Stochastic Signal Processing

IC 1407 ACCREDIT

STOCHASTIC SIGNAL PROCESSING

Prof., Dr. Yury Kuznetsov Moscow Aviation Institute (National Research University) Russian Federation COST Training School: EMC for Emerging Technologies

Department of Physics - Faculty of Science University of Malta

MOSCOW 19.04.2018 AVIATION INSTITUTE Contents

 Motivation and the theory of statistical signal processing (SP)  Non-stationary  Stationary  Cyclostationary  Characterization of measurement data and localization of the spatial distribution  Example: Intel Galileo Board  Separation of two distinct cyclostationary sources and localization of their spatial distributions  Example: Artix Training Board

MOSCOW AVIATION INSTITUTE 2 Motivation of statistical SP

 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 INSTITUTE 3 Classification of random processes

Random process

Time sequence of random The set of random signals variables = , : Ω , = , = 1, 𝒮𝒮 𝑡𝑡 is an𝑆𝑆 elementary𝑡𝑡 𝜔𝜔 𝜔𝜔 ∈ event𝑡𝑡 ∈ ℝ 𝑖𝑖 Ω is a sample space is𝒮𝒮 a random𝑡𝑡 𝑆𝑆𝑡𝑡 variable𝑖𝑖 at𝑛𝑛 = 𝜔𝜔 is a set of real numbers 𝑡𝑡𝑖𝑖 𝑖𝑖 𝑆𝑆 The stochastic phenomena𝑡𝑡 can be𝑡𝑡 mathematically characterized by the model of random ℝ 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 INSTITUTE 4 Cumulative Distribution Function

 The cumulative distribution function of the stochastic process is:

, = , = : , d = : , Ω 𝒮𝒮 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 𝐹𝐹 𝑠𝑠 𝑡𝑡 𝑃𝑃 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤1,𝑠𝑠 : �, 𝐈𝐈 𝑃𝑃 𝜔𝜔 𝐸𝐸 𝐈𝐈  indicator function = : , 0, : , > 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤ 𝑠𝑠 𝐈𝐈 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 ≤𝑠𝑠 �  is the operator of statistical expectation𝜔𝜔 𝑆𝑆 𝑡𝑡 or𝜔𝜔 ensemble𝑠𝑠 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 MOSCOW A. Napolitano, Generalizations of Cyclostationary Signal Processing: Spectral Analysis and AVIATION Applications INSTITUTE . John Wiley & Sons Ltd - IEEE Press, 2012.5 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 INSTITUTE 6 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  where , is a spectral1 correlation2 function2 with bounded variation.𝒮𝒮 1 2 It is also known that a 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜔𝜔 𝑆𝑆 𝑡𝑡 𝜔𝜔 �ℝ 𝑒𝑒 𝒢𝒢 𝑓𝑓 𝑓𝑓 stochastic process is harmonizable if and only if its covariance function is harmonizable 𝒢𝒢𝒮𝒮 𝑓𝑓1 𝑓𝑓2

 The Fourier transform of the realization , of the harmonizable stochastic process can be expressed as: 𝑆𝑆 𝑡𝑡 𝜔𝜔 , = , d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  , ̂ where can contain Dirac𝑆𝑆 delta𝑓𝑓 𝜔𝜔 functions�ℝ 𝑆𝑆 𝑡𝑡 𝜔𝜔 𝑒𝑒 𝑡𝑡 𝑆𝑆̂ 𝑓𝑓 𝜔𝜔

