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Lecture 7 Linear time invariant systems

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1 Lecture 7 Linear time invariant systems
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 Poisson Process Bernoulli and Binomial Processes The Random Walk Wiener Processes The Markov Process

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 𝑆π‘₯π‘₯ 𝑓 = π‘™π‘–π‘š π‘‡β†’βˆž ⁑𝐸 βˆ’π‘‡ 𝑇 𝑠 𝑑 𝑒 βˆ’π‘—2πœ‹π‘“π‘‘ 𝑑𝑑 2 2𝑇

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 Ο‰r is a unifom distributed random variable wiht a range from 0-Ο€ What is the PSD for the process and The power sepctrum for a single realization X 𝑑 =sin⁑( πœ” π‘Ÿ 𝑑)

10 Properties of the PSD Sxx(f) is real and nonnegative
The average power in X(t) is given by: 𝐸 𝑋 2 (𝑑) =𝑅π‘₯π‘₯ 0 = βˆ’βˆž ∞ 𝑆π‘₯π‘₯ 𝑓 𝑑𝑓 If X(t) is real Rxx(Ο„) and Sxx(f) are also even 𝑆π‘₯π‘₯ βˆ’π‘“ =𝑆π‘₯π‘₯ 𝑓 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-Khinchin Two 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)] Ξ΄(f)

16 Cross spectral densities example
1 Hz Sinus curves in white noise Where w(t) is Gaussian noise 𝑋 𝑑 = sin 2πœ‹ 𝑑 +3 𝑀(𝑑) π‘Œ 𝑑 = sin 2πœ‹ 𝑑+ πœ‹ 𝑀(𝑑)

17 The periodogram The estimate of the PSD
The PSD can be estimate from the autocorrelation Or directly from the signal 𝑆π‘₯π‘₯ Ο‰ = π‘š=βˆ’π‘+1 π‘βˆ’1 𝑅π‘₯π‘₯ [π‘š] 𝑒 βˆ’π‘—Ο‰π‘š 𝑆π‘₯π‘₯ Ο‰ = 1 𝑁 𝑛=0 π‘βˆ’1 π‘₯ [𝑛] 𝑒 βˆ’π‘—Ο‰π‘› 2

18 Bias in the estimates of the autocorrelation
𝑅π‘₯π‘₯ π‘š = 𝑛=0 π‘βˆ’ π‘š βˆ’1 π‘₯ 𝑛 π‘₯[𝑛+π‘š]

19 Variance in the PSD The variance of the periodogram is estimated to the power of two of PSD π‘‰π‘Žπ‘Ÿ 𝑆π‘₯π‘₯ πœ” = 𝑆π‘₯π‘₯(πœ”) 2

20 Averaging Divide the signal into K segments of M length
π‘₯𝑖=π‘₯ π‘–βˆ’1 𝑀+1:𝑖 𝑀 ≀𝑖≀𝐾 Calculate the periodogram of each segment 𝑆𝑖π‘₯π‘₯ Ο‰ = 1 𝑀 𝑛=0 π‘€βˆ’1 π‘₯ 𝑖[𝑛] 𝑒 βˆ’π‘—Ο‰π‘› 2 Calculate the average periodogram 𝑆 π‘₯π‘₯[Ο‰]= 1 𝐾 𝑖=0 𝐾 𝑆𝑖π‘₯π‘₯[Ο‰]

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 log10Β { PSD(f) Β } .

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

27 Recap: Linear time invariant systems (LTI)
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 70 years 45 years 20 years A non Time invariant

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

29 The impulse response The output of a system if Dirac delta is input

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 Ξ‘ is a delay factor like Ο„

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? 𝐸 π‘Œ 𝑑 =𝐸 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό β„Ž 𝛼 𝑑𝛼 π‘Œ(𝑑)= βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό β„Ž 𝛼 𝑑𝛼 𝐸 π‘Œ 𝑑 = βˆ’βˆž ∞ 𝐸 𝑋 π‘‘βˆ’π›Ό β„Ž 𝛼 𝑑𝛼 If mean of x(t) is defined as mx(t) 𝐸 π‘Œ 𝑑 = βˆ’βˆž ∞ π‘šπ‘₯(π‘‘βˆ’π›Ό)β„Ž 𝛼 𝑑𝛼

