Report

Stochastic processes Lecture 6 Power spectral density (PSD) 1 Random process 2 1st order Distribution & density function First-order distribution First-order density function 3 2end order Distribution & density function 2end order distribution 2end order density function 4 EXPECTATIONS • Expected value • The autocorrelation 5 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 Single pulse Single pulse with random amplitude and arrival time: X (t) = A S(t −Θ) A and Θ are statistically independent Amplitude Amplitude Nerve spike Amplitude Deterministic pulse: S(t): Deterministic function. Random variables: A: gain a random variable Θ: arrival time. 2 0 -2 0 0.5 1 1.5 t (ms) 2 2.5 3 0 0.5 1 1.5 t (ms) 2 2.5 3 0 0.5 1 1.5 t (ms) 2 2.5 3 2 0 -2 2 0 -2 7 Multiple pulses Single pulse with random amplitude and arrival time: x = =1 ( − ) Deterministic pulse: S(t): Deterministic function. Random variables: Ak: gain a random variable Θk: arrival time. n: number of pulses Multiple Nerve spikes 2 0 -2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 2 0 -2 2 0 Ak and Θk are statistically independent -2 8 Periodic Random Processes • A process which is periodic with T x = + n is an integrer Signal 2 1.5 1 T=100 X(t) 0.5 0 -0.5 -1 -1.5 -2 0 100 200 300 400 x = 500 t 2 50 600 700 + Θ + 800 2 100 +Θ 900 1000 9 The Gaussian Process • X(t1),X(t2),X(t3),….X(tn) are jointly Gaussian fro all t and n values • Example: randn() in Matlab Histogram of Gaussian process Gaussian process 700 5 4 600 3 500 2 1 400 0 300 -1 200 -2 100 -3 -4 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 -4 -3 -2 -1 0 1 2 3 4 5 10 The Poisson Process • Typically used for modeling of cumulative number of events over time. – Example: counting the number of phone call from a phone = = ! −() 11 Alternative definition Poisson points • The number of events in an interval N(t1,t2) 1, 2 = = 2 − 1 0, 2 2 − 1 = = ! −(2−1) − = = = = ! 12 Bernoulli Processes • A process of zeros and ones X=[0 0 1 1 0 1 0 0 1 1 1 0] 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 13 Binomial process Summed Bernoulli Processes Where X[n] is a Bernoulli Processes 14 Random walk • For every T seconds take a step (size Δ) to the left or right after tossing a fair coin Random walks 6 4 x[n] 2 0 -2 -4 -6 -8 0 5 10 15 20 25 n 30 35 40 45 50 15 The Markov Process • 1st order Markov process – The current sample is only depended on the previous sample Density function Expected value 16 The frequency of earth quakes • Statement the number large earth quakes has increased dramatically in the last 10 year! 17 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? 18 The frequency of earth quakes • Data • http://earthquake.usgs.gov/earthquakes/eqar chives/year/graphs.php 19 Agenda (Lec 16) • Power spectral density – Definition and background – Wiener-Khinchin – Cross spectral densities – Practical implementations – Examples 20 Fourier transform recap 1 Transform between time and frequency domain Fourier transform Invers Fourier transform 21 Fourier transform recap 2 • Assumption: The signal can be reconstructed from sines and cosines functions. e(jw n) Real Imaginary 1 Amplitude 0.5 −2 0 -0.5 = cos 2 − sin 2 -1 0 20 40 60 80 100 n • Requirement: absolute integrable x[n] n 22 Fourier transform of a stochastic process • A stationary stochastic process is typical not absolute integrable • There the signal is truncated • Before Fourier transform 23 What is power? • In the power spectrum density power is related to electrical power = 2 24 Power of a signal • The power of a signal is calculated by squaring the signal. ()2 • The average power in e period is : 25 Parseval's theorem • The power of the squared absolute Fourier transform is equal the power of the signal 26 Power of a stochastic process • Thereby can the expected power can be calculated from the Fourier spectrum 27 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 28 PSD Example Fourier transform PSD 1 Sxx(f) 0.8 0.6 0.4 0.2 0 -15 -10 -5 0 5 10 15 29 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 X = sin( ) – 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 30 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) 31 Properties of the PSD 1. Sxx(f) is real and nonnegative 2. The average power in X(t) is given by: 2 () = 