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The coherent gain of a discrete window function is its mean coefficient:It describes the amplitude scaling of a coherent complex tone under the chosen DFT normalization. Window coefficients and their normalization must be specified; a window's name alone does not fix every finite-length convention.^[ref]

For $x_n=A{e}^{i(2\pi k_{0}n/N+\varphi )}$, multiply by $w_n$ and evaluate the unnormalized discrete Fourier transform at the exact tone bin:Thus $|X_w[k_0]|/(N|G_c|)$ recovers $A$ when $G_c\ne 0$. This result is exact for this complex-tone model. A real cosine also has a negative-frequency component; the usual doubled one-sided amplitude requires that its contribution to the chosen positive-frequency bin be absent or separately accounted for.

A rectangular window has $G_c=1$. The periodic Hann window $w_n=(1-\cos (2\pi n/N))/2$ has $G_c=1/2$ for $N>1$. Its coherent bin therefore has half the unwindowed amplitude before correction. The symmetric endpoint convention $w_n=(1-\cos (2\pi n/(N-1)))/2$, for $N>2$, instead gives $G_c=(N-1)/(2N)$. At $N=64$ this is $0.4921875$, so blindly multiplying by two is not the exact correction for that convention.

Coherent gain does not correct every amplitude error. An off-bin tone samples the window's frequency response away from its maximum, and a DC-removing window with zero sum has no usable gain correction at all. Coherent sampling, leakage and window normalization are distinct conditions.

Noise uses a different scaling. For independent equal-variance samples, a weighted DFT bin's noise variance is proportional to ${\sum }_n{w}_n^2$, not $({\sum }_n{w}_n{)}^2$. This motivates equivalent noise bandwidth in bins,which is $3/2$ for a periodic Hann window with $N>2$. The manufacturer's table lists both coherent gain and noise bandwidth rather than treating one as a substitute for the other.^[ref]

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.

Spectral leakage bias in harmonic distortion measurements occurs when a finite record spreads a strong component into bins interpreted as distinct harmonics. A waveform can have zero physical total harmonic distortion while a naive FFT-bin ratio reports otherwise. Confining a sinusoid to a finite interval creates spectral sidelobes; the pinned primary guide derives this leakage from the finite-time Fourier transform.^[ref]

For an exact discrete calculation, take $x_m=A\cos (2\pi \nu m/N+\varphi )$, where $\nu$ is the number of cycles in the record. With the unscaled forward discrete Fourier transform, a finite geometric sum giveswherewhenever the denominator is nonzero, with the finite-sum limit elsewhere. Integer-cycle coherent sampling makes the unwanted integer bins cancel; a noninteger cycle count generally does not. The negative-frequency contribution also matters because the input is real.

For an original pure-tone example, let $A=1$, $\varphi =0$, $N=128$ and $\nu =5.3$. Its true higher-harmonic amplitudes are zero. Yet choosing the nearest bins $5$, $11$, and $16$ for the fundamental, second and third harmonic givesA naive result would be about $6.58\%$ distortion from this undistorted sinusoid. This is an exactly specified record calculation, not an instrument's measured distortion floor.

A window reduces some sidelobes at the cost of a broader main lobe and a changed amplitude normalization. Its coherent gain of a window must be treated consistently, and one cannot assume that one selected bin captures a noncoherent tone's full amplitude. Zero padding samples the same finite-record spectrum more densely; it does not restore missing observation time or remove leakage.

A measurement can instead synchronize the source and acquisition for an integer number of cycles, fit a justified sinusoidal model, or use a validated window/band-integration method. The finite-record phasor extraction with coherent sampling gives a particularly transparent exact case. Calibrated receiver response and actual harmonic coverage remain necessary, as in the primary Anritsu apparatus.^[ref] Power spectral density and squared-amplitude spectrum outputs also have different units; an arbitrary square root of a density bin is not a calibrated harmonic amplitude.^[ref]