HilbertTransform

The Hilbert transform is a linear operator that assigns to a real-valued function x(t) another real-valued function H{x}(t) defined by the principal value integral (1/π) PV ∫_{-∞}^{∞} x(τ)/(t−τ) dτ. It can be interpreted as a 90-degree phase shift of the Fourier components of x, and is commonly defined for functions in suitable function spaces such as L^p with 1<p<∞.

In the frequency domain, the Hilbert transform corresponds to multiplication by -i sgn(ω): F{H{x}}(ω) = -i sgn(ω) X(ω). Consequently H is linear, and applying it twice yields H{H{x}} = -x (under appropriate conditions).

Physically, the Hilbert transform is used to form the analytic signal x_a(t) = x(t) + i H{x}(t). The envelope is A(t) = sqrt(x(t)^2 + H{x}(t)^2) and the instantaneous phase is φ(t) = arctan2(H{x}(t), x(t)). This representation is central to amplitude and frequency modulation analysis.

Applications include envelope detection, synchronous demodulation, quadrature signal generation, estimation of instantaneous frequency, and feature extraction in communications and signal processing. The transform is also used in time-frequency analysis and as a building block in wavelet and empirical mode decomposition contexts.

Practical computation uses FFT-based discrete Hilbert transforms or analytic filtering; edge effects and boundary handling must be managed. For real, finite-length signals, the transform approximates the ideal operator and is most well-behaved for band-limited or smoothly varying inputs.