What is an AMDF?

I've never seen the word "Formula" with "AMDF". My understanding of the definition of AMDF is

$$ Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \Big| x[n+n_0] - x[n+n_0+k] \Big| $$

$n_0$ is the neighborhood of interest in $x[n]$. Note that you are summing up only non-negative terms. So $Q_x[k,n_0] \ge 0$. We call "$k$" the "lag". clearly if $k=0$, then $Q_x[0,n_0]=0$. Also, if $x[n]$ is periodic with period $P$ (and let's pretend for the moment that $P$ is an integer) then $Q_x[P,n_0]=0$ and $Q_x[mP,n_0]=0$ for any integer $m$.

Now even if $x[n]$ is not precisely periodic, or if the period is not precisely an integer number of samples (at the particular sampling rate you are using), we would expect $Q_x[k,n_0] \approx 0$ for any lag $k$ that is close to the period or any integer multiple of the period. In fact, if $x[n]$ is nearly periodic, but the period is not at an integer number of samples, we expect to be able to interpolate $Q_x[k,n_0]$ between integer values of $k$ to get an even lower minimum.

My favorite is not the AMDF but the "ASDF" (guess what the "S" stands for?)

$$ Q_x[k,n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \big( x[n+n_0] - x[n+n_0+k] \big)^2 $$

Turns out you can do calculus with that because the square function has continuous derivatives, but the absolute value function does not.

Here's another reason i like ASDF better than AMDF. If $N$ is very large and we play a little fast-and-loose with the limits of summation:

$$\begin{align} Q_x[k] & = \frac{1}{N} \left( \sum_n \big( x[n] - x[n+k] \big)^2 \right) \\ & = \frac{1}{N} \left( \sum_n (x[n])^2 + (x[n+k])^2 - 2 x[n] x[n+k] \right) \\ & = \frac{1}{N} \left( \sum_n (x[n])^2 + \sum_n (x[n+k])^2 - 2 \sum_n x[n] x[n+k] \right) \\ & = \frac{1}{N} \sum_n (x[n])^2 + \frac{1}{N}\sum_n (x[n+k])^2 - \frac{2}{N} \,\sum_n x[n] x[n+k] \\ & = \overline{x^2[n]} + \overline{x^2[n]} - 2\, R_x[k] \\ & = 2 \left( \overline{x^2[n]} - R_x[k] \right) \\ \end{align}$$

where

$$\begin{align} R_x[k] & \triangleq \frac{1}{N} \sum_n x[n] x[n+k] \\ & = \overline{x^2[n]} - \tfrac{1}{2} Q_x[k] \\ & = R_x[0] - \tfrac{1}{2} Q_x[k] \\ \end{align}$$

is normally identified as the "autocorrelation" of $x[n]$.

So we expect the autocorrelation function to be an upside-down (and offset) replica of the ASDF. Wherever the autocorrelation peaks is where the ASDF (and usually also the AMDF) has a minimum.