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**Lecture 7 Linear time invariant systems**

Stochastic processes Lecture 7 Linear time invariant systems

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Random process

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**1st order Distribution & density function**

First-order distribution First-order density function

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**2end order Distribution & density function**

2end order distribution 2end order density function

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EXPECTATIONS Expected value The autocorrelation

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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

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**Recap: Power spectrum density**

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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 𝑆𝑥𝑥 𝑓 = 𝑙𝑖𝑚 𝑇→∞ 𝐸 −𝑇 𝑇 𝑠 𝑡 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑡 2 2𝑇

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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 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 X 𝑡 =sin( 𝜔 𝑟 𝑡)

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**Properties of the PSD Sxx(f) is real and nonnegative**

The average power in X(t) is given by: 𝐸 𝑋 2 (𝑡) =𝑅𝑥𝑥 0 = −∞ ∞ 𝑆𝑥𝑥 𝑓 𝑑𝑓 If X(t) is real Rxx(τ) and Sxx(f) are also even 𝑆𝑥𝑥 −𝑓 =𝑆𝑥𝑥 𝑓 If X(t) has periodic components Sxx(f)has impulses Independent on phase

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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

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**Wiener-Khinchin Two method for estimation of the PSD**

Fourier Transform |X(f)|2 X(t) Sxx(f) Fourier Transform Autocorrelation

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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

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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

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**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)

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**Cross spectral densities example**

1 Hz Sinus curves in white noise Where w(t) is Gaussian noise 𝑋 𝑡 = sin 2𝜋 𝑡 +3 𝑤(𝑡) 𝑌 𝑡 = sin 2𝜋 𝑡+ 𝜋 𝑤(𝑡)

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**The periodogram The estimate of the PSD**

The PSD can be estimate from the autocorrelation Or directly from the signal 𝑆𝑥𝑥 ω = 𝑚=−𝑁+1 𝑁−1 𝑅𝑥𝑥 [𝑚] 𝑒 −𝑗ω𝑚 𝑆𝑥𝑥 ω = 1 𝑁 𝑛=0 𝑁−1 𝑥 [𝑛] 𝑒 −𝑗ω𝑛 2

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**Bias in the estimates of the autocorrelation**

𝑅𝑥𝑥 𝑚 = 𝑛=0 𝑁− 𝑚 −1 𝑥 𝑛 𝑥[𝑛+𝑚]

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Variance in the PSD The variance of the periodogram is estimated to the power of two of PSD 𝑉𝑎𝑟 𝑆𝑥𝑥 𝜔 = 𝑆𝑥𝑥(𝜔) 2

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**Averaging Divide the signal into K segments of M length**

𝑥𝑖=𝑥 𝑖−1 𝑀+1:𝑖 𝑀 ≤𝑖≤𝐾 Calculate the periodogram of each segment 𝑆𝑖𝑥𝑥 ω = 1 𝑀 𝑛=0 𝑀−1 𝑥 𝑖[𝑛] 𝑒 −𝑗ω𝑛 2 Calculate the average periodogram 𝑆 𝑥𝑥[ω]= 1 𝐾 𝑖=0 𝐾 𝑆𝑖𝑥𝑥[ω]

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**Illustrations of Averaging**

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**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) } .

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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

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**Focus continuous signals and system**

Continuous signal: Discrete signal:

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Systems

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**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] Ö 20 log(x[n])

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**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

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**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)

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

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Convolution 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

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**Fourier transform of LTI systems**

Convolution corresponds to multiplication in the frequency domain Time domain * = Frequency domain x =

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Causal systems Independent on the future signal

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**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

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**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.

