β: This English translation is in beta — the Traditional-Chinese original is the authoritative version.

Math Toolbox — Math Identities

See also: notation (authoritative symbols and units), glossary (Chinese–English term intuitions), convolution_derivation (LTV convolution uses the tools on this page), white_noise_to_phase_noise (application site for Parseval/integrator response)

ISF theory looks very "circuits," but its skeleton is really a handful of standard math tools combined repeatedly. This page collects them, gives each a short proof or citation, and marks which page on the site uses it. Come back here when a derivation stalls — it beats memorizing formulas by rote.

How to use this page: each section first gives "one line on what it says," then "why it holds" (proof or citation), and finally "where on the site it's used." Every quantity carries units — doing a dimension check is always the fastest way to catch mistakes.

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1. Fourier series and Parseval (harmonic decomposition of the ISF)

One line: any $2\pi$-periodic real function can be split into DC plus a set of harmonics; its "total energy" equals the sum of each harmonic's energy. This is the mathematical basis for splitting the ISF $\Gamma(\omega_0\tau)$ into $c_0,c_1,c_2,\dots$.

The ISF is a dimensionless, $2\pi$-periodic function, written as a cosine series ([P1] Eq.(12), p.183):

$$\Gamma(\omega_0\tau)=\frac{c_0}{2}+\sum_{n=1}^{\infty}c_n\cos(n\omega_0\tau+\theta_n)$$

Write the argument as $x=\omega_0\tau$. Parseval's relation ([P1] Eq.(20), p.185):

$$\sum_{n=0}^{\infty}c_n^2=\frac{1}{\pi}\int_0^{2\pi}|\Gamma(x)|^2\,dx=2\,\Gamma_{rms}^2$$

Why it holds (short proof): square the series and integrate from $0$ to $2\pi$. Cosines of different harmonics are mutually orthogonal,

$$\int_0^{2\pi}\cos(mx+\theta_m)\cos(nx+\theta_n)\,dx=\pi\,\delta_{mn}\quad(m,n\ge1),$$

so the cross terms all vanish, leaving only the square terms: each $c_n\cos(\cdot)$ term contributes $c_n^2\cdot\pi$, and the DC term $\left(\frac{c_0}{2}\right)^2$ contributes $\left(\frac{c_0}{2}\right)^2\cdot 2\pi=\frac{c_0^2}{2}\pi$. Dividing both sides by $\pi$ gives $\sum_{n=1}^{\infty}c_n^2+\frac{c_0^2}{2}$. Hajimiri–Lee write the DC term as $\frac{c_0^2}{2}$ (note the $n=0$ term is counted as only half in the sum), and rearranging gives exactly the form above.

rms definition consistency check: $\Gamma_{rms}^2=\frac{1}{2\pi}\int_0^{2\pi}|\Gamma(x)|^2\,dx$ (rms is just "root mean square": square first, average, then take the root). Substituting into the right-hand side of Parseval: $\frac{1}{\pi}\int=2\cdot\frac{1}{2\pi}\int=2\Gamma_{rms}^2$ ✓.

Numerical feel: for the ideal LC, $\Gamma(\theta)=-\sin\theta$, only $c_1=1$ (all others 0). $\Gamma_{rms}^2=\frac{1}{2\pi}\int_0^{2\pi}\sin^2\theta\,d\theta=\frac12$, so $\Gamma_{rms}=1/\sqrt2\approx0.707$; the right-hand side of Parseval, $2\times\frac12=1=c_1^2$ ✓.

Used on this site: fourier_series_of_isf (splitting the ISF into harmonics, explaining how noise near $n\omega_0$ downconverts), rms_isf, lab_05 (numerically verifying $\sum c_n^2=2\Gamma_{rms}^2$).

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2. Convolution and the LTI vs LTV distinction (why the oscillator is "time-variant")

One line: an LTI (linear time-invariant) system's impulse response depends only on "how long ago" $t-\tau$; an LTV (linear time-variant) system also depends on "at what instant it's kicked" $\tau$. The oscillator's response to noise is LTV — this is the core of ISF theory.

