2009年4月7日 星期二

A Short Note about Spectral Representation

One of the most prominent feature of weakly stationary process is the spectral representation. This representation resembles the familiar Fourier transform of deterministic functions. It doesn’t sounds very exciting for engineers, because they’re already used to perform their favorite transformation on whatever the object is. However, intuition could go wrong when dealing with the infinities. Convergence is a subtle but crucial issue, and must be treated carefully.

This note tries to fill the blank of the textbook `Probability and Random Processes’ as the author only gives the proof for discrete WSS process.

There are two parts of the Spectral representation theorem, one regards of the autorrelation function and the other takes care of the process itself. First we make the notations and definition clear:

A random process X(t) is called wide-sense stationary (WSS) if
\mathbb{E}[\overline{X(s)}X(t)]=R_{XX}(t-s)

Thm. Spectral representation for autocovariance function
H1. The process X(t) has finite variance
H2. Its autocovariance function is continuous at 0
C. The autocovariance function can be expressed as
R_{XX}(\tau)=\int S_{XX}(f)e^{-2\pi ft}df
where Sxx(f) is called its power spectral density function, because it has the unit power per Hertz ([X]^2/Hz).

Thm. Spectral representation for X(t)
H. X(t) satisfies the above assumptions and has zero mean.
C. X(t) can be expressed as
X(t)=\int e^{i2\pi ft}\widehat{X}(f)df
where ^X(f) is defined in a distributional sense.

Remark. the original theorem states the result as

where S is a complex random process with independent increments. I preffer to use the oridinary frequency to stick to the engineer’s convention

Sketch of the proof:
We’ll prove the original formula, and defined the spectral process by
\int\phi(\nu)dS(\nu)=\int\phi(\nu)\widehat{X}(\nu)d\nu

Define the inner product space generated by the random process X(t):
\mathcal{H}_X=\text{span}\{X_t\}_{t\in\mathbb{R}}
the inner product is defined by
\langle X,Y\rangle=\mathbb{E}[X^*Y]
It’s easy to see that it is a subspace of the L2 space
\mathcal{H}_X\subset L^2(\Omega,P)

Now define the other inner product space
\mathcal{H}_{\mu}=\text{span}\{e^{i2\pi ft}\}_{t\in\mathbb{R}}\subset L^2(\mathbb{R},\mu)
where the measure is defined by
\mu((-\infty,\nu])=\int_{-\infty}^{\nu}S_{XX}(f)df
this is well-defined because of the spectral representation for autocovariance function.
The inner product is defined by
\langle \phi_i,\phi_j\rangle=\int\phi_i^*(f)\phi_j(f)S_{XX}(f)df

There is a natural way to identify each element in H_mu with H_X. We can define a linear mapping T from H_mu to H_X by
T(e^{i2\pi ft})=X(t)
It is an isometry(at least formally), because
\begin{align*}\langle T(e^{i2\pi fs}),T(e^{i2\pi fs})\rangle&=\langle X(s),X(t)\rangle=R_{XX}(t-s)\\&=\int e^{i2\pi f(t-s)}S_{XX}(f)df\\&=\langle e^{i2\pi fs},e^{i2\pi ft}\rangle\end{align*}
This enables us to extend the domain of T from H_mu to its closure, because we’re sure T is continuous by this isometry.

Define the spectral process S by
S(\nu)=T(1_{(-\infty,\nu]}(f))
It’s easy to see S(f) is zero mean, it has orthogonal increments and has variance equals to Sxx(f). i.e
\mathbb{E}[|S(f)|^2]=S_{XX}(f)

But we need to make sure the indicator function belongs to the closure of H_mu, otherwise the process S is not well-defined. This is true because we can approximate the indicator function by the Fourier series of the truncated indicator function. By making the window of truncation longer and longer, the series converges both pointwisely and in L^2 because it is a finite measure space. Therefore, the closure of H_mu must equal to L^2, because H_mu contains all the step functions, and step functions are L^2 dense in L^2.

Define the stochastic integral of a simple function wrt S by
I(\phi)=\sum_i\phi(t_i)(S(t_i)-S(t_{i-1}))=\int\phi(\nu)dS(\nu)
This integration is a linear mapping from the step functions to H_X. In fact, this integration agrees with the mapping T on simple functions, that is
I(\phi)=T(\phi)
therefore, we can happily extend this integral to all the function on L^2(R,\mu), and get the result
I(e^{i2\pi ft})=X(t)=\int e^{i2\pi ft}dS(f)

As we said before, the `Fourier transform’of X(t) is defined as
\int\phi(\nu)dS(\nu)=\int\phi(\nu)\widehat{X}(\nu)d\nu
^X(f) has the following property that’s used a lot in engineering literatures:
\mathbb{E}[\widehat{X}^*(\nu')\widehat{X}(\nu)]=S_{XX}(\nu)\delta(\nu-\nu')
Because
\begin{align*}\langle \int\phi_i\widehat{X}d\nu',\int\phi_j\widehat{X}d\nu\rangle&=\langle\int\phi_idS,\int\phi_jdS\rangle\\&=\langle T(\phi_i),T(\phi_j)\rangle\\&=\langle \phi_i,\phi_j\rangle\\\end{align*}
which motivates us to write formally that

and
\int\mathbb{E}[\widehat{X}^*(\nu')\widehat{X}(\nu)]d\nu'd\nu=S_{XX}(\nu)d\nu
which leads to the concludsion that
\mathbb{E}[\widehat{X}^*(\nu')\widehat{X}(\nu)]=S_{XX}(\nu)\delta(\nu-\nu')

Take a geometric way of interpreting the spectral representation: The process X(t) can be seen as a L^2 uniformlly continuous path on the sphere of radious R_XX(0). On the other hand, the spectral process S starts form the origin of the L^2 space at frequency -\infty, and `growth’by orthogonal increments to the surface of the sphere at frequency \infty.

The weak derivative of S seems to be at the platform of white noise anasis. If we define the probability space by ^X, then we get the white noise space? That’s something to be explore in the furture.

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