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**Expected Value of Y(t) (1/2)**

How do we estimate the mean of the output? 𝐸 𝑌 𝑡 =𝐸 −∞ ∞ 𝑋 𝑡−𝛼 ℎ 𝛼 𝑑𝛼 𝑌(𝑡)= −∞ ∞ 𝑋 𝑡−𝛼 ℎ 𝛼 𝑑𝛼 𝐸 𝑌 𝑡 = −∞ ∞ 𝐸 𝑋 𝑡−𝛼 ℎ 𝛼 𝑑𝛼 If mean of x(t) is defined as mx(t) 𝐸 𝑌 𝑡 = −∞ ∞ 𝑚𝑥(𝑡−𝛼)ℎ 𝛼 𝑑𝛼

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**Expected Value of Y(t) (2/2)**

If x(t) is wide sense stationary 𝑚𝑥 𝑡−𝛼 =𝑚𝑥 𝑡 =𝑚𝑥 (𝑚𝑥 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) 𝑚𝑦=𝐸 𝑌 𝑡 = −∞ ∞ 𝑚𝑥(𝑡−𝛼)ℎ 𝛼 𝑑𝛼 −∞ ∞ 𝑚𝑥ℎ 𝛼 𝑑𝛼 𝑚𝑦=𝐸 𝑌 𝑡 =𝑚𝑥 −∞ ∞ (𝑡−𝛼)ℎ 𝛼 𝑑𝛼 Alternative estimate: At 0 Hz the transfer function is equal to the DC gain −∞ ∞ ℎ 𝛼 𝑑𝛼=𝐻(0) Therefor: 𝑚𝑦=𝐸 𝑌 𝑡 =𝑚𝑥 𝐻(0)

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**Expected Mean square value (1/2)**

𝑌(𝑡)= −∞ ∞ 𝑋 𝑡−𝛼 ℎ 𝛼 𝑑𝛼 𝐸 𝑌 𝑡 2 =𝐸 𝑌 𝑡 𝑌 𝑡 𝐸 𝑌 𝑡 2 =𝐸 −∞ ∞ 𝑋 𝑡−𝛼1 ℎ 𝛼1 𝑑𝛼1 −∞ ∞ 𝑋 𝑡−𝛼2 ℎ 𝛼2 𝑑𝛼2 𝐸 𝑌 𝑡 2 =𝐸 −∞ ∞ −∞ ∞ 𝑋 𝑡−𝛼1 𝑋 𝑡−𝛼2 ℎ 𝛼1 ℎ 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝐸 𝑋 𝑡−𝛼1 𝑋 𝑡−𝛼2 ℎ 𝛼1 ℎ 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝑡−𝛼1,𝑡−𝛼2) ℎ 𝛼1 ℎ 𝛼2 𝑑𝛼1𝑑𝛼2 𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝛼1,𝛼2) ℎ 𝑡−𝛼1 ℎ 𝑡−𝛼2 𝑑𝛼1𝑑𝛼2

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**Expected Mean square value (2/2)**

𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝛼1,𝛼2) ℎ 𝑡−𝛼1 ℎ 𝑡−𝛼2 𝑑𝛼1𝑑𝛼2 𝛼=𝑡−𝛼1 𝛽=𝑡−𝛼2 By substitution: 𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝑡−𝛼,𝑡−𝛽)ℎ 𝛼 ℎ 𝛽 𝑑𝛼1𝑑𝛼2 If X(t)is WSS 𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝛼−𝛽) ℎ 𝛼 ℎ 𝛽 𝑑𝛼1𝑑𝛼2 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 𝑅𝑦𝑥 𝑡+𝜏,𝑡 =𝐸 𝑌 𝑡+𝜏 𝑋∗(𝑡) 𝑅𝑦𝑥 𝑡+𝜏,𝑡 =𝐸 −∞ ∞ 𝑋 𝑡−𝛼+𝜏 ℎ 𝛼 𝑑𝛼 𝑋 ∗ (𝑡) 𝑅𝑦𝑥 𝑡+𝜏,𝑡 =𝐸 −∞ ∞ 𝑋 𝑡−𝛼+𝜏 𝑋 ∗ (𝑡)ℎ 𝛼 𝑑𝛼 𝑅𝑥𝑥 𝜏 =𝐸 𝑋 𝑡+𝜏 𝑋 (𝑡) 𝑅𝑦𝑥 𝜏 = −∞ ∞ 𝑅𝑥𝑥 𝜏−𝛼 ℎ 𝛼 𝑑𝛼=𝑅𝑥𝑥 𝜏 ∗ℎ(𝜏) 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)**

