## Friday, May 10, 2019

### MIL-STD-188-110D mini-probe base sequences

The mini-probes specified in MIL-STD-188-110D, Appendix D have interesting structures. Let us start with the two exceptions, $$n = 13$$ and $$n = 19$$:

### $$n = 13$$: (TABLE D-XXII)

This is the well-known Barker-13 sequence, i.e., $$[+1, +1, +1, +1, +1, −1, −1, +1, +1, −1, +1, −1, +1]\;.$$

### $$n = 19$$: (TABLE D-XXIV)

This sequence is based on Legendre-19: $$[1,\, -\mathsf{Legendre}(k/19)]\;.$$

### $$n = m^2$$: (TABLES D-XXIII, D-XXV - D-XXXVI)

All other mini-probe base sequences have lengths $$n=m^2$$ for $$4\le m \le 17$$ and can be obtained from the following formula, $$MP(k;m) = \exp\left\{-2\pi i m \left\lfloor{\frac{k}{m}}\right\rfloor \frac{k}{n} \right\}\;,$$ where $$\left\lfloor{x}\right \rfloor$$ denotes floor function which returns the greatest integer $$\leq x$$. Curiously, the mini-probe base sequences for $$m\ge 14$$ are the complex conjugate of the formula above, so I wonder if this is by design or it is an error in the standard.

Note that $$MP(k;4)$$ is the (complex conjugate of the) length-16 Frank-Heimiller sequence contained in TABLE C-VIII, i.e., $$\exp\big\{2\pi i\, [0, 0, 0, 0,\; 0, 2, 4, 6,\; 0, 4, 0, 4,\; 0, 6, 4, 2]/8\big\}\;.$$ Many pages could have been saved by describing the mini-probe base sequences in terms of the formula above.