Figure 1 Both positive and negative times and frequencies of a real causal response (top) and real (mid) and imaginary (bottom) parts of its FT.
Half of the values in Figure 1 convey no information: these are the zero values at negative time, and the negative frequencies of the FT. In other words, the right half of Figure 1 is redundant, and is generally not shown. Likewise, the bottom plot, which is the imaginary part, is generally not shown, because it is derivable in a simple way from given information. Computation of the unseen imaginary part is called ``Hilbert transform.'' Here we will investigate details and applications of the Hilbert transform. These are surprisingly many, including phase-shift filtering, envelope functions, the instantaneous frequency function, and relating amplitude spectra to phase spectra.
Ordinarily a function is specified entirely in the time domain or entirely in the frequency domain. The Fourier transform then specifies the function in the other domain. The Hilbert transform arises when half the information is in the time domain and the other half is in the frequency domain. (Algebraically speaking, any fractional part could be given in either domain.)