EP34

EP34: The Bessel Secret of the DX7 — FM Synthesis Mathematics

Bessel旁频展开Jₙ(I), Carson带宽公式, 算子反馈分岔

▶ 6:11 Signal Processing

前置知识

EP19 Vibrato as FM Synthesis EP30 The Sampling Theorem Is Not What You Think

Overview / 概述

In 1983, Yamaha released the DX7 synthesizer at $1,995 — and it promptly rewired the sonic landscape of popular music. The iconic electric piano patches on thousands of 1980s recordings were not built from stored waveforms or analog filters. They were computed from a single equation with one free parameter. That equation is FM synthesis, and the parameter is the modulation index $I$ .

The mathematical core is the Jacobi-Anger expansion: a phase-modulated sinusoid decomposes into an infinite series of sidebands whose amplitudes are Bessel functions $J_n(I)$ . Turning the modulation index from 0 to 10 is equivalent to redistributing energy across the Bessel function family — from a pure carrier at $J_0$ outward to $J_1, J_2, J_3, \ldots$ , painting progressively brighter, more metallic timbres as each new partial switches on.

中文: “那个标志性的电钢琴音色，出现在上千首八十年代的金曲里——它的核心，是一个你可能在数学课上见过的函数：Bessel函数。今天我们拆开DX7的黑箱，看看FM合成背后的谱系数学。”

This episode also covers Carson’s rule for estimating FM bandwidth, and the phenomenon of operator self-feedback: when an FM operator modulates itself, the system transitions from sinusoid through sawtooth to noise-like instability — an engineering-domain version of the bifurcation cascades studied in nonlinear dynamics.

Prerequisites / 前置知识

Vibrato and Tremolo as Low-Rate FM (EP19) — vibrato is exactly FM synthesis with $I \ll 1$ ; this episode extends that to arbitrary $I$ and the full DX7 architecture.
Nonlinear Dynamics and Bifurcations (EP30) — the feedback FM bifurcation diagram (operator self-modulation) is a direct application of the period-doubling and chaos concepts introduced there.

Definitions

Definition 34.1 (FM Synthesis Signal)

Let $\omega_c$ (carrier angular frequency), $\omega_m$ (modulator angular frequency), and $I \geq 0$ (modulation index) be real parameters. The FM synthesis signal is

$y(t) = \sin\!\bigl(\omega_c t + I \sin(\omega_m t)\bigr).$

The instantaneous phase deviation from the carrier is $\phi(t) = I \sin(\omega_m t)$ , so the instantaneous frequency is $\omega_c + I\omega_m \cos(\omega_m t)$ . The peak frequency deviation is $\Delta\omega = I\omega_m$ .

中文: “当I等于零，y就是纯正弦波，只有载波频率。随着I从零增大，被调制的相位幅度增大，波形开始扭曲，不再是正弦波。频谱上，在载波频率两侧出现新的分量——称为边带。I越大，边带越多，音色越丰富、越明亮、越金属感。”

Definition 34.2 (Bessel Function of the First Kind)

The Bessel function of the first kind of order $n$ is defined by the integral

$J_n(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{i(x\sin\theta - n\theta)}\,d\theta, \qquad n \in \mathbb{Z},\; x \in \mathbb{R}.$

Key symmetry: $J_{-n}(x) = (-1)^n J_n(x)$ .

Selected values at small argument: $J_0(0)=1$ , $J_n(0)=0$ for $n\neq 0$ . For large $n$ with $n > x$ , $J_n(x) \approx 0$ (the function is exponentially small beyond the “turning point”), which is why only finitely many sidebands are audibly significant at any finite modulation index.

Definition 34.3 (FM Sideband)

For an FM signal with carrier

\omega_c

and modulator

\omega_m

, the n-th order sideband is the spectral component at angular frequency

\omega_c + n\omega_m

, for

n \in \mathbb{Z}

. Positive-

n

sidebands lie above the carrier; negative-

n

sidebands lie below. The

n=0

component is the carrier itself (modified in amplitude). The amplitude of the

n

-th sideband is

J_n(I)

Definition 34.4 (Carson Bandwidth)

The Carson bandwidth of an FM signal with modulation index $I$ and modulator frequency $f_m = \omega_m/(2\pi)$ is

$B \approx 2(I + 1)\,f_m.$

Carson’s rule captures the empirical observation that sidebands beyond order $\lfloor I\rfloor + 1$ have negligible energy ( $|J_n(I)| < 0.01$ for $n > I + 1$ approximately), so the occupied bandwidth is approximately $2(I+1)$ sideband spacings.

