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).
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(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.