Time domain analysis

As noted above, periodic signals are best analysed in the frequency domain while stochastic signals are usually more profitably analysed in the time domain. The analysis in the time domain often involves the comparison of two different signals while in the frequency domain analyses usually concern only one signal. The expectation value of the covariance function of uncorrelated signals is zero. The expected value of the autocorrelation function (E{ACF}, Sect. 12.3.2) of white noise also is zero everywhere except for 1 at zero lag. The expected ACF of a stochastic signal of correlation length l vanishes outside a range $\pm l$ about the lags. The ACF of a deterministic function does not vanish at infinity. In particular the ACF of a function with period P has the same value, P. Signals of intermediate or mixed type with an ACF which has several maxima spaced evenly by l and a correlation length $L\gg l$ is called a quasiperiodic oscillation. Its power is significantly above the noise in the $1/l\pm 1/L$ range of frequencies and its correlation length L is called the coherence length.