# The Hilbert Transform

*The Kramers–Kronig relations, stripped of physical dressing, are simply the statement that χ′ and χ″ are a Hilbert-transform pair. Everything a signal-processing engineer already knows about Hilbert transforms therefore applies verbatim to K–K.*

## Definition

The Hilbert transform of a real function u(t) on the real line is the
principal-value convolution with 1/(πt):

```
H[u](t) = (1/π)  P ∫_{−∞}^{∞}  u(τ) / (t − τ)  dτ
```

or, equivalently, H[u](t) = (1/π) P ∫ u(t − τ)/τ dτ. The principal
value handles the singularity at τ = t; without it the integral is
divergent in the ordinary sense. H is a linear operator, bounded on
L²(ℝ), and on L^p for 1 < p < ∞ (the Riesz theorem).

Comparing to Chapter 12's two-sided K–K,

```
χ′(ω) =  (1/π) P ∫ χ″(ω′) / (ω′ − ω) dω′  =  −H[χ″](ω)    (!)
χ″(ω) = −(1/π) P ∫ χ′(ω′) / (ω′ − ω) dω′  =   H[χ′](ω)
```

Careful with the sign: the K–K kernel is `1/(ω′ − ω)`, whereas the
standard Hilbert kernel is `1/(t − τ)`. Swap the argument order and a
minus sign appears. With the conventional sign choice above, **the K–K
pair (χ′, χ″) satisfies χ″ = H[χ′] and χ′ = −H[χ″]**. Equivalently,
applying H twice inverts the sign:

```
H[H[u]] = −u     (on L², modulo the DC component)
```

So H² = −I, and H is an isometry of L² with H⁻¹ = −H. This is the
"90-degree phase shifter" property in disguise.

## Fourier-domain multiplier

Because H is a convolution with 1/(πt), its action in the frequency
domain is multiplication by the Fourier transform of 1/(πt), which (as
a tempered distribution) is `−i sgn(Ω)`:

```
ℱ{H[u]}(Ω) = −i sgn(Ω) · ℱ{u}(Ω)
```

Multiplication by `−i sgn(Ω)` is exactly a 90° phase shift: positive-
frequency components are rotated by −90°, negative-frequency components
by +90°, and DC (Ω = 0) is killed. This is why H² = −I: applying the
multiplier twice gives `(−i sgn Ω)² = − sgn²(Ω) = −1` almost everywhere.

Two basis-function sanity checks:

```
H[cos(ωt)] = sin(ωt)
H[sin(ωt)] = −cos(ωt)
H[e^{iωt}] = −i sgn(ω) · e^{iωt}
```

Each is a rigid 90° rotation in the complex plane.

## Analytic signal

Given a real signal u(t), its **analytic signal** is

```
u_a(t) = u(t) + i H[u](t).
```

In the frequency domain, `ℱ{u_a}(Ω) = (1 + sgn Ω) · ℱ{u}(Ω)`, which
equals `2·ℱ{u}(Ω)` for Ω > 0 and 0 for Ω < 0. So u_a is a complex
signal containing only positive-frequency components — its spectrum is
supported in the upper half-plane of frequency, which is precisely the
geometric dual of "χ analytic in the upper half-plane of ω" from Chapter
11.

The analytic signal is the tool used to define instantaneous amplitude
`|u_a(t)|` and instantaneous phase `arg u_a(t)` for a real signal, and
it is the engineering object whose very existence encodes K–K-type
causality in reverse: a spectrum supported only on Ω > 0 implies the
imaginary part of the time signal is the Hilbert transform of the real
part.

## Minimum-phase systems

A stable, causal LTI system with no zeros in the right-half s-plane is
called **minimum-phase**. For such systems, the log-magnitude and phase
of the transfer function are themselves a Hilbert-transform pair:

```
arg H(jω) = − H_ω [ ln|H(jω)| ]
```

(where the Hilbert transform acts on the frequency variable). This is
the **Bode gain–phase relation** used throughout control theory, and it
is a cousin of K–K: causality plus minimum-phase ⇒ log|H| is the real
part of an analytic function of ln(s), and its phase is determined by a
Hilbert transform of the log-magnitude. K–K dresses the same fact in
the natural variables of dielectric response; Bode dresses it in the
natural variables of feedback amplifier design.

Chapter 16 (DSP slant) picks this up in detail; here we just flag the
equivalence.

## The K–K relations as a Hilbert-transform identity

Gathering the threads:

| Object | Statement |
|---|---|
| **K–K (physics)** | χ″ = H[χ′],  χ′ = −H[χ″] |
| **Bode (control)** | arg H = −H_ω[ln|H|] for minimum-phase systems |
| **Analytic signal (DSP)** | u_a = u + i H[u] has one-sided spectrum |
| **Hardy space (math)** | boundary value of f ∈ H²(UHP) satisfies Im f = H[Re f] |

These are four statements of one theorem: the boundary values of an
H²-class analytic function on the upper half-plane have real and
imaginary parts that are Hilbert transforms of each other. K–K is the
physics-response flavour; the other rows are the same theorem wearing
different clothes.

## Practical properties worth remembering

- **Linearity.** H[au + bv] = aH[u] + bH[v].
- **Shift and scale.** H commutes with time shifts and with positive
  scalings of the argument.
- **Parseval.** ∫|H[u]|² dt = ∫|u|² dt (H is an L² isometry).
- **Orthogonality.** ∫ u(t) · H[u](t) dt = 0 — a function is
  orthogonal to its Hilbert transform.
- **Differentiation.** H commutes with d/dt.
- **DC kill.** H[constant] = 0; the Hilbert transform has a zero mode.
  This is why the "H² = −I" is only true modulo DC, and why K–K cannot
  recover the asymptotic χ(∞) — subtracted dispersion relations
  (Chapter 15) are needed when that asymptote is non-zero.

Every one of these properties has a direct K–K analogue, and checking
them against a known response function (e.g. the Lorentz oscillator of
Chapter 1) is a good sanity exercise before moving on to sum rules.

## See also

- [08_principal_value.md](08_principal_value.md)
- [12_derivation_KK.md](12_derivation_KK.md)
- [14_sum_rules.md](14_sum_rules.md)
- [19_minimum_phase_DSP.md](19_minimum_phase_DSP.md)