We call this wide-sense stationary because the mean and covariance do not change as the process evolves. In a strict-sense stationary process, the distribution of each random variable in the process would not change.
Definition 6
Spectral Density
Recall that the Discrete Time Fourier Transform is given by
The Inverse Discrete Time Fourier Transform is given by
Since the DTFT is an infinite summation, it may or may not converge.
Definition 7
This class covers most real-world signals.
Theorem 5
Definition 8
Theorem 6
Tempered distributions like the Dirac Delta function are other functions which are important for computing the DTFT, and they arise from the theory of generalized functions.
Suppose we want to characterize the signal using its DTFT.
Definition 9
Definition 10
We call the DTFT of the autocorrelation the energy spectral density because, by the Inverse DTFT,
Definition 11
The Power Spectral Density of a Wide-Sense Stationary random process is given by
Theorem 7 (Cramer-Khinchin)
Besides giving us a decomposition of a WSS random process, Theorem 7 tells a few important facts.
Z-Spectrum
Recall that the Z-transform converts a discrete-time signal into a complex representation. It is given by
It is a special type of series called a Laurent Series.
Theorem 8
A Laurent Series will converge absolutely on an open annulus
In some cases, it can be useful to only compute the Z-transform of the right side of the signal.
Definition 12
If the Z-transform of the sequence is a rational function, then we can quickly compute what the unilateral Z-transform will be by leveraging its partial fraction decomposition.
Theorem 9
Definition 13
Using this definition, we can see that
Definition 14
Markov Processes
Definition 15
Mathematically, Markov triplets satisfy three properties.
Theorem 10
Definition 16
Because of the conditional independence property, we can write the joint distribution of all states in the Markov process as
Definition 17
Definition 18
All Wide-Sense Markov models have a very succint representation.
Theorem 11
Hidden Markov Processes
Definition 19
Hidden Markov Models can be represented by undirected graphical models. To create an undirected graphical model,
Create a node for each random variable.
Draw an edge between two nodes if a factor of the joint distribution contains both nodes.
Undirected graphical models of Hidden Markov Processes are useful because they let us derive additional Markov dependepencies between groups of variables.
Theorem 12
State-Space Models
Suppose we have a discrete-time random process which evolves in a recursive fashion, meaning the current state depends in some way on the previous state. We can express this recursion with a set of equations.
Definition 20
with initial condition
$$\langle \boldsymbol{X}_i, \boldsymbol{X}_j \rangle = \begin{cases} \Phi_{i,j}\Pi_j & i \geq j \\ \Pi_i \Phi_{j,i}^* & i \leq j \end{cases}$$
$$\langle \boldsymbol{Y}_i, \boldsymbol{Y}_j \rangle = \begin{cases} H_i \Phi_{i,j+1}N_j & i > j\\ R_i + H_i\Pi_iH_i^* & i=j \\ N_i^*\Phi^*_{j,i+1}H_j^* & i < j \end{cases} \text{ where } N_i=F_i\Pi_iH_i^*+G_iS_i$$
A Discrete-Time Random Process is a countably infinite collection of random variables on the same probability space {Xn:n∈Z}.
Discrete Time Random Processes have a mean function μn=E[Xn]and an auto-correlation function RX(n1,n2)=E[Xn1Xn2∗]
A Wide-Sense Stationary Random Process is a disrete-time random process with constant mean, finite variance, and an autocorrelation function that can be re-written to only depend on n1−n2.
A WSS process Z∼WN(0,σ2) is a white noise process with variance σ2 if and only if E[Zn]=0 and E[ZnZm∗]=σ2δ[n,m].
X(ejω)=∑n=−∞∞x[n]e−jωn.
x[n]=2π1∫−ππX(ejω)ejωndω.
A signal x[n] belongs to the l1 class of signals if the series converges absolutely. In other words,
∑k=−∞∞∣x[k]∣<∞.
If x[n] is a l1 signal, then the DTFT X(ejω) converges uniformly and is well-defined for every ω. X(ejω)is also a continuous function.
A signal x[n] belongs to the l2 class of signals if it is square summable. In other words,
∑k=−∞∞∣x[k]∣2<∞.
The l2 class contains important functions such as sinc.
If x[n] is a l2 signal, then the DTFT X(ejω) is defined almost everywhere and only converges in the mean-squared sense:
limN→∞∫−ππ(∑k=−NNx[k]e−jωn)−X(ω)2dω=0
The energy of a deterministic, discrete-time signal x[n] is given by
∑n∈Z∣x[n]∣2.
The autocorrelation of x[n], given by a[n]=x[n]∗x∗[−n], is closely related to the energy of the signal since a[0]=∑n∈Z∣x(n)∣2.
The Energy Spectral Density x[n] with auto-correlation a[n] is given by
A(ejω)=∑n∈Za[n]e−jωn
a[0]=2π1∫−ππA(ejω)dω.
Since summing over each frequency gives us the energy, we can think of A(ejω) as storing the energy density of each spectral component of the signal. We can apply this same idea to wide-sense stationary stochastic processes.
SX(ejω)=∑k∈ZRX(k)e−jωk.
