Doppler tolerance, complementary code sets, and generalised Thue–Morse sequences

Abstract

The authors generalise the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki–Calderbank–Moran–Howard to complementary code sets having more than two codes, which they call Doppler-null codes. This is accomplished by exploiting number-theoretic results involving the sum-of-digits function and a generalisation to more than two symbols of the classical two-symbol Thue–Morse sequence. Two approaches are taken to establish higher-order nulls of the composite ambiguity function: one by rewriting it in terms of equal sums of powers (ESP) and the other by factoring it in product form to reveal a higher-order zero, analogous to spectral-null codes. They conclude by describing an application of minimal ESP sets to multiple-input–multiple-output radar.