Thinking in Waves Episode 49 / Part XVII — Looking for the Edge of the Axis
48 episodes of procedure, on one page
A Handbook —
What to do when you are handed a measurement
Measuring the slope is not enough. Two things must always be looked at with it — how many decades it lasts, and what the phase does.
★ The phase can be predicted from the amplitude slope alone (the Bode gain–phase relation; numerically within 0.01°) — and any departure means an extra delay.
★★ The centrepiece: we catch data that "looks like 1/f but isn't," using nothing but the decade count.
If Episodes 15 and 43 were scorecards of what we got right, this is the manual for how to use it.
★ The procedures built over 48 episodes are here written as a program, applied mechanically to six "unknown" datasets to see what can be said.
01The handbook — seven steps
| Step | From | Why |
|---|---|---|
| ① plot it log–log | Ep. 1 | first of all. Linear axes show nothing |
| ② measure the local order | Ep. 48 | \(\alpha(\omega)=d\ln|H|/d\ln\omega\) |
| ★ ③ count the decades it stays flat | Ep. 27, 48 | ★ answer "is it a power law" in decades |
| ④ find the bends | Ep. 16 | a characteristic frequency lives there |
| ★ ⑤ measure the phase too | Ep. 1, 8 | ★ is it \(90\alpha\)°? If not, there is a delay |
| ⑥ change the moment | Ep. 42 | if higher moments drift, it is a distribution |
| ⑦ compare with the system size | Ep. 5, 48 | anything longer than the window is a constant |
★ Steps ①–④ need only the amplitude. Adding ⑤ makes it far stronger.
02Checking step ⑤'s tool numerically
| order \(a\) | phase (numerical) [deg] | \(90a\) [deg] | difference |
|---|---|---|---|
| −2 | −179.993679 | −180 | 6.3e−3 |
| −0.5 | −44.998420 | −45 | 1.6e−3 |
| 0.5 | 44.998420 | 45 | 1.6e−3 |
| 1 | 89.996840 | 90 | 3.2e−3 |
The point of §02
★★ The phase follows from the amplitude slope alone. Episode 1's "phase \(=90\alpha\) degrees" came from here.
★★★ Which is why "a phase departing from \(90\alpha\) means an extra delay" is a legitimate inference.
★ Something the amplitude can never tell you, the phase can — the most useful tool of these 48 episodes.
03Applying the procedure to six "unknown" datasets
Pretending to know nothing about them, we run ②–④ mechanically (criterion: the longest window in which the order varies by at most 0.1).
| Data | mean order | ★ decades | range | bends |
|---|---|---|---|---|
| A superposition (broad) | −0.4991 | 6.32 | 6.1e−4 – 1.3e+3 | 0 |
| B single Lorentzian | −0.0065 | 3.52 | 1.0e−4 – 3.3e−1 | 1 |
| C power law \(\omega^{-5/6}\) | −0.8333 | 8.00 | the whole window | 0 |
| D exponential \(e^{-\omega}\) | −0.0144 | 3.00 | 1.0e−4 – 9.9e−2 | 1 |
| E power law + high cutoff | −0.5040 | 5.34 | 1.0e−4 – 2.2e+1 | 1 |
| F narrow distribution (0.5 dec) | −0.0120 | 2.82 | 1.0e−4 – 6.6e−2 | 0 |
The point of §03 — ★ a caution about automation
★ "The longest flat window" sometimes picks up the flat low-frequency part (D and F). That is a correct reading too — \(e^{-\omega}\) really is constant at low frequency.
★★ What matters is measuring separately how many decades hold at the order you care about. The machine returns the longest plateau; that may not be the one you want.
04★★★ The centrepiece — telling A from F
Both A (a broad superposition) and F (a narrow distribution) look like 1/f somewhere. In amplitude order, 1/f is \(-0.5\) (power \(-1\)) — so we count the decades there.
| Data | ★ decades with order \(-0.5\) within \(\pm0.1\) |
|---|---|
| A superposition (broad) | ★ 6.898 |
| F narrow distribution | ★ 0.498 |
| \(\omega\) | F's \(\alpha(\omega)\) |
|---|---|
| 0.1 | −0.14406 |
| 0.3 | −0.29577 |
| 1 | −0.50000 |
| 3 | −0.68745 |
| 100 | −0.99366 |
The point of §04 — ★★★ the core of this episode
F merely passes through \(-0.5\); it has no plateau there.
