Four-phase radar probe codebook · n = 21

Formal definition

• ContainerFive sequences of length N over the alphabet {0, 1, 2, 3}, representing the four unit-modulus phases 1, i, −1, −i.$$x_{a,t}=i^p,p\in \{0,1,2,3\}$$
• SubmittedFive strings, each of length N and starting with 0.
• Scored eventsEach sequence against itself at all N − 1 non-zero aperiodic shifts, and each pair of distinct sequences at all 2N − 1 shifts, totalling 5(N − 1) + 10(2N − 1) = 25N − 15 sidelobe events.
• CorrelationThe aperiodic correlation at shift k is given by the sum below over all indices t where both positions lie in [0, N − 1]. Because every symbol lies in {±1, ±i}, R is a Gaussian integer and |R|² is a non-negative integer.$$R(a,b,k)=\sum _t{x}_{a,t+k}\overline{x_{b,t}}$$
• Score components𝒜 is the set of 25N − 15 scored events. M is the peak squared modulus, C is the number of events attaining M, and E is the total energy across all events; all three are computed in exact integers.$$M\left(P\right)=\underset{e\in 𝒜}{\max }{\left|R_e\right|}^2;C\left(P\right)=\#\left\{e\in 𝒜:{\left|R_e\right|}^2=M\left(P\right)\right\};E\left(P\right)=\sum _{e\in 𝒜}{\left|R_e\right|}^2$$
• ObjectiveMinimize (M, C, E) lexicographically: compare M first, then C if M is tied, and finally E.$$\underset{P}{\mathrm{lex}\min }\left(M\right(P),C(P),E(P\left)\right)$$