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ECE 6010
Lecture 6 – Basic Concepts of Random Processes
Basic definitions and concepts
Definition 1 A random process (or stochastic process on a probability space (Λ, F, P ) is an indexed collection of random variables {Xt, t ∈ T } each defined on (Ω, F, P ), where T is an indexing set of real numbers. 2
- If T is a singleton (one element) then {Xt, t ∈ T } is a r.v.
- If T = {t 1 , t 2 }, then {Xt, t ∈ T } is a bivariate r.v.
- If T consists of a finite number of elements, then {Xt, t ∈ T } is a random vector.
- If T is countable, then {Xt, t ∈ T } is a random sequence.
For most applications we think of t as “time.” In some cases, T is multidimensional. Then Xt is called a random field.
Three interpretations of a r.p.
- A collection of waveforms that occur randomly. That is, it is defined on some prob- ability space. For each ω ∈ Ω there is a corresponding waveform {Xt(ω), t ∈ T } as a function of t with ω fixed. Think of having a big bag of waveforms. We reach into the bag and pick out a waveform — a function of t — at random.
- A collection of random variables. In this case, that is, for each fixed t ∈ T , we have a random variable Xt.
- A real-valued function of two variables Xt : Ω × T → R.
Definition 2 A function {Xt(ω), t ∈ T } assumed by {Xt, t ∈ T } for a fixed ω ∈ Ω is called a realization of the process. (Also known as a sample function or sample path. 2 A realization is just a function. It does not exhibit the randomness.
Definition 3 If T contains a continuum of values (e.g., T = R or T = [0, 1]) then {Xt, t ∈ T } is a continuous-time random process. 2
Definition 4 If T contains only countably many values (e.g. T = Z, T = Z+) then {Xt, t ∈ T } is a discrete-time random process. 2
Definition 5 Let n be a positive integer and {Xt, t ∈ T } a random process. The set of n-dimensional distributions of {Xt, t ∈ T } is a collection of all multivariable distributions of collections (Xt 1 , Xt 2 ,... , Xtn ), where ti ∈ T. The set of all n-dimensional distributions for all orders n is called the set of finite- dimensional distributions (f.d.d.s) of {Xt, t ∈ T }. 2 We will assume this set completely characterizes the statistical distribution of the pro- cess.
Definition 6 T is closed under addition if T 1 , T 2 ∈ T implies T 1 + T 2 ∈ T. 2
Definition 7 Suppose T is closed under addition. The random process {Xt, t ∈ T } is sta- tionary to order k if for all t 1 , t 2 ,... , tk ∈ T , the distribution of (Xt 1 +h, Xt 2 +h,... , Xtk +h does not depend on h for h ∈ T.
If this is true for all orders k, the r.p. is (strictly) stationary. 2 Strict stationarity is a fairly strong condition, and we don’t necessarily need it always.
Example 1 Stationarity to order 1 means that FXt is the same for every t ∈ T. Stationarity to order 2 means that FXt ,Xs depends only on the difference between t and s. 2
Ergodicity
Assume throughout that {Xt} is stationary. Loosely speaking, a random process {Xt} is ergodic if time averages are equal to ensemble averages. That is, averages over ω — expectations — are the same as averages over t. That is, ensemble averages are the same as sample averages.
∫ Here is an example: Suppose^ {Xn}^ is an i.i.d. sequence. The ensemble mean is^ μ^ = X(ω)P (dω). The sample mean is
1 n
∑^ n
k=
Xk (ω).
By the S.L.L.N. we have 1 n
∑^ n
k=
Xk(ω) → μ.
This is an example of an ergodic property.
Means and Autocorrelations
Definition 8 The mean function of a r.p. {Xt, t ∈ T } is
μX (t) = E{Xt}, t ∈ T.
2
Definition 9 The autocorrelation function of a r.p. {Xt, t ∈ T } is
RX (t, s) = E[XtXs], t, s, ∈ T.
2
Definition 10 A random process is second order if E[X (^2) t ] < ∞ for all t ∈ T. 2 For a second order r.p., |μx(t)| < ∞ and |RX (t, s)| < ∞ for all t, s, ∈ T.
Properties of Autocorrelation functions
- RX (t, t) = E[X t^2 ]. (This is the second moment)
- |RX (t, s)|^2 ≤ RX (t, t)RX (s, s) (Schwartz inequality)
- RX (t, s) = RX (s, t) (symmetric)
We observe that this process is ergodic in the mean — a time average is equal to the en- semble average.
RX (t, s) = E[XtXs] = E[A^2 sin(ω 0 t + θ) sin(ω 0 s + θ)] = E[A^2 ]E[sin(ω 0 t + θ) sin(ω 0 s + θ)]
= E[A^2 ]
2 · 2 π
∫ (^) π
−π
cos(ω 0 (t − s)) + cos(ω 0 (t + s) + 2θ) dθ
= E[A^2 ]
cos(ω 0 (t − s)).
