 Finite-record phasor extraction with coherent sampling

Finite-record phasor extraction with coherent sampling recovers a sinusoid's complex amplitude by arranging its frequency on a discrete Fourier bin. Let samples be taken at $t_m=t_0+m\Delta t$, with $N\Delta t$ containing exactly $k$ cycles, where $k$ is an integer and $0<k<N/2$. Forthe constant baseline cancels from bin $k$. The peak phasor versus positive-frequency Fourier coefficient relation then gives the estimateThe factor $e^{-i\omega t_0}$ assigns wave phase to the stated physical time origin; without it the coefficient refers to the record's start. The default forward FFT is unscaled, while other FFT normalization modes require different prefactors.^[ref]

For an original example, sample at $80\mathrm{kHz}$ with $N=80$ and use a $7\mathrm{kHz}$ tone, so $k=7$ and the cyclic record duration is $1\mathrm{ms}$. Let its physical peak phasor be $2e^{i\pi /6}\mathrm{V}$ and let $t_0=50\mu \mathrm{s}$. The record-start wave phase is $30^{\circ }+126^{\circ }=156^{\circ }$, and the unscaled bin isApplying both $2/N$ and the time-origin correction recovers the physical $2e^{i30^{\circ }}\mathrm{V}$ phasor. Its RMS amplitude is $\sqrt{2}\mathrm{V}$. Omitting the start-time correction would give a correct magnitude but a different wave phase reference.

With independent zero-mean real noise of variance ${\sigma }^2$ on each sample, the extracted phasor remains unbiased. Orthogonality gives noise variance $2{\sigma }^2/N$ in each real and imaginary component and zero covariance between them. For $\sigma =0.01\mathrm{V}$ here, each component's standard deviation is about $1.581\mathrm{mV}$. These are complex-amplitude uncertainties; taking a magnitude is a nonlinear operation and need not retain unbiasedness.

Coherent sampling must describe the actual source/clock relationship, not just the rounded frequency entered into software. Frequency drift, extra tones, aliasing or a changed window invalidate the simple exact-bin identity; spectral leakage bias in harmonic distortion measurements shows one consequence. A coherent gain of a window can correct an on-bin amplitude scale in suitable models, but cannot in general turn a noncoherent one-bin reading into an exact phasor.