“Introduction:

“The Hilbert Transform is used as a digital demodulation technique. Discussions of the Hilbert Transform contain such mathematical terms as analytical signal, complex domain, and convolution. Difficult mathematical concepts are sometimes best presented in graphic form. This article has to do with visualizing use of the Hilbert Transform in demodulating an amplitude-modulated signal.

Use of the Hilbert Transform is first explained and then followed by a demodulation example. A pure (non-distorted) harmonic waveform (Figure 1) is used to describe steadystate harmonic vibratory motion. It is a two-dimensional plot of amplitude vs time.

Think of the waveform as the side view of a coiled spring (Figure 2). If the coil were turned out of the page and viewed from the end, it would appear to be a circle like the end of a coil spring. From the end view, one could then see, like the cross-hairs of a riflescope, another time-amplitude plane in the horizontal direction. As drawn in Figure 1, this imaginary plane is sticking out of the paper along the 0 amplitude line; edge-on it cannot be seen. In a vibration analysis the complete waveform must be analyzed to obtain the correct solution.

If the waveform in the imaginary plane could be flipped up so that it appeared in the real plane shown in Figure 1, the imaginary waveform would look identical except it would be offset from the actual data along the time axis by ¼ wavelength. From the circular end-view of the coil spring, the transition would be equivalent to ¼ circle, or 90°. The time axis shift is known as a 90° phase difference.

The Hilbert Transform uses a Fourier Transform and an inverse Fourier Transform to produce the 90° phase shifted waveform. The imaginary waveform is the real waveform (see Figure 1) shifted along the time axis by ¼ waveform, or 90°.The Hilbert Transform in effect creates a duplicate waveform, shifted 90°.The real and imaginary waveforms are overlaid in the same display (Figure 3). The real waveform leads the imaginary waveform.

If the two waveforms are added, specifically if the two are added together in quadrature, their sum will be the line shown along the tops of the waveform peaks in Figure 3.

Quadrature means square the real part, square the imaginary part, add the two squared parts, and take the square root of the total. Quadrature addition is used to obtain the long side of a right triangle; i.e. the square root of the sum of the squares of the two sides, or the vector sum.”