We observe that RX (t, s) depends only on the time difference t − s. Hence, the r.p. is WSS. Let τ = t − s. We can write
RX (τ ) = E[A^2 ]
cos(ω 0 τ ) 2
Checking the properties, observe that we have a (local) maximum at τ = 0, and that the function is symmetric.
The Homogeneous Poisson Counting Process
Let T = [0, ∞). Suppose events occur randomly in time in the following fashion:
- The number of events occurring in non-overlapping intervals of time are independent.
- The probability of one event exactly in any interval of length ∆t is equal to λ∆t + o(∆t) for ∆t sufficiently small.
o(∆t) ∆t
→ 0 as ∆t → 0.
(That is, o(∆t) is the generic term for terms of order higher than ∆t.) Also, the probability of more than one event occurring during an interval of length ∆t = o(∆t).
Now define a r.p. {Xt, t ∈ T } by Xt as the number of events occurring in the interval [0, t]. Then Xt has the following properties:
- Xt − Xs is Poisson with parameter λ(t − s):
P (Xt − Xs = k) =
(λ(t − s))k^ e−λ(t−s) k!
- (Xt 1 − Xs 1 ) and (Xt 2 − Xs 2 ) are independent r.v.s for all nonoverlapping intervals [s 1 , t 1 ] and [s 2 , t 2 ].
The parameter λ is called the rate of Xt. Property 2 follows from the first assumption. We say that such a process has independent increments. Such a process is called a Poisson counting process (PCP) with rate λ. These two prop- erties complete determine a PCP. All finite-dimensional distributions (fdds) of the process can be determined from these two properties. How do we show the Poisson distribution property? Pick t > s ≥ 0. Let
pk (t, s) = Pr(exactly k occurrences in [s, t])
for k ≥ 0. Then
pk (t + ∆t, s) = Pr(k occurrences in [s, t + ∆t]) = Pr(k occurrences in [s, t]) Pr(0 in [t, t + ∆t])+ Pr(k − 1 occurrences in [s, t]) Pr(1 occurrence in [t, t + ∆t])+ Pr(fewer than k − 1 occurrences in [s, t]) Pr(all the rest)
By assumption 2,
pk(t + ∆t, s) = pk(t, s)(1 − λ∆t + o(∆t)) + pk− 1 (t, s)(λ∆t + o(∆t)) + (?)(o(∆t)) = ∆tλ(pk− 1 (t, s) − pk (t, s)) + pk (t, s) + o(∆t).
Now ∂ ∂t
pk (t, s) = lim ∆t→ 0
pk(t + ∆t, s) − pk(t, s) ∆t
= λ[pk− 1 (t, s) − pk (t, s)] + lim ∆t→ 0
o(∆t) ∆t
So ∂ ∂t
pk(t, s) = λ[pk− 1 (t, s) − pk (t, s)] t ≥ s.
Now p− 1 (t, s) = 0. When k = 0 we get
∂ ∂t
p 0 (t, s) = −λp 0 (t, s). (*)
so p 0 (t, s) = C(s)eλt
We have another boundary condition: p 0 (s, s) = 1, giving
p 0 (t, s) = e−λ(t−s)
Now we could proceed solve the set of equations for k = 1, 2 ,.... For example, when k = 1: ∂ ∂t
p 1 (t, s) = λ(p 0 (t, s) − p 1 (t, s))
This could be solved, e.g., using Laplace transforms. In general we would find
pk(t, s) =
e−λ(t−s)(λ(t − s))k k!
k = 0, 1 ,... , t > s ≥ 0.
As stated, the properties allow us to find all finite dimensional distributions. For exam- ple, suppose we want to find the joint distribution of Xt 1 and Xt 2 for t 1 < t 2.
P (Xt 1 = i, Xt 2 = j) = P (Xt 1 = i, Xt 2 − Xt 1 = j − i) = P (Xt 1 = i)P (Xt 2 − Xt 1 = j − i)
= (λt 1 )ie−λt^1 i!
(λ(t 2 − t 1 ))j−ie−λ(t^2 −t^1 ) (j − i)!
where the factorization occurs because of independent increments. Draw a typical sample path... The process is called homogeneous because the rate at which the events occur does not depend on t. Let us work out the mean and autocorrelation functions.
μX (t) = E[Xt] = E[Xt − X 0 ] = λt
That is, the PSD is the Fourier transform of the autocorrelation function:
SX (ω) =
−∞
e−iωτ^ RX (τ ) dτ,
assuming the transform exists. Suppose T = Z. The power spectral density (or spectrum) is
SX = F(RX )
where the discrete-time Fourier transform is used,
SX (ω) =
∑^ ∞
k=−∞
e−iωkRX (k),
assuming the transform exists. 2 A sufficient condition for the existence of SX is that
−∞ |RX^ (τ^ )|dτ <^ ∞^ or^
k=−∞ |RX^ (k)|^ < ∞.