MOSCOW AVIATION INSTITUTE 7 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 INSTITUTE 8 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 , = , 𝜏𝜏d 𝑡𝑡 − 𝑡𝑡 −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 spectrum for stochastic processes: 𝑡𝑡1 𝑡𝑡 𝜏𝜏 𝑡𝑡2 𝑡𝑡 − 𝜏𝜏 , = + /2 /2 d = + /2 /2 d −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 𝒮𝒮 ̂ ̂ 𝒲𝒲 𝑡𝑡 𝑓𝑓 �ℝ 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝑆𝑆 𝑡𝑡 − 𝜏𝜏 𝑒𝑒 𝜏𝜏 �ℝ 𝐸𝐸 𝑆𝑆 𝑓𝑓 𝜈𝜈 𝑆𝑆 𝑓𝑓 − 𝜈𝜈 𝑒𝑒 𝜈𝜈

MOSCOW AVIATION INSTITUTE 9 Characteristics of RP

 Distribution of the probability over RVs of the random process  Ensemble averaging of the random process by using expectation operator  Statistical mean and 2D-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 INSTITUTE 10 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 𝒮𝒮 1 2 1 2 = 𝒢𝒢 𝑓𝑓 𝑓𝑓 , d d𝑓𝑓 =−𝑓𝑓 , d d 𝑗𝑗2𝜋𝜋 𝑓𝑓1𝑡𝑡1+𝑓𝑓2 𝑡𝑡1−𝜏𝜏 𝑗𝑗2𝜋𝜋𝑓𝑓1𝜏𝜏 1 1 2 𝒮𝒮 1 2 1 2 2 𝒮𝒮 1 1 1 2 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝑆𝑆 𝑡𝑡 − 𝜏𝜏 �ℝ 𝒢𝒢 𝑓𝑓 𝑓𝑓 𝑒𝑒 𝑓𝑓 𝑓𝑓 �ℝ 𝒢𝒢 𝑓𝑓 −𝑓𝑓 𝑒𝑒 𝑓𝑓 𝑓𝑓

MOSCOW AVIATION INSTITUTE 11 Stationary process  Random process realization

0𝑆𝑆 𝑡𝑡

𝑡𝑡  Autocorrelation function

𝑅𝑅𝒮𝒮 𝜏𝜏

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

𝑉𝑉𝒮𝒮 𝑓𝑓

0 𝑓𝑓 MOSCOW AVIATION INSTITUTE 12 𝛼𝛼 𝛼𝛼 − 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 Discrete Time Fourier𝑘𝑘=− ∞Transform (DTFT) of the realization of the harmonizable discrete stochastic process can be expressed as: ∆ 𝑆𝑆 𝑛𝑛

= ∞ −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  = 𝑆𝑆̂ 𝜑𝜑 � 𝑆𝑆𝑛𝑛𝑒𝑒 where is a normalizes frequency. The𝑛𝑛 inverse=−∞ DTFT gives the initial realization of the discretized stochastic process: 𝜑𝜑 𝑓𝑓∆ / = 1 2 / 𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 MOSCOW 𝑆𝑆 𝑛𝑛 � 𝑆𝑆̂ 𝜑𝜑 𝑒𝑒 AVIATION −1 2 INSTITUTE 13 Discrete-time stationary process  Discrete time process

𝒮𝒮 𝑛𝑛

 Discrete autocorrelation function𝑛𝑛

[ ]

𝑅𝑅𝒮𝒮 𝑚𝑚

1 0 1 𝑚𝑚  Periodical −power spectral density

𝑉𝑉𝒮𝒮 𝑓𝑓

MOSCOW 2 1 0 1 2 AVIATION 𝜑𝜑 INSTITUTE 14 − − 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 spectral correlation𝐸𝐸 𝑆𝑆 function𝑛𝑛 𝑆𝑆 𝑛𝑛 (Loève� bifrequency−1 2 1 2 𝑒𝑒 spectrum) of the𝒢𝒢 harmonizable𝜑𝜑 𝜑𝜑 discretized stochastic process is defined as: , =

