[Sequences with] flat spectra

Any impulse sequence Join[{1}, Table[0, {n}]] will yield a flat spectrum. With odd n the same turns out to be true for sequences Exp[2 π Mod[Range[n]^{2}, n]/n]—a fact used in the design of acoustic diffusers (see page 1183). For sequences involving only two distinct integers flat spectra are rare; with ±1 those equivalent to {1, 1, 1, -1} seem to be the only examples. ({r^{2}, r s, s^{2}, -r s} works for any r and s, as do all lists obtained working modulo x^{n} - 1 from p[x]/p[1/x] where p[x] is any invertible polynomial.) If one ignores the first component of the spectrum the remainder is flat for a constant sequence, or for a random sequence in the limit of infinite length. It is also flat for maximal length LFSR sequences (see page 1084) and for sequences JacobiSymbol[Range[0, p - 1], p] with prime p (see page 870). By adding a suitable constant to each element one can then arrange in such cases for the whole spectrum to be flat. If Mod[p, 4] 1 JacobiSymbol sequences also satisfy Fourier[list] list. Sequences of 0's and 1's that have the same property are {1, 0, 1, 0}, {1, 0, 0, 1, 0, 0, 1, 0, 0} or in general Flatten[Table[{1, Table[0, {n - 1}]}, {n}]]. If -1 is allowed, additional sequences such as {0, 1, 0, -1, 0, -1, 0, 1} are also possible. (See also pages 911.)