intro
Both positive and negative times and frequencies
of a real causal response (top) and real (mid) and
imaginary (bottom) parts of its FT.
Figure 1 |

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.)

- A Z-transform view of Hilbert transformation
- The quadrature filter
- The analytic signal
- Instantaneous envelope
- Instantaneous frequency

10/21/1998