37 Expected Value of Y(t) (2/2)
If x(t) is wide sense stationary π‘šπ‘₯ π‘‘βˆ’π›Ό =π‘šπ‘₯ 𝑑 =π‘šπ‘₯ (π‘šπ‘₯ 𝑖𝑠 π‘Ž π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘) π‘šπ‘¦=𝐸 π‘Œ 𝑑 = βˆ’βˆž ∞ π‘šπ‘₯(π‘‘βˆ’π›Ό)β„Ž 𝛼 𝑑𝛼 βˆ’βˆž ∞ π‘šπ‘₯β„Ž 𝛼 𝑑𝛼 π‘šπ‘¦=𝐸 π‘Œ 𝑑 =π‘šπ‘₯ βˆ’βˆž ∞ (π‘‘βˆ’π›Ό)β„Ž 𝛼 𝑑𝛼 Alternative estimate: At 0 Hz the transfer function is equal to the DC gain βˆ’βˆž ∞ β„Ž 𝛼 𝑑𝛼=𝐻(0) Therefor: π‘šπ‘¦=𝐸 π‘Œ 𝑑 =π‘šπ‘₯ 𝐻(0)

38 Expected Mean square value (1/2)
π‘Œ(𝑑)= βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό β„Ž 𝛼 𝑑𝛼 𝐸 π‘Œ 𝑑 2 =𝐸 π‘Œ 𝑑 π‘Œ 𝑑 𝐸 π‘Œ 𝑑 2 =𝐸 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό1 β„Ž 𝛼1 𝑑𝛼1 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό2 β„Ž 𝛼2 𝑑𝛼2 𝐸 π‘Œ 𝑑 2 =𝐸 βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό1 𝑋 π‘‘βˆ’π›Ό2 β„Ž 𝛼1 β„Ž 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝐸 𝑋 π‘‘βˆ’π›Ό1 𝑋 π‘‘βˆ’π›Ό2 β„Ž 𝛼1 β„Ž 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(π‘‘βˆ’π›Ό1,π‘‘βˆ’π›Ό2) β„Ž 𝛼1 β„Ž 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(𝛼1,𝛼2) β„Ž π‘‘βˆ’π›Ό1 β„Ž π‘‘βˆ’π›Ό2 𝑑𝛼1𝑑𝛼2

39 Expected Mean square value (2/2)
𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(𝛼1,𝛼2) β„Ž π‘‘βˆ’π›Ό1 β„Ž π‘‘βˆ’π›Ό2 𝑑𝛼1𝑑𝛼2 𝛼=π‘‘βˆ’π›Ό1 𝛽=π‘‘βˆ’π›Ό2 By substitution: 𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(π‘‘βˆ’π›Ό,π‘‘βˆ’π›½)β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼1𝑑𝛼2 If X(t)is WSS 𝐸 π‘Œ 𝑑 2 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(π›Όβˆ’π›½) β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼1𝑑𝛼2 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 𝑅𝑦π‘₯ 𝑑+𝜏,𝑑 =𝐸 π‘Œ 𝑑+𝜏 π‘‹βˆ—(𝑑) 𝑅𝑦π‘₯ 𝑑+𝜏,𝑑 =𝐸 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό+𝜏 β„Ž 𝛼 𝑑𝛼 𝑋 βˆ— (𝑑) 𝑅𝑦π‘₯ 𝑑+𝜏,𝑑 =𝐸 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›Ό+𝜏 𝑋 βˆ— (𝑑)β„Ž 𝛼 𝑑𝛼 𝑅π‘₯π‘₯ 𝜏 =𝐸 𝑋 𝑑+𝜏 𝑋 (𝑑) 𝑅𝑦π‘₯ 𝜏 = βˆ’βˆž ∞ 𝑅π‘₯π‘₯ πœβˆ’π›Ό β„Ž 𝛼 𝑑𝛼=𝑅π‘₯π‘₯ 𝜏 βˆ—β„Ž(𝜏) 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)
𝑅𝑦𝑦 𝜏 =𝑅𝑦𝑦 𝑑+𝜏,𝑑 =𝐸 π‘Œ 𝑑+𝜏 π‘Œ(𝑑) Y(t) and Y(t+Ο„) is : π‘Œ(𝑑+𝜏)= βˆ’βˆž ∞ 𝑋 𝑑+πœβˆ’π›Ό β„Ž 𝛼 𝑑𝛼 π‘Œ(𝑑)= βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›½ β„Ž 𝛽 𝑑𝛽 𝑅𝑦𝑦 𝜏 =𝐸 βˆ’βˆž ∞ 𝑋 𝑑+πœβˆ’π›Ό β„Ž 𝛼 𝑑𝛼 βˆ’βˆž ∞ 𝑋 π‘‘βˆ’π›½ β„Ž 𝛽 𝑑𝛽 𝑅𝑦𝑦 𝜏 =𝐸 βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑋 𝑑+πœβˆ’π›Ό 𝑋 π‘‘βˆ’π›½ β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼𝑑𝛽 𝑅𝑦𝑦 𝜏 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝐸[𝑋 𝑑+πœβˆ’π›Ό 𝑋 π‘‘βˆ’π›½ ]β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼𝑑𝛽 𝑅𝑦𝑦 𝜏 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝑅π‘₯π‘₯(πœβˆ’π›Ό+𝛽)β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼𝑑𝛽

