Spectral Analysis of Random Processes

Transcription

1 Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all realizations of the same random process are different. Intuitively, many random processes possess some internal timescales, that are visible from realizations. One wants to analyse frequency content of a random process in a way similar to the one for deterministic functions. here are two general ways to introduce a spectral characteristic of a random process. NB: Although the process is random, its autocorrelation function is a deterministic function!

2 Power Spectral Density: Wiener-Khinchine heorem If Xt) is a wide-sense stationary random process with autocorrelation function K XX τ), its power spectral density Sω) can be introduced as a Fourier ransform of K XX τ):# Sω) = IK XX τ )) = K XX τ )e jωτ dτ hen autocorrelation function K XX τ) is then an inverse F of power spectral density Sω): K XX τ ) = I 1 Sω)) = 1 π Sω)e jωτ dω Wiener-Khinchine heorem: Power spectral density and autocorrelation function of a wide-sense stationary random process are related via Fourier ransform. Power of a WSS Process Consider F of autocorrelation: K XX τ ) = 1 π If τ=, then: K XX ) = 1 π K XX ) = Xt)Xt +τ ) τ = = X = P Sω)e jωτ dω Sω)e dω = 1 π Sω)dω = P he integral over the power spectral density is a power P of a Wide-Sense Stationary process. i.e. power P is equal to the value of autocorrelation function K XX τ) at τ=. Sω) P = K XX ) = 1 π Sω)dω ω ω+dω) # Meaning of Power Spectral Density Sω) : he value of Sω) is the power concentrated in a frequency interval [ω; ω+dω]. ω

3 Calculation of Power Calculation from mean and variance If one knows mean and variance of the random process, the power P can be calculated using the following formulae: P = X ) σ X = X X P = σ X + X) Calculation from power spectral density If one knows power spectral density S of the process in terms of radial frequency ω, or of plain frequency f, the power P is an integral over S: P = K XX ) = 1 Sω)dω = S f )df π NB: he factor 1/π) vanishes if the spectrum is known in terms of plain frequency f. 1) Sω) Properties of Power Spectral Density of Wide-Sense Stationary Process PSD Sω) is positive, because the meaning of Sω) is the power inside the frequency interval [ω,ω+δω]. ) Sω) is real, because power is real. 3) Sω) is an even function of frequency ω. Let us prove this. Consider S ω) = Sω) Property of ACF : K XX τ ) = K XX τ ) Sω) = K XX τ )e jωτ dτ = K XX τ ) cosωτ ) jsinωτ ))dτ = = K XX τ )cosωτ )dτ j K XX τ )sinωτ )dτ = K XX τ )cosωτ )dτ; = S ω) = K XX τ )e jωτ dτ = K XX τ ) cosωτ )+ jsinωτ ))dτ = = K XX τ )cosωτ )dτ = Sω). 4) he wider the spectrum, the narrower the autocorrelation function is, and vice versa.

4 Mini-test 1. he mean value of a wide sense stationary random process is 4, and the variance is 7. What is the power of the process?. Which of the following functions could be valid power spectrum density of a stationary process - a) S ω) = j ω + 4 ω 4 +8 c) S ω) = ω + 4 ω 4 8 b) S ω) = ω + 4 ω 4 +8 d) S ω) = ω ω 6 8 Example: Cosine with Random Phase PS., Q. 1) Given: Xt)=Acosω t+ϕ)=fϕ,t), where A,ω are constants, ϕ is a random variable uniformly distributed in the interval [-π,π]. An example of a real system where such a process can be realized is a pendulum without friction whose amplitude of oscillations is small. Initial push with velocity v together with the initial angle α define the phase ϕ. α# v Different realizations of Xt) are shown. he process Xt) is wide sense stationary. t Covariance of Xt) found earlier reads: Ψ XX t,τ ) = A cosω τ ) = Ψ XX τ )

5 Example: Cosine with Random Phase PS., Q. 1) continued ) Ψ XX t,τ) = A cosω τ) = Ψ XX τ) A natural time scale present in the process is frequency ω. Autocorrelation ACF) is equal to covariance. ACF oscillates with frequency ω. Power spectral density PSD) of this process is π Sω) = A δω ω )+δω +ω )) I.e. PSD contains a component at the frequency ω. Power Spectral Density of a Random Process: from ACF and Directly from Realizations

