic autocorrelation function of the ACS stochastic process posses the periodicity in time and can be expressed by the Fourier series expansion1: , = + = , 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗�  where is ℛthe𝒮𝒮 𝑡𝑡operator𝜏𝜏 𝐸𝐸 𝑆𝑆 of𝑡𝑡 ensemble𝑆𝑆 𝑡𝑡 𝜏𝜏 averaging:� 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝛼𝛼∈𝒜𝒜 1 𝐸𝐸 � + = lim + + + + 1 𝑁𝑁 𝑇𝑇0 0 0 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝑁𝑁→∞ � 𝑆𝑆 𝑡𝑡 𝑛𝑛𝑇𝑇 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝑛𝑛𝑇𝑇 2𝑁𝑁 𝑛𝑛=−𝑁𝑁  The Fourier coefficients of , are called cyclic autocorrelation functions and can be defined by a time domain cyclic averaging for each ℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 known cyclic frequency: 1 / , = lim , d 𝑇𝑇 2 / −𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝒮𝒮 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 𝑇𝑇→∞ � ℛ 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝑡𝑡 1 W. A. Gardner, Introduction to Random Processes𝑇𝑇 −𝑇𝑇 2 with Applications to Signals and Systems. Macmillan, New York, 1985 (2nd Edition McGraw-Hill, New York, 1990). MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 20 INSTITUTE Example of cyclostationary process  Pulse amplitude modulated (PAM) signal

𝑆𝑆 𝑡𝑡 / 0 1 2 3 4 5 6 7 8 9 𝑡𝑡 Δ  Periodic ACF  Cyclic ACF , ,

𝑅𝑅𝒮𝒮 𝑡𝑡 𝜏𝜏 / 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 1 0 1 / 2 5 𝜏𝜏 Δ 4 − 𝜏𝜏 Δ − 1 − 3 − 2 − 0 − 1 − 0 1 − 1 2 2 3 4 3 5 1 0 1 /  −where is a cycle of PAM signal𝛼𝛼 ⋅ Δ 𝑡𝑡 Δ MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 21 INSTITUTE Δ Cyclic autocorrelation function  The magnitude and phase of , represent amplitude and phase of the additive complex harmonic component at frequency for time lag 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 contained in the autocorrelation function of the ACS stochastic process 𝛼𝛼 𝜏𝜏 , .  = 0 Forℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 the cyclic autocorrelation function reduces to the autocorrelation function of the stationary random process . 𝛼𝛼  = = 0 For a zero-mean stochastic process the𝑅𝑅𝒮𝒮 𝜏𝜏magnitude of the cyclic autocorrelation functions , 0 as . If the mean 𝑚𝑚𝒮𝒮 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 function of the stochastic process = , 0, then some 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 → 𝜏𝜏 → ∞ , contain additive sinusoidal functions of , which arise from the 𝑚𝑚𝒮𝒮 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≠ products of sinusoidal terms in . Such ACS processes are called 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 𝜏𝜏 unpure and need to be processed accounting on such property of the 𝑚𝑚𝒮𝒮 𝑡𝑡 process1.

1 J. Antoni, “Cyclostationarity by examples,” Mechanical Systems and Signal Processing 23(4), 2009, pp. 987–1036.

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 22 INSTITUTE Cyclic autocovariance function

 For cyclo-ergodic stochastic process the pure cyclic autocorrelation function called cyclic autocovariance function can be evaluated by synchronize removing the deterministic mean function from realization of the ACS process: 1 / , = lim + + d 𝑇𝑇 2 / −𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝒮𝒮 𝑆𝑆 𝑆𝑆 𝐶𝐶 𝛼𝛼 𝜏𝜏 𝑇𝑇→∞ � 𝐸𝐸 𝑆𝑆 𝑡𝑡 − 𝜇𝜇 𝑡𝑡 � 𝑆𝑆 𝑡𝑡 𝜏𝜏 − 𝜇𝜇 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝑡𝑡  where periodic𝑇𝑇 mean−𝑇𝑇 2 function 1 = = lim + 2 + 1 𝐾𝐾 𝑆𝑆 0 𝜇𝜇 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝐾𝐾→∞ � 𝑆𝑆 𝑡𝑡 𝑘𝑘𝑇𝑇 𝐾𝐾 𝑘𝑘=−𝐾𝐾