Definition 34.5 (DX7 Operator and Algorithm)

A DX7 operator is an FM oscillator with controllable frequency, amplitude envelope, and feedback amount. The Yamaha DX7 contains six operators. An algorithm is a directed acyclic graph (with optional self-loops on one designated feedback operator) on the six operators, specifying which operators modulate which. The DX7 provides 32 preset algorithms. In graph terms:

Carriers are operators with no outgoing modulation edges (they feed the audio output directly).
Modulators are operators whose output feeds into a carrier’s or another modulator’s phase input.
Algorithm 1 is a fully serial chain: OP1 → OP2 → OP3 → OP4 → OP5 → OP6 → output.
Algorithm 32 is fully parallel: six independent carriers summed to output (additive synthesis).

Main Theorems / 主要定理

Theorem 34.1 (Jacobi-Anger FM Expansion)

Let $y(t) = \sin(\omega_c t + I\sin(\omega_m t))$ . Then

$y(t) = \sum_{n=-\infty}^{\infty} J_n(I)\,\sin\!\bigl((\omega_c + n\omega_m)t\bigr),$

where $J_n$ is the Bessel function of the first kind of order $n$ , and the series converges absolutely and uniformly in $t$ .

Proof.

Strategy: rewrite the FM signal using the complex exponential, apply the Jacobi-Anger expansion, then recover the sine series by taking the imaginary part.

Step 1 — Complex form. Write

$\sin(\omega_c t + I\sin(\omega_m t)) = \operatorname{Im}\!\left[e^{i(\omega_c t + I\sin(\omega_m t))}\right] = \operatorname{Im}\!\left[e^{i\omega_c t}\cdot e^{iI\sin(\omega_m t)}\right].$

Step 2 — Jacobi-Anger identity. The classical Jacobi-Anger identity states that for any real $x$ and $\theta$ ,

$e^{ix\sin\theta} = \sum_{n=-\infty}^{\infty} J_n(x)\,e^{in\theta}.$

This follows from the generating-function definition of $J_n$ : $e^{x(z - 1/z)/2} = \sum_{n=-\infty}^\infty J_n(x)z^n$ evaluated at $z = e^{i\theta}$ .

Setting $x = I$ and $\theta = \omega_m t$ :

$e^{iI\sin(\omega_m t)} = \sum_{n=-\infty}^{\infty} J_n(I)\,e^{in\omega_m t}.$

Step 3 — Combine and extract imaginary part.

$e^{i\omega_c t}\cdot e^{iI\sin(\omega_m t)} = \sum_{n=-\infty}^{\infty} J_n(I)\,e^{i(\omega_c + n\omega_m)t}.$

Taking the imaginary part,

$y(t) = \sum_{n=-\infty}^{\infty} J_n(I)\,\sin\!\bigl((\omega_c + n\omega_m)t\bigr).$

Convergence. The Bessel functions satisfy $\sum_{n=-\infty}^\infty J_n(I)^2 = 1$ (see Theorem 34.2), so $\sum_n |J_n(I)| < \infty$ by the Cauchy-Schwarz inequality applied to the sequence $(|J_n(I)|\cdot 1)$ ; absolute convergence of the series follows, and uniformity in $t$ follows from the uniform bound $|\sin((\omega_c + n\omega_m)t)| \leq 1$ . $\square$

中文: “Jn of I就是第n阶第一类Bessel函数，参数是调制指数I。第零阶J0表示载波本身的幅度。第一阶J1表示距载波加减omega-m处的边带幅度。第二阶J2表示距载波加减2 omega-m处的边带幅度，以此类推。”

Theorem 34.2 (Bessel Energy Conservation (Parseval for FM))

For any modulation index $I \geq 0$ ,

$\sum_{n=-\infty}^{\infty} J_n(I)^2 = 1.$

In particular, the total sideband power of an FM signal with unit amplitude equals the carrier power of an unmodulated sinusoid. FM synthesis redistributes energy across sidebands but does not create or destroy it.

Proof.

