The analytic signal

The so-called analytic signal can be constructed from a real-valued time series u_t and itself $90^{\circ}$ phase shifted, i.e., v_t can be found using equation (5). The analytic signal is g_t, where

$$G(Z) \eq U(Z) + i V(Z) \eq [1+ iQ(Z)] \ U(Z)$$ (11)

In the time domain, the filter [1+ iQ(Z)] is $\delta_t+iq_t$,where $\delta_t$ is an impulse function at time t=0. The filter $1+iQ(Z)= 1 +\omega/ \vert\omega\vert$vanishes for negative $\omega$.Thus it is a real step function in the frequency domain. The values all vanish at negative frequency.

We can guess where the name ``analytic signal'' came from if we think back to Z-transforms and causal functions. Causal functions are free of poles inside the unit circle, so they are ``analytic'' there. Their causality is the Fourier dual to the one-sidedness we see here in the frequency domain.

A function is ``analytic'' if it is one-sided in the dual (Fourier) domain.