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 23 INSTITUTE Time-frequency relation  The almost-periodic time-variant spectrum of the ACS process is the Fourier transform of the cyclic autocorrelation function , :

, = , ℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗�  where the cyclic spectrum𝒱𝒱𝒮𝒮 𝑡𝑡 𝑓𝑓 correlation� 𝑉𝑉𝒮𝒮 𝛼𝛼 function𝑓𝑓 𝑒𝑒 , can be defined 𝛼𝛼∈𝒜𝒜 by the Fourier transform of the autocorrelation function , : 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 , = , d 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝒮𝒮 𝒮𝒮 𝑉𝑉 𝛼𝛼 𝑓𝑓 �ℝ 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝜏𝜏 Re , Im , ,

𝒱𝒱𝒮𝒮 𝑡𝑡 𝑓𝑓 𝒱𝒱𝒮𝒮 𝑡𝑡 𝑓𝑓 5 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 0 0 4 𝑓𝑓 ⋅ Δ 𝑓𝑓 ⋅ Δ 3 𝑓𝑓 ⋅ Δ 1 1 − 2 − 1 2 2 − 0 − 1 3 3 − 2 3 4 4 4 5 5 5 / /

MOSCOW 𝑡𝑡 Δ 𝑡𝑡 Δ 𝛼𝛼 ⋅ Δ AVIATION COST IC 1407 Training School – Prague, 2019 24 INSTITUTE Cyclic spectrum

 The alternative approach for the evaluation of the cyclic spectrum correlation function is the averaging of the short-time Fourier transforms (STFT) of the stochastic process realizations :

/ 𝑆𝑆 𝑡𝑡 , = d / 𝑡𝑡+1 ∆𝑓𝑓 / −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 1 ∆𝑓𝑓 𝑋𝑋 𝑡𝑡 𝑓𝑓 �𝑡𝑡−1 ∆𝑓𝑓 𝑆𝑆 𝜉𝜉 𝑒𝑒 𝜉𝜉  The averaging of / , can be implemented by two strictly ordered successive limits: 𝑋𝑋1 ∆𝑓𝑓 𝑡𝑡 𝑓𝑓 / , = lim lim , , d 𝑇𝑇 2 / / / ∆𝑓𝑓 1 ∆𝑓𝑓 1 ∆𝑓𝑓 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 ∆𝑓𝑓→0 𝑇𝑇→∞ � 𝐸𝐸 𝑋𝑋 𝑡𝑡 𝑓𝑓 𝑋𝑋 𝑡𝑡 𝛼𝛼 − 𝑓𝑓 𝑡𝑡 𝑇𝑇 −𝑇𝑇 2

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 25 INSTITUTE Wigner-Ville cyclic spectrum

,  The Wigner-Ville spectrum for ACS stochastic processes: 𝑊𝑊𝒮𝒮 𝑡𝑡 𝑓𝑓 0 , = , + /2 𝑓𝑓 ⋅ Δ 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 1 𝒮𝒮 𝒮𝒮  The𝑊𝑊 spectral𝑡𝑡 𝑓𝑓 � correlation𝑉𝑉 𝛼𝛼 𝑓𝑓 function𝛼𝛼 𝑒𝑒 𝛼𝛼∈𝒜𝒜 2 (Loève bi-frequency spectrum) of the ACS stochastic process is defined as: 3 4 , = , + 5 𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2 � 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓1 𝛿𝛿 𝑓𝑓2 𝑓𝑓1 − 𝛼𝛼 / 𝛼𝛼∈𝒜𝒜 𝑡𝑡 Δ

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 26 INSTITUTE Cross-correlation function

𝑆𝑆 𝑡𝑡 𝑌𝑌1 𝑡𝑡 1 𝑡𝑡 𝑆𝑆 𝑡𝑡 ℎ 𝑡𝑡 𝑌𝑌2 𝑡𝑡 ℎ2 𝑡𝑡 , = = =

1 2 𝑌𝑌 𝑌𝑌 1 2 1 1 2 2 2 1 1 1 2 2 2 1 2 1 2 ℛ 𝑡𝑡 𝑡𝑡 𝐸𝐸 𝑌𝑌 𝑡𝑡 𝑌𝑌 𝑡𝑡 �ℝ ℎ 𝑡𝑡 − 𝜏𝜏 ℎ 𝑡𝑡, − 𝜏𝜏 𝐸𝐸 𝑆𝑆 𝜏𝜏 𝑆𝑆 𝜏𝜏 𝑑𝑑𝜏𝜏 𝑑𝑑𝜏𝜏