LTI superposition is standard convolution:

$$y(t)=\int_{-\infty}^{\infty}h(t-\tau)\,x(\tau)\,d\tau .$$

• Core feature: $h$ depends only on the difference $t-\tau$ (time-invariant). Delay the input, and the output is delayed unchanged.

The oscillator's excess-phase impulse response, however, is ([P1] Eq.(10), p.182):

$$h_\phi(t,\tau)=\frac{\Gamma(\omega_0\tau)}{q_{max}}\,u(t-\tau).$$

It simultaneously depends on $\tau$ (via $\Gamma(\omega_0\tau)$ — at what phase of the waveform it's kicked) and on $t-\tau$ (via the step $u(t-\tau)$ — it persists permanently after being kicked). Superposing all past noise ([P1] Eq.(11), p.182):

$$\phi(t)=\int_{-\infty}^{\infty}h_\phi(t,\tau)\,i_n(\tau)\,d\tau=\frac{1}{q_{max}}\int_{-\infty}^{t}\Gamma(\omega_0\tau)\,i_n(\tau)\,d\tau .$$

Why it's LTV, not LTI: because $\Gamma(\omega_0\tau)$ is a periodic function of the absolute instant of injection. The same impulse, kicked at the peak ($\Gamma\approx0$), barely changes phase; kicked at the zero crossing ($|\Gamma|$ maximal), changes it the most. This is exactly [P1] Sec. III's claim C1: "the oscillator's response to noise is linear time-variant, not LTI."

Unit check: $h_\phi$ has units of $1/\text{C}$ ($\Gamma$ dimensionless, $q_{max}$ in C, $u$ dimensionless), $\int h_\phi\, i_n\, d\tau$ has units $=(1/\text{C})\cdot\text{A}\cdot\text{s}=(1/\text{C})\cdot\text{C}=$ dimensionless $=$ rad ✓.

Used on this site: impulse_to_phase_shift, convolution_derivation, lab_04 (lti_vs_ltv_impulse_response.png: LTI's step height is fixed, LTV's step height varies with injection phase).

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3. The integrator's frequency response $1/(j\omega)$ (why white noise → 1/f² phase noise)

One line: phase is the "integral" of noise current (the upper limit of Eq.(11) is $t$); an integrator in the frequency domain is $1/(j\omega)$, and in power terms $1/\omega^2$. This is the source of the $-20$ dB/dec slope.

For an ideal integrator $y(t)=\int_{-\infty}^{t}x(\tau)\,d\tau$, with a single tone $x(t)=e^{j\omega t}$:

$$\int^{t}e^{j\omega\tau}\,d\tau=\frac{1}{j\omega}e^{j\omega t}\;\Rightarrow\;H(j\omega)=\frac{1}{j\omega}.$$

• Magnitude: $|H(j\omega)|=1/\omega$. Power (PSD multiplier): $|H|^2=1/\omega^2$.
• dB slope: $10\log_{10}(1/\omega^2)=-20\log_{10}\omega$, $-20$ dB per decade — exactly the $1/f^2$-region slope of phase noise.

Connecting to the ISF: taking $\Gamma$'s rms as an effective gain, the noise-current PSD $S_i$, after "multiplying by $\Gamma_{rms}/q_{max}$ then integrating," gives the phase PSD:

$$S_\phi(\Delta\omega)=\frac{\Gamma_{rms}^2}{q_{max}^2}\cdot\frac{S_i}{\Delta\omega^2}.$$

• Unit check (using $\text{C}=\text{A}\cdot\text{s}$):

$$\frac{1}{\text{C}^2}\cdot\frac{\text{A}^2/\text{Hz}}{(\text{rad/s})^2}=\frac{\text{A}^2/\text{Hz}}{\text{C}^2\cdot\text{s}^{-2}}=\frac{\text{A}^2/\text{Hz}}{\text{A}^2}=\frac{1}{\text{Hz}}=\text{rad}^2/\text{Hz}\ \checkmark$$

