Stochastic processes Lecture 7 Linear time invariant systems
- Slides: 59
Stochastic processes Lecture 7 Linear time invariant systems 1
Random process 2
1 st order Distribution & density function First-order distribution First-order density function 3
2 end order Distribution & density function 2 end order distribution 2 end 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
Recap: Power spectrum density 7
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 8
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 9
Properties of the PSD • 10
Wiener-Khinchin 1 • If the X(t) is stationary in the wide-sense the PSD is the Fourier transform of the Autocorrelation 11
Wiener-Khinchin Two method for estimation of the PSD Fourier Transform |X(f)|2 X(t) Sxx(f) Fourier Transform Autocorrelation 12
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 13
Cross spectral densities • If X(t) and Y(t) are two jointly wide-sense stationary processes, is the Cross spectral densities • Or 14
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) 15
Cross spectral densities example • 1 Hz Sinus curves in white noise Where w(t) is Gaussian noise 16
The periodogram The estimate of the PSD • The PSD can be estimate from the autocorrelation • Or directly from the signal 17
Bias in the estimates of the autocorrelation N=12 18
Variance in the PSD • The variance of the periodogram is estimated to the power of two of PSD 19
Averaging • 20
Illustrations of Averaging 21
PSD units • Typical units: 2/Hz or d. B V/Hz • Electrical measurements: V. • Sound: Pa 2/Hz or d. B/Hz • How to calculate d. B I a power spectrum: PSDd. B(f) = 10 log 10 { PSD(f) } 22
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 23
Focus continuous signals and system Continuous signal: Discrete signal: 24
Systems 25
Recap: Linear time invariant systems (LTI) • What is a Linear system: – The system applies to superposition Linear system 20 18 20 16 15 14 10 y(t) 12 y(t) Nonlinear systems 25 10 5 8 0 6 -5 4 -10 x[n] 2 -15 Öx[n] 2 0 0 1 2 3 x(t) 4 5 -20 20 log(x[n]) 0 1 2 3 x(t) 4 5 26
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: Then: y 2(t)=f[x 1(t)] y 2(t+t 0)=f[x 1(t+t 0)] The same to Day tomorrow and in 1000 years A non Time invariant 20 years 45 years 70 years 27
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)=3 t x(t) 28
The impulse response The output of a system if Dirac delta is input T{∙}
Convolution • The output of LTI system can be determined by the convoluting the input with the impulse response 30
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 31
Fourier transform of LTI systems • Convolution corresponds to multiplication in the frequency domain Time domain * = x = Frequency domain 32
Causal systems • Independent on the future signal 33
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 34
Statistical estimates of output • The specific distribution function f. X(x, t) is difficult to estimate. Therefor we stick to – Mean – Autocorrelation – PSD – Mean square value. 35
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) 36
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: 37
Expected Mean square value (1/2) 38
Expected Mean square value (2/2) By substitution: If X(t)is WSS Thereby the Expected Mean square value is independent on time 39
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 40 the auto-correlation of x(t) and the impulse response
Autocorrelation of the output (1/2) Y(t) and Y(t+τ) is : 41
Autocorrelation of the output (2/2) By substitution: α=-β 42
Spectrum of output • Given: • The power spectrum is x = 43
Filter examples 44
Typical LIT filters • FIR filters (Finite impulse response) • IIR filters (Infinite impulse response) – Butterworth – Chebyshev – Elliptic 45
Ideal filters • Highpass filter • Band stop filter • Bandpassfilter
Filter types and rippels 47
Analog lowpass Butterworth filter • Is ”all pole” filter – Squared frequency transfer function • N: filter order • fc: 3 d. B cut off frequency • Estimate PSD from filter
Chebyshev filter type I • Transfer function • Where ε is relateret to ripples in the pass band • Where TN is a N order polynomium
Transformation of a low pass filter to other types (the s-domain) Filter type Transformation New Cutoff frequency Lowpas>Lowpas>Highpas Lowpas>Stopband Old Cutoff frequency Lowest Cutoff frequency New Cutoff frequency Highest Cutoff frequency
Discrete time implantation of filters • A discrete filter its Transfer function in the zdomain or Fourier domain – Where bk and ak is the filter coefficients • In the time domain: 51
Filtering of a Gaussian process • Gaussian process – X(t 1), X(t 2), X(t 3), …. 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. 52
The Gaussian Process • X(t 1), X(t 2), X(t 3), …. X(tn) are jointly Gaussian for all t and n values • Example: randn() in Matlab
The Gaussian Process and a linear time invariant systems • Out put = convolution between input and impulse response Gaussian input Gaussian output
Example • x(t): • h(t): Low pass filter • y(t):
Filtering of a Gaussian process example 2 Band pass filter 56
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. 57
Modeling example • Estimated 3 th order model 58
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 59
- A first course in stochastic processes
- Stochastic process
- Stochastic processes
- Stochastic processes
- Linear time-invariant system solved examples
- Linear position invariant degradation
- Linear position invariant degradation
- Concurrent processes are processes that
- Time invariant system
- Time invariant system
- 01:640:244 lecture notes - lecture 15: plat, idah, farad
- Linear regression lecture
- Integrated business processes with erp systems
- Stochastic rounding
- Stochastic programming
- Stochastic process model
- Wan optimization tutorial
- Deterministic and stochastic inventory models
- Put call formula
- Stochastic vs dynamic
- Absorbing stochastic matrix
- Regressors meaning
- Non stochastic theory of aging
- Stochastic process introduction
- Stochastic progressive photon mapping
- Known vs unknown environment
- Discrete variable
- Logistic regression stochastic gradient descent
- Stochastic process modeling
- Stochastic process
- Stochastic process
- Stochastic process
- Stochastic process
- Guided, stochastic model-based gui testing of android apps
- Stochastic specification of prf
- Stochastic uncertainty
- Components of time series analysis
- Stochastic vs probabilistic
- Stochastic vs probabilistic
- Stochastic calculus
- Stationary stochastic process
- Stochastic vs probabilistic
- Fast stochastic
- Stochastic gradient descent
- Stochastic gradient langevin dynamics
- Pca vs umap
- Stochastic rounding
- Deterministic and stochastic inventory models
- Stochastic regressors
- Operating systems lecture notes
- Articulators
- Lecture sound systems
- What is elapsed time
- Pmp time management
- Describe fully the single transformation
- Věta o transpozici vztažného bodu
- Quicksort invariant
- What is the correctness of algorithm
- Sebutkan contoh varian dan contoh invariant
- What is loop variant