 The relation between the autocorrelation𝒢𝒢𝒮𝒮 𝜑𝜑1 function𝜑𝜑2 𝐸𝐸 and𝑆𝑆̂ 𝜑𝜑the1 𝑆𝑆spectral̂ 𝜑𝜑2 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 MOSCOW 𝒮𝒮 1 2 1 2 AVIATION 𝒢𝒢 𝜑𝜑 𝜑𝜑 � � 𝐸𝐸15𝑆𝑆 𝑛𝑛 𝑆𝑆 𝑛𝑛 𝑒𝑒 INSTITUTE 𝑛𝑛1=−∞ 𝑛𝑛2=−∞ Characteristics of stationary RP

 Distribution of the probability over RVs 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 RP gives the periodic PSD and discrete ACF

MOSCOW AVIATION INSTITUTE 16 Cyclostationary random process  The cyclostationary random process is a non-stationary stochastic process whose statistical properties are periodically vary with respect to time. The period is called a cycle, and its inverse = 1/ is a cyclic frequency. More generally, the𝒮𝒮 process𝑡𝑡 is almost-cyclostationary (ACS) if its statistical properties can 0 0 be represented by a superposition of periodic functions𝑇𝑇 with distinct cyclic frequencies 𝛼𝛼 . 𝑇𝑇

 The autocorrelation function of the ACS stochastic process posses the periodicity in time𝛼𝛼 and∈ 𝒜𝒜 can be expressed by the Fourier series expansion1: , = + = , 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗�  where is the operatorℛ of𝒮𝒮 ensemble𝑡𝑡 𝜏𝜏 𝐸𝐸 averaging:𝑆𝑆 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝜏𝜏 � 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝛼𝛼∈𝒜𝒜 1 𝐸𝐸 � + = lim + + + + 1 𝑁𝑁 𝑇𝑇0 0 0  The Fourier coefficients𝐸𝐸 𝑆𝑆 of𝑡𝑡 𝑆𝑆 𝑡𝑡 , 𝜏𝜏 are called𝑁𝑁→∞ cyclic autocorrelation� 𝑆𝑆 𝑡𝑡 𝑇𝑇 functions𝑆𝑆 𝑡𝑡 𝜏𝜏 and𝑇𝑇 can be defined by a 2𝑁𝑁 𝑛𝑛=−𝑁𝑁 time domain cyclic averaging for each known cyclic frequency: 𝒮𝒮 ℛ 𝑡𝑡 𝜏𝜏 1 / , = lim , d 𝑇𝑇 2 / −𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝒮𝒮 𝒮𝒮 1 W. A. Gardner, Introduction𝑅𝑅 𝛼𝛼 to𝜏𝜏 Random𝑇𝑇→∞ Processes� ℛwith𝑡𝑡 Applications𝜏𝜏 𝑒𝑒 to𝑡𝑡 Signals and Systems. MOSCOW 𝑇𝑇 −𝑇𝑇 2 AVIATION INSTITUTE Macmillan, New York, 1985 (2nd Edition McGraw17 -Hill, New York, 1990). Pulse Amplitude Modulation  Pulse amplitude modulated (PAM) signal

𝑆𝑆 𝑡𝑡 /

0 1 2 3 4 5 6 7 8 9 𝑡𝑡 ∆  Autocorrelation function  Cyclic autocorrelation function , ,

𝒮𝒮 𝑅𝑅 𝑡𝑡 𝜏𝜏 / 𝑅𝑅𝒮𝒮 𝛼𝛼 𝜏𝜏 1 0 1 / 2 5 𝜏𝜏 ∆ 4 − 𝜏𝜏 ∆ − 1 − 3 − 2 − 0 − 1 − 0 1 − 1 2 2 3 4 3 MOSCOW 5 AVIATION 1 0 1 INSTITUTE / 18 − 𝛼𝛼 ⋅ ∆ 𝑡𝑡 ∆ Cyclic correlation function