2 1 1 1 2 2 2 𝒮𝒮 1 2 1 2  Two different ACS� stochasticℝ ℎ 𝑡𝑡 − 𝜏𝜏processesℎ 𝑡𝑡 − 𝜏𝜏 ℛ and𝜏𝜏 𝜏𝜏 𝑑𝑑𝜏𝜏 𝑑𝑑are𝜏𝜏 said to be jointly correlated if the second-order cross-correlation function 𝑌𝑌1 𝑡𝑡 𝑌𝑌2 𝑡𝑡 , = + = , 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝑌𝑌1𝑌𝑌2 1 2 𝑌𝑌1𝑌𝑌2 at cycle frequenciesℛ 𝑡𝑡 𝜏𝜏 𝐸𝐸 𝑌𝑌is𝑡𝑡 defined𝑌𝑌 𝑡𝑡 by𝜏𝜏 non-zero� cyclic𝑅𝑅 cross𝛼𝛼 𝜏𝜏 -𝑒𝑒correlation 𝛼𝛼∈𝒜𝒜12 functions 12 𝛼𝛼 ∈ 𝒜𝒜 1 / , = lim , d 𝑇𝑇 2 / −𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝑌𝑌1𝑌𝑌2 𝑌𝑌1𝑌𝑌2 MOSCOW 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑇𝑇→∞ �−𝑇𝑇 2ℛ 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝑡𝑡 AVIATION COST IC 1407 Training𝑇𝑇 School – Prague, 2019 27 INSTITUTE Cross-correlation spectrum

 The cyclic spectrum cross-correlation , can be defined by the Fourier transform of the cyclic cross-correlation function , : 𝑉𝑉𝑌𝑌1𝑌𝑌2 𝛼𝛼 𝑓𝑓 , = , d 𝑅𝑅𝑌𝑌1𝑌𝑌2 𝛼𝛼 𝜏𝜏 −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  It can be independently𝑌𝑌1𝑌𝑌2 evaluated by𝑌𝑌1 𝑌𝑌the2 averaging of the short-time Fourier 𝑉𝑉 𝛼𝛼 𝑓𝑓 �ℝ 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝜏𝜏 transforms (STFT) of the stochastic process realizations and : / , = lim lim , ,1 d 2 𝑇𝑇 2 , / , / 𝑌𝑌 𝑡𝑡 𝑌𝑌 𝑡𝑡 / 𝑌𝑌1𝑌𝑌2 ∆𝑓𝑓 1 1 ∆𝑓𝑓 2 1 ∆𝑓𝑓 where 𝑉𝑉 𝛼𝛼 𝑓𝑓 ∆𝑓𝑓→0 𝑇𝑇→∞ �−𝑇𝑇 2𝐸𝐸 𝑋𝑋 𝑡𝑡 𝑓𝑓 𝑋𝑋 𝑡𝑡 𝛼𝛼 − 𝑓𝑓 𝑡𝑡 𝑇𝑇 / , = d ; = 1,2 , / 𝑡𝑡+1 ∆𝑓𝑓 / −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  𝑖𝑖 1 ∆𝑓𝑓 𝑖𝑖 For the estimation𝑋𝑋 of 𝑡𝑡second𝑓𝑓 �-𝑡𝑡order−1 ∆𝑓𝑓 statistical𝑌𝑌 𝜉𝜉 𝑒𝑒 functions𝜉𝜉 𝑖𝑖 of ACS stochastic process need to have finite or “effectively finite” memory. It means that ACS process need to be a zero-mean stochastic process = = 0.