Example 2 Suppose RX (τ ) = σ^2 e−β|τ^ |. Then
SX (ω) =
2 βσ^2 β^2 + ω^2
Such a process is called a wide-sense Markov process. Comment on changes with β. A G.R.P. with this spectrum is called an Orstein-Uhlenbeck process. It is sometimes used as a model for wide-band (e.g., nearly white) noise which has been lowpass filtered. 2
Example 3 The Ideal Low Pass Process. Suppose
SX (ω) =
S 0 |ω| < ω 0 0 otherwise.
This is a model for wideband noise in the passband of a system of interest.
RX (τ ) =
S 0 ω 0 π
sin(ω 0 τ ) ω 0 τ
Observe (from the sinc function) that there are delays where the signals are uncorrelated, π/ω 0 , 2 π/ω 0 , etc. If the signal were Gaussian, it would also be independent. 2
Example 4 Let RX (k) = σ^2 r|k|^ for |r| < 1 and k ∈ Z. Then
SX (ω) =
σ^2 (1 − r^2 ) 1 − 2 r cos ω + r^2
Properties of Spectra
- Symmetry:
SX (ω) =
∑−∞^ cos(ωτ^ )RX^ (τ^ )^ dτ^ T^ =^ R ∞ k=−∞ cos(kω)RX^ (k)^ T^ =^ Z.
- SX (ω) = SX (−ω). SX (ω) = S∗ X (ω).
- Inverse: RX (τ ) =
2 π
−∞
eiωτ^ SX (ω) dω T = R.
2 π
−∞
cos(ωτ )SX (ω) dω.
RX (τ ) =
2 π
∫ (^) π
−π
eiωτ^ SX (ω) dω T = Z.
2 π
−∞
cos(ωτ )SX (ω) dω.
- If T = R, RX (0) = E[X t^2 ] =
2 π
−∞
SX (ω) dω.
if T = Z:, RX (0) =
2 π
∫ (^) π
−π
SX (ω) dω.
- SX (ω) ≥ 0 for all ω. (This follows from the non-negative definiteness of RX. Any symmetric non-negative definite function having a finite integral is a legitimate spectral density. Observe that being nnd and finite integral is analogous to a probability density, so that makes RX analogous to a characteristic function.
Cases when RX (τ ) does not have a transform
Example 5 Recall the random sinusoid had
RX (τ ) =
E[A^2 ]
cos(ω 0 τ ).
This is a periodic function. As a result, it can be described using a Fourier series. However, if we restrict ourselves to conventional functions (as opposed to δ functions), there is no Fourier transform. 2 We examine this and other cases that are WSS but do not have a Fourier transform (in the conventional sense).
- Suppose T = R. Then RX (τ ) is continuous and is the autocorrelation function of a WSS r.p. if and only if there is a c.d.f. GX satisfying GX (b) = 1 − GX (b) such that
RX (τ )/RX (0) =
−∞
eiωτ^ dGX (ω).
This transform is called the Fourier-Stieltjes transform.
- Suppose T = Z. Then RX (k) is the autocorrelation function of a WSS r.p. if and only if there exists a GX satisfying GX (b) = 1 − GX (b) such that
RX (k)/RK (0) =
∫ (^) π
−π
eiωk^ dGX (ω)
Definition 19 If Xt and Yt are jointly WSS random processes, the cross power spectrum is defined as
SXY (ω) = F[RXY (τ )] =
−∞ e
−iωτ (^) RXY (τ ) dτ T = R ∑∞ k=−∞ e
−iωkRXY (k) T = Z.
2 Properties of Spectra:
- SXY (ω) = S∗ XY (ω).
- If Xt and Yt are individually and jointly WSS then
SXY (ω)|^2 ≤ SX (ω)SY (ω).
- Re SXY (ω) = Re SXY (−ω) and Im(SXY (ω)) = − Im(SXY (ω)).
Uncorrelated and independent
Definition 20 Two random processes are uncorrelated if CXY (t, s) = 0 for all s, t ∈ T. 2
Definition 21 The random processes Xt and Yt are independent if (Xt 1 ,... , Xtn ) and (Ys 1 ,... , Ysm ) are independent random vectors for all
n, m ∈ Z+
and all t 1 ,... , tn, s 1 ,... , sm ∈ T. 2 Note: Independence implies uncorrelated. The converse is not true.
Definition 22 The random processes Xt and Yt are jointly Gaussian if (Xt 1 ,... , Xtn , Ys 1 ,... , Ysm ) is a Gaussian random vector for all n, m ∈ Z+^ and all t 1 ,... , tn, s 1 ,... , sm ∈ T. 2 For jointly Gaussian random processes, we can characterize by a mean vector and a covariance matrix. All f.d.d.s are determined by μx(t), μy (t), RX (t, s), RY (t, s) and RXY (t, s). For this case, it is true that uncorrelated implies independence.