 The magnitude and phase of , represent amplitude and phase of the additive complex harmonic component at frequency for time lag contained in the autocorrelation function of the ACS stochastic 𝒮𝒮 process , . For = 0 the𝑅𝑅 cyclic𝛼𝛼 𝜏𝜏 autocorrelation function reduces to the autocorrelation function of the stationary random process𝛼𝛼 . 𝜏𝜏 ℛ𝒮𝒮 𝑡𝑡 𝜏𝜏 𝛼𝛼  For a zero-mean stochastic process = = 0 the magnitude of the cyclic autocorrelation 𝑅𝑅𝒮𝒮 𝜏𝜏 functions , 0 as . If the mean function of the stochastic process = , 0, 𝒮𝒮 then some , contain additive𝑚𝑚 sinusoidal𝑡𝑡 𝐸𝐸 functions𝑆𝑆 𝑡𝑡 of , which arise from the products of sinusoidal 𝒮𝒮 𝒮𝒮 terms in 𝑅𝑅 𝛼𝛼. Such𝜏𝜏 → ACS processes𝜏𝜏 → ∞ are called unpure and need to be processed accounting𝑚𝑚 𝑡𝑡 𝐸𝐸 on𝑆𝑆 such𝑡𝑡 𝜔𝜔 ≠ 𝒮𝒮 property of𝑅𝑅 the𝛼𝛼 process𝜏𝜏 1. 𝜏𝜏 𝑚𝑚𝒮𝒮 𝑡𝑡  For cyclo-ergodic stochastic process the pure cyclic autocorrelation function called 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 1 𝐶𝐶 𝛼𝛼 𝜏𝜏 𝑇𝑇→∞ �−𝑇𝑇 2 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝜏𝜏 − 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝑡𝑡 MOSCOW J. Antoni, “Cyclostationarity𝑇𝑇 by examples,” Mechanical Systems and Signal Processing 23(4), AVIATION INSTITUTE 2009, pp. 987–1036. 19 Time-frequency Relation  The almost-periodic time-variant spectrum of the ACS process is the Fourier transform of the cyclic autocorrelation function , with respect to the lag parameter :

ℛ𝒮𝒮 𝑡𝑡 ,𝜏𝜏 = , 𝜏𝜏 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗�  where the cyclic spectrum correlation𝒱𝒱𝒮𝒮 𝑡𝑡 𝑓𝑓 function� 𝑉𝑉𝒮𝒮 𝛼𝛼, 𝑓𝑓 𝑒𝑒can be defined by the Fourier transform 𝛼𝛼∈𝒜𝒜 of the autocorrelation function , with respect to the lag parameter : 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 𝑅𝑅𝒮𝒮 ,𝛼𝛼 𝜏𝜏 = , d 𝜏𝜏 −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋 Re , 𝒮𝒮 Im , 𝒮𝒮 , 𝑉𝑉 𝛼𝛼 𝑓𝑓 �ℝ 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝜏𝜏 𝒮𝒮 𝒮𝒮 𝒱𝒱 𝑡𝑡 𝑓𝑓 𝒱𝒱 𝑡𝑡 𝑓𝑓 𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓 5 0 0 4 𝑓𝑓 ⋅ ∆ 𝑓𝑓 ⋅ ∆ 3 𝑓𝑓 ⋅ ∆ 1 1 − 2 − − 1 2 2 − 0 − 1 3 3 2 3 4 4 4 5 5 5 MOSCOW / / AVIATION INSTITUTE 20 𝑡𝑡 ∆ 𝑡𝑡 ∆ 𝛼𝛼 ⋅ ∆ 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 INSTITUTE 21 Wigner-Ville Cyclic Spectrum

 The Wigner-Ville spectrum for ACS stochastic processes:

, = , + /2 , 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝒮𝒮 𝒮𝒮 The Wigner-Ville𝑊𝑊 spectrum𝑡𝑡 𝑓𝑓 � of the𝑉𝑉 ACS𝛼𝛼 𝑓𝑓 stochastic𝛼𝛼 𝑒𝑒 process can be 𝒮𝒮 𝛼𝛼∈𝒜𝒜 𝑊𝑊 𝑡𝑡 𝑓𝑓 expressed by the Fourier series expansion over the time t with 0 frequencies and Fourier-series coefficients , + /2 . 𝑓𝑓 ⋅ ∆  1 The spectral𝛼𝛼 ∈ correlation𝒜𝒜 function (Loève bifrequency𝑉𝑉𝒮𝒮 𝛼𝛼 𝑓𝑓spectrum)𝛼𝛼 of the ACS stochastic process is defined as: 2

, = , + 3 4 It concentrated𝒢𝒢𝒮𝒮 in𝑓𝑓1 the𝑓𝑓2 countable� 𝑉𝑉𝒮𝒮 set𝛼𝛼 𝑓𝑓of1 lines𝛿𝛿 𝑓𝑓2 with𝑓𝑓1 slope− 𝛼𝛼 +1 on the 𝛼𝛼∈𝒜𝒜 bi-frequency plane. It means that ACS process have distinct 5 spectral components that are correlated only if the spectral / separation belongs to a countable set of cycle frequencies. 𝑡𝑡 ∆

MOSCOW AVIATION INSTITUTE 22 Cross-Correlation Function

𝑌𝑌1 𝑡𝑡

𝑆𝑆 𝑡𝑡 ℎ1 𝑡𝑡 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  For two different ACS stochastic�ℝ ℎ 𝑡𝑡 processes− 𝜏𝜏 ℎ 𝑡𝑡 − 𝜏𝜏andℛ 𝜏𝜏 𝜏𝜏are𝑑𝑑 said𝜏𝜏 𝑑𝑑 𝜏𝜏to be jointly correlated if the second-order cross-correlation function 𝒮𝒮1 𝑡𝑡 𝒮𝒮2 𝑡𝑡 , = + = , 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� 𝑌𝑌1𝑌𝑌2 1 2 𝑌𝑌1𝑌𝑌2 at cycle frequencies ℛ is𝑡𝑡 defined𝜏𝜏 𝐸𝐸 by𝑌𝑌 non𝑡𝑡 𝑌𝑌-zero𝑡𝑡 cyclic𝜏𝜏 cross�-correlation𝑅𝑅 𝛼𝛼 𝜏𝜏 functions𝑒𝑒 𝛼𝛼∈𝒜𝒜12 1 / 12 , = lim , d 𝛼𝛼 ∈ 𝒜𝒜 𝑇𝑇 2 MOSCOW / −𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗� AVIATION 𝑌𝑌1𝑌𝑌2 𝑌𝑌1𝑌𝑌2 INSTITUTE 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑇𝑇→∞ 23� ℛ 𝑡𝑡 𝜏𝜏 𝑒𝑒 𝑡𝑡 𝑇𝑇 −𝑇𝑇 2 Cross-Correlation Spectrum