𝑚𝑚𝒮𝒮 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 28 INSTITUTE Relation between characteristics

Cyclic Classical Autocorrelation autocorrelation autocorrelation function function function = 0 , , Fourier𝑡𝑡 ↔ 𝛼𝛼 𝛼𝛼 ℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 series 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 𝑅𝑅𝒮𝒮 𝜏𝜏 Fourier Fourier Fourier transform transform transform 𝜏𝜏 ↔ 𝑓𝑓 𝜏𝜏 ↔ 𝑓𝑓 𝜏𝜏 ↔ 𝑓𝑓 = 0 , , Fourier 𝑡𝑡 ↔ 𝛼𝛼 𝛼𝛼 𝑊𝑊𝒮𝒮 𝑡𝑡 𝑓𝑓 series 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 𝑉𝑉𝒮𝒮 𝑓𝑓 Wigner-Ville Spectral Power spectrum correlation spectral density density

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 29 INSTITUTE Characteristics of cyclostationary process

 Mean function and periodic autocorrelation function of the cyclostationary process can be expressed by a superposition of periodic functions with different periods  Cyclic autocorrelation function can be obtained by the time- domain cyclic averaging of the non-linear time-shift transformation  To obtain the pure cyclic ACF the mean function need to be removed from the realizations of the random process  Cyclic spectral correlation function can be evaluated by the frequency-domain averaging of the frequency-shifted Fourier transforms of the measured realizations

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 30 INSTITUTE Outline

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 31 INSTITUTE Degree of cyclostationarity  The instantaneous probabilistic autocorrelation function of a wide-sense nonstationary stochastic process , = + 𝑋𝑋 𝑡𝑡 2 2  𝜏𝜏 𝜏𝜏 , For a wide-sense stationaryℛ𝑋𝑋 𝑡𝑡 𝜏𝜏 stochastic𝐸𝐸 𝑋𝑋 𝑡𝑡 process𝑋𝑋 𝑡𝑡 − is independent of the time-location parameter 𝑋𝑋 , = ℛ 𝑡𝑡 𝜏𝜏 𝑡𝑡  As a measure of the degree of cyclostationarity (DCS)1 of a process ℛ𝑋𝑋 𝑡𝑡 𝜏𝜏 ℛ𝑋𝑋 𝜏𝜏 can be used normalized time-averaged lag-integrated squared difference between the periodic autocorrelation function , and the 𝑋𝑋 𝑡𝑡 closest stationary autocorrelation : ℛ𝑋𝑋 𝑡𝑡 𝜏𝜏 ℛ𝑋𝑋, 𝜏𝜏 DCS ∞ 2 ∫−∞ 𝐸𝐸 ℛ𝑋𝑋 𝑡𝑡 𝜏𝜏 − ℛ𝑋𝑋 𝜏𝜏 𝑑𝑑𝜏𝜏 ≜ ∞ 2 −∞ 𝑋𝑋 1 Goran D. Živanović, William A. Gardner,∫ “Degreesℛ 𝜏𝜏of cyclostationarity𝑑𝑑𝑑𝑑 and their application to signal detection and estimation,” Signal Processing, vol. 22, N 3, 1991, pp. 287–297.

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 32 INSTITUTE Pulse-amplitude modulated signal  Model of signal  Autocorrelation function

 Cyclic ACF  Periodic ACF

 DCS = 0.5

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 33 INSTITUTE Measurement setup

Oscilloscope Scanning plane

Near-field probe

 Langer EMV-Technik RF-R 50-1 magnetic field probes  Frequency band from 30 MHz up to 3 GHz  13 GHz Serial Data Analyzer LeCroy SDA 813Zi-A

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 34 INSTITUTE Stationary stochastic process  Measured signal  Autocorrelation function

 Cyclic ACF  Periodic ACF

 DCS = 7⋅10-3

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 35 INSTITUTE Cyclostationary stochastic process  Measured signal  Autocorrelation function

 Cyclic ACF  Periodic ACF

Cyclic frequency α = 1/Tbit  DCS = 7.8

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 36 INSTITUTE Cyclostationary stochastic process  Measured signal  Auto-covariation function

 Cyclic ACF  Periodic ACF

Residue signal  DCS = 8.2

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 37 INSTITUTE Degree of cyclostationarity

 Dependence of DCS on the relative cyclic frequency

 Eye-diagram  DCS

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 38 INSTITUTE Outline

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 39 INSTITUTE Cross-talk characterization

 Cross-talk between neighbor transmission lines causing due to the increasing of the circuit speed and reducing of the distance between traces on the surface of the PCB  The possible error occurring during transmission of the digital data along the trace of the PCB can be characterized by jitter disturbing the eye diagram of the received signal  The cyclic averaging of the digital jitter gives the instantaneous power of interference signal during the bit interval  Crosstalk influence on the BER caused by the nearby transmission line can be evaluated by the corresponding cyclostationary correlation functions of the the total jitter