Factor-of-2 note: using the equation above (a clean time-domain derivation) gives $\mathcal{L}=\frac12 S_\phi$, corresponding to a denominator of $2\Delta\omega^2$; whereas [P1] Eq.(21), p.185 writes $4\Delta\omega^2$. The factor-of-2 difference is an SSB bookkeeping convention, a well-known minor point of contention in the literature, and does not affect the $\Gamma_{rms}^2/q_{max}^2$ scaling or the $-20$ dB/dec slope. See white_noise_to_phase_noise for details.

Used on this site: white_noise_to_phase_noise, lab_06.

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4. The PSD of a stochastic process and the Wiener–Khinchin theorem

One line: a stationary stochastic process's "power spectral density" (PSD) is the Fourier transform of its "autocorrelation function." This connects time-domain noise (autocorrelation) with frequency-domain noise (PSD), and is the foundation of every phase-noise calculation.

For a stationary stochastic process $x(t)$, the autocorrelation is $R_x(\tau)=\langle x(t)\,x(t+\tau)\rangle$. Wiener–Khinchin theorem:

$$S_x(\omega)=\int_{-\infty}^{\infty}R_x(\tau)\,e^{-j\omega\tau}\,d\tau .$$

• White-noise special case: $R_x(\tau)=\frac{N_0}{2}\delta(\tau)$ (completely uncorrelated at different instants) $\Rightarrow S_x(\omega)=\frac{N_0}{2}$ (frequency-independent — that's what "white" means). This site uses the single-sided PSD $S_i=\overline{i_n^2}/\Delta f$ (A²/Hz).
• Variance (total power):

$$\sigma_x^2=R_x(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_x(\omega)\,d\omega=\int_0^{\infty}S_x^{(1\text{-side})}(f)\,df .$$

This is where the phase-variance formula $\sigma_\phi^2=\int_{f_1}^{f_2}S_\phi(f)\,df$ comes from: phase variance is the integral of the phase PSD over frequency ([P1]'s usage convention; a standard stochastic-process result).

Unit check: $S_\phi$ is in rad²/Hz, $\int S_\phi\, df$ has units $=\text{rad}^2/\text{Hz}\cdot\text{Hz}=\text{rad}^2$ ✓.

External source: Wiener–Khinchin is a standard stochastic-process theorem (not among the five downloaded PDFs, standard textbook material, e.g. Papoulis; [P1] uses its conclusion directly).

Used on this site: psd_phase_noise_jitter, stochastic_processes_recap, lab_06 (estimating PSD via Welch's method), lab_08 (integrating PSD to get jitter).

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5. $10\log_{10}$ / dB and dBc/Hz conversion

One line: dB is a "log scale of a power ratio"; dBc/Hz is the phase-noise unit "power relative to the carrier, per Hz of bandwidth." Remembering "$\times10$ power $=+10$ dB, $\times2$ power $\approx+3$ dB" is enough to get by.

Definition (power ratio):

$$X_{\text{dB}}=10\log_{10}\!\left(\frac{P}{P_{ref}}\right).$$

• Voltage/amplitude ratio: because power $\propto$ amplitude$^2$, $X_{\text{dB}}=20\log_{10}(V/V_{ref})$ (an extra factor of 2).
• Inverse conversion: $P/P_{ref}=10^{X_{\text{dB}}/10}$.
• SSB phase noise $\mathcal{L}(\Delta f)$'s unit is dBc/Hz: "c" = relative to carrier, "/Hz" = per unit bandwidth. Its relation to the phase PSD (small-angle approximation, see Section 6):

$$\mathcal{L}(\Delta f)\approx\frac12 S_\phi(\Delta f)\quad\Longrightarrow\quad \mathcal{L}_{\text{dBc/Hz}}=10\log_{10}\!\left(\tfrac12 S_\phi\right),\;\; S_\phi=2\cdot10^{\mathcal{L}_{\text{dBc/Hz}}/10}\ \text{rad}^2/\text{Hz}.$$

Numerical feel (canonical example C): $\mathcal{L}=-100$ dBc/Hz $\Rightarrow 10^{-100/10}=10^{-10}$, $S_\phi=2\times10^{-10}$ rad²/Hz. 10× better in power ($-110$ dBc/Hz) = $\sqrt{10}\approx3.16\times$ better in amplitude (rms).

