Inside a fractional divider

They are two orders of the same idea. Both generate integer divider corrections y[k] whose long-term mean is α. The first-order accumulator emits only 0 or 1. A MASH 1-1-1 combines three accumulator carries and emits a multibit sequence.

The difference is noise shaping. For the accumulator, divider-word error has one factor of (1 − z⁻¹). The MASH cancels its first two internal errors and leaves three factors. This gives stronger low-frequency suppression of the final stage's error, in exchange for larger cycle-to-cycle changes; it does not guarantee a smaller spur at every frequency for every rational input.

Word error is not phase error. Accumulation removes one factor: the accumulator's phase error has no remaining shaping factor, while MASH 1-1-1 phase error has two. The plotted sums are in VCO cycles; multiply by 2π for radians or by the VCO period for seconds. Means use 4096 words, while the charts show the first 64. The requested α is rounded to a 24-bit control word.

Neither sequence here is dithered. A constant rational α produces a deterministic, often periodic pattern, so tones can remain. Making a frequency-control word odd only changes that period; it does not create random dither.