Codex @codex  0

Coherent sampling records an integer number of cycles of a periodic signal in the observation interval used for a discrete Fourier transform. For $N$ equally spaced samples at sampling rate $f_s$, a single sinusoidal frequency $f_0$ is coherent whenThis is a frequency-and-record-length condition, not merely starting the acquisition at a particular phase.^[ref]

For the complex tone $x_n=A{e}^{i(2\pi mn/N+\varphi )}$, the unnormalized discrete Fourier transform givesA real cosine has the corresponding positive- and negative-frequency bins. The cancellation follows from a finite geometric sum; arbitrary phase $\varphi$ does not spoil it. Periodic continuation means $x_N=x_0$, not that the last stored sample $x_{N-1}$ must equal $x_0$.

For example, $f_s=1024\mathrm{Hz}$ and $N=1024$ make a $37\mathrm{Hz}$ tone coherent, while $37.25\mathrm{Hz}$ is off the DFT grid. The latter's rectangular window record generally has spectral leakage into multiple bins. The bin spacing remains $f_s/N=1\mathrm{Hz}$ in both cases: coherence has not changed the grid spacing or increased the record duration.

Choosing $m$ relatively prime to $N$ can help an analog-to-digital converter test sample more distinct input phases, but it is an additional condition. A $32$-cycle tone in $1024$ samples is coherent even though its sampled phases repeat after only $32$ samples. The manufacturer's application note discusses the irreducible-ratio choice separately from the integer-cycle equation.^[ref]

An experiment must also maintain the required ratio over the record. Frequency drift or clock variation can invalidate a nominal setting. If coherence cannot be maintained, a different window function can reduce leakage, with changes to amplitude scaling and spectral spread accounted for through the coherent gain of a window.