Common conversion table:

Power ratio	dB	Voltage ratio	Intuition
$\times2$	$+3.01$ dB	$\times\sqrt2$	double the power
$\times10$	$+10$ dB	$\times\sqrt{10}\approx3.16$	one order of magnitude
$\times100$	$+20$ dB	$\times10$	two orders of magnitude
$\times\tfrac12$	$-3.01$ dB	$\times1/\sqrt2$	halve it

Used on this site: psd_phase_noise_jitter, numerical_feeling (Example 3, the full set of conversions), white_noise_to_phase_noise.

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6. The small-angle PM approximation (where $\mathcal{L}\approx\frac12 S_\phi$ comes from)

One line: when phase jitter is small, the single-sideband power of a phase modulation is about half the phase PSD. This is the bridge that converts "phase PSD" into the "dBc/Hz" on a datasheet.

Consider a phase-modulated carrier: $v(t)=A\cos\!\big(\omega_0 t+\phi(t)\big)$. When $|\phi(t)|\ll1$ rad (small-angle), expand trigonometrically and approximate $\cos\phi\approx1$, $\sin\phi\approx\phi$:

$$v(t)=A\big[\cos\omega_0 t\cos\phi-\sin\omega_0 t\sin\phi\big]\approx A\cos\omega_0 t-A\,\phi(t)\sin\omega_0 t .$$

• The first term is the pure carrier; the second term $-A\,\phi(t)\sin\omega_0 t$ is the "phase sideband" — it moves the baseband $\phi(t)$ to either side of the carrier.
• For a phase component at a single offset frequency $\Delta f$, the sideband power is proportional to the power of $\phi$; converting to the single-sideband (SSB) density relative to the carrier gives

$$\mathcal{L}(\Delta f)\approx\frac12 S_\phi(\Delta f).$$

That $\frac12$ comes from the bookkeeping of "total phase power split evenly between the upper and lower sidebands."

Applicability condition (important): holds only for $\sigma_\phi\ll1$ rad. If the integration bandwidth is wide enough that $\sigma_\phi$ approaches or exceeds 1 rad, this approximation fails (the carrier gets "smeared out"), and a more complete treatment is needed. Canonical example C's $\sigma_\phi=14.07$ mrad $\ll1$, so it is safe.

Used on this site: psd_phase_noise_jitter, numerical_feeling, Eq.(16) of the spec ($\mathcal{L}\approx\frac12 S_\phi$).

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7. Trigonometric identities (the workhorse behind deriving Eq.(15)–(18))

One line: when multiplying "some harmonic of the ISF" by "an injected single tone" and integrating over time, you use product-to-sum identities to split the product into a slow term (survives) and a fast term (vanishes after integration).

For an injected single tone $i(t)=I_0\cos(\Delta\omega\,t)$ near DC, the phase response ([P1] Eq.(13), p.183's $c_0$ term) requires computing $\int^{t}I_0\cos(\Delta\omega\,\tau)\,d\tau=\dfrac{I_0\sin(\Delta\omega\,t)}{\Delta\omega}$, directly giving [P1] Eq.(15), p.183:

$$\phi(t)\approx\frac{I_0\,c_0\sin(\Delta\omega\,t)}{2q_{max}\,\Delta\omega}.$$

When the tone is near $n\omega_0$, use product-to-sum:

$$\cos(n\omega_0\tau+\theta_n)\cos\big((n\omega_0+\Delta\omega)\tau\big) =\frac12\cos\big((2n\omega_0+\Delta\omega)\tau+\theta_n\big)+\frac12\cos\big(\Delta\omega\,\tau-\theta_n\big).$$

• The first term has frequency $\approx2n\omega_0$ (fast); after integration its amplitude $\propto1/(2n\omega_0)$, tiny, negligible.
• The second term has frequency $\Delta\omega$ (slow, near DC); after integration its amplitude $\propto1/\Delta\omega$, survives.

