1.14 Karhunen-loeve expansion

The representation of the process, $X(t)$ , is the sequence of random variables $X_i$ . The choice basis of $\{{}_i(t)\}$ is unrestricted. Of particular interest is to restrict the basis functions to those which make the $\{X_i\}$ uncorrelated random variables. When this requirement is satisfied, the resulting representationof $X(t)$ is termed the Karhunen-Love expansion. Mathematically, we require $\forall i, j, i\neq j\colon (X_i{X}_j)=(X_i)(X_j)$ . This requirement can be expressed in terms of the correlation function of $X(t)$ . $$(X_i{X}_j)=(\int_0^T X(){}_i()\,d \int_0^T X(){}_j()\,d )=\int_0^T \int_0^T {}_i(){}_j()R_X(, )\,d \,d$$ As $(X_i)$ is given by $$(X_i)=\int_0^T m_X(){}_i()\,d$$ our requirement becomes

The approach to solving for the eigenfunction and eigenvalues of $K_X(t, u)$ is to convert the integral equation into an ordinary differential equation which can be solved. This approach isbest illustrated by an example.

The Karhunen-Love expansion has several important properties.

• The eigenfunctions of a positive-definite covariance function constitute a complete set. One can easily show thatthese eigenfunctions are also mutually orthogonal with respect to both the usual inner product and with respect to the innerproduct derived from the covariance function.
• If $X(t)$ Gaussian, $X_i$ are Gaussian random variables. As the random variables $\{X_i\}$ are uncorrelated and Gaussian, the $\{X_i\}$ comprise a sequence of statistically independent random variables.
• Assume $K_X(t, u)=\frac{N_0}{2}(t-u)$ : the stochastic process $X(t)$ is white. Then $$\int \frac{N_0}{2}(t-u)(u)\,d u=(t)$$ for all $(t)$ . Consequently, if ${}_i=\frac{N_0}{2}$ , this constraint equation is satisfied no matter what choice is made for the orthonormal set $\{{}_i(t)\}$ . Therefore, the representation of white, Gaussian processes consists of a sequence of statisticallyindependent, identically-distributed (mean zero and variance $\frac{N_0}{2}$ ) Gaussian random variables. This example constitutes the simplest case of the Karhunen-Love expansion.