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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]

Correlation of neighboring periodogram powers describes dependence between squared spectral magnitudes estimated from the same random record. Their dependence is not determined by the sign of amplitude covariance. A negative correlation between complex Fourier amplitudes can produce a positive correlation between their powers, changing the precision obtained by averaging nearby spectral values.

Let $Z_k$ be zero-mean jointly Gaussian complex Fourier estimates. Define their complex covariance and nonconjugated second moment byand put $I_k=|Z_k{|}^2$. Applying Isserlis's theorem to the four factors gives three pairings:Subtracting the product of the means therefore yieldsIn particular, Gaussian power covariance is nonnegative even when a complex amplitude covariance is negative. Also $\mathrm{Var}(I_k)=C_{kk}^2+|P_{kk}{|}^2$; ignoring the pseudo-covariance of a complex random variable changes the answer when that term is present.

Consider independent sample noise with independent Gaussian real and imaginary components of equal variance. Its nonconjugated moments vanish, so $P=0$ after any deterministic complex linear transform. If the normalized bin variances are all $V$, thenFor the periodic Hann window with $N\ge 5$ on the original $N$-point Fourier grid, noise covariance between windowed Fourier bins gives amplitude covariance ratios $-2/3$ at separation one and $1/6$ at separation two. The corresponding power correlations are $4/9$ and $1/36$. More distant nonconjugate bins have zero covariance in that particular model.

An original calculation averages $M$ consecutive powers without wrapping around the Fourier grid. Assume $N\ge M+2$ so that the block does not acquire additional cyclic neighbor pairs. Counting the $M-1$ adjacent pairs and $\max (M-2,0)$ second-neighbor pairs givesFor $M=2$, the variance is $13V^2/18$, rather than the independent-value result $V^2/2$. For $M=10$, it is $0.18444V^2$, rather than $0.10V^2$. A variance-equivalent independent-power count is the reciprocal of this normalized variance: about $5.42$ for those ten bins. For $M\ge 2$, the bracket simplifies to $35M/18-1$, approaching roughly $18M/35$ independent powers for large blocks in sufficiently long records.

Real-valued Gaussian input has conjugate Fourier symmetry and can have nonzero $P_{kl}$, especially for mirrored frequencies and endpoint bins. A deterministic spectral component also invalidates the zero-mean calculation unless its additional mean-dependent terms are included. Colored noise, taper convention and detrending change $C$ and $P$; the formulas above do not make all measured spectra follow one correlation pattern.

A periodogram may be expressed as a squared spectrum or a power spectral density. SciPy documents their window-dependent conversion, one-sided folding and optional detrending explicitly.^[ref] A common positive scaling does not change the correlation coefficient, but combining unequal scalings or folding conjugate values requires carrying those transformations through the covariance.

A proposed check generates independent records under the stated noise model, calculates bin powers with the same window, and compares empirical pair covariances and block-average variance with the predictions. Independent records provide a different averaging resource from additional bins or zero-padded frequency points in one record. Report which values were averaged and their covariance before assigning an independent-sample precision gain.

The discrete-time Fourier transform represents a sequence indexed by integer sample number using a continuous, periodic frequency variable. In cycles per sample, the transform pair isFor an absolutely summable sequence, the sum converges uniformly to a continuous function, and the inverse formula follows by integrating the exponentials term by term. Their integral over one frequency period is one when the integer exponent is zero and zero otherwise. The primary text gives the same pair in angular frequency $\omega =2\pi \nu$, where the inverse integral carries a factor $1/(2\pi )$.^[ref]

Because $e^{-2\pi in}=1$, $X(\nu +1)=X(\nu )$. Discrete time therefore does not imply discrete frequency: the DTFT is defined at every frequency in one period. With sampling frequency $f_s$, a physical frequency $f$ corresponds to $\nu =f/f_s$; frequencies separated by integer multiples of $f_s$ give identical sampled complex exponentials. This is the basic frequency ambiguity behind aliasing.