Keeping only the slow term and integrating gives [P1] Eq.(16)/(17), p.183:

$$\phi(t)\approx\frac{I_0\,c_n\sin(\Delta\omega\,t)}{2q_{max}\,\Delta\omega}.$$

That $\frac12$ is exactly what falls out of the product-to-sum step. Physical meaning: the ISF's $n$-th harmonic acts like a mixer, "downconverting" noise near $n\omega_0$ into low-frequency phase modulation near the carrier — this is the frequency-translation picture of phase noise.

Common identity quick reference:

Identity	Use
$\cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)]$	separating slow/fast terms (Eq.16/17)
$\sin^2\theta=\tfrac12(1-\cos2\theta)$	computing $\Gamma_{rms}$ (rms of $-\sin$ is $1/\sqrt2$)
$\int^{t}\cos(\omega\tau)d\tau=\sin(\omega t)/\omega$	integrator $1/\omega$ (Eq.15)

Used on this site: fourier_series_of_isf, convolution_derivation, the orthogonality used in Section 1's Parseval.

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8. Random walk variance (the $\sqrt{\Delta t}$ law of accumulated jitter)

One line: an open-loop oscillator has no absolute time reference, so each period's phase error accumulates independently like a "drunkard's walk"; the variance of the error grows linearly, so the standard deviation grows as $\sqrt{\Delta t}$.

Suppose each period injects an independent, zero-mean phase error $\delta_k$ with variance $\sigma_1^2$ (the result of white noise integrated over one period). After $M$ periods the total phase error is $\Phi_M=\sum_{k=1}^{M}\delta_k$. Because each $\delta_k$ is independent, the variances add (cross-correlation terms have zero expectation):

$$\mathrm{Var}(\Phi_M)=\sum_{k=1}^{M}\mathrm{Var}(\delta_k)=M\,\sigma_1^2 .$$

• The measurement interval is $\Delta t=M\cdot T$, so $M=\Delta t/T$, $\mathrm{Var}(\Phi_M)=\dfrac{\sigma_1^2}{T}\,\Delta t\propto\Delta t$.
• Converting to time jitter ($\sigma_t=\sigma_\phi/(2\pi f_0)$) and taking the square root:

$$\sigma_{\Delta t}=\kappa\,\sqrt{\Delta t}\qquad([P2]\ \text{Eq.}(10),\ \text{p.793}).$$

$\kappa$ is a proportionality constant specific to each device, with units $\sqrt{\text{s}}$; it is determined by the same $\Gamma_{rms}^2/q_{max}^2$ ratio ([P2] Eq.(12), p.793: $\kappa=\frac{\Gamma_{rms}}{q_{max}}\sqrt{\tfrac12\tfrac{\overline{i_n^2}}{\Delta f}}$, verified verbatim).

Key intuition: variance (power) grows linearly, standard deviation (rms) grows as a square root. This is the hallmark of a random walk; it appears whenever phase errors accumulate independently with no restoring force (contrast: with a PLL locked, there is a restoring force and jitter is suppressed, no longer growing without bound). Corresponds to claim C6.

Numerical feel: at a spacing of 1000 periods (at 5 GHz, $\Delta t=200$ ns), $\sigma_{\Delta t}$ is $\sqrt{1000/10}=\sqrt{100}=10\times$ that at a spacing of 10 periods ($\Delta t=2$ ns) — a 100× longer interval gives only 10× more jitter.

Used on this site: lab_03 (ring_oscillator_timing_noise_accumulation.png: $\sigma_{\Delta t}=\sigma\sqrt{\Delta N}$ random walk), psd_phase_noise_jitter, serdes_clocking_connection.

