Hilbert Transform and Analytic Signal
Rotate harmonics into quadrature and trace the amplitude of an analytic signal.
Free theory articleRotate harmonics into quadrature and trace the amplitude of an analytic signal.
Concept & interpretation
The Hilbert transform rotates every harmonic component of a signal by 90 degrees (π/2 radians), turning a real signal into its quadrature component. Combined, they form a complex-valued analytic signal which allows us to define the instantaneous amplitude and phase.
Experimenting with the Module
Open the Hilbert Transform and Analytic Signal module to explore how the transform affects different types of signals.
- Select the Cosine signal and change the Phase to pi/4. Observe the Signal value and Hilbert value are approximately 0.707.
- Change the Signal to Sine. The Hilbert transform of a sine wave is negative cosine, making the Hilbert value roughly -0.707 at phase pi/4.
- Switch to the Two harmonics signal. The amplitude trace is no longer a constant line, but a varying envelope bounding the signal.
- Select DC plus cosine. The DC offset (a constant term) passes through without rotation in the signal, while the Hilbert transform evaluates to 0.0 at phase pi.