1. SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung

2. lecture 2 Ali Karimpour Nov 2009 2 Lecture 2 Time-invariant linear systems Topics to be covered include: v Impulse responses, disturbances and transfer functions. v Frequency domain expression. v Signal spectra

3. lecture 2 Ali Karimpour Nov 2009 3 Impulse responses Sampling It is well known that a linear, time-invariant, causal system can be described as: Most often, the input signal u(t) is kept constant between the sampling instants: So

4. lecture 2 Ali Karimpour Nov 2009 4 Impulse responses For ease of notation assume that T is one time unit and use t to enumerate the sampling instants Also define the notation: And for simplicity:

5. lecture 2 Ali Karimpour Nov 2009 5 Disturbances There are always signals beyond our control that also affect the system. We assume that such effects can be lumped into an additive term v(t) at the output So + u(t)u(t)y(t)y(t) v(t)v(t) There are many sources and causes for such a disturbance term. Measurement noise. Uncontrollable inputs. ( a person in a room produce 100 W/person)

6. lecture 2 Ali Karimpour Nov 2009 6 Disturbances Characterization of disturbances Its value is not known beforehand. Making qualified guesses about future values is possible. It is natural to employ a probabilistic framework to describe future disturbances. We put ourselves at time t and would like to know disturbance at t+k, k ≥ 1 so we use the following approach. Where e(t) is a white noise. This description does not allow completely general characteristic of all possible probabilistic disturbances, but it is versatile enough.

7. lecture 2 Ali Karimpour Nov 2009 A realization of v(t) for propose e(t) 7 Disturbances Consider for example, the following PDF for e(t): Small values of μ are suitable to describe classical disturbance patterns, steps, pulses, sinuses and ramps.

8. lecture 2 Ali Karimpour Nov 2009 8 Disturbances On the other hand, the PDF: Often we only specify the second-order properties of the sequence {e(t)} that is the mean and variances. A realization of v(t) for propose e(t)

9. lecture 2 Ali Karimpour Nov 2009 9 Disturbances Mean: Covariance: We will assume that e(t) has zero mean and variance λ. Now we want to know the characteristic of v(t) :

10. lecture 2 Ali Karimpour Nov 2009 10 Disturbances We will assume that e(t) has zero mean and variance λ. Now we want to know the characteristic of v(t) : Mean: Covariance: Since the mean and covariance are do not depend on t, the process is said to be stationary.

11. lecture 2 Ali Karimpour Nov 2009 11 Transfer functions Define forward and backward shift operator q and q -1 as Now we can write output as: G(q) is the transfer operator or transfer function Similarly for disturbance we have So the basic description for a linear system with additive disturbance is:

12. lecture 2 Ali Karimpour Nov 2009 12 Transfer functions Some terminology We shall say that the transfer function G(q) is stable if G(q) is the transfer operator or transfer function This means that G(z) is analytic on and outside the unit circle. We shall say the filter H(q) is monic if h(0)=1:

13. lecture 2 Ali Karimpour Nov 2009 13 Frequency-domain expressions Let It will convenient to write Now we can write output as: So we have

14. lecture 2 Ali Karimpour Nov 2009 14 Periodograms of signals over finite intervals Fourier transform (FT) Discrete Fourier transform (DFT) Exercise: Show that u(t) can be derived by putting U N (ω) in u(t).

15. lecture 2 Ali Karimpour Nov 2009 15 Periodograms of signals over finite intervals Some property of U N (ω) The function U N (ω) is therefore uniquely defined by its values over the interval [ 0, 2π ]. It is, however, customary to consider U N (ω) for the interval [ - π, π ]. So U N (ω)can be defined as The number U N (ω) tell us the weight that the frequency ω carries in the decomposition. So Is known as the periodogram of the signal u(t), t= 1, 2, 3, ….. Parseval’s relationship:

16. lecture 2 Ali Karimpour Nov 2009 Periodograms of signals over finite intervals Example: Periodogram of a sinusoid

17. lecture 2 Ali Karimpour Nov 2009 17 Signal Spectra 17 Discrete Fourier transform (DFT) The periodogram defines, in a sense, the frequency contents of a signal over a finite time interval. But we seek for a definition of a similar concept for signals over the interval [1, ∞). But this limits fail to to exist for many signals of practical interest. So we shall develop a frame work for describing signals and their spectra that is applicable to deterministic as well as stochastic signals.