0 = ∞ −∞ 3. If X(t) is real Rxx(τ) and Sxx(f) are also even − = 4. If X(t) has periodic components Sxx(f)has impulses 5. Independent on phase 32 Wiener-Khinchin 1 • If the X(t) is stationary in the wide-sense the PSD is the Fourier transform of the Autocorrelation Proof: page33175 Wiener-Khinchin Two method for estimation of the PSD Fourier Transform X(f) |X(f)|2 X(t) Sxx(f) X(t) f Fourier Transform t Rxx() Sxx(f) Autocorrelation f 34 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 35 Cross spectral densities • If X(t) and Y(t) are two jointly wide-sense stationary processes, is the Cross spectral densities • Or 36 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) 37 Cross spectral densities example • 1 Hz Sinus curves in white noise = sin 2 + 3 () = sin 2 + 2 + 3 () Welch Cross Power Spectral Density Estimate Where w(t) is Gaussian noise 5 0 X(t) 10 0 -10 0 5 10 t (s) Signal Y(t) 15 20 Y(t) 10 -5 -10 -15 -20 -25 0 -10 Power/frequency (dB/Hz) Signal X(t) -30 0 5 10 t (s) 15 20 0 5 10 15 Frequency (Hz) 20 25 38 Implementations issues • The challenges includes – Finite signals – Discrete time 39 The periodogram The estimate of the PSD • The PSD can be estimate from the autocorrelation −1 [] −ω ω = =−+1 • Or directly from the signal ω = 1 2 −1 [] −ω =0 40 The discrete version of the autocorrelation Rxx(τ)=E[X1(t) X(t+τ)]≈Rxx[m] m=τ where m is an integer − −1 = [ + ] =0 N: number of samples Normalized version: 1 = − −1 [ + ] =0 41 Bias in the estimates of the autocorrelation N=12 − −1 = [ + ] =0 Autocorrelation Autocorrelation M=-10 M=0 M=4 222 888 111 666 000 444 -1-1 -1 -2-2 -2 -10 -10 -10 -5 -5 -5 000 555 nnn 10 10 10 15 15 15 20 20 20 222 000 222 111 -2 -2 -2 000 -4 -4 -4 -1-1 -1 -2-2 -2 -10 -10 -10 -5 -5 -5 000 555 n+m n+m n+m 10 10 10 15 15 15 20 20 20 -6 -6 -6 -15 -15 -15 -10 -10 -10 -5 -5 0 5 10 15 42 Bias in the estimates of the autocorrelation • The edge effect correspond to multiplying the true autocorrelation with a Bartlett window [ ] = []_ [] − || = 0 < ℎ w[m] 1 0.5 0 -15 -10 -5 0 m 5 10 15 43 Alternative estimation of autocorrelation • The unbiased estimate 1 = − || − −1 [ + ] =0 Unbiased 0.6 0.6 0.4 0.4 0.2 0.2 Rxx[m] Rxx[m] Biased 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -15 -10 -5 0 m 5 10 15 -15 -10 -5 0 m 5 10 15 • Disadvantage: high variance when |m|→N 44 Influence at the power spectrum • Biased version: a Bartlett window is applied ∞ ω = =−∞ []_ [] −ω • Unbiased version: a Rectangular window is applied ∞ ω = =−∞ []_[] −ω = 1 0 < ℎ 45 Example Autocorrelation biased and unbiased Unbiased 0.6 0.4 0.4 0.2 0.2 Rxx[m] 0.6 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -15 -10 -5 0 m 5 10 Estimated PSD’s 15 -15 -10 -5 0 m 5 10 15 estimated PSD 5 Unbiased Biased 4 Sxx() Rxx[m] Biased 3 2 1 0 0 1 2 3 4 5 6 7 46 Variance in the PSD • The variance of the periodogram is estimated to the power of two of PSD = () 2 Realization 1 10 0 5 -5 True PSD 1 0 0.8 Sxx(f) PSD: Realization 1 5 5 t (s) Realization 2 10 0 5 10 0 5 0 50 100 150 f (Hz) PSD: Realization 2 200 50 200 0.6 0.4 -5 0.2 0 0 50 100 f (Hz) 150 200 0 5 t (s) Realization 3 10 0 5 10 0 5 -5 0 5 t (s) 10 0 0 0 100 150 f (Hz) PSD: Realization 3 50 100 f (Hz) 150 200 47 Averaging • Divide the signal into K segments of M length = − 1 + 1: 1≤≤ • Calculate the periodogram of each segment 1 ω = 2 −1 [] −ω =0 • Calculate the average periodogram 1 [ω] = [ω] =0 48 Illustrations of Averaging X(t) 2 0 -2 -4 0 1 2 10 3 4 5 4 6 7 10 8 9 0 100 10 6 4 5 2 5 2 0 0 100 200 0 0 100 200 0 0 100 200 0 200 10 5 0 0 50 100 f (Hz) 150 200 49 Effect of Averaging • The variance is decreased = 1 () 2 • But the spectral resolution is also decreased 50 Additional options The Welch method • Introduce overlap between segment = − 1 + 1: − 1 + 1≤≤ – Where Q is the length between the segments • Multiply the segment's with windows 1 ω = 2 −1 [] [] −ω =0 51 Example • Heart rate variability • http://circ.ahajournals.org/cgi/content/full/93 /5/1043#F3 • High frequency component related to Parasympathetic nervous system ("rest and digest") • Low frequency component related to sympathetic nervous system (fight-or-flight) 52 Agenda (Lec 16) • Power spectral density – Definition and background – Wiener-Khinchin – Cross spectral densities – Practical implementations – Examples 53