42 Autocorrelation of the output (2/2)
𝑅𝑦𝑦 𝜏 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝐸[𝑋 𝑑+πœβˆ’π›Ό 𝑋 π‘‘βˆ’π›½ ]β„Ž 𝛼 β„Ž 𝛽 𝑑𝛼𝑑𝛽 By substitution: Ξ±=-Ξ² 𝑅𝑦𝑦 𝜏 = βˆ’βˆž ∞ βˆ’βˆž ∞ 𝐸[𝑋 𝑑+πœβˆ’π›Ό 𝑋 𝑑+𝛼 ]β„Ž 𝛼 β„Ž βˆ’π‘Ž 𝑑𝛼𝑑𝛼 Remember: 𝑅𝑦π‘₯ 𝜏 =𝑅π‘₯π‘₯ 𝜏 βˆ—β„Ž 𝜏 = βˆ’βˆž ∞ 𝑅π‘₯π‘₯ πœβˆ’π›Ό β„Ž 𝛼 𝑑𝛼 𝑅𝑦𝑦 𝜏 =𝑅𝑦π‘₯ 𝜏 βˆ—β„Ž(βˆ’πœ) 𝑅𝑦𝑦 𝜏 =𝑅π‘₯π‘₯ 𝜏 βˆ—β„Ž(𝜏)βˆ—β„Ž(βˆ’πœ)

43 Spectrum of output Given: The power spectrum is
𝑅𝑦𝑦 𝜏 =𝑅π‘₯π‘₯ 𝜏 βˆ—β„Ž 𝜏 βˆ—β„Ž(βˆ’πœ) |𝐻 𝑓 | 2 =𝐻 𝑓 𝐻 βˆ— (𝑓) 𝑆𝑦𝑦 𝑓 =𝑆π‘₯π‘₯ 𝑓 𝐻 𝑓 𝐻 βˆ— (𝑓) 𝑆𝑦𝑦 𝑓 =𝑆π‘₯π‘₯ 𝑓 |𝐻 𝑓 | 2 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 Ξ΅ 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)
Filter type Transformation New Cutoff frequency Lowpas>Lowpas Lowpas>Highpas Lowpas>Stopband Old Cutoff frequency Lowest Cutoff frequency New Cutoff frequency Highest Cutoff frequency

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

52 Filtering of a Gaussian process
X(t1),X(t2),X(t3),….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.

53 The Gaussian Process X(t1),X(t2),X(t3),….X(tn) are jointly Gaussian for all t and n values Example: randn() in Matlab

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

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

56 Filtering of a Gaussian process example 2
Band pass filter

57 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.

58 Modeling example Estimated 3th order model

59 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


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