An original two-sample example takes $x_0=x_1=1/2$ and all other entries zero. ThenAt zero frequency its value is one; at $\nu =1/4$ its magnitude is $1/\sqrt{2}$ with phase $-\pi /4$; at $\nu =1/2$ it vanishes. As the impulse response of a two-tap moving average, it preserves a constant input and removes an alternating-sign input. SciPy’s frequency-response definition evaluates precisely such sums when its denominator is one; its frequency units depend on the fs parameter.^[ref]

For a finite record $x_0,\dots ,x_{N-1}$ extended by zero, the unnormalized discrete Fourier transform gives samples of the same continuous function:Zero padding the record to length $M>N$ instead evaluates $X(k/M)$ on a denser grid. It does not change $X(\nu )$ or restore unobserved time samples. The rectangular window and Hann window demonstrate how the selected finite record shapes that continuous response.

Convergence assumptions matter. For $x_n=a^n$ when $n\ge 0$, zero otherwise, and $|a|<1$, the geometric sum givesAt $a=1$, that ordinary convergent-sum argument fails. Square-summable sequences admit an $L^2$ interpretation, while perpetual tones require a generalized interpretation; neither should be treated as an absolutely convergent version of the same series without stating the interpretation.

A useful check of numerical transform conventions is to evaluate a finite sum at arbitrary frequencies, compare its DFT-grid values with the FFT output, and repeat after zero-padding. This tests the exponent sign, normalization and frequency units independently. In a physical acquisition, one must also state sampling rate, record duration and the response before sampling; dense numerical frequency points alone do not validate those experimental inputs.

The equivalent noise bandwidth of a finite window is the width of an ideal rectangular frequency response that passes the same white-noise power when both responses have the same peak gain. It quantifies noise collected by a spectral amplitude estimate, rather than the separation needed to distinguish two tones. For real coefficients $w_n$ with nonzero sum and sampling frequency $f_s$,$B_e$ is in hertz; $b_e$ is in bins of an $N$-sample record. The primary software documentation gives $B_e$ as the exact conversion factor between its squared spectrum and power-spectral-density scalings.^[ref]

An original variance derivation explains the formula. Let independent zero-mean sample noise ${ϵ}_n$ have variance ${\sigma }^2$, and form the complex amplitude estimate at frequency ${\nu }_0$,Cross terms vanish by the independence and zero means, soThe flat two-sided input noise spectral density is ${\sigma }^2/f_s$. Integrating it over the peak-normalized window response gives the same result by Parseval identity. A one-sided real-signal convention folds negative-frequency power into positive frequencies; factors of two must be handled consistently, rather than changing the window’s bandwidth ratio.

For a rectangular window, $\sum w_n=N$ and $\sum w_n^2=N$, giving $b_e=1$. For the periodic Hann window with $N>2$, the cosine sums give $\sum w_n=N/2$ and $\sum w_n^2=3N/8$, hence $b_e=1.5$. These agree with the instrument manual’s tabulated values.^[ref]

For an original numerical comparison, take $f_s=4096\mathrm{Hz}$ and $N=256$, so the original frequency grid spacing is $16\mathrm{Hz}$. Rectangular and periodic Hann bandwidths are respectively $16$ and $24\mathrm{Hz}$. With independent sample noise of standard deviation $0.20\mathrm{V}$, the RMS magnitude of the zero-mean complex noise estimate is $0.0125\mathrm{V}$ for rectangular weighting and about $0.01531\mathrm{V}$ for Hann weighting. The latter variance is 1.5 times greater, despite Hann’s lower distant sidelobes. It is the RMS complex-error magnitude, not automatically the standard deviation of a positive spectral-magnitude display.

Zero padding leaves $f_s$, the actual window coefficients and $B_e$ unchanged. If a 256-sample Hann record is padded to 1024 points, the display spacing becomes $4\mathrm{Hz}$ and the same $24\mathrm{Hz}$ bandwidth spans six displayed bins. Replacing $N$ by the padded FFT length in a formula while leaving the coefficient sum unchanged confuses these two bin units.

The noise covariance between windowed Fourier bins also matters when combining several spectral values: a taper can correlate neighboring estimates. Bandwidth alone does not specify their joint statistics, and colored noise requires weighting its actual spectral density. A proposed check repeats the same estimator on independent synthetic noise records with known variance, then on measured noise with a characterized acquisition response, comparing empirical mean squared complex error with the formula. Report window convention, detrending, one-sided/two-sided scaling and record duration so the test can be reproduced.

