Thinking in Waves Episode 58 / Part XIX — Back to Physics, with the Tools
Window sidelobes and particle creation are the same formula
What is outside the window leaks in ──
the smoothness of the cut sets the price
How fast a window’s sidelobes fall is set by how many derivatives vanish at its ends, and that is exactly this series’ order \(\alpha\) ── dB/oct \(=-6\alpha\).
★ That \(\alpha\) sets the ceiling on the \(\beta\) you can measure (ceiling \(=2\alpha\): 2 for a rectangular window, 6 for Hann).
★★★ And the same rule governs Episode 45’s Bogoliubov coefficient ── spectral leakage and particle creation are the same calculation.
Episode 57 concluded that an edge is set by its leading correction term. ★ Now we turn it around ── what does a window’s edge bring inside?
01Sidelobes are set by how smooth the edge is
Take \(w_p(x)=\cos^p(\pi x)\) on \(x\in[-\tfrac12,\tfrac12]\) ── a window that vanishes to order \(p\) at its ends.
| \(p\) | window | measured dB/oct | predicted \(-6(p{+}1)\) | \(\alpha=p{+}1\) |
|---|---|---|---|---|
| 0 | rectangular | −5.934 | −6.021 | 1 |
| 1 | cosine | −11.990 | −12.041 | 2 |
| 2 | Hann | −17.876 | −18.062 | 3 |
| 3 | \(\cos^3\) | −23.985 | −24.082 | 4 |
| 4 | \(\cos^4\) | −29.873 | −30.103 | 5 |
| 5 | \(\cos^5\) | −36.097 | −36.124 | 6 |
The core of §01
dB/oct \(=-6\alpha\), with \(\alpha=\) how many derivatives vanish at the edge
★★★ Episode 1’s yardstick applies to window sidelobes unchanged. There \(\alpha\) was “how many times differentiated”; here it is “how smooth the cut is.”
★ It is also the window-side face of Episode 57’s \(n\) ── whichever way you look at an edge, its character collapses onto a single number.
02The common windows on the same yardstick
| window | value at the edge | measured dB/oct | \(\alpha\) | first sidelobe |
|---|---|---|---|---|
| rectangular | 1.0000 | −5.934 | 0.99 | −13.3 dB |
| Bartlett | 0.0000 | −11.685 | 1.94 | −26.5 dB |
| Hann | 0.0000 | −17.880 | 2.97 | −31.5 dB |
| Hamming | 0.0800 | −5.840 | 0.97 | −42.7 dB |
| Blackman | 0.0000 | −17.789 | 2.95 | −58.1 dB |
The core of §02 ── Hamming’s two faces
★★ Hamming’s first sidelobe is 11 dB below Hann’s (−42.7 against −31.5), yet its sidelobes fall like a rectangular window’s, \(\alpha\approx1\). Because 0.08 is left at the edge.
★★★ “Low first sidelobe” and “falls far” are different things. The near field is set by tuning coefficients; the far field is set by edge smoothness alone ── exactly Episode 57’s division of labour, where \(A\) moves the intercept and \(n\) the slope.
03The leakage floor ── what a strong low tone leaves up high
A single tone at \(f_0=4.3\) bins (off the grid). What is left at high frequency (dB relative to the peak)?
| window | \(f=16\) | \(f=64\) | \(f=256\) | \(f=512\) | slope dB/oct |
|---|---|---|---|---|---|
| rectangular | −33.1 | −47.3 | −59.7 | −65.7 | −6.47 |
| cosine | −59.4 | −87.7 | −112.7 | −125.1 | −13.05 |
| Hann | −75.8 | −118.3 | −155.8 | −174.1 | −19.51 |
| \(\cos^3\) | −94.9 | −151.7 | −201.7 | −226.1 | −26.03 |
| \(\cos^4\) | −106.2 | −177.3 | −239.8 | −270.2 | −32.55 |
★ The leakage floor falls at \(-6\alpha\) dB/oct too ── the same quantity as §01, because the window’s sidelobes become the measurement’s floor.
04★★★ There is a ceiling on the \(\beta\) you can measure ── \(2\alpha\)
If the true spectrum is \(S\propto f^{-\beta}\), what you measure is \(S(f)+\)(leakage from low frequency). The leakage goes as \(f^{-2\alpha}\) in power, so for \(\beta>2\alpha\) it overtakes the signal.
| true \(\beta\) | rect (\(\alpha{=}1\)) | cosine (2) | Hann (3) | \(\cos^4\) (5) |
|---|---|---|---|---|
| 1 | 0.994 | 0.978 | 0.979 | 0.979 |
| 2 | 2.002 | 1.979 | 1.979 | 1.979 |
| 3 | 1.986 | 3.099 | 2.980 | 2.979 |
| 4 | 1.981 | 4.020 | 3.983 | 3.979 |
| 5 | 1.981 | 4.021 | 5.521 | 4.980 |
| 6 | 1.981 | 4.021 | 5.998 | 5.988 |
★ The ceiling it sticks to is exactly \(2\alpha\) ── rectangular 2.0 (measured 1.981), cosine 4.0 (4.021), Hann 6.0 (5.998).
