Derivations for acf->psd, the spectral ar form, and lorentzian bias

[ryanhammonds]

I finished the math connecting ACF <-> PSD <-> AR(1) in the context of timescales. A summary of the added notebook:

1. ACF as an AR(1)

The ACF can be represented in terms of the AR(1) coefficient, $\varphi$.

$$ \begin{align} A(t) &= e^{\frac{-t}{\tau f_s}} \\ &= \varphi^t \\ \end{align} $$

2. PSD is equal the the abs(fft(A(t))^2, giving the AR(1) spectral form

The AR(1) PSD is the the spectral representation of an exponentially decaying ACF, $A(t)$, confirmed using the Fourier transform and Wiener–Khinchin theorem.

$$ \begin{align} FFT(f) &= \sum_{t=-\infty}^{\infty} \left(\varphi e^{-i 2 \pi f \frac{1}{f_s}}\right)^t \\ &= \frac{1}{1 - \varphi e^{-i 2 \pi f \frac{1}{f_s}}} \\\ P(f) &= |FFT(f)|^2 \\ &= \frac{1}{|1-\varphi e^{-i2\pi f \frac{1}{f_s}}|^2} \end{align} $$

3) Lorentzian Bias

The common Lorentzian spectral form is:

$$ P(f) = \frac{1}{f_k^2 f^2} $$

The notebook shows how to derive this from the AR(1) spectral form. This requires two approximations that introduce bias to timescale estimation: a cosine approximation and an approximation of $\frac{1-\varphi}{\sqrt{\varphi}}$. This bias increases as $\varphi$ becomes smaller, timescales become shorter, or as the knee frequency approaches the Nyquist frequency. Instead, it's recommended to learn $\varphi$ and $\tau$ from the unbiased AR(1) spectral form:

$$ \begin{align} P(f) &= \frac{1}{|1-\varphi e^{-i2\pi f \frac{1}{f_s}}|^2} \\ &= \frac{1}{1 - 2 \varphi \cos{(2 \pi f \frac{1}{f_s})} + \varphi^2} \end{align} $$

where

$$ \begin{align} \tau &= -\frac{1}{\ln(\varphi) f_s} \end{align} $$

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