Hilbert transforms are used extensively in the analysis of modulation systems, such as echo detection.
Consider the time signal of the form
and its Hilbert transform
where A is the amplitude, is the natural resonant frequency, and
is the time decay constant.
The natural logarithm of the magnitude of the analytic signal is given by
which has the form of a line with slope
Thus, you can extract the time constant of the system by graphing
Consider the echo signal shown in the following graph. The echo signal is difficult to locate because the time delay between the source and the echo signal is short relative to the time decay constant of the system and because the echo amplitude is small compared to the source.
You can make the echo signal visible by plotting the magnitude of xA(t) on a logarithmic scale. The discontinuity that now appears in the following graph indicates the location of the time delay of the echo.
The following block diagram shows an echo detector. All the necessary nodes are either functions or analysis VIs.
This example generates the analytic signal using the Fast Hilbert Transform VI, finds its magnitude with the 1D Rectangular To Polar VI, and computes the natural log to detect the presence of an echo.