Recursive filters with uniform power distribution

Recursive linear digital filters which operate without roundoff errors, known as cyclotomic filters, have been shown to be useful as tone generators and detectors applicable to pseudo-random noise generation. Touch Tone®, f.s.k. and broadband frequency detection. These filters are modelled by periodic linear recursions with integer feedback coefficients. In certain applications, such as tone detection, it is desirable to have considerable differentiation between the detected and the rejected frequency amplitudes. Other applications such as pseudorandom noise generation require a more uniform power distribution throughout the spectrum. It is shown that the number N(m) of distinct nonzero spectral amplitude levels for such a filter of period m is exactly N(m)=¿(m0)/2 when in m0>2, where m0 is the largest square-free divisor of m and ¿(m0) is the number of positive integers less than and sharing no prime factor with in m0; N(m)=1 when m0¿2. In particular, a uniform power distribution is effected over the entire spectrum of resonating frequencies (i.e. N(m)=1) if and only if m=2a3b for some non-negative integers a and b.