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) = Aetcos(2πf0t) (A)
xH(t) = –Aetsin(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:

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.