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Hilbert Transform

Create analytic signals for envelope and instantaneous frequency detection

50 Hz
5 Hz

Original Signal x(t)

Hilbert Transform x̂(t)

Envelope |z(t)| = √(x² + x̂²)

Instantaneous Phase φ(t)

Analytic Signal: z(t) = x(t) + j·x̂(t) = A(t)·e^(jφ(t))

Hilbert Transform Applications

Envelope Detection: Extract the amplitude envelope of modulated signals (AM radio, vibration analysis).

Instantaneous Frequency: Track how frequency changes over time (radar, speech analysis).

Single-Sideband Modulation: Create SSB radio signals by removing one sideband.

Analytic Signal: The complex combination x(t) + j·x̂(t) has no negative frequency components.

90° Phase Shift: The Hilbert transform shifts all frequency components by 90 degrees.