App-4
IM DLM3054-01EN
Hilbert Function (HLBT)
Normally, when we analyze real-time signals, it is useful to think of these signals as the real part of functions of
complex variables, and to carry out the actual signal analysis using such functions.
If the real-time signal is considered to be the real part of the function, the imaginary part can be determined with
the Hilbert transform of the real part.
The Hilbert transform does not change the order of the individual variables. Hilbert transform of a time signal
results in another time signal.
The Hilbert transform procedure is as follows.
When a time-domain signal is transformed, the signal is first transformed into the frequency domain through
Fourier transform. Next, the phase of each frequency component is shifted by –90 degrees if the frequency is
positive and +90 degrees if the frequency is negative. Lastly, taking the inverse Fourier transform completes the
Hilbert transform.
Example
The Hilbert transform can be used to analyze an envelope waveform.
AM (amplitude modulation): SQRT(C1 × C1 + HLBT(C1) × HLBT(C1))
Demodulation of an FM Signal: DIF(PH(C1, HLBT(C1)))
Phase Function (PH)
Phase function PH(X1, Y1) computes tan
-1
(X1/Y1).
The phase function takes the phase of the previous point into consideration and continues to sum even when the
value exceeds ±π (the ATAN function reflects at ±π).
The unit is radians.
Previous point
θ2
θ2
θ2 = θ1 + Δθ2 θ2 = θ1 – Δθ2
Δθ2
Δθ2
θ1
θ1
Appendix
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