 The cyclic spectrum cross-correlation , can be defined by the Fourier transform of the cyclic cross-correlation function , with respect to the lag parameter : 𝑉𝑉𝑌𝑌1𝑌𝑌2 𝛼𝛼 𝑓𝑓 𝑅𝑅𝑌𝑌1𝑌𝑌2 𝛼𝛼 𝜏𝜏 , = , d 𝜏𝜏 −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  It can be independently evaluated𝑌𝑌 1by𝑌𝑌2 the averaging𝑌𝑌 1of𝑌𝑌2 the short-time Fourier transforms (STFT) of the 𝑉𝑉 𝛼𝛼 𝑓𝑓 �ℝ 𝑅𝑅 𝛼𝛼 𝜏𝜏 𝑒𝑒 𝜏𝜏 stochastic process realizations and : / , = 1lim lim 2 , , d 𝑌𝑌 𝑡𝑡 𝑌𝑌 𝑡𝑡 𝑇𝑇 2 , / , / ∆𝑓𝑓 / where 𝑌𝑌1𝑌𝑌2 1 1 ∆𝑓𝑓 2 1 ∆𝑓𝑓 𝑉𝑉 𝛼𝛼 𝑓𝑓 ∆𝑓𝑓→0 𝑇𝑇→∞ �−𝑇𝑇 2𝐸𝐸 𝑋𝑋 𝑡𝑡 𝑓𝑓 𝑋𝑋 𝑡𝑡 𝛼𝛼 − 𝑓𝑓 𝑡𝑡 𝑇𝑇 / , = d ; = 1,2 , / 𝑡𝑡+1 ∆𝑓𝑓 / −𝑗𝑗2𝜋𝜋𝜋𝜋𝜋𝜋  For the estimation of second𝑖𝑖-1order∆𝑓𝑓 statistical functions𝑖𝑖 of ACS stochastic process need to have finite or 𝑋𝑋 𝑡𝑡 𝑓𝑓 �𝑡𝑡−1 ∆𝑓𝑓 𝑌𝑌 𝜉𝜉 𝑒𝑒 𝜉𝜉 𝑖𝑖 “effectively finite” memory. It means that ACS process need to be a zero-mean stochastic process = = 0.

𝒮𝒮 MOSCOW𝑚𝑚 𝑡𝑡 𝐸𝐸 𝑆𝑆 𝑡𝑡 AVIATION INSTITUTE 24 Relation between ACS Characteristics

Cyclic Classical Autocorrelation autocorrelation autocorrelation function function function = 0 , , Fourier 𝑡𝑡series↔ 𝛼𝛼 𝛼𝛼 𝒮𝒮 𝒮𝒮 𝒮𝒮 Fourierℛ 𝑡𝑡 𝜏𝜏 Fourier𝑅𝑅 𝛼𝛼 𝜏𝜏 Fourier𝑅𝑅 𝜏𝜏 transform transform transform

𝜏𝜏 ↔ 𝑓𝑓 𝜏𝜏 ↔ 𝑓𝑓 = 0 𝜏𝜏 ↔ 𝑓𝑓 , , Fourier 𝑡𝑡series↔ 𝛼𝛼 𝛼𝛼 Wigner𝒮𝒮 -Ville 𝒮𝒮Spectral Power𝒮𝒮 𝑊𝑊spectrum𝑡𝑡 𝑓𝑓 𝑉𝑉correlation𝛼𝛼 𝑓𝑓 𝑉𝑉spectral𝑓𝑓 density density

MOSCOW AVIATION INSTITUTE 25 Characteristics of CS Process

 Mean function and 2D autocorrelation function of the CS process can be expressed by a superposition of periodic functions with different periods  Cyclic autocorrelation function (ACF) 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 INSTITUTE 26 Device under test The Intel® Galileo Board

400MHz 32-bit Intel® Pentium processor 10/100 Ethernet connector Full PCI Express* mini-card slot USB 2.0 Host connector MOSCOW AVIATION INSTITUTE 27 Device under test

 Test modes

Memory test OFF Memory test ON. Memory intensive process where random integer numbers are generated and will be saved in a random element in a large array allocated in the memory

MOSCOW AVIATION INSTITUTE 28 Near-field measurement setup

Langer near-field 10 mm probe

Two polarization of the probe: HX and HY Scanning area 75 x 85 mm 5 mm scanning step 4 mm distance between PCB and probe 13 GHz Oscilloscope LeCroy SDA 813Zi-A 2.5 GSa/s sampling frequency 5 MSa data length

MOSCOW AVIATION INSTITUTE 29 Power hot spots of the DUT Mesh grid Memory  HX polarization  HY polarization

2 2 Clock SMPS  Power level 63 mV  Power level 27 mV

MOSCOW AVIATION INSTITUTE 30 Memory hot spot

Measured signals are nonperiodic Memory test signals are random  Maximum of the PS at 118 MHz