The half-bin amplitude response of a window is the magnitude response to an isolated tone halfway between neighboring discrete Fourier transform frequencies, divided by the response to a bin-centered tone of the same amplitude. It quantifies a particular off-grid amplitude error after correcting the coherent gain of a window. For coefficients $w_0,\dots ,w_{N-1}$ with nonzero sum, defineA loss convention reports $L_{1/2}=-20{\log }_{10}{r}_{1/2}$ in decibels. The primary instrument manual tabulates approximately $3.92\mathrm{dB}$ for rectangular weighting and $1.42\mathrm{dB}$ for Hanning weighting.^[ref]

To derive the ratio, take a complex tone $x_n=A{e}^{2\pi i(k+1/2)n/N}$. Evaluating the windowed transform at bin $k$ gives $AW(-1/(2N))$. For real window coefficients, the responses at opposite offsets have equal magnitude. Dividing by $|W(0)|$ removes the on-bin scaling and leaves $|A|r_{1/2}$. This calculation isolates the frequency-grid effect from noise and other spectral components.

For a rectangular window, summing the geometric series yieldsFor $N=64$, the ratio is about $0.63668$. The large-$N$ value is about $0.63662$, explaining the familiar $3.92\mathrm{dB}$ scalloping loss rather than an exact finite-record constant.

For the periodic Hann window, $w_n=(1-\cos (2\pi n/N))/2$. Its finite response iswhere $W_N$ is the rectangular response and $W_H(0)=N/2$ for $N>1$. Substitution gives about $0.84883$ at $N=64$. A useful limiting derivation replaces the normalized sum by its integral:with continuous limiting values at $x=\pm 1$. At $x=1/2$ its magnitude is $8/(3\pi )\approx 0.84883$. For a short $N=4$ periodic window, the coefficients $(0,1/2,1,1/2)$ instead give $r_{1/2}=(1+\cos (\pi /4))/2\approx 0.85355$. The finite definition avoids treating the asymptotic value as exact.

The symmetric Hann window uses denominator $N-1$, not $N$; it has a different finite response and coefficient sum. SciPy explicitly distinguishes the symmetric filter-design convention from the periodic spectral convention.^[ref] The exact half-bin ratio should therefore use the coefficients actually applied.

For an original acquisition example, a $64$-sample record at $6400\mathrm{Hz}$ has $100\mathrm{Hz}$ bin spacing. A complex tone at $1050\mathrm{Hz}$ lies halfway between bins 10 and 11. An amplitude-one tone gives a corrected nearest-bin magnitude about $0.637$ with rectangular weighting or $0.849$ with periodic Hann weighting. Zero padding changes the sampled grid of the same discrete-time Fourier transform; if the padded grid contains $1050\mathrm{Hz}$, its value there has no half-bin error relative to that tone.

The half-bin response is not automatically a universal worst-case bound for every conceivable window. Nor does the isolated-complex-tone calculation cover every real-sinusoid measurement: near zero frequency or the sampling limit, the mirrored spectral component can overlap and make the observed magnitude phase-dependent. A proposed test sweeps a constant-amplitude tone through a bin interval, varies phase for real signals, and compares the complete finite-sum response before quoting a single off-bin accuracy number.

A Hann window tapers a record with a raised cosine. For the endpoint-symmetric convention with $N>2$,It vanishes at the two stored endpoints. A periodic convention instead uses denominator $N$, corresponding to sampling one cosine period without storing its repeated endpoint. SciPy's primary documentation distinguishes these with sym=True and sym=False; the default is symmetric.^[ref] The adjacent Hanning window alias also supports the terminology used in older instrument documentation.

An original short-array example makes the difference concrete. At $N=4$, the symmetric coefficients are $(0,3/4,3/4,0)$, while the periodic coefficients are $(0,1/2,1,1/2)$. Their means are $3/8$ and $1/2$. A symmetric even-length array need not contain a coefficient equal to one, despite the nominal raised-cosine peak.

The coherent gain of a window is $G_c=N^{-1}{\sum }_n{w}_n$. It equals $1/2$ for the periodic Hann convention with $N>1$, and $(N-1)/(2N)$ for the symmetric convention with $N>2$. A coherent complex tone gives windowed DFT magnitude $N|A|G_c$ at its exact bin, so amplitude recovery must use the chosen coefficient sum.