Strategy: Parseval’s theorem applied to the $2\pi$ -periodic function $\theta \mapsto e^{iI\sin\theta}$ .

Step 1. From the Jacobi-Anger identity, the Fourier coefficients of $f(\theta) = e^{iI\sin\theta}$ are

$\hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{iI\sin\theta} e^{-in\theta}\,d\theta = J_n(I).$

Step 2. By Parseval’s identity for $L^2(-\pi,\pi)$ ,

$\sum_{n=-\infty}^{\infty} |J_n(I)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(\theta)|^2\,d\theta.$

Step 3. Since $|f(\theta)| = |e^{iI\sin\theta}| = 1$ for all $\theta$ ,

$\frac{1}{2\pi}\int_{-\pi}^{\pi} 1\,d\theta = 1.$

Since $J_n(I)$ is real, $|J_n(I)|^2 = J_n(I)^2$ , giving the stated identity. $\square$

中文: “当I很小，只有J0接近1，其他接近零——几乎纯正弦。随着I增大，能量从J0流向J1、J2、J3。不同阶的Bessel函数在不同I值处到达峰值，有的甚至变号——对应边带相位翻转。这就是调制指数控制音色的数学机制：I是在Bessel函数族中分配能量的旋钮。”

Prop 34.1 (Sideband Symmetry)

For an FM signal with real modulation index $I$ , the $n$ -th and $(-n)$ -th sidebands satisfy $J_{-n}(I) = (-1)^n J_n(I)$ . Therefore:

Odd-order negative sidebands are phase-inverted relative to their positive counterparts.
Even-order negative sidebands are in phase with their positive counterparts.
The amplitude spectrum (plotting $|J_n(I)|$ ) is symmetric about $n=0$ .

Proof.

Substitute $n \to -n$ in the integral definition:

$J_{-n}(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(x\sin\theta + n\theta)}\,d\theta.$

Substitute $\theta \to -\theta$ (changing sign on both the argument and the $d\theta$ measure):

$= \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(-x\sin\theta + n\theta)}\,d\theta = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(x\sin\theta - n\theta)\cdot(-1)+i(x\sin\theta - n\theta+x\sin\theta - n\theta)}\,d\theta.$

More directly: since $\sin(-\theta)=-\sin\theta$ ,

$J_{-n}(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(-x\sin\phi - n\phi)}(-d\phi) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i(x\sin\phi + n\phi)}\,d\phi.$

Using the identity $e^{-i(x\sin\phi)} = e^{i(x\sin\phi)} \cdot e^{-2ix\sin\phi}$ is more complex; instead use the generating function directly. From $e^{x(z-1/z)/2} = \sum_n J_n(x)z^n$ , replace $z \to 1/z$ :

$e^{x(1/z - z)/2} = \sum_n J_n(x) z^{-n} = \sum_n J_{-n}(x)z^n.$

But $e^{x(1/z-z)/2} = e^{-x(z-1/z)/2} = \sum_n J_n(x)(-1)^n z^n$ (substitute $x \to -x$ in the generating function, noting $J_n(-x)=(-1)^n J_n(x)$ by the same argument applied to the integral with $\phi \to \phi + \pi$ ). Equating coefficients of $z^n$ gives $J_{-n}(x) = (-1)^n J_n(x)$ . $\square$

Theorem 34.3 (Carson's Rule for FM Bandwidth)

Let $\epsilon > 0$ be a small threshold (typically 1% of total power). Define the essential bandwidth $B_\epsilon$ to be the smallest bandwidth containing all sidebands with $|J_n(I)| \geq \epsilon$ . Then

$B_\epsilon \;\approx\; 2(I+1)\,f_m$

for the conventional choice $\epsilon \approx 0.01$ , and this approximation is tight for $I \geq 1$ .

Proof.

Strategy: bound the index $N$ beyond which $|J_n(I)| < \epsilon$ using the asymptotic behavior of Bessel functions.

Step 1 — Upper bound for large order. For integer $n > 0$ and $n > I$ , the Bessel function satisfies

$\left|J_n(I)\right| \leq \frac{1}{n!}\left(\frac{I}{2}\right)^n.$

This follows from the series representation $J_n(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!\,(n+k)!}\left(\frac{x}{2}\right)^{2k+n}$ ; bounding by the first term gives $|J_n(x)| \leq \frac{(x/2)^n}{n!}$ .

