What is the uHz-rotator algorithm? According to this page, the Baudline software tool implements


  ...the uHz-rotator algorithm that operates in the complex domain to
  calculate an extremely accurate estimate of the frequency.


Furthermore


  Under good conditions (strong signal and low noise) the frequency
  accuracy can exceed 8 significant digits (~.000001 Hz).


I tested it with a signal generator (ToneGen). For 82.41 Hz (E2) it displays 82.40999 with averaging and 82.4 without (the rest of the digits keep spinning). The FFT size is only 2048 and the sample rate is the standard 44100Hz. 

To compare, an implementation I'm working on, which employs quadratic interpolation, has a poor estimate of anywhere between 84 and 86Hz. The phase difference estimator isn't doing much better. In both cases I only get the precision to the first decimal when I increase the FFT size to 8192.

What is this uHz algorithm and how do you get such precision?
From the (limited) description the uHz rotator algorithm sounds like one of the phase-weighted averages from this site, but it's not an algorithm I am familiar with.

The Cramér–Rao lower bound$^1$ for estimating the frequency of sinusoid with amplitude $A$ in white noise with variance $\sigma^2$ is given by:

$$
\mathrm{var}(\hat{f}) \ge \frac{12}{(2\pi)^2\eta N(N^2-1)}
$$

where $N$ is the number of samples and $\eta = A^2/(2\sigma^2)$ (taken from Kay's Fundamentals of Statistical Signal Processing, Estimation Theory equation 3.41).  

So, assuming $A =1$, $\sigma^2 = 0.1$, and assuming that frequency accuracy can exceed 8 significant digits means that $\mathrm{var}(\hat{f}) &lt; (10^{-8})^2$ we get:
$$
\begin{array}
\ (10^{-8})^2 &amp;\ge&amp; \frac{12}{(2\pi)^2 5 N(N^2-1)}\\
\frac{20}{12} \pi^2 \times 10^{-16} &amp;\ge&amp; \frac{1}{N(N^2-1)}\\
N(N^2-1) &amp;\ge&amp; 6.079271 \times 10^{14}\\
N &amp;\ge&amp;  84713
\end{array}
$$

This means that your data length would need to be 84713 samples to achieve this accuracy.

For different values of $\sigma^2$ see the following table.

$$
\begin{array}
\ \sigma^2 &amp; \eta &amp; 10\log_{10}(\eta) &amp; N\\
\\
0.1&amp;    5.\ \ &amp;    6.9897\ \ &amp;    84714.\\  
\\
    0.01&amp;    50.\ \ &amp;    16.9897\ \ &amp;    39321.\\  
\\
    0.001&amp;    500.\ \ &amp;    26.9897\ \ &amp;    18252.\\  
\\
    0.0001&amp;    5000.\ \ &amp;    36.9897\ \ &amp;    8472.\\  
\end{array}
$$

PS: I have emailed the author of that web-site and asked for their comment. Let's see if they respond.



$^1$ The Cramér–Rao lower bound is the lowest achievable variance of an unbiased estimator.