Perhaps the results can be summarised by: (1+x)2 = 1+2x + x2 For |x| << 1 the term in x2 is negligible. For FM the term in x determines the amplitude of the (2fc-fm), (2fc+fm) components whereas the x2 term determines the amplitude of the (2fc-2fm) and (2fc+2fm) components. The 1 term determines the amplitude of the dc and 2fc components. When the carrier amplitude is much larger than the sideband components the amplitudes of the 2(fc - fm) and 2(fc+fm) components are much smaller than the amplitudes of the (2fc-fm) and the (2fc+fm) components in the doubler output spectrum. Similarly for frequency multiplication by N the amplitudes of the (Nfc - fm) and (Nfc + fm) components are much larger than all other components other than the Nfc component. In this case (1+x)^N = 1 + Nx + (N(N-1)/2)x2 + .... The real lesson is that speculation is no substitute for analysis, its just too easy to arrive at an incorrect conclusion. Bruce