6 Power Spectral Density from experimental data Problems arising when estimating from a time series: 1. In order to introduce power spectral density by means of Wiener- Khinchine heorem, one first has to estimate autocorrelation function of the process. However, with experimental data this can be a separate problem requiring a lot of calculation.. Whatever is measured during experiment, will have finite duration. herefore, any estimate will be made with a certain mistake. Question: Can power spectral density for a random process be introduced straightforwardly for the experimental signal? Power Spectral Density from Realizations of a Random Process We have introduced F for a deterministic function of time. Note, that F exists only for functions that are absolutely integrable f t) dt < How can a function be absolutely integrable? If it is nonzero on a finite time interval only, or if it asymptotically tends to zero as time t tends to plus or minus infinity. Some exceptions are mentioned in Lecture 1. - Now, since we study random processes, we want to be able to estimate the spectral content of it. We can introduce F of a realization of random process by analogy to the F of a deterministic function of time. However, most random processes of our interest are NO absolutely integrable! herefore, their F most probably does not exist. Can we do anything about it?

7 Power Spectral Density from Realizations of a Random Process cont ) Consider random process Xt) with finite variance, and its particular realization xt). Define x t) to be a portion of xt) inside the time interval [-; ]:!# xt), - < t < x t) = " $#, elsewhere Because is finite, the integral over x t) will be finite: x t) dt = x t) dt < herefore, x t) will have F, which we denote F ω): F ω) = x t)e - jωt dt = xt)e - jωt dt - Energy and power of the process Consider energy W and power P of xt) inside a finite time interval [-; ] W = xt) dt, P = 1 Note: Power is energy per time unit. hen: Let tend to infinity. Condition: Let for almost all realizations xt) of random process Xt) the power P of the whole process be finite, i.e. 1 P = lim < P < xt) dt xt) dt - Processes with finite power can be analysed by means of Fourier ransform. Remark: he energy of the infinitely long process can be infinite, but its power can be finite!

8 Let: Xt) be a wide-sense stationary random process. Random Spectrum For a particular realization of a random process of duration one can introduce Fourier ransform, or Fourier Spectrum another term) as follows: F ω) = xt)e - jωt dt Note: For different realizations of the same random process Fourier Spectrum will be different. In fact, it will randomly change from one realization to another. herefore, spectrum F ω) of xt) can be called a random spectrum. Summary: Power Spectral Density from Realizations of a Random Process For a random process Xt) a power spectral density Sω) can be introduced as follows: 1) Calculate the Fourier ransform F ω) for a realization xt) of duration as and F ω) as: F ω) = F ω) = F ω)f * ω) Sω) = lim P xt)e - jωt dt ) Average F ω) over the ensemble of realizations of the process to obtain 1 3) Introduce power P F ω) witihin a finite time interval [-;]: 4) ake the limit of P ω) as tends to infinity to obtain power spectral density Sω): his is the power of the process in the frequency band [ω; ω+dω], i.e. 1 ω) = lim Fω) F ω) ω) = P ω)

9 PSD of cosine with random phase PS 8, Q. 3) Given: random process Xt)=A cosω t+ϕ)=fϕ,t), where A, ω are constants, # ϕ is a random variable uniformly distributed in the interval [-π,π]. Find: power spectral density Sω) of Xt) by two methods: using Wiener-Khinchine theorem and from periodograms. Sketch power spectral density. Hint: Autocorrelation function of Xt) was found within PS, Q. 1 to be: K XX t,τ ) = A cosω τ PSD of cosine with random phase from Wiener-Khintchine theorem PS 8, Q. 3) Solution: 1) Find power spectral density Sω) using Wiener-Khinchine theorem, i.e cos ω t) = exp jω t)+ exp jω t) Sω) = K XX τ )e jωτ dτ = A cosω τ )e jωτ dτ = A e jω τ e jωτ dτ + e jω τ e jωτ dτ * δt) = 1 e jωt dω ) π = A [ 4 πδω ω )+ πδω +ω ) ] = A π [ δω ω )+δω +ω )]