★★ But the ceiling is not a cliff. Sweeping \(\beta\) finely with a Hann window (\(\alpha=3\), ceiling 6):
| true \(\beta\) | 3 | 4 | 4.5 | 5 | 5.5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| estimate | 2.967 | 3.968 | 4.567 | 5.605 | 6.000 | 5.999 | 6.000 | 6.000 |
| difference | −0.03 | −0.03 | +0.07 | +0.61 | +0.50 | 0.00 | −1.00 | −2.00 |
The core of §04 ── an overshoot just below the ceiling
★★★ As \(\beta\) approaches \(2\alpha\) the estimate first overshoots (\(+0.61\) at \(\beta=5\)), then sticks to \(2\alpha\).
★ Why: the leakage floor \(f^{-2\alpha}\) is steeper than the signal \(f^{-\beta}\), so it lifts only the low-frequency side of the fitting band and makes the slope look steeper.
★★ So “\(\beta<2\alpha\)” is a rule of thumb; in practice leave a margin of about 1 ── with Hann, \(\beta\lesssim5\) is the safe zone.
★ Episode 40 needed to cut at an integer number of periods precisely to dodge the rectangular window’s \(\alpha=1\) (ceiling \(\beta=2\)).
05★★★ The same rule governs Episode 45’s particle creation
What if the switching profile \(f(x)\) is smooth to order \(p\) at its ends?
| \(p\) (edge smoothness) | \(T=8\) | \(T=16\) | \(T=64\) | ★ slope | predicted \(-(p{+}1)\) |
|---|---|---|---|---|---|
| 0 (linear ramp) | 2.77e−2 | 1.82e−2 | 4.83e−3 | −1.060 | −1 |
| 1 | 1.46e−2 | 2.88e−3 | 1.63e−4 | −1.984 | −2 |
| 2 | 1.06e−2 | 9.79e−4 | 1.51e−5 | −3.025 | −3 |
| 3 | 1.28e−2 | 3.70e−4 | 1.52e−6 | −3.962 | −4 |
The core of §05 ── the two are one thing
★★★ \(|\beta|\propto T^{-(p+1)}\) ── exactly the rule for window sidelobes. The smoother the switch at its ends, the fewer particles are made.
★ \(|\alpha|^2-|\beta|^2=1\) holds to 8.3e−14 (Episodes 36 and 45’s \(\det=1\)).
★★ Why they are the same: \(\beta\) is the Fourier component of \(d\omega/dt\) at frequency \(2\omega\), so the smoothness of the switch’s edges sets how fast it falls ── the very same integral that sets a window’s sidelobes.
★★★ Episode 45’s “the vacuum depends on the observer” turns out to be the physics-side phrasing of “what is outside the window leaks in.”
06Four faces side by side
| setting | what the “edge” is | its effect | what \(\alpha\) means |
|---|---|---|---|
| window functions (Ep. 40) | the window’s ends | sidelobes \(f^{-\alpha}\) | edge smoothness |
| periodogram (Ep. 51) | the record’s ends | the leakage floor | ceiling \(\beta=2\alpha\) |
| particle creation (Ep. 45) | before and after the switch | \(|\beta|\propto T^{-\alpha}\) | switch smoothness |
| superposition (Ep. 56) | the ends of \(\tau\) | decades lost | the correction’s power |
★★★ All four are the price of having cut something off at a finite point, and the size of that price is set by the smoothness of the cut alone.
Exercises
- Is the Hamming window a bad window?
Show answer
It depends on the job ── it is bad only far away.
★ Excellent up close: its first sidelobe is −42.7 dB, 11 dB below Hann’s −31.5. For separating two closely spaced lines, Hamming is the better choice.
★★ Among the worst far away: its sidelobes fall with \(\alpha\approx1\), no better than a rectangular window. For a steep continuous spectrum (§04) its ceiling is \(\beta=2\).
★★★ So for “separating lines” and “measuring a continuum’s slope,” the preferred window is reversed. To add to Episode 49’s handbook: decide what you are measuring before you choose a window. - §04 found an overshoot below the ceiling. Which of Episode 51’s two errors is it?
Show answer
Systematic (a bias) ── not statistical. Averaging 32 realizations leaves 5.521 unmoved; more samples will not remove it.