𝑅𝑦𝑦 𝜏 =𝑅𝑦𝑦 𝑡+𝜏,𝑡 =𝐸 𝑌 𝑡+𝜏 𝑌(𝑡) Y(t) and Y(t+τ) is : 𝑌(𝑡+𝜏)= −∞ ∞ 𝑋 𝑡+𝜏−𝛼 ℎ 𝛼 𝑑𝛼 𝑌(𝑡)= −∞ ∞ 𝑋 𝑡−𝛽 ℎ 𝛽 𝑑𝛽 𝑅𝑦𝑦 𝜏 =𝐸 −∞ ∞ 𝑋 𝑡+𝜏−𝛼 ℎ 𝛼 𝑑𝛼 −∞ ∞ 𝑋 𝑡−𝛽 ℎ 𝛽 𝑑𝛽 𝑅𝑦𝑦 𝜏 =𝐸 −∞ ∞ −∞ ∞ 𝑋 𝑡+𝜏−𝛼 𝑋 𝑡−𝛽 ℎ 𝛼 ℎ 𝛽 𝑑𝛼𝑑𝛽 𝑅𝑦𝑦 𝜏 = −∞ ∞ −∞ ∞ 𝐸[𝑋 𝑡+𝜏−𝛼 𝑋 𝑡−𝛽 ]ℎ 𝛼 ℎ 𝛽 𝑑𝛼𝑑𝛽 𝑅𝑦𝑦 𝜏 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝜏−𝛼+𝛽)ℎ 𝛼 ℎ 𝛽 𝑑𝛼𝑑𝛽

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**Autocorrelation of the output (2/2)**

𝑅𝑦𝑦 𝜏 = −∞ ∞ −∞ ∞ 𝐸[𝑋 𝑡+𝜏−𝛼 𝑋 𝑡−𝛽 ]ℎ 𝛼 ℎ 𝛽 𝑑𝛼𝑑𝛽 By substitution: α=-β 𝑅𝑦𝑦 𝜏 = −∞ ∞ −∞ ∞ 𝐸[𝑋 𝑡+𝜏−𝛼 𝑋 𝑡+𝛼 ]ℎ 𝛼 ℎ −𝑎 𝑑𝛼𝑑𝛼 Remember: 𝑅𝑦𝑥 𝜏 =𝑅𝑥𝑥 𝜏 ∗ℎ 𝜏 = −∞ ∞ 𝑅𝑥𝑥 𝜏−𝛼 ℎ 𝛼 𝑑𝛼 𝑅𝑦𝑦 𝜏 =𝑅𝑦𝑥 𝜏 ∗ℎ(−𝜏) 𝑅𝑦𝑦 𝜏 =𝑅𝑥𝑥 𝜏 ∗ℎ(𝜏)∗ℎ(−𝜏)

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**Spectrum of output Given: The power spectrum is**

𝑅𝑦𝑦 𝜏 =𝑅𝑥𝑥 𝜏 ∗ℎ 𝜏 ∗ℎ(−𝜏) |𝐻 𝑓 | 2 =𝐻 𝑓 𝐻 ∗ (𝑓) 𝑆𝑦𝑦 𝑓 =𝑆𝑥𝑥 𝑓 𝐻 𝑓 𝐻 ∗ (𝑓) 𝑆𝑦𝑦 𝑓 =𝑆𝑥𝑥 𝑓 |𝐻 𝑓 | 2 x =

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Filter examples

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**Typical LIT filters FIR filters (Finite impulse response)**

IIR filters (Infinite impulse response) Butterworth Chebyshev Elliptic

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Ideal filters Highpass filter Band stop filter Bandpassfilter

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**Filter types and rippels**

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**Analog lowpass Butterworth filter**

Is ”all pole” filter Squared frequency transfer function N:filter order fc: 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 TN is a N order polynomium

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**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

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**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:

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**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.

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The Gaussian Process X(t1),X(t2),X(t3),….X(tn) 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**

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.

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Modeling example Estimated 3th order model

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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

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