18. lecture 2 Ali Karimpour Nov 2009 18 Signal Spectra A Common Framework for Deterministic and Stochastic Signals In this course, we shall frequently work with signals that are described as stochastic processes with deterministic components. y(t) is not a stationary process To deal with this problem, we introduce the following definition: Quasi-stationary signals: A signal {s(t)} is said to be quasi-stationary if it is subject to

19. lecture 2 Ali Karimpour Nov 2009 Signal Spectra Quasi-stationary signals: A signal {s(t)} is said to be quasi-stationary if it is subject to If {s(t)} is a deterministic sequence Quasi-stationary means {s(t)} is a bounded sequence and If {s(t)} is a stationary stochastic process (i) and (ii) are trivially satisfied since

20. lecture 2 Ali Karimpour Nov 2009 20 Signal Spectra Notation: The notation means that the limit exists. Quasi-stationary signals: A signal {s(t)} is said to be quasi-stationary if it is subject to Sometimes with some abuse of notation, we call it Covariance function of s. Exercise: Show that sometime it is exactly covariance function of s.

21. lecture 2 Ali Karimpour Nov 2009 21 Signal Spectra Uncorrelated Two signals {s(t)} and {w(t)} are jointly quasi-stationary if: 1- They both are quasi-stationary, 2- the cross-covariance function exist.

22. lecture 2 Ali Karimpour Nov 2009 22 Definition of Spectra When following limits exists: We define the (power) spectrum of {s(t)} as and cross spectrum between {s(t)} and {w(t)} as

23. lecture 2 Ali Karimpour Nov 2009 Definition of Spectra Exercise Spectra of a Periodic Signal: Consider a deterministic, periodic signal with period M, i.e., s(t)=s(t+M) Show that Where And finally show that

24. lecture 2 Ali Karimpour Nov 2009 Definition of Spectra Show that Exercise Spectra of a Sinusoid: Consider

25. lecture 2 Ali Karimpour Nov 2009 Definition of Spectra Example Stationary Stochastic Processes: Consider We will assume that e(t) has zero mean and variance λ. We show that : The spectrum is: Where

26. lecture 2 Ali Karimpour Nov 2009 26 Signal Spectra Spectrum of Stationary Stochastic Processes The stochastic process described by v(t)= H(q)e(t), where {e(t)} is a sequence of independent random variables with zero mean values and covariances λ, has the spectrum

27. lecture 2 Ali Karimpour Nov 2009 27 Spectrum of a Mixed Deterministic and Stochastic Signal deterministic Stochastic: stationary and zero mean Signal Spectra Exercise: Prove it.

28. lecture 2 Ali Karimpour Nov 2009 28 Transformation of Spectra by Linear Systems Then {s(t)} is also quasi-stationary and Proof can be found in standard texts on stochastic processes or stochastic control such as Astrom, 1970. Suppose that is a rational function of cosω (or e iω ). Then there exist a monic rational function of z, R(z), with no poles and no zeros on or outside the unit circle such that Theorem: Let{w(t)} be a quasi-stationary with spectrum, and let G(q) be a stable transfer function. Let Important note

29. lecture 2 Ali Karimpour Nov 2009 29 Spectral Factorization ARMA, AR and MA models Example: ARMA Processes Let {v(t)} be a stationary processes with a rational spectrum We can represent it by AR MA

30. lecture 2 Ali Karimpour Nov 2009 30 Simulation part Exercise: ARMA Processes: Let Suppose that e(t) is: a) Derive v(t) and plot it for µ=0.02 for t=0 till 100. b) Derive v(t) and plot it for µ=1 for t=0 till 100. c) Derive the periodogram of the realization part b. d) Derive the spectrum of the realization part b.