★ This is exactly the systematic side of Episode 51’s “split the error into statistical and systematic.” And its sign is fixed (always towards steeper), which makes it worse.
★★ How to spot it: if the estimate moves when you change the window, the window is the cause. Reading the \(\beta=5\) row of §04 across gives cosine 4.021, Hann 5.521, \(\cos^4\) 4.980 ── all different, which is the evidence that the window is to blame.
★★★ A step worth adding to the handbook: measure the same data with two or more windows; if they disagree, you are touching the ceiling. - §05 concludes that a smooth switch makes no particles. How does that bear on cosmology?
Show answer
It bears on how inflation ends (Episode 47). A sharp transition from inflation to radiation domination makes particles in quantity (preheating); a smooth one does not.
★ In §05’s language ── the yield is set by the order of the transition.
★★ But this episode computed a simple monotone switch \(\omega_1=1\to\omega_2=2\), while in the real universe \(\omega\) oscillates (parametric resonance ── Episode 40’s Floquet). With resonance the story changes: even a smooth switch can grow exponentially.
★★★ So this conclusion applies only in the absence of resonance ── in Episode 55’s classification, a value conditional on a model. It should be written that way.
Summary — leakage and particle creation are the same calculation
Turning Episode 57’s “an edge is its leading correction term” around, we measured what a window’s edge brings inside.
★★★ Window sidelobes ── dB/oct \(=-6\alpha\),
with \(\alpha=\) how many derivatives vanish at the edge.
★ Episode 1’s yardstick works on the smoothness of a cut, unchanged. Rectangular \(\alpha=1\), Bartlett 2, Hann 3, \(\cos^p\) gives \(p+1\) ── measured within 0.2 dB of prediction.
★★ Hamming’s first sidelobe is 11 dB below Hann’s, yet its sidelobes fall like a rectangular window’s (0.08 is left at the edge). “Low first sidelobe” and “falls far” are different things ── the same division of labour as Episode 57’s “\(A\) sets the intercept, \(n\) the slope.”
★★★ There is a ceiling on the measurable \(\beta\), and it is exactly \(2\alpha\) ── rectangular 2.0, cosine 4.0, Hann 6.0 (measured 1.981 / 4.021 / 5.998). ★ But it is not a cliff: the estimate overshoots below it (\(+0.61\) at \(\beta=5\) with Hann). Leave a margin of about 1 in practice; this is a fixed-sign systematic error, not statistical scatter. Episode 40 had to cut at an integer number of periods precisely to avoid the rectangular ceiling \(\beta=2\).
★★★ And the same rule governs Episode 45’s particle creation ── a switch smooth to order \(p\) at its ends gives \(|\beta|\propto T^{-(p+1)}\) (measured −1.06 / −1.98 / −3.03 / −3.96). Because \(\beta\) is the Fourier component of \(d\omega/dt\), it is literally the same integral that sets a window’s sidelobes. Episode 45’s “the vacuum depends on the observer” was the physics-side phrasing of “what is outside the window leaks in.”
★ Next we exploit that ── measure with many different windows and infer the true value from how the estimate moves.
★★ Episode 51’s split of the errors, Episode 55’s “count the assumptions,” and this episode’s \(\alpha\) ── together they should make a procedure for measuring the systematic error itself.
This document is Episode 58 of the “Thinking in Waves” series. The computations are in kenshou/mado.js (own FFT, own RK4, zero dependencies). On the numbers in this episode: the window sidelobe slopes in dB/oct, the \(\alpha\) and first sidelobe of the named windows, the leakage floor, the \(\beta\) estimates and their ceiling, and the \(T\) dependence of the Bogoliubov coefficient are all computed here. That a window’s sidelobe decay follows from the order of its endpoint discontinuity, that Hamming trades far-field decay for a lower first sidelobe, and that a sudden switch creates particles are all well-known results and are not discoveries of this episode ── what this episode did was unify them under Episode 1’s yardstick “dB/oct \(=-6\alpha\),” show numerically that the measurable \(\beta\) has a ceiling at \(2\alpha\), and verify on the same yardstick that window sidelobes and particle creation obey the same power law. The §04 estimates use 4096 points averaged over 32 realizations, fitted over \(k=32\)–512; changing the band or the length changes the size of the overshoot. That the estimate reads 0.98 even at \(\beta=1\) is the finite-band fitting bias of Episode 51. §05 is limited to a monotone switch \(\omega_1=1\to\omega_2=2\) and does not apply when there is parametric resonance (in Episode 55’s classification, a value conditional on a model). This episode also uses synthetic data only, and has not been applied to real measurements. ── To print, use your browser’s Print → Save as PDF.