Note that when considering stochastic signals, the metric changes from energy to power. This is because if Xn is Wide-Sense Stationary, then
E[∑n∈Z∣Xn∣2]=∞,
so energy doesn’t even make sense. To build our notion of power, let AT(ω) be a truncated DTFT of the auto-correlation of a wide-sense stationary process, then
The DTFT of the auto-correlation function naturally arises out of taking the energy spectral density and normalizing it by time (the truncated sequence is made of 2T+1 points). In practice, this means to measure the PSD, we need to either use the distribution of the signal to compute RX, or estimate the PSD by averaging multiple realizations of the signal.
The inverse DTFT formula tells us that we can represent a deterministic, discrete-time signal x[n] as a sum of complex exponentials weighted by 2πX(ejω)dω. This representation has an analog for stochastic signals as well.
For a complex-valued WSS stochastic process Xn with power spectral density SX(ω), there exists a unique right-continuous stochastic process F(ω),ω∈(−π,π] with square-integrable, orthogonal increments such that
Xn=∫−ππejωndF(ω)
where for any interval [ω1,ω2],[ω3,ω4]⊂[−π,π],
For two jointly WSS processes Xn,Yn, the z-cross spectrum is the Z-Transform of the correlation function RYX(k)=E[YnXn−k∗].
SYX(z)=∑k∈ZRYX(k)z−k
SXY(z)=SYX∗(z−∗).
We can also look at the Z-transform of the auto-correlation function of a WSS process X to obtain SX(z).
For a rational function SX(z) with finite power (∫−ππSX(ejω)dω<∞) and is strictly positive on the unit circle, the canonical spectral factorization decomposes SX(z) into a product of a re>0 and the transfer function of a minimum phase system L(z) with L(∞)=1
SX(z)=L(z)reL∗(z−∗)
Because L(z) is minimum phase and L(∞)=1, it must take the form
L(z)=1+∑i=1∞l[i]z−i
since minimum phase systems are causal. Using Definition 14, we can express SX(z) as the product of a right-sided and left-sided process.
SX(z)=(reL(z))(reL∗(z−∗))=SX+(z)SX−(z)
Note that SX−(ejω)=(SX+(ejω))∗. Using the assumptions built into Definition 14, we can find a general form for L(z) since we know SY(z) takes the following form
If we let the z−αi and z−βi terms be part of L(z), then
L(z)=zn−m∏i=1n(z−βi)∏i=1m(z−αi).
We say that random variables X,Y,Z form a Markov Triplet X \textemdash Y \textemdash Z if and only if X and Z are conditionally independent on Y
p(x,z∣y)=p(x∣y)p(z∣y)
p(z∣x,y)=p(z∣y)
p(x∣y,z)=p(x∣y)
Because of these rules, the joint distribution can be written as p(x,y,z)=p(x)p(y∣x)p(z∣y).
Random variables X,Y,Z form a Markov triplet if and only if there exist ϕ1,ϕ2 such that p(x,y,z)=ϕ1(x,y)ϕ2(y,z).
To simplify notation, we can define Xmn=(Xm,Xm+1,⋯,Xn) and Xn=X1n.
A Markov Process is a Discrete Time Random Process {Xn}n≥1 where X_n \textemdash X_{n-1} \textemdash X^{n-2} for all n≥2
p(xn)=∏t=1np(xt∣xt−1)=∏t=1np(xt∣xt−1).
The requirement for X \textemdash Y \textemdash Z to satisfy p(x,y,z)=p(x)p(y∣x)p(z∣y) is a very strict requirement. If we wanted to create a “wider” requirement of Markovity, then we could settle for X^(Y)=X^(Y,Z) where X^ is the best linear estimator of X since this property is satisfied by all Markov triplets, but does not imply a Markov Triplet.
Random variables X,Y,Z form a Wide Sense Markov Triplet X \textemdash Y \textemdash Z if and only if the best linear estimator of X given Y is identical to the best linear estimator of X given Y and Z.
X^(Y)=X^(Y,Z)
A stochastic process {Yi}i=0n is a Wide-Sense Markov Process if and only if for any 1≤i≤n−1, Y_{i+1} \textemdash Y_i \textemdash Y^{i-1}forms a Wide-Sense Markov Triplet.
A process X is Wide-Sense Markov if and only if Xi+1=FiXi+GiUi and
⟨[UiX0],[UjX0]⟩=[Qiδ[i−j]00Π0]
If {Xn}n≥1 is a Markov Process, then {Yn}n≥1 is a Hidden Markov Process if we can factorize the conditional probability density
p(yn,xn)=∏i=1np(yi∣xi)
We can think of Y as a noisy observation of an underlying Markov Process. The joint distribution of {Xn}n≥1 and {Yn}n≥1 can be written as
For 3 disjoint sets S1,S2,S3 of notes in a graphical model, if any path from S1 to S3 passes through a node in S2, then S_1 \textemdash S_2 \textemdash S_3.
The standard state space model describes random processes which describe the evolution of state vectors Xi and observation vectors Yi according to the equations
From Theorem 11, we can easily see that state space models are Wide-Sense Markov. Note that Ui and Vi are white noise, and that the dynamics of the system can change at every time step. From these equations, we can derive six different properties. Let Πi=⟨Xi,Xi⟩and Φi,j=∏k=ji−1Fk and Φi,u=I.