Hilbert transform
E825432
The Hilbert transform is an integral transform that produces the harmonic conjugate of a real-valued function, playing a central role in signal processing, harmonic analysis, and the theory of analytic signals.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | integral transform ⓘ |
| actsOn |
L2 function
ⓘ
real-valued function ⓘ tempered distribution ⓘ |
| appliesTo |
spatial signals
ⓘ
time-domain signals ⓘ |
| definition |
(Hf)(x) = (1/π) p.v. ∫_{−∞}^{∞} f(t)/(x−t) dt
ⓘ
principal value convolution with 1/(πt) ⓘ |
| domain |
L2(R)
ⓘ
Lp(R) for 1 < p < ∞ ⓘ |
| field |
Fourier analysis
NERFINISHED
ⓘ
complex analysis ⓘ harmonic analysis ⓘ signal processing ⓘ |
| FourierMultiplier | −i·sgn(ξ) ⓘ |
| generalization |
discrete-time Hilbert transform
ⓘ
fractional Hilbert transform ⓘ multidimensional Hilbert transform NERFINISHED ⓘ |
| hasKernel | 1/(πt) ⓘ |
| inverseOf | negative Hilbert transform ⓘ |
| namedAfter | David Hilbert NERFINISHED ⓘ |
| produces |
harmonic conjugate
ⓘ
quadrature component of a signal ⓘ |
| property |
Fourier multiplier operator
ⓘ
H^2 = −I on L2(R) modulo constants ⓘ bounded on L2(R) ⓘ convolution operator ⓘ isometry on L2(R) up to a constant factor ⓘ scale invariant ⓘ singular integral operator ⓘ skew-adjoint on L2(R) ⓘ translation invariant ⓘ unitary on L2(R) after suitable normalization ⓘ |
| range |
L2(R)
ⓘ
Lp(R) for 1 < p < ∞ ⓘ |
| relatedConcept |
Bedrosian theorem
NERFINISHED
ⓘ
Cauchy integral NERFINISHED ⓘ Hardy spaces NERFINISHED ⓘ Kramers–Kronig relations NERFINISHED ⓘ Riesz transforms NERFINISHED ⓘ analytic signal ⓘ |
| usedFor |
Kramers–Kronig relations
NERFINISHED
ⓘ
causal filter design ⓘ construction of analytic signal ⓘ dispersion analysis ⓘ envelope detection ⓘ instantaneous frequency estimation ⓘ phase analysis of signals ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.