  Complex demodulation of evolutionary spectra is formulated as a
two-dimensional kernel smoother in the time-frequency domain. In the first
stage, a tapered Fourier transform, $y_{nu}(f,t)$, is calculated. Second, the
log-spectral estimate, $\hat{\theta}_{\nu}(f,t) \equiv \ln(|y_{nu}(f,t)|^2$, is
smoothed. As the characteristic widths of the kernel smoother increase, the
bias from temporal and frequency averaging increases while the variance
decreases. The demodulation parameters, such as the order, length, and
bandwidth of spectral taper and the kernel smoother, are determined by
minimizing the expected error. For well-resolved evolutionary spectra, the
optimal taper length is a small fraction of the optimal kernel half-width. The
optimal frequency bandwidth, $w$, for the spectral window scales as $w^2
\approx \lambda_F/ \tau $, where $\tau$ is the characteristic time, and
$\lambda_F$ is the characteristic frequency scale-length. In contrast, the
optimal half-widths for the second stage kernel smoother scales as $h \approx
1/(\tau \lambda_F)^{1 \over ( p+2) }$, where $p$ is the order of the kernel
smoother. The ratio of the optimal frequency half-width to the optimal time
half-width satisfies $h_F / h_T ~ (|\partial_t ^p \theta | / |\partial_f^p
\theta|)$. Since the expected loss depends on the unknown evolutionary spectra,
we initially estimate $|\partial_t^p \theta|^2$ and $|\partial_f^p \theta|^2$
using a higher order kernel smoothers, and then substitute the estimated
derivatives into the expected loss criteria.
