Echo Detection and Hilbert Transform

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:

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.


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