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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 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.

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.