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., \begin{equation} [+1, +1, +1, +1, +1, −1, −1, +1, +1, −1, +1, −1, +1]\;. \end{equation}

\(n = 19\): (TABLE D-XXIV)

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

\(n = m^2\): (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, \begin{equation} MP(k;m) = \exp\left\{-2\pi i m \left\lfloor{\frac{k}{m}}\right\rfloor \frac{k}{n} \right\}\;, \end{equation} 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., \begin{equation} \exp\big\{2\pi i\, [0, 0, 0, 0,\; 0, 2, 4, 6,\; 0, 4, 0, 4,\; 0, 6, 4, 2]/8\big\}\;. \end{equation} Many pages could have been saved by describing the mini-probe base sequences in terms of the formula above.

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