# Echo Detection

Echo detection using Hilbert transforms is a common measurement for the analysis of modulation systems.

Equation A describes a time-domain signal. Equation B yields the Hilbert transform of the time-domain signal.

x(t) = Ae–t/τcos(2πf0t) | (A)

xH(t) = –Ae–t/τsin(2πf0t) | (B)

where A is the amplitude, f0 is the natural resonant frequency, and τ is the time decay constant.

Equation C yields the natural logarithm of the magnitude of the analytic signal xA(t).

| (C)

The result from Equation C 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 in the following figure:

In the previous figure, the discontinuity is plainly visible and indicates the location of the time delay of the echo.