The modulo-30 wheel as a sampling pattern

We test the eight residues of the modulo-30 wheel, W = {1, 7, 11, 13, 17, 19, 23, 29}, asa multicoset sampling pattern: of every L Nyquist samples only those at the positions Ware kept. With L = 30 the pattern fails. All its elements are odd, so the spectral slices land l + 15 cannot be told apart, and blind recovery of multiband signals succeeds in only0–2% of trials. The cause is a general rule. If a pattern takes fewer than d residues moduloa divisor d of L, then the d slices spaced L/d apart are linearly dependent. The wheel omitsthe residue 0 modulo 2, 3 and 5 by its very definition. With the prime period L = 31the same eight numbers succeed in every trial, by Chebotar¨ev’s theorem on the minors of aprime-order Fourier matrix. Taking the period 31 mixes the parities of the sampling times,which is what the period 30 forbids. We also study the construction of the wheel from theeven vectors 2 + 30n and 14 + 30n by repeated halving. The odd parts of 2 + 30n alwaysfall on {17, 19, 23, 31}, the kernel of the character χ15, and those of 14 + 30n on the otherhalf {7, 11, 13, 29}. More generally, removing all factors ℓ ∈ {2, 3, 5} from a + 30n leaveso ≡ a ℓ−e(mod 30/ℓ), where e is the number of divisions. The three constructions of thewheel, by 2, 3 and 5, therefore realise its three halves. Division by 3 splits the result by χ5according to the parity of e, division by 5 splits it by χ3, and division by 2 splits it by χ15according to the starting vector. As sampling patterns, the residues 2, 3, 5 of the parentvectors supply exactly what the wheel lacks: the wheel together with {2, 3, 5} recovers theoccupied slices in 100%, 99% and 95% of trials for 3, 4 and 5 slices, with the period still30. A final test of Ramanujan filter banks for detecting the waves that run through a vectorshows no gain over the Fourier harmonic comb for these prime periods; their advantage isthe integer coefficients.