The Kramers Kronig Relations

The Kramers Kronig relations are used to get the odd part of a causal function of time given the even part and vice versa. In the frequency domain, the parts are imaginary and real, respectively.

Any causal function can be broken into even and odd components,

\[x(t) = x_0(t) + x_e(t).\]

where the definitions of even and odd are

\[\begin{split}x_e(-t) = x_e(t) \\ x_o(-t) = -x_o(t)\end{split}\]

An arbitrary x(t) is shown in Figure 1 along with its even and odd parts. The even part is the original signal, halved and mirrored across the y-axis. The odd part is the same as the even part but inverted for negative time, i.e. the even part multipled by the signum function.

Figure 1. Any arbitrary causal function can be expressed as a sum of even and odd parts.

The signum relationship between the even and odd functions means that x(t) can be expressed using only one of the two parts,

\[\begin{split}x &= x_e + x_o \\ &= x_e(t) + x_e(t)*sgn(t) \\ &= x_o(t) + x_o(t)*sgn(t) \\ &= g(t)[1+sgn(t)].\end{split}\]

The frequency domain result is found by applying the convolution operator,

\[\begin{split}X(\omega) &= F\{g(t)[1+sgn(t)]\} \\ &= G(\omega)+\frac{1}{2\pi}\int_{-\infty}^{\infty}G(\omega')\frac{2}{j(\omega-\omega')}d\omega' \\ &= G(\omega)-\frac{j}{\pi}\int_{-\infty}^{\infty}\frac{G(\omega')}{(\omega-\omega')}d\omega'\end{split}\]

where the Fourier transform pair of sgn(t) is 2/j/ω and care must be taken when ω = ω’. If G(ω) is imaginary then

\[\begin{split}X(\omega) = j Im\{X(\omega)\}-\frac{j}{\pi}\int_{-\infty}^{\infty}\frac{j Im\{X(\omega')\}}{(\omega-\omega')}d\omega' \\ ∴ Re\{X(\omega)\} = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{Im\{X(\omega')\}}{(\omega-\omega')}d\omega'\end{split}\]

The relation appears in the literature as an integration over positive frequencies,

\[\begin{split}Re\{X(\omega)\} &= \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{Im\{X(\omega')\}}{(\omega-\omega')}\left(\frac{\omega+\omega'}{\omega+\omega'}\right)d\omega' \\ &= \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\omega Im\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega' + \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\omega' Im\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega' \\ &= \frac{2}{\pi}\int_{0}^{\infty}\frac{\omega'Im\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega'\end{split}\]

where the first term dropped away because it is an odd function being integrated from \(-\infty\) to \(\infty\). For the second term, the value is doubled and the limits halved because it is even. If G(ω) is real then

\[\begin{split}X(\omega) = Re\{X(\omega)\}-\frac{j}{\pi}\int_{-\infty}^{\infty}\frac{Re\{X(\omega')\}}{(\omega-\omega')}d\omega' \\ ∴ Im\{X(\omega)\} = -\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{Re\{X(\omega')\}}{(\omega-\omega')}d\omega'\end{split}\]

\[\begin{split}Im\{X(\omega)\} &= \frac{-1}{\pi}\int_{-\infty}^{\infty}\frac{Re\{X(\omega')\}}{(\omega-\omega')}\left(\frac{\omega+\omega'}{\omega+\omega'}\right)d\omega' \\ &= \frac{-1}{\pi}\int_{-\infty}^{\infty}\frac{\omega Re\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega' + \frac{-1}{\pi}\int_{-\infty}^{\infty}\frac{\omega' Re\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega' \\ &= \frac{-2\omega}{\pi}\int_{0}^{\infty}\frac{Re\{X(\omega')\}}{(\omega^2-\omega'^2)}d\omega'\end{split}\]

where the second term dropped away because it is an odd function being integrated from \(-\infty\) to \(\infty\). For the first term, the value is doubled and the limits halved because it is even.

Phase Angle from Reflectance

\[\begin{split}R(\omega) &= |R|e^{j \phi (\omega)} \\ ln(R(\omega)) &= ln(|R|) + j \phi (\omega) \\ j \phi (\omega) &= ln(R(\omega)) - ln(|R|) \\ \phi (\omega) &= Im\{ ln(R(\omega))\} \\\end{split}\]