What is the Hilbert transform and what are its applications?

The Hilbert transform shifts all frequency components by 90 degrees (pi/2): H{x(t)} produces the quadrature component. Combined with original signal, forms the analytic signal: xa(t) = x(t) + jH{x(t)}, which has zero negative frequency components. Applications: Envelope detection (|xa(t)| gives instantaneous envelope), Instantaneous frequency (derivative of phase), Single-sideband modulation (generate SSB from DSB), and Creating quadrature signals for I/Q processing. Implemented via FIR filter approximation or via FFT (zero negative frequencies, double positive, inverse FFT).