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9. Large-angle PM: the Bessel sideband ladder and where the small-angle approximation fails

One line: Section 6's small-angle PM result ($\mathcal{L}\approx\frac12 S_\phi$) is only the first-order approximation of "large-angle phase modulation" valid for $\beta\ll1$; the full solution is an infinite ladder of Bessel sidebands, with the carrier and each sideband having amplitude $J_0(\beta)$ and $J_n(\beta)$ respectively. This section builds the full ladder and quantifies exactly when the small-angle approximation starts to break down.

9.1 The Jacobi–Anger expansion (external literature, not in this site's 5 PDFs)

For any real $\beta$ (the modulation index) and angle $\Omega t$, the Jacobi–Anger expansion (a standard special-function identity, e.g. Abramowitz & Stegun 9.1.42; external literature, not in this site's 5 PDFs):

$$e^{j\beta\sin\Omega t}=\sum_{n=-\infty}^{\infty}J_n(\beta)\,e^{jn\Omega t}$$

where $J_n(\beta)$ is the Bessel function of the first kind, order $n$. Why it holds (one-line provenance): $e^{j\beta\sin\Omega t}$ is $2\pi$-periodic in $\Omega t$; expanding it in a Fourier series, the integral defining its coefficients $\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x-nx)}dx$ is exactly one of the standard integral representations of $J_n(\beta)$ — that integral representation itself is not re-derived here, only its result is cited.

Properties (used below): $J_{-n}(\beta)=(-1)^n J_n(\beta)$ (symmetric for even $n$, antisymmetric for odd $n$), and the Bessel Parseval-like identity:

$$\sum_{n=-\infty}^{\infty}J_n(\beta)^2=1\qquad(\forall\beta)$$

This can later be used to check "the sideband ladder's total power is conserved" — a complete analogy to the Parseval relation for the ISF's Fourier coefficients in Section 1.

9.2 The sideband ladder for single-tone PM

Consider a carrier phase-modulated by a single audio tone $\Omega$ (writing Section 6's $\phi(t)=\phi_p\sin\Omega t$ as $\phi(t)=\beta\sin\Omega t$, where $\beta$ is the modulation index, i.e. the peak phase deviation, units rad):

$$V(t)=\cos\!\big(\omega_0 t+\beta\sin\Omega t\big)$$

Writing this as $\mathrm{Re}\{e^{j\omega_0 t}\,e^{j\beta\sin\Omega t}\}$ and substituting the Jacobi–Anger expansion:

$$V(t)=\mathrm{Re}\left\{e^{j\omega_0 t}\sum_{n=-\infty}^{\infty}J_n(\beta)e^{jn\Omega t}\right\} =\sum_{n=-\infty}^{\infty}J_n(\beta)\cos\big((\omega_0+n\Omega)t\big)$$

This is the full Bessel sideband ladder:

• Carrier ($n=0$): amplitude $J_0(\beta)$, located at $\omega_0$.
• $n$-th sideband ($n=\pm1,\pm2,\dots$): amplitude $J_n(\beta)$, located at $\omega_0+n\Omega$.
• Because $J_{-n}=(-1)^nJ_n$, the upper and lower sidebands have equal magnitude ($|J_{-n}|=|J_n|$), differing only in sign (phase).

Unit check: $\beta=\phi_p$ is the peak phase, arising from the integral $\int\phi(t)$, with units rad; $\Omega t$ and $\omega_0 t$ are both dimensionless phase angles (rad), and $J_n(\beta)$ itself is dimensionless (an amplitude ratio), so substituting into $\cos(\cdot)$ raises no unit issue; $V(t)$ has the same units as the original $\cos(\omega_0t+\phi(t))$ (normalized amplitude) ✓.

