[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 I 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.)