# Causality and LTI Systems

*How the time-domain requirement h(t) = 0 for t < 0 translates into a precise analytic constraint on the frequency-domain response, via Titchmarsh's theorem.*

## Linear, time-invariant, causal

A system that maps an input signal x(t) to an output y(t) is **linear** if
superposition holds and **time-invariant** if a time shift in the input
produces the same time shift in the output. Every such LTI system admits
a convolution representation,

```
y(t) = (h ∗ x)(t) = ∫_{−∞}^{∞} h(t − τ) x(τ) dτ
```

where h(t) is the impulse response. Taking Fourier transforms with the
physics convention `e^{−iωt}` (see Chapter 1 on sign choices) turns
convolution into multiplication,

```
Y(ω) = χ(ω) X(ω),    χ(ω) = ∫_{−∞}^{∞} h(t) e^{iωt} dt
```

so χ(ω) — the transfer function, susceptibility, admittance, compliance,
or whatever the physical application calls it — is the Fourier transform
of h(t).

The system is **causal** if the output at time t depends only on inputs at
times τ ≤ t. Equivalently, the impulse response vanishes for negative
time:

```
h(t) = 0     for all t < 0
```

or, compactly, h(t) = h(t)·Θ(t) where Θ is the Heaviside step. This is
the entire physical content of "effects cannot precede causes" for an LTI
system. Everything that follows in this series is a mathematical
consequence of that single equation.

## Titchmarsh's theorem

Titchmarsh (1948) proved the equivalence theorem that powers the whole
K–K machinery. Let χ(ω) be square-integrable on the real line,
χ ∈ L²(ℝ). The following three statements are equivalent:

1. **Analytic extension.** χ(ω) is the boundary value, as Im ω → 0⁺, of a
   function χ(z) that is analytic in the open upper half-plane
   Im z > 0 and satisfies a uniform L² bound
   `sup_{y>0} ∫ |χ(x + iy)|² dx < ∞` (the Hardy-space H²(UHP)
   condition).
2. **Hilbert-transform pair.** The real and imaginary parts of χ on the
   real axis are Hilbert transforms of one another:
   ```
   χ′(ω) = (1/π) P ∫_{−∞}^{∞} χ″(ω′) / (ω′ − ω) dω′
   χ″(ω) = −(1/π) P ∫_{−∞}^{∞} χ′(ω′) / (ω′ − ω) dω′
   ```
3. **Vanishing past.** The inverse Fourier transform h(t) of χ(ω)
   satisfies h(t) = 0 for all t < 0.

The three are not loosely related — they are logically the same
statement, wearing three different disguises. (1) is the complex-analytic
face, (2) is the integral-equation face, and (3) is the physical
time-domain face. The K–K relations **are** statement (2), and the
purpose of Chapters 11–12 is to carry out the (3) ⇒ (1) ⇒ (2)
implication explicitly using Cauchy's theorem.

## Why L² matters

Titchmarsh's theorem as stated above lives in L². Physical response
functions often fail to be L² without modification:

- A constant DC term χ(∞) ≠ 0 is not square-integrable.
- A delta-function contribution at t = 0 in h(t) (an instantaneous
  response) corresponds to a constant in χ(ω).
- A refractive index n(ω) with n(∞) = 1 does not decay.

In these cases one either subtracts the asymptote (work with
n(ω) − 1 instead of n(ω)) or uses a **subtracted dispersion relation**
(Chapter 15). The core logic is unchanged; only the convergence
bookkeeping differs.

## Real impulse response ⇒ crossing symmetry

If the physical system is real-valued — real input produces real output —
then h(t) ∈ ℝ for all t. Taking the complex conjugate of the defining
integral,

```
χ*(ω) = ∫_{−∞}^{∞} h(t) e^{−iωt} dt = χ(−ω)
```

so χ satisfies the **crossing relation** χ(−ω) = χ*(ω). Splitting into
real and imaginary parts,

```
χ′(−ω) =  χ′(ω)     (even function of ω)
χ″(−ω) = −χ″(ω)     (odd function of ω)
```

The dispersive part is an even function of frequency; the absorptive part
is odd. This parity is what allows the symmetric two-sided K–K integrals
(statement 2 above) to be folded onto the positive half-axis, producing
the `(2/π) P ∫₀^∞ …` one-sided form quoted in Chapter 1.

Note the mental model: the two-sided form is mathematically more natural
(it is a plain Hilbert transform), the one-sided form is experimentally
more natural (measurements are only made for ω ≥ 0), and the bridge
between them is crossing symmetry, which rides on reality of h(t).

## What causality does **not** give

Causality alone does not force χ(ω) to be bounded, analytic at infinity,
or free of resonant poles on the real axis. What it forces is: no poles
in the open upper half-plane, and analyticity of the extension there.
Dissipation (χ″ > 0) is what pushes the poles into the **lower** half-
plane strictly, giving stability and the usual damped-oscillator picture.
A lossless causal system can still have poles **on** the real axis, which
is why the principal-value prescription appears in K–K: it handles such
marginal cases cleanly.

## Causality vs passivity

It is worth separating two constraints that often get conflated:

- **Causality** ⇒ χ(ω) analytic in UHP ⇒ K–K relates χ′ and χ″.
- **Passivity** (no energy created internally) ⇒ additionally ω·χ″(ω) ≥ 0
  for ω > 0 (physics convention). Equivalently, in engineering terms,
  the transfer function is *positive-real* and has all poles in the LHP.

K–K survives without passivity — an active device with internal gain
(a maser, an amplifier) is still causal and still K–K-consistent. What
active media lose is the sign positivity of the imaginary part: χ″ can
become negative, indicating gain instead of loss. Passivity adds sum
rules (Chapter 14) and is what lets dielectric-spectroscopy software
flag "implausibly negative loss" as a calibration error. Keep the two
labels separate when reading across disciplines — RF and circuits use
"passive" to mean Re Z ≥ 0, optics uses it to mean α ≥ 0; both are
stronger than causality but imply it.

Chapter 11 now takes the (3) ⇒ (1) direction of Titchmarsh and proves it
directly by writing χ(ω) as a Fourier integral restricted to t ≥ 0.

## See also

- [09_jordan_lemma.md](09_jordan_lemma.md)
- [11_analyticity_upper_half_plane.md](11_analyticity_upper_half_plane.md)
- [12_derivation_KK.md](12_derivation_KK.md)