9.3 The small-$\beta$ limit: recovering Section 6's small-angle approximation

Small-argument expansion of the Bessel functions (standard external-literature result):

$$J_0(\beta)\approx1-\frac{\beta^2}{4},\qquad J_1(\beta)\approx\frac{\beta}{2}\qquad(\beta\ll1)$$

Substituting back into the sideband ladder: the carrier amplitude $J_0(\beta)\approx1-\beta^2/4\approx1$ (barely attenuated at first order), and the first-sideband amplitude $J_1(\beta)\approx\beta/2$. The single-sideband relative power:

$$\mathcal{L}(\Delta f)\approx J_1(\beta)^2\approx\left(\frac{\beta}{2}\right)^2$$

This is exactly Section 6's small-angle PM result $\mathcal{L}\approx\left(\frac{\phi_p}{2}\right)^2=\frac12 S_\phi$ ($\beta=\phi_p$ is the peak phase, $S_\phi=\phi_p^2/2$ is its power) — the Bessel sideband ladder automatically converges to the small-angle PM formula as $\beta\ll1$; the two are the first-order and full versions of the same thing. The higher-order sidebands $J_2(\beta)\approx\beta^2/4,\,J_3(\beta)\approx\beta^3/48,\dots$ vanish rapidly as $\beta\ll1$ ($J_n(\beta)\sim(\beta/2)^n/n!$), which is why it is reasonable for the small-angle approximation to keep only the first-order sideband.

9.4 Failure boundary: where does the small-angle approximation err by 1 dB?

The small-angle approximation replaces $J_1(\beta)$ with the first-order term $\beta/2$; as $\beta$ grows, this approximation drifts from the true value. Define the error (comparing the amplitude ratio of the two, in dB):

$$\text{err}_{\text{dB}}(\beta)=20\log_{10}\!\left(\frac{J_1(\beta)}{\beta/2}\right)$$

($\text{err}_{\text{dB}}\to0$ as $\beta\to0$; as $\beta$ grows, $J_1$'s growth slows, the ratio drops below 1, and $\text{err}_{\text{dB}}<0$, i.e. the small-angle approximation overestimates the sideband.) Solving numerically for $\text{err}_{\text{dB}}(\beta)=-1$ dB (root-finding scipy.optimize.brentq on scipy.special.jv, see the Python block below):

$$\beta_{1\text{dB}}\approx0.950\ \text{rad}\quad(\approx54.5^\circ)$$

In plain terms: as long as the peak phase deviation $\beta\lesssim0.95$ rad, the small-angle approximation $\mathcal{L}\approx(\beta/2)^2$ is accurate to within 1 dB; beyond this range (e.g. wideband FM, large-index PM, or an integration bandwidth wide enough that $\sigma_\phi$ approaches 1 rad), the full Bessel sideband ladder must be used — the small-angle formula no longer applies. This echoes Section 6's qualitative statement that it "holds only for $\sigma_\phi\ll1$ rad," now with a quantitative boundary.

9.5 Carrier-suppression null and Carson bandwidth (external literature, not in this site's 5 PDFs)

Carrier suppression: as $\beta$ grows to the first root of $J_0(\beta)=0$, the carrier vanishes completely and all power is transferred to the sidebands — a classic phenomenon in FM/PM systems:

$$J_0(\beta)=0\text{'s first root: }\beta=2.405\quad(\text{numerical solution, see the Python block below: }J_0(2.405)\approx-9\times10^{-5}\approx0)$$

Carson's bandwidth rule (external literature, an engineering rule of thumb, not in this site's 5 PDFs): in practice the sideband ladder has infinitely many terms, but $J_n(\beta)$ decays rapidly to negligible size once $n>\beta+1$, so an effective bandwidth is defined ($f_m=\Omega/2\pi$):

$$BW_{\text{Carson}}\approx2(\beta+1)f_m$$

At $\beta=2.405$, $BW\approx6.81\,f_m$; at small-angle $\beta=0.1$, $BW\approx2.2\,f_m$ (barely more than a single sideband pair, consistent with the picture that "small-angle PM has only the $\pm1$ sidebands").