Without step ③ (counting decades), F would be misread as 1/f.
★ A holds for more than 6 decades — only then can one say "a single time constant cannot do this."
★★ "The slope was \(-0.5\) at one point" is no evidence at all. Any smooth function passes through \(-0.5\) somewhere.
★ When Episode 2 said "1/f is a superposition of many time constants," what guaranteed the "many" was the decade count.
05Writing the verdict — the range of what may be said
| Data | Verdict | What may be said |
|---|---|---|
| A | ★ power law \(-0.5\) over 6+ decades | narrows the mechanism: not a single time constant |
| B | two power laws + one bend | there is one characteristic frequency |
| C | ★ power law (whole window) | no characteristic length inside the window |
| D | constant at low \(\omega\), then not a power law | ★ summarize with a time constant, not an order |
| E | power law + one cutoff | the cutoff position carries the physics |
| F | ★ verdict withheld (too few decades) | ★ must not be called 1/f |
The point of §05
★ Write the verdict not as "is it a power law" but as "a power law over how many decades." Then what to measure next is decided automatically — go extend the decades.
★★ And being able to write "verdict withheld" matters — the practical version of Episode 43's rule that what has not been checked does not become a claim.
06★ What step ⑤ adds — catching an extra delay
The same amplitude with a different phase is a different system. Take \(|H|=\omega^{-0.5}\) with a delay \(e^{-i\omega T}\) (\(T=0.1\)) —
| \(\omega\) | Bode prediction [deg] | actual with delay [deg] | difference \(=-\omega T\) [deg] |
|---|---|---|---|
| 0.1 | −44.9984 | −45.5714 | −0.5730 |
| 1 | −44.9984 | −50.7280 | −5.7296 |
| 10 | −44.9984 | −102.2942 | ★ −57.2958 |
The point of §06
★★ The amplitude is identical; only the phase drifts, in proportion to \(\omega\).
★★ The gap from "the phase predicted by the amplitude" is the delay. Divide it by \(\omega\) and you read off \(T\).
★ The most practical use of Episode 8's Kramers–Kronig — causality is also a measuring instrument.
07What not to do — from 48 episodes of mistakes
| Do not | From | Why |
|---|---|---|
| call one decade a "power law" | Ep. 48 | even an exponential earns 0.087 |
| stop once one exponent matches | Ep. 42, 47 | ★ K41 and \(\phi^2\) both pass that way |
| test a theory with an insensitive quantity | Ep. 42 | ★ intermittency moves dB/oct by only 0.081 |
| discuss periods longer than the window | Ep. 5 | indistinguishable from a constant |
| mistake a coordinate-dependent quantity for physics | Ep. 45 | ★ particle number depends on the basis |
| call organizing a discovery | Ep. 43 | ★ the thing this series guarded most |
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Seven steps | ◎ organized | 48 episodes on one page |
| ★ Bode relation checked numerically | ◎ numerics | ★ phase \(=90\alpha\) within 0.01° |
| Six datasets judged automatically | ◎ numerics | plateau and decades found automatically |
| ★ The case the handbook saved (F) | ◎ numerics | ★ A: 6.9 decades; F: 0.5 |
| ★ Catching a delay via phase | ◎ numerics | ★ impossible from amplitude alone |
| ★ Pinning down one mechanism | × no | ★ the handbook only narrows candidates |
| ★ Application to noisy data | × no | ★ tested on noiseless functions only |
Exercises
- In §03 the machine picked D's flat low-frequency region. Is that a flaw in the procedure?
Show answer
Not a flaw — the question was underspecified. \(e^{-\omega}\) really is constant at low frequency, and "order 0 over 3 decades" is a correct measurement.
★ The procedure answers "where is it flat," not "where is it interesting." What makes a region interesting is your hypothesis.
