S tochastic processes

1. Stochastic processes Lecture 7 Linear time invariant systems

2. Random process

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

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

5. EXPECTATIONS • Expected value • The autocorrelation

6. Some random processes • Single pulse • Multiple pulses • Periodic Random Processes • The Gaussian Process • The PoissonProcess • Bernoulli and Binomial Processes • The RandomWalk Wiener Processes • The MarkovProcess

7. Recap: Power spectrum density

8. 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

9. 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

10. Properties of the PSD • Sxx(f) is real and nonnegative • The average power in X(t) is given by: • If X(t) is real Rxx(τ) and Sxx(f) are also even • If X(t) has periodic components Sxx(f)has impulses • Independent on phase

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

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

13. 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

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

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

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

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

18. Bias in the estimates of the autocorrelation N=12

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

20. Averaging • Divide the signal into K segments of M length • Calculate the periodogram of each segment • Calculate the average periodogram

21. Illustrations of Averaging

22. PSD units • Typical units: • Electrical measurements: V2/Hz or dB V/Hz • Sound: Pa2/Hz or dB/Hz • How to calculate dB I a power spectrum: PSDdB(f) = 10 log10 { PSD(f)  } .

23. 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

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

25. Systems

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

27. 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: y2(t)=f[x1(t)] Then: y2(t+t0)=f[x1(t+t0)] The same to Day tomorrow and in 1000 years A non Time invariant 20 years 45 years 70 years

28. 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)=3tx(t)

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

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

31. 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

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

33. Causal systems • Independent on the future signal

34. 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

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

36. 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)

37. 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:

38. Expected Mean square value (1/2)

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

40. 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 the auto-correlation of x(t) and the impulse response

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

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

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

44. Filter examples

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

46. Ideal filters • Highpass filter • Band stop filter • Bandpassfilter

47. Filter types and rippels

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

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

50. Transformation of a low pass filter to other types (the s-domain) Old Cutoff frequency Lowest Cutoff frequency New Cutoff frequency Highest Cutoff frequency