Spectral envelop

Let sigma(theta) denote a spectral measure of the stochastic process y_t , the envelop of maximal spectral power is defined as

$rho(theta) := sup_{sigma} left{lim_{epsilon rightarrow 0} (sigma(theta+epsilon)-sigma(theta-epsilon)) ~mid~ dsigma geq0, R=frac{1}{2pi} int_{-pi}^{pi} G(e^{itheta})dsigma(theta) G(e^{itheta})^* right},$

where represents the state covariance and G(z) is the input-to-state filter. In other words, rho(theta) represents the maximal spectral “mass” located at theta which is consistent with the covariance matrix. It can also be shown that

$rho(theta) =maxleft{ rho ~mid~ R-rho G(e^{itheta})G(e^{itheta})^* geq 0right},$

where $G(e^{itheta})G(e^{itheta})^*$ represents the state covariance for the sinusoidal input $e^{iktheta}$ . The optimal solution is

$rho(theta)=frac{1}{G(e^{itheta})^* R^{-1}G(e^{itheta})},$

and implemented in envlp.m. This generalizes the Capon spectral estimation method which applies to being Toeplitz matrix and G(z) the tapped delay line transfer function. The Capon method may be motivated by a noise suppression formulation aimed at antenna array applications [P. Stoica, R.L. Moses, 1997].

For real valued processes with a symmetric spectrum with respect to the origin, a symmetrized version of the spectral envelop [1] can be similarly obtained:

$rho_{mbox{real}}(theta)=maxleft{rho ~mid~ R-frac{rho}{2} left(G(e^{itheta})G(e^{itheta})^*+G(e^{-itheta})G(e^{-itheta})^* right)right}.$

specenvlp

% spectral envelop
rhohalf = envlp(R,A,B,th);
rho = rhohalf.^2;
plot(th,rho);