9.6 Python check: the full sideband ladder at three values of $\beta$

import numpy as np
from scipy.special import jv, jn_zeros
from scipy.optimize import brentq

for beta in [0.1, 1.0, 2.405]:
    print(f"beta={beta}")
    for n in range(5):
        print(f"  J_{n}({beta}) = {jv(n, beta):.6f}")
    # -> beta=0.1: J0=0.997502, J1=0.049938, J2=0.001249, J3=0.000021, J4≈0
    # -> beta=1.0: J0=0.765198, J1=0.440051, J2=0.114903, J3=0.019563, J4=0.002477
    # -> beta=2.405: J0=-0.000091(≈0), J1=0.519110, J2=0.431783, J3=0.199032, J4=0.064763

# carrier null (classic 2.405)
z0 = jn_zeros(0, 1)[0]
print(f"first zero of J0: {z0:.6f}")
# -> first zero of J0: 2.404826

# small-angle approximation vs exact
for beta in [0.1, 0.3, 0.5, 1.0]:
    J0, J1 = jv(0, beta), jv(1, beta)
    print(f"beta={beta}: J0={J0:.6f} vs 1-b^2/4={1-beta**2/4:.6f} | "
          f"J1={J1:.6f} vs b/2={beta/2:.6f}")
    # -> beta=0.1: J0=0.997502 vs 0.997500 | J1=0.049938 vs 0.050000
    # -> beta=1.0: J0=0.765198 vs 0.750000 | J1=0.440051 vs 0.500000

# 1 dB failure boundary: 20log10(J1(beta)/(beta/2)) = -1 dB
f = lambda b: 20*np.log10(jv(1, b)/(b/2)) - (-1.0)
beta_1dB = brentq(f, 0.5, 1.5)
print(f"beta_1dB = {beta_1dB:.4f}")
# -> beta_1dB = 0.9505

# Parseval check: sum J_n^2 = 1
for beta in [0.1, 1.0, 2.405]:
    s = sum(jv(n, beta)**2 for n in range(-50, 51))
    print(f"beta={beta}: sum J_n^2 = {s:.8f}")
    # -> all beta give 1.00000000

Used on this site: Section 6's small-angle PM (this section is its full version), white_noise_to_phase_noise (the small-angle premise behind the single-tone sideband Eq.(16)–(18)), psd_phase_noise_jitter (the applicability boundary $\sigma_\phi\ll1$ rad, now with a quantitative number $\approx0.95$ rad).

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Quick-reference summary table

Tool	Core formula	Main use on this site
Fourier/Parseval	$\sum c_n^2=2\Gamma_{rms}^2$	ISF harmonic decomposition, $\Gamma_{rms}$
LTV convolution	$\phi=\frac{1}{q_{max}}\int^{t}\Gamma\,i_n\,d\tau$	phase superposition, LTV vs LTI
integrator $1/(j\omega)$	$\vert H\vert ^2=1/\omega^2$	white noise → $1/f^2$ ($-20$ dB/dec)
Wiener–Khinchin	$S_x=\mathcal{F}\{R_x\}$	PSD ↔ autocorrelation, variance
dB / dBc/Hz	$X_{\text{dB}}=10\log_{10}(P/P_{ref})$	unit conversion
small-angle PM	$\mathcal{L}\approx\frac12 S_\phi$	dBc/Hz ↔ phase PSD
product-to-sum	$\cos A\cos B=\tfrac12[\cdots]$	Eq.(15)–(18) mixing
random walk	$\sigma_{\Delta t}=\kappa\sqrt{\Delta t}$	accumulated jitter
Bessel sideband ladder	$J_n(\beta)$, small-angle $J_1\approx\beta/2$, fails for $\beta\gtrsim0.95$	large-angle PM, carrier suppression at $\beta=2.405$

Further reading

• Symbol and unit summary table: notation
• Stochastic-process recap: stochastic_processes_recap
• Numerical-feel exercises: numerical_feeling
• Equation index (each equation → derivation page → source): equation_index
• Full literature list: references