For the periodic convention, its discrete-time Fourier transform is obtained from the rectangular window transform by modulation:The first central zeros lie at $\nu =\pm 2/N$ for the usual $N>3$ case, twice as far from the center as the rectangular window's first zeros. Smooth tapering reduces distant sidelobes while widening the central feature. Nearby frequencies and a weak component far from a strong one can therefore favor different window choices.

Noise normalization is another distinction. For $N>2$, the periodic Hann coefficients obey ${\sum }_n{w}_n=N/2$ and ${\sum }_n{w}_n^2=3N/8$, giving equivalent noise bandwidthin DFT bins. The manufacturer's table lists both coherent gain and noise bandwidth.^[ref] Correcting a coherent amplitude does not make noise behave as though the rectangular weights had been used.

The original comparison below evaluates the finite 64-sample periodic Hann and rectangular coefficients at a dense frequency grid, normalizing each response by its own coefficient sum. The extra frequency samples display the same finite-record transform more finely; they do not create a longer observation.

Noise covariance between windowed Fourier bins describes the statistical dependence created when several spectral estimates use the same noisy record. A window changes not only each estimate’s equivalent noise bandwidth, but also the overlap of their frequency responses. Uncorrelated time samples therefore need not produce uncorrelated spectral bins after tapering.

Let independent zero-mean sample errors ${ϵ}_n$ satisfy $\mathbb{E}[{ϵ}_n\overline{{ϵ}_m}]={\sigma }^2{\delta }_{nm}$. Use real coefficients $w_n$, nonzero sum $S={\sum }_n{w}_n$, and coherent-amplitude-normalized estimates on the original $N$-point grid,Expanding their complex covariance and removing the vanishing cross terms givesThus the covariance is the Fourier transform of the squared window, evaluated at the bin difference. Its diagonal agrees with the equivalent noise bandwidth variance formula. SciPy’s spectrum/density scaling likewise uses the coefficient-square sum separately from the squared coherent sum.^[ref]

For a rectangular window, the exponential orthogonality sum gives $C_{kl}=({\sigma }^2/N){\delta }_{kl}$ on this unpadded grid. This is a statement about covariance; independence additionally follows for an appropriate Gaussian noise model, not for every possible input distribution.

For the periodic Hann window, with $N\ge 5$,Its original-grid covariance has only five nonzero cyclic diagonals:with bin indices modulo $N$. Normalizing by the common diagonal gives adjacent-bin covariance ratio $-2/3$ and second-neighbor ratio $1/6$. The negative sign follows from this window’s placement in the record; changing the time placement can change covariance phases without changing their magnitudes. The periodic and symmetric coefficient conventions must not be interchanged in this finite calculation.^[ref]

Correlation of neighboring periodogram powers is different from the sign of this complex covariance. Suppose each input sample has independent Gaussian real and imaginary parts, each of variance ${\sigma }^2/2$, with all components independent across time. The nonconjugated second moments vanish. Gaussian fourth-moment pairing, as in Isserlis's theorem, then givesConsequently neighboring Hann-bin powers are positively correlated despite the negative covariance of their complex amplitudes. If $V=C_{kk}$, their individual power variances are $V^2$, and the variance of their arithmetic mean isTreating them as independent would incorrectly give $V^2/2$. For real Gaussian input, a further term $|\mathbb{E}[Z_k{Z}_l]{|}^2$ enters the power covariance; the pseudo-covariance of a complex random variable and conjugate-frequency symmetry therefore need attention near paired bins and the real-valued endpoint bins.

Zero padding creates more closely spaced evaluations of the same record, not more independent noise realizations. Even a rectangular record generally has correlated estimates on that denser grid. Detrending, colored input noise and overlapping records change the covariance calculation too. SciPy exposes detrending and FFT-length choices explicitly; its default constant detrending must be disabled when testing the unmodified independent-sample model above.^[ref]

A proposed numerical check generates many independent records with the stated noise distribution, computes the estimates with fixed coefficients, and compares empirical complex covariance and power covariance separately with the formulas. A physical noise test additionally needs a characterized acquisition bandwidth and an assessment of time-sample correlation. Averaging several correlated spectral values does not supply the same precision gain as averaging independent records.

