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.
The following figure shows a section of the block diagram of the VI used to produce the previous figures of the time-domain signal and the visible echo signal.
The VI in the previous figure completes the following steps to detect an echo.
- Processes the input signal with the Fast Hilbert Transform VI to produce the analytic signal xA(t).
 |
Note You can use the Fast Hilbert Transform VI only in the LabVIEW Full and Professional Development Systems. |
- Computes the magnitude of xA(t) with the Re/Im To Polar function.
- Computes the natural log of xA(t) to detect the presence of an echo.
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