 Measured signals  Power spectrum

MOSCOW AVIATION INSTITUTE 31 Memory test on

Bit duration is 5.2 ns The shape of pulses is identical Memory test process is cyclostationary

MOSCOW AVIATION INSTITUTE 32 Memory test off

Pulse duration is 5.2 ns Sequence of single pulses Period of signal is 7.7 mks

MOSCOW AVIATION INSTITUTE 33 Cyclic auto-correlation cumulant functions

 Memory test on  Memory test off

 Power level 165 mV2  Power level 25 mV2

MOSCOW AVIATION INSTITUTE 34 Cyclic auto-correlation cumulant functions

 Power spectrum  Cyclic CCCF

 Maximum of cyclic CCCF corresponds to the cyclic frequency 190.5 MHz  Cyclic frequency is suppressed in the power spectrum

MOSCOW AVIATION INSTITUTE 35 Cyclic cross-correlation cumulant functions

 Cyclic frequency α = 0  Cyclic frequency α = 190.5 MHz

Correlation interval corresponds to the pulse duration 5.2 ns Both slices are nearly identical MOSCOW AVIATION INSTITUTE 36 Spatial distribution of the cyclic CCCF

Reference probe

MOSCOW AVIATION INSTITUTE 37 Spatial distribution of the cyclic CCCF

MOSCOW AVIATION INSTITUTE 38 Conclusion Arduino

 Frequency, time and spatial characterization of the physical radiated sources have been obtained  Characterization of the random data signals reveals hidden cyclic frequencies of the sequence  Localization of the physical radiated sources of the DUT was performed

MOSCOW AVIATION INSTITUTE 39 Device under test  Xilinx FPGA Development Board Artix-7 XC7A35T

Bottom side Top side MOSCOW AVIATION INSTITUTE 40 Experimental results

 Near-field measured signal

Bit frequencies are 166.67 MHz and 156.25 MHz

MOSCOW AVIATION INSTITUTE 41 Experimental results

 Amplitude spectrum of the measured signal

MOSCOW AVIATION INSTITUTE 42 Experimental results  Cyclic autocorrelation cumulant functions

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

MOSCOW AVIATION INSTITUTE 43 Experimental results

 Cyclic cross-correlation cumulant functions

α1 = 166.67 MHz α2 = 156.25 MHz

MOSCOW AVIATION INSTITUTE 44 Experimental results  Spatial distribution of cyclic CCCF

α1 = 166.67 MHz α2 = 156.25 MHz

MOSCOW AVIATION INSTITUTE 45 Far-field measurement setup Z

Y ϕ R X

Direction of rotation Direction of rotation

 Distance 1 m  Distance 4 m

MOSCOW AVIATION INSTITUTE 46 Far-field measurements

 α1 = 166.67 MHz  α2 = 156.25 MHz

 Distance 1 m  Horizontal orientation of antenna

MOSCOW AVIATION INSTITUTE 47 Far-field measurements

 α1 = 166.67 MHz  α2 = 156.25 MHz

 Distance 4 m  Horizontal orientation of antenna

MOSCOW AVIATION INSTITUTE 48 Far-field measurements

 α1 = 166.67 MHz  α2 = 156.25 MHz

 Distance 1 m  Vertical orientation of antenna

MOSCOW AVIATION INSTITUTE 49 Far-field measurements  α = 156.25 MHz  α1 = 166.67 MHz 2

 Distance 4 m  Vertical orientation of antenna

MOSCOW AVIATION INSTITUTE 50 Conclusion Artix

 Cyclic cross-correlation cumulant functions can be used for separation of two different random bit sequences with different cyclic frequencies  Special-time distribution was used for the localization of the transmission lines over the DUT surface  For cyclostationary source separation the position of the reference probe need to be chosen for sensing radiations of both sources

MOSCOW AVIATION INSTITUTE 51 Questions?

MOSCOW AVIATION INSTITUTE 52