Echo detection using Hilbert transforms is a common measurement for the analysis of modulation systems.
The following equation describes a time-domain signal.
where A is the amplitude, f0 is the natural resonant frequency, and T is the time decay constant.
The following equation yields the Hilbert transform of the time-domain signal.
The following equation yields the natural logarithm of the magnitude of the analytic signal xA(t).
The result from the previous equation has the form of a line with slope m = -1/τ. Therefore, you can extract the time constant of the system by graphing ln|xA(t)|.
The following figure shows a time-domain signal containing an echo signal.
The following conditions make the echo signal difficult to locate in the previous figure:
- The time delay between the source and the echo signal is short relative to the time decay constant of the system.
- 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, as shown on the following figure.
In the previous figure, the discontinuity is plainly visible and indicates the location of the time delay of the echo.