A rectangular window keeps all samples in a selected finite record with equal weight and sets the signal outside it to zero. For $N$ discrete samples, $w_n=1$ for $0\le n<N$ and zero otherwise. Calling this no taper does not remove the effect of having a finite record. SciPy's primary implementation documentation identifies the boxcar and rectangular window as this same all-ones choice.^[ref]

Its discrete-time Fourier transform at frequency $\nu$ in cycles per sample is a finite geometric sum:At integer $\nu$, the limiting value is $N$. The first zeros surrounding the central maximum are at $\nu =\pm 1/N$, so the central zero-to-zero width is $2/N$. The phase factor depends on the record's time origin; the magnitude response does not.

For a continuous centered record of duration $T$, $w(t)=1$ on $[-T/2,T/2]$ has transformwhere the normalized sinc function is $\mathrm{sinc}(x)=\sin (\pi x)/(\pi x)$. Multiplication by the finite window spreads a sharp spectral component according to this response, producing spectral leakage.

An original finite-record example quantifies an off-bin amplitude error. A complex tone of amplitude $A$ lying half a DFT bin from the nearest sampled frequency has normalized responseAt $N=64$ the ratio is about $0.6367$, or $-3.92\mathrm{dB}$ relative to the coherent peak. The coherent gain of a window is still one; it does not correct this scalloping loss. Zero-padding samples the same frequency response more densely rather than lengthening the measured record.

For independent equal-variance sample noise, rectangular weighting has equivalent noise bandwidth of one DFT bin under the usual coherent-amplitude normalization. A Hann window suppresses distant sidelobes but changes the central width and noise scaling. The primary analysis manual reports these separate window properties rather than one universal best window.^[ref]

A proposed test records a known sinusoidal input with an integer and then a non-integer number of cycles in the same record length, holding the sampling rate and amplitude fixed. Compare the raw DFT bins and the continuous window response. This separates finite-record leakage and bin placement from aliasing, which involves a different sampling ambiguity.

Scalloping loss is the underestimate of an isolated sinusoidal amplitude caused by measuring it only at the nearest sampled discrete Fourier transform frequency, after correcting the window’s coherent amplitude scale. An off-grid tone is evaluated away from the maximum of the window response. This differs from the coherent gain of a window, which describes the response at the correctly centered frequency. The instrument manual treats amplitude scaling and off-bin errors separately.^[ref]

Let $w_n$ be a finite real window with positive sum, and let $x_n=A{e}^{2\pi i(k+\delta )n/N}$, where $k$ is the nearest unpadded DFT bin and $|\delta |\le 1/2$. Write $W(\nu )={\sum }_{n=0}^{N-1}{w}_n{e}^{-2\pi in\nu }$. The coherent-gain-corrected bin magnitude relative to $|A|$ isHere $L$ is a positive loss when $r<1$. The half-bin amplitude response of a window evaluates this ratio at the largest nearest-bin offset. Other tones, noise and the negative-frequency image of a real sinusoid can also affect a measured peak; the derivation isolates one complex tone.

For a rectangular window, its geometric-sum transform givesFor $N=64$, the half-bin ratio is about $0.63668$ and the loss about $3.92\mathrm{dB}$. An amplitude-one tone exactly centered on a bin is recovered as one, while the same tone shifted by half a bin gives only about $0.637$ at either nearest bin. Dividing by coherent gain cannot remove this offset dependence.

For the periodic Hann window, usewhere $W_N$ is the rectangular response. At $N=64$, this gives a half-bin ratio about $0.84883$, or $1.42\mathrm{dB}$ loss. The smaller nearest-bin error comes with a wider main lobe and a larger equivalent noise bandwidth; window choice affects several measurement properties at once.

Zero padding reduces the nearest-grid offset without changing the measured record. Padding to $M=pN$ points makes the nearest offset at most $1/(2p)$ in original-bin units. With $p=4$ and $N=64$, the rectangular response at that worst offset is about $0.97450$, a loss of only $0.224\mathrm{dB}$. This is denser sampling of the same discrete-time Fourier transform, not extra physical frequency resolution. A fitted peak can reduce a grid error too, but its signal model and noise sensitivity must be stated.

A proposed analyzer test holds tone amplitude fixed while sweeping frequency through one bin interval, retaining record length, sampling rate and window. Divide out coherent gain, plot the measured peak versus fractional-bin offset, and compare with the exact response above. Include phase and nearby-tone checks for a real signal. Summing spectral powers instead answers an energy question and requires its own bandwidth normalization; it should not silently replace this single-bin amplitude definition.