Stochastic processes Lecture 7 Linear time invariant systems

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Stochastic processes Lecture 7 Linear time invariant systems 1

Stochastic processes Lecture 7 Linear time invariant systems 1

Random process 2

Random process 2

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

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

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

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

EXPECTATIONS • Expected value • The autocorrelation 5

EXPECTATIONS • Expected value • The autocorrelation 5

Some random processes • • Single pulse Multiple pulses Periodic Random Processes The Gaussian

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

Recap: Power spectrum density 7

Recap: Power spectrum density 7

Power spectrum density • Since the integral of the squared absolute Fourier transform contains

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

Power spectrum density • The PSD is a density function. – In the case

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 ωr 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

Properties of the PSD • 10

Properties of the PSD • 10

Wiener-Khinchin 1 • If the X(t) is stationary in the wide-sense the PSD is

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

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

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

The inverse Fourier Transform of the PSD • Since the PSD is the Fourier

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

Cross spectral densities • If X(t) and Y(t) are two jointly wide-sense stationary processes,

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

Properties of Cross spectral densities 1. Since is 2. Syx(f) is not necessary real

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

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

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

The periodogram The estimate of the PSD • The PSD can be estimate from

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

Bias in the estimates of the autocorrelation N=12 18

Bias in the estimates of the autocorrelation N=12 18

Variance in the PSD • The variance of the periodogram is estimated to the

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

Averaging • 20

Averaging • 20

Illustrations of Averaging 21

Illustrations of Averaging 21

PSD units • Typical units: 2/Hz or d. B V/Hz • Electrical measurements: V.

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

Agenda (Lec. 7) • Recap: Linear time invariant systems • Stochastic signals and LTI

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

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

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

Systems 25

Systems 25

Recap: Linear time invariant systems (LTI) • What is a Linear system: – The

Recap: Linear time invariant systems (LTI) • What is a Linear system: – The system applies to superposition Linear system 20 18 20 16 15 14 10 y(t) 12 y(t) Nonlinear systems 25 10 5 8 0 6 -5 4 -10 x[n] 2 -15 Öx[n] 2 0 0 1 2 3 x(t) 4 5 -20 20 log(x[n]) 0 1 2 3 x(t) 4 5 26

Recap: Linear time invariant systems (LTI) • Time invariant: • A time invariant systems

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: Then: y 2(t)=f[x 1(t)] y 2(t+t 0)=f[x 1(t+t 0)] The same to Day tomorrow and in 1000 years A non Time invariant 20 years 45 years 70 years 27

Examples • A linear system y(t)=3 x(t) • A nonlinear system y(t)=3 x(t)2 •

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

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

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

Convolution • The output of LTI system can be determined by the convoluting the

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

Fourier transform of the impulse response • The Transfer function (System function) is the

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

Fourier transform of LTI systems • Convolution corresponds to multiplication in the frequency domain

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

Causal systems • Independent on the future signal 33

Causal systems • Independent on the future signal 33

Stochastic signals and LTI systems • Estimation of the output from a LTI system

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

Statistical estimates of output • The specific distribution function f. X(x, t) is difficult

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

Expected Value of Y(t) (1/2) • How do we estimate the mean of the

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

Expected Value of Y(t) (2/2) If x(t) is wide sense stationary Alternative estimate: At

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

Expected Mean square value (1/2) 38

Expected Mean square value (1/2) 38

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

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

Cross correlation function between input and output • Can we estimate the Cross correlation

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

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

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

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

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

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

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

Filter examples 44

Filter examples 44

Typical LIT filters • FIR filters (Finite impulse response) • IIR filters (Infinite impulse

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

Ideal filters • Highpass filter • Band stop filter • Bandpassfilter

Ideal filters • Highpass filter • Band stop filter • Bandpassfilter

Filter types and rippels 47

Filter types and rippels 47

Analog lowpass Butterworth filter • Is ”all pole” filter – Squared frequency transfer function

Analog lowpass Butterworth filter • Is ”all pole” filter – Squared frequency transfer function • N: filter order • fc: 3 d. B cut off frequency • Estimate PSD from filter

Chebyshev filter type I • Transfer function • Where ε is relateret to ripples

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

Transformation of a low pass filter to other types (the s-domain) Filter type Transformation

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

Discrete time implantation of filters • A discrete filter its Transfer function in the

Discrete time implantation of filters • A discrete filter its Transfer function in the zdomain or Fourier domain – Where bk and ak is the filter coefficients • In the time domain: 51

Filtering of a Gaussian process • Gaussian process – X(t 1), X(t 2), X(t

Filtering of a Gaussian process • Gaussian process – X(t 1), X(t 2), X(t 3), …. X(tn) 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

The Gaussian Process • X(t 1), X(t 2), X(t 3), …. X(tn) are jointly

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

The Gaussian Process and a linear time invariant systems • Out put = convolution

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

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

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

Filtering of a Gaussian process example 2 Band pass filter 56

Filtering of a Gaussian process example 2 Band pass filter 56

Intro to system identification • Modeling of signals using linear Gaussian models: • Example:

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

Modeling example • Estimated 3 th order model 58

Modeling example • Estimated 3 th order model 58

Agenda (Lec. 7) • Recap: Linear time invariant systems • Stochastic signals and LTI

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