10 PSD of cosine with random phase from Wiener-Khintchine theorem PS 8, Q. 3) Sω) = A π δω ω ) + A π δω +ω ) π A π δω + ω ) A δω ω ) Sω) his is the power spectral density of cosine with random phase. PSD of cosine with random phase from realizations ) Now introduce power spectral density from realizations of random process Xt)=A cosω t+ϕ), where ϕ is a random variable uniformly distributed between -π and +π..1 First, we need to find the finite time Fourier ransform of a single realization, i.e. F ω) F ω) = A cosω t +ϕ)e - jωt dt = A e jω t+ jϕ e - jωt dt + A e jω t jϕ e - jωt dt = A # e jϕ e jω ω)t dt + e jϕ e jω +ω)t dt $ = A # e jϕ 1 jω ω) e jω ω)t + e jϕ 1 jω +ω) e jω +ω)t $

11 PSD of cosine with random phase from realizations cont ) ) F ω) = A e jϕ e jω ω) e jω ω) ) +e jϕ e jω +ω) e jω +ω) = A e jϕ sin ω ω) [ ] [ ] = A e jϕ sin ω ω) ω ω) j ) + j j)ω +ω) * j j jω ω) + ake into account that sine can be represented through exponents: sin ω t) = exp jω t) exp jω t) j ω ω) + e jϕ sin ω +ω) [ ] [ ] + A e jϕ sin ω +ω) ω +ω) ) + = ω +ω) * PSD of cosine with random phase from realizations cont ) Now, find the square of the magnitude of F ω): F ω) = F ω)f * ω) Denote: F ω) = c, A sin [ ω ω) ] ω ω) = a, A sin [ ω +ω) ] ω +ω) = b Calculate cc*: c = ae jϕ + be jϕ = acosϕ + jasinϕ + bcosϕ jbsinϕ c * = acosϕ jasinϕ + bcosϕ + jbsinϕ = ae jϕ + be jϕ ) ae jϕ + be jϕ ) = a + b + abe jϕ + abe cc * = c = ae jϕ + be jϕ jϕ

12 PSD of cosine with random phase from realizations cont ). We now need to find mean of c averaged over the ensemble of realizations of this random process, i.e. in fact over ϕ. a, b and their combinations do not depend on ϕ, and thus do not participate in averaging over ϕ: they are treated as constants. c = a + b + abe jϕ + abe jϕ = a + b + abe jϕ jϕ + abe e jϕ = e jϕ p 1 ϕ)dϕ = 1 π e jϕ dϕ = 1 π cosϕ)dϕ + j 1 π sinϕ)dϕ = π π π e jϕ = π π π hus, average over the ensemble of realizations of F ω) is F ω) = A $ ) sin [ω ω)] ω ω) ) $ ) sin [ω +ω)] + A ω +ω) ) lim PSD of cosine with random phase from realizations cont ).3 Now we have to find the following limit as observation time tends to infinity 1 F ω) = lim *, = lim+ -, 1 A *, $ A ) sin ω ω) + -, ω ω) $ sin ω ω) [ ] [ ] ω ω) ) ) + A $ ) sin [ω +ω) ] + A = A πδω ω)+ A πδω +ω) = Sω) [ ] ω +ω) $ sin ω +ω) ω +ω)., ) /,., ) /, Conclusion: hus, power spectral densities introduced by two ways for the same process cosine of random phase, coincide! lim sinα) α ) * = πδα) One can prove the validity of the above limit, but here we will take it for granted.

13 Why Did we Have to Learn the Previous Material? In experimental situation you are going to consider single realizations of random processes. Question: If power spectral density is introduced from an ensemble of realizations, or from autocorrelation function of the process, what is the use of it if we do not have ensemble of realizations? Answer: here is a class of random processes that are ergodic. For them all or some of their important statistical characteristics can be estimated from averaging over time. We thus do not need an ensemble of realizations to estimate autocorrelation function of an ergodic process, only one realization will be sufficient. If one realization contains full information about the autocorrelation function ACF), then it must contain full information about the Fourier ransform of ACF. he latter is power spectral density!