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 40 INSTITUTE Device under test

 Zigzag line

2 mm

Matched load

The microstrip lines were fabricated on a 1.6-mm-thick FR4 substrate with a characteristic impedance of 50 Ω and a length of 50 mm

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 41 INSTITUTE Measurement setup

Victim signal

Victim Aggressor signal

Aggressor Oscilloscope

Victims and aggressor PRBS signals were generated by Xilinx FPGA Artix-7 Development Board

Repetition rate Fbit = 250 MHz

Sampling frequency Fs = 20 GSa/s

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 42 INSTITUTE Experimental results  Signal in the victim line

 Signal in the near-field of aggressor line

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 43 INSTITUTE Experimental results  Signal in the victim line with the mutual crosstalk between aggressor and victim

 Noise without crosstalk  Noise with crosstalk

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 44 INSTITUTE Experimental results  Periodic ACF of noise  Periodic ACF of noise without crosstalk with crosstalk

𝔼𝔼

 Averaged instantaneous periodic power of the random process = ( ) = , 0 𝑋𝑋 𝑡𝑡 2 𝑃𝑃𝑇𝑇𝑏𝑏 𝑡𝑡 𝔼𝔼 𝑋𝑋 𝑡𝑡 𝑅𝑅𝑋𝑋 𝑡𝑡

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 45 INSTITUTE Experimental results  Averaged instantaneous periodic power

 Histograms of the crossing  Bit error plot for the victim signal points for the victim signal

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 46 INSTITUTE Outline

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 47 INSTITUTE Cyclostationary EMI sources separation

 Cyclic averaging with the known cyclic frequency extracts the spatial positions containing the random signals with the defined cyclostationary properties  Separating random signals with different cyclic frequencies and also exclude the existing stationary noise from the obtained characteristics.  Experimental setup

Oscilloscope Scanning points Scanning Scanning plane probe h

DUT

Reference probe

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 48 INSTITUTE Measurement setup

Reference probe

Scanning probe

 Langer EMV-Technik RF-R 50-1 magnetic field probes  Frequency band from 30 MHz up to 3 GHz  13 GHz Serial Data Analyzer LeCroy SDA 813Zi-A

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 49 INSTITUTE Experimental results

 Signal paths  Signal measured in near-field

 Bit frequencies are 166.67 MHz and 156.25 MHz

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 50 INSTITUTE Experimental results

 Amplitude spectrum of the measured signal

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 51 INSTITUTE Experimental results

 Cyclic autocorrelation cumulant functions

 α1 = 1/Tbit1 = 166.67 MHz  α2 = 1/Tbit2 = 156.25 MHz

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 52 INSTITUTE Experimental results

 Cyclic cross-correlation cumulant functions

 α1 = 1/Tbit1 = 166.67 MHz  α2 = 1/Tbit2 = 156.25 MHz

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 53 INSTITUTE Spatial distribution of cyclic CCCF

 α1 = 166.67 MHz  α2 = 156.25 MHz

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 54 INSTITUTE Experimental results

 Mean power of EMI in near-field of DUT

 The sampling frequency 10 GS/s  Number of samples 5 million  Scanning step 2 mm step and the height h = 2 mm

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 55 INSTITUTE Experimental results

 Spatial distribution of maxima for synchronized cross sections of the two-dimensional autocovariance functions

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 56 INSTITUTE Conclusion

 The presented theoretical background for the characterization of the second order wide sense cyclostationary stochastic process allowed to formulate algorithms for synchronous averaging of the measured near- field emissions caused by random data transferring over the surface of the PCB.  The cross-talk between the nearby traces with the data sequences possessing similar cyclostationary properties can be characterized by the total jitter resulting in significant increasing of the BER  Separation of different random data sequences on the surface of the PCB can be implemented by synchronous cyclic averaging of the measured near field emissions using corresponding cross or auto covariance functions.

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 57 INSTITUTE Acknowledgment

This work was supported by COST Action IC 1407 ACCREDIT Advanced Characterization and Classification of Radiated Emissions in Densely Integrated Technologies

THANK YOU FOR YOUR KIND ATTENTION!

MOSCOW AVIATION COST IC 1407 Training School – Prague, 2019 58 INSTITUTE