= Coherent sampling
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 when
$$
\frac{f_0}{f_s}=\frac{m}{N},\qquad m\in\mathbb Z.
$$
This is a frequency-and-record-length condition, not merely starting the acquisition at a particular phase.<Coherent Sampling vs. Window Sampling by Analog Devices>[What Defines Coherent Sampling?, integer-cycle relation]{ref}
For the complex tone $x_n=Ae^{i(2\pi mn/N+\phi)}$, the unnormalized <discrete Fourier transform> gives
$$
X_k=Ae^{i\phi}\sum_{n=0}^{N-1}e^{2\pi i(m-k)n/N}
=\begin{cases}NAe^{i\phi},&k\equiv m\pmod N,\\0,&\text{otherwise}.\end{cases}
$$
A real cosine has the corresponding positive- and negative-frequency bins. The cancellation follows from a finite geometric sum; arbitrary phase $\phi$ 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.<Coherent Sampling vs. Window Sampling by Analog Devices>[Figure 1 explanation, cycle count and irreducible ratio]{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>.
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