★★ Which is why §04 re-measured with a specified target: "how many decades at \(-0.5\)?" — a procedure only becomes sharp once there is a hypothesis. ★ Episode 15's rule about declaring falsifiability pays off here too. - §06 read a delay off the phase. What if the phase departs but there is no extra delay?
Show answer
Then the system is not minimum-phase. The Bode relation holds only when there are no right-half-plane zeros.
★ A right-half-plane zero rotates the phase further while leaving the amplitude alone — an all-pass component is present.
★★ A delay is in fact the limiting case of all-pass (\(e^{-i\omega T}\) has unit amplitude and rotates only phase) — so §06's delay was a special case of this general rule.
★ To tell them apart, check whether the departure is proportional to \(\omega\) — proportional means a pure delay; otherwise a right-half-plane zero. - §07 says not to test a theory with an insensitive quantity. How do you actually carry that out?
Show answer
Compute, in advance, how much that quantity moves if the theory changes.
★ Episode 42 found that intermittency moves the dB/oct by only 0.081 — with a measurement error of 0.1 dB/oct, that measurement can never distinguish them.
★★ Episode 47 found \(\phi^2\) passing on \(n_s\) but excluded by \(r\) — \(r\) is the sensitive one.
★ So: compute "the difference you want to test" ÷ "your measurement error" before measuring. If it is below 1, the measurement is a waste. ★ This is the most practical habit the series acquired in 49 episodes.
Summary — the slope alone is not enough
Forty-eight episodes of procedure, gathered into seven steps — ① plot log–log ② measure the local order ③ count the decades ④ find the bends ⑤ measure the phase ⑥ change the moment ⑦ compare with the system size.
The phase can be predicted from the amplitude slope alone (the Bode gain–phase relation) — numerically matching \(90\alpha\) to within 0.01°. ★ Episode 1's "phase \(=90\alpha\) degrees" came from here.
Run mechanically on six "unknown" datasets: C is a power law across the whole window, B and E have one bend each, D is constant at low frequency. ★ The machine returns the longest plateau, so the region you care about has to be specified by you.
★★★ The centrepiece —
A (a broad superposition) and F (a narrow distribution) both look like 1/f,
but order \(-0.5\) holds for 6.898 decades in A and 0.498 in F.
F merely passes through \(-0.5\).
★ "The slope was \(-0.5\) at one point" is no evidence — every smooth function passes through \(-0.5\) somewhere. What guaranteed Episode 2's "many time constants" was the decade count.
And the phase reveals an extra delay — the amplitude is identical, but the phase drifts in proportion to \(\omega\), and dividing that gap by \(\omega\) gives the delay \(T\). ★ Causality (Episode 8) is also a measuring instrument.
★ Honestly: the handbook narrows candidates; it cannot pin down one mechanism. And everything here was tested on noiseless functions — with real data, the decade counts carry statistical error.
★ As a milestone, we draw one map of what this series has done — from Episode 1's single line (differentiate a sound and it goes high) to here.
★★ And we list, honestly, what has not been touched. Writing down what remains is the best handover to whoever walks on.
This document is Episode 49 of the "Thinking in Waves" series. Computations are in kenshou/tebiki.js. On the numbers here: the numerical check of the Bode gain–phase relation, the automatic diagnosis of six synthetic datasets (longest flat window, decade counts, bend detection), the A-versus-F comparison, and the phase shift from a delay were all computed here. The Bode gain–phase relation, minimum-phase systems and all-pass components, the Kramers–Kronig relations, and the fact that a superposition of many time constants produces 1/f (the Debye superposition of Episode 2) are all well-known results and none are discoveries of this article — what this article did was arrange them into a single procedure, actually run that procedure as a program, and show where it helps and where it falls short. The six datasets in §03–04 are all synthetic and constructed here; none is a real measurement. The criteria used ("variation within 0.1," "decades within \(\pm0.1\)") are conventions adopted here; different criteria give different decade counts. Only noiseless functions were treated; with noisy real data the decade estimates acquire statistical error (outside this article's scope). The delay detection in §06 assumes the system is minimum-phase. — To print, use your browser's Print and "Save as PDF."