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Stochastic processes Lecture 7 Linear time invariant systems 1

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Random process 2

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1 st order Distribution & density function First-order distribution First-order density function 3

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2 end order Distribution & density function 2 end order distribution 2 end order density function 4

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EXPECTATIONS Expected value The autocorrelation 5

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Some random processes Single pulse Multiple pulses Periodic Random Processes The Gaussian Process The Poisson Process Bernoulli and Binomial Processes The Random Walk Wiener Processes The Markov Process 6

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Recap: Power spectrum density 7

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Power spectrum density Since the integral of the squared absolute Fourier transform contains the full power of the signal it is a density function. So the power spectral density of a random process is: Due to absolute factor the PSD is always real 8

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Power spectrum density The PSD is a density function. – In the case of the random process the PSD is the density function of the random process and not necessarily the frequency spectrum of a single realization. Example – A random process is defined as – Where ω r is a unifom distributed random variable wiht a range from 0-π – What is the PSD for the process and – The power sepctrum for a single realization 9

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Properties of the PSD 10

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Wiener-Khinchin 1 If the X(t) is stationary in the wide-sense the PSD is the Fourier transform of the Autocorrelation 11

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Wiener-Khinchin Two method for estimation of the PSD X(t) Fourier Transform |X(f)| 2 Sxx(f) Autocorrelation Fourier Transform 12

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The inverse Fourier Transform of the PSD Since the PSD is the Fourier transformed autocorrelation The inverse Fourier transform of the PSD is the autocorrelation 13

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Cross spectral densities If X(t) and Y(t) are two jointly wide-sense stationary processes, is the Cross spectral densities Or 14

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Properties of Cross spectral densities 1.Since is 2.S yx (f) is not necessary real 3.If X(t) and Y(t) are orthogonal S xy (f)=0 4.If X(t) and Y(t) are independent S xy (f)=E[X(t)] E[Y(t)] δ(f) 15

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Cross spectral densities example 1 Hz Sinus curves in white noise Where w(t) is Gaussian noise 16

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The periodogram The estimate of the PSD The PSD can be estimate from the autocorrelation Or directly from the signal 17

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Bias in the estimates of the autocorrelation N=12 18

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Variance in the PSD The variance of the periodogram is estimated to the power of two of PSD 19

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Averaging 20

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Illustrations of Averaging 21

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PSD units Typical units: Electrical measurements: V 2 /Hz or dB V/Hz Sound: Pa 2 /Hz or dB/Hz How to calculate dB I a power spectrum: PSD dB (f) = 10 log 10 { PSD(f) } 22.

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Agenda (Lec. 7) Recap: Linear time invariant systems Stochastic signals and LTI systems – Mean Value function – Mean square value – Cross correlation function between input and output – Autocorrelation function and spectrum output Filter examples Intro to system identification 23

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Focus continuous signals and system Continuous signal: Discrete signal: 24

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Systems 25

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Recap: Linear time invariant systems (LTI) What is a Linear system: – The system applies to superposition 26 012345 0 2 4 6 8 10 12 14 16 18 20 Linear system x(t) y(t) 012345 -20 -15 -10 -5 0 5 10 15 20 25 Nonlinear systems x(t) y(t) x[n] 2 20 log(x[n])

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Recap: Linear time invariant systems (LTI) Time invariant: A time invariant systems is independent on explicit time – (The coefficient are independent on time) That means If: y 2 (t)=f[x 1 (t)] Then: y 2 (t+t 0 )=f[x 1 (t+t 0 )] The same to Day tomorrow and in 1000 years 27 70 years45 years 20 years A non Time invariant

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Examples A linear system y(t)=3 x(t) A nonlinear system y(t)=3 x(t) 2 A time invariant system y(t)=3 x(t) A time variant system y(t)=3 t x(t) 28

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The impulse response T{ ∙ } The output of a system if Dirac delta is input

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Convolution 30 The output of LTI system can be determined by the convoluting the input with the impulse response

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Fourier transform of the impulse response The Transfer function (System function) is the Fourier transformed impulse response The impulse response can be determined from the Transfer function with the invers Fourier transform 31

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Fourier transform of LTI systems Convolution corresponds to multiplication in the frequency domain 32 Time domain Frequency domain * = x =

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Causal systems Independent on the future signal 33

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Stochastic signals and LTI systems Estimation of the output from a LTI system when the input is a stochastic process 34 Α is a delay factor like τ

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Statistical estimates of output The specific distribution function f X (x,t) is difficult to estimate. Therefor we stick to – Mean – Autocorrelation – PSD – Mean square value. 35

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Expected Value of Y(t) (1/2) How do we estimate the mean of the output? 36 If mean of x(t) is defined as m x (t)

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Expected Value of Y(t) (2/2) 37 If x(t) is wide sense stationary Alternative estimate: At 0 Hz the transfer function is equal to the DC gain Therefor:

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Expected Mean square value (1/2) 38

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Expected Mean square value (2/2) 39 By substitution: If X(t)is WSS Thereby the Expected Mean square value is independent on time

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Cross correlation function between input and output Can we estimate the Cross correlation between input and out if X(t) is wide sense stationary 40 Thereby the cross-correlation is the convolution between the auto-correlation of x(t) and the impulse response

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Autocorrelation of the output (1/2) 41 Y(t) and Y(t+τ) is :

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Autocorrelation of the output (2/2) 42 By substitution: α=-β

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Spectrum of output Given: The power spectrum is 43 x =

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Filter examples 44

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Typical LIT filters FIR filters (Finite impulse response) IIR filters (Infinite impulse response) – Butterworth – Chebyshev – Elliptic 45

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Ideal filters Highpass filter Band stop filter Bandpassfilter

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Filter types and rippels 47

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Analog lowpass Butterworth filter Is ”all pole” filter – Squared frequency transfer function N:filter order f c : 3dB cut off frequency Estimate PSD from filter

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Chebyshev filter type I Transfer function Where ε is relateret to ripples in the pass band Where T N is a N order polynomium

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Transformation of a low pass filter to other types (the s-domain) Filter typeTransformationNew Cutoff frequency Lowpas>Lowpas Lowpas>Highpas Lowpas>Stopband Lowest Cutoff frequency Highest Cutoff frequencyNew Cutoff frequency Old Cutoff frequency

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Discrete time implantation of filters A discrete filter its Transfer function in the z- domain or Fourier domain – Where b k and a k is the filter coefficients In the time domain: 51

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Filtering of a Gaussian process Gaussian process – X(t 1 ),X(t 2 ),X(t 3 ),….X(t n ) are jointly Gaussian for all t and n values Filtering of a Gaussian process – Where w[n] are independent zero mean Gaussian random variables. 52

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The Gaussian Process X(t 1 ),X(t 2 ),X(t 3 ),….X(t n ) are jointly Gaussian for all t and n values Example: randn() in Matlab

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The Gaussian Process and a linear time invariant systems Out put = convolution between input and impulse response Gaussian input Gaussian output

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Example x(t): h(t): Low pass filter y(t):

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Filtering of a Gaussian process example 2 56 Band pass filter

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Intro to system identification Modeling of signals using linear Gaussian models: Example: AR models The output is modeled by a linear combination of previous samples plus Gaussian noise. 57

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Modeling example Estimated 3 th order model 58

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Agenda (Lec. 7) Recap: Linear time invariant systems Stochastic signals and LTI systems – Mean Value function – Mean square value – Cross correlation function between input and output – Autocorrelation function and spectrum output Filter examples Intro to system identification 59

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