The Hilbert transform H{x(t)} = (1/π) P.V. ∫ x(τ)/(t-τ) dτ shifts every frequency component by -90°: cosines become sines and vice versa. Combined with the original signal, it creates the analytic signal z(t) = x(t) + i·H{x(t)}, whose magnitude |z(t)| is the instantaneous amplitude (envelope) and whose argument arg(z(t)) gives instantaneous phase. The instantaneous frequency f_i(t) = (1/2π)·d/dt[arg(z(t))] is the local frequency — for a chirp signal this linearly increases over time. This underpins AM demodulation, radar Doppler processing, EEG analysis, and single-sideband radio.