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**Lecture 6 Power spectral density (PSD)**

Stochastic processes Lecture 6 Power spectral density (PSD)

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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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**Single pulse X (t) = A S(t −Θ)**

Single pulse with random amplitude and arrival time: Deterministic pulse: S(t): Deterministic function. Random variables: A: gain a random variable Θ: arrival time. A and Θ are statistically independent X (t) = A S(t −Θ)

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Multiple pulses Single pulse with random amplitude and arrival time: Deterministic pulse: S(t): Deterministic function. Random variables: Ak: gain a random variable Θk: arrival time. n: number of pulses Ak and Θk are statistically independent x 𝑡 = 𝑘=1 𝑛 𝐴 𝑘 𝑆(𝑡− 𝛩 𝑘 )

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**Periodic Random Processes**

A process which is periodic with T x 𝑡 =𝑥 𝑡+𝑛𝑇 n is an integrer x 𝑡 =𝑠𝑖𝑛 2𝜋𝑡 50 +Θ +𝑠𝑖𝑛 2𝜋𝑡 100 +Θ

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

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The Poisson Process Typically used for modeling of cumulative number of events over time. Example: counting the number of phone call from a phone 𝑃 𝑋 𝑡 =𝑘 = 𝜆 𝑡 𝑘 𝑘! 𝑒 −𝜆(𝑡)

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**Alternative definition Poisson points**

The number of events in an interval N(t1,t2) 𝑃 𝑁 𝑡1,𝑡2 =𝑘 =𝑃 𝑋 𝑡2 −𝑋 𝑡1 =𝑘 = 𝜆 𝑡2−𝑡1 𝑘 𝑘! 𝑒 −𝜆(𝑡2−𝑡1) 𝑃 𝑁 0,𝑡2 =𝑘 =𝑃 𝑋 𝑡 =𝑘 = 𝜆𝑡 𝑘 𝑘! 𝑒 −𝜆𝑡

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**Bernoulli Processes A process of zeros and ones**

X=[ ] Each sample must be independent and identically distributed Bernoulli variables. The likelihood of 1 is defined by p The likelihood of 0 is defined by q=1-p

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**Binomial process Summed Bernoulli Processes**

Where X[n] is a Bernoulli Processes

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Random walk For every T seconds take a step (size Δ) to the left or right after tossing a fair coin

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**The Markov Process 1st order Markov process**

The current sample is only depended on the previous sample Density function Expected value

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**The frequency of earth quakes**

Statement the number large earth quakes has increased dramatically in the last 10 year!

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**The frequency of earth quakes**

Is the frequency of large earth quakes unusual high? Which random processes can we use for modeling of the earth quakes frequency?

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**The frequency of earth quakes**

Data

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**Agenda (Lec 16) Power spectral density Definition and background**

Wiener-Khinchin Cross spectral densities Practical implementations Examples

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**Fourier transform recap 1**

Transform between time and frequency domain Fourier transform Invers Fourier transform

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**Fourier transform recap 2**

Assumption: The signal can be reconstructed from sines and cosines functions. Requirement: absolute integrable 𝑒 −𝑗2𝜋𝑓𝑡 = cos 2𝜋𝑓𝑡 −𝑗sin 2𝜋𝑓𝑡

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**Fourier transform of a stochastic process**

A stationary stochastic process is typical not absolute integrable There the signal is truncated Before Fourier transform

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What is power? In the power spectrum density power is related to electrical power 𝑃= 𝑉 2 𝑅

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Power of a signal The power of a signal is calculated by squaring the signal. 𝑥(𝑡) 2 The average power in e period is :

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Parseval's theorem The power of the squared absolute Fourier transform is equal the power of the signal

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**Power of a stochastic process**

Thereby can the expected power can be calculated from the Fourier spectrum

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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

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PSD Example Fourier transform |X(f)|2

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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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**PSD of random process versus spectrum of deterministic signals**

In the case of the random process the PSD is usual the expected value E[Sxx(f)] In the case of deterministic signals the PSD is exact (There is still estimation error)

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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 Proof: page 175

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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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**Implementations issues**

The challenges includes Finite signals Discrete time

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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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**The discrete version of the autocorrelation**

Rxx(τ)=E[X1(t) X(t+τ)]≈Rxx[m] m=τ where m is an integer N: number of samples Normalized version: 𝑅𝑥𝑥 𝑚 = 𝑛=0 𝑁− 𝑚 −1 𝑥 𝑛 𝑥[𝑛+𝑚] 𝑅𝑥𝑥 𝑚 = 1 𝑁 𝑛=0 𝑁− 𝑚 −1 𝑥 𝑛 𝑥[𝑛+𝑚]

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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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**Bias in the estimates of the autocorrelation**

The edge effect correspond to multiplying the true autocorrelation with a Bartlett window 𝐸[𝑅𝑥𝑥 𝑚 ]=𝑤[𝑚]𝑅𝑥𝑥_𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑[𝑚] 𝑤𝑏 𝑚 = 𝑁−|𝑚| 𝑁 𝑚 <𝑁 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

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**Alternative estimation of autocorrelation**

The unbiased estimate Disadvantage: high variance when |m|→N 𝑅𝑥𝑥 𝑚 = 1 𝑁−|𝑚| 𝑛=0 𝑁− 𝑚 −1 𝑥 𝑛 𝑥[𝑛+𝑚]

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**Influence at the power spectrum**

Biased version: a Bartlett window is applied Unbiased version: a Rectangular window is applied 𝑆𝑥𝑥 ω = 𝑚=−∞ ∞ 𝑤𝑟[𝑚]𝑅𝑥𝑥_𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑[𝑚] 𝑒 −𝑗ω𝑚 𝑆𝑥𝑥 ω = 𝑚=−∞ ∞ 𝑤𝑏[𝑚]𝑅𝑥𝑥_𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑[𝑚] 𝑒 −𝑗ω𝑚 𝑤𝑟 𝑚 = 1 𝑚 <𝑁 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

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Example Autocorrelation biased and unbiased Estimated PSD’s

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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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**Effect of Averaging The variance is decreased**

But the spectral resolution is also decreased 𝑉𝑎𝑟 𝑆𝑥𝑥 𝜔 = 1 𝐾 𝑆𝑥𝑥(𝜔) 2

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**Additional options The Welch method**

Introduce overlap between segment 𝑥𝑖=𝑥 𝑖−1 𝑄+1: 𝑖−1 𝑄+𝑀 ≤𝑖≤𝐾 Where Q is the length between the segments Multiply the segment's with windows 𝑆𝑖𝑥𝑥 ω = 1 𝑀 𝑛=0 𝑀−1 𝑤[𝑛]𝑥 𝑖[𝑛] 𝑒 −𝑗ω𝑛 2

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**Example Heart rate variability**

High frequency component related to Parasympathetic nervous system ("rest and digest") Low frequency component related to sympathetic nervous system (fight-or-flight)

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**Agenda (Lec 16) Power spectral density Definition and background**

Wiener-Khinchin Cross spectral densities Practical implementations Examples

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