Integer-N vs fractional-N

Integer-N. The feedback divider is one integer N, so f_out can only move in steps of f_ref. A requested frequency between two channels must be rounded.

Fractional-N. The divider changes among nearby integers. Its long-term average is N + α, so the average output can land between integer channels. The modulator is part of how that average is produced; it is not a different frequency formula.

The plots below show the system view. Compare phase-detector timing in the time domain and the resulting output spectrum. The divider-word sequence, accumulated phase error and noise-shaping order are separated into Inside a fractional divider.

Model assumptions. A reference-rate behavioral loop with a linear phase detector, type-II PI filter and two extra poles at 6 MHz is tuned to a 1 MHz closed-loop −3 dB bandwidth. White reference/PD noise uses a −228 dBc/Hz normalized floor; free-running VCO noise follows 1/f² with −120 dBc/Hz at 1 MHz. The undithered 24-bit MASH has no DTC or charge-pump mismatch here. These are illustrative assumptions, not predictions for a particular PLL circuit.

Frequency and noise readouts. Fractional resolution is f_ref/2²⁴ (up to 5.96 Hz here), with at most half a step of rounding error. RMS jitter is the detrended time-record rms, including deterministic tones, over 32768 reference samples; the listed band is the record's nominal FFT span, not a brick-wall integration filter. Spectral levels average the two sidebands and are normalized to the measured carrier. “Not detected” means no tone passed the 18 dB local-floor threshold above 10 kHz; it does not prove zero spurs.