Step 2 — When does the bound drop below $\epsilon$ ? We need $\frac{(I/2)^n}{n!} < \epsilon$ . By Stirling’s approximation, $n!$ grows faster than $(I/2)^n$ for $n > I/2 \cdot e \approx 1.36I$ . For the 1% threshold, numerical evaluation of Bessel zeros shows that the last significant sideband occurs at approximately $n \approx \lfloor I\rfloor + 1$ , independent of $\epsilon$ over a wide range.

Step 3 — Bandwidth. With at most $N \approx I + 1$ significant sidebands on each side of the carrier, the total occupied bandwidth (distance from lowest to highest significant sideband) is

$B \approx 2(I+1)f_m.$

This is Carson’s rule. The approximation $\approx$ reflects that the exact cutoff order depends weakly on $I$ and $\epsilon$ ; the formula is exact for $I = 0$ (bandwidth = $2f_m$ , the two first-order sidebands) and becomes increasingly accurate as $I$ grows. $\square$

Theorem 34.4 (Feedback FM Fixed-Point Equation)

For the simplified feedback FM model

$y = \sin(\omega t + I \cdot y),$

the equation has a unique solution $y^*(t)$ for each $t$ when $I < 1$ , and this solution can be obtained as the limit of the fixed-point iteration $y_{k+1} = \sin(\omega t + I\cdot y_k)$ . For $I \geq 1$ , the fixed-point mapping need not be a contraction and the iteration may diverge or cycle.

Proof.

Step 1 — Contraction mapping. Define $g(y) = \sin(\omega t + I\cdot y)$ . Then

$|g(y_1) - g(y_2)| = |\sin(\omega t + Iy_1) - \sin(\omega t + Iy_2)| \leq |I|\,|y_1 - y_2|,$

where we used $|\sin a - \sin b| \leq |a - b|$ (mean value theorem, since $|\cos\xi| \leq 1$ ).

Step 2 — Banach fixed-point theorem. When $|I| < 1$ , the Lipschitz constant of $g$ is $|I| < 1$ , so $g$ is a contraction on $\mathbb{R}$ (restricted to the bounded range $[-1,1]$ , which is invariant). By the Banach contraction mapping theorem, there exists a unique fixed point $y^*$ and the iteration $y_{k+1} = g(y_k)$ converges to it for any starting value $y_0 \in [-1,1]$ .

Step 3 — Failure for $I \geq 1$ . When $I = 1$ , the Lipschitz constant equals 1 and the contraction property fails. For $I > 1$ , $g$ is an expansion near the fixed point: a small perturbation grows under iteration, destabilizing the periodic solution. In discrete implementations (such as the DX7’s one-sample delay feedback), this transition corresponds to the onset of a rich spectrum that empirically resembles a sawtooth wave for moderate $I$ and noise-like instability for large $I$ . $\square$

中文: “当I很小时，反馈量微弱，输出接近纯正弦。I稍大，波形开始向锯齿波倾斜，产生奇次和偶次谐波。I继续增大到某个临界值，系统产生类噪声的不稳定频谱——更接近工程意义上的不稳定，而非严格数学上的分岔混沌。”

Numerical Examples

Example 1 — Spectral Profile at I = 1 (Electric Piano Region)

Compute $J_n(1)$ for $n = 0, 1, 2, 3$ :

Order $n$	$J_n(1)$	Sideband frequency	Power fraction
0	0.7652	$\omega_c$ (carrier)	58.6%
±1	0.4401	$\omega_c \pm \omega_m$	2 × 19.4% = 38.7%
±2	0.1149	$\omega_c \pm 2\omega_m$	2 × 1.3% = 2.6%
±3	0.0196	$\omega_c \pm 3\omega_m$	negligible

Check: $0.7652^2 + 2(0.4401^2) + 2(0.1149^2) + \cdots \approx 0.586 + 0.387 + 0.026 \approx 0.999 \approx 1$ . Energy conservation holds. Carson bandwidth: $B = 2(1+1)f_m = 4f_m$ .

Example 2 — Spectral Profile at I = 5 (Brass Region)

At $I = 5$ , Carson bandwidth is $B = 2(5+1)f_m = 12f_m$ . The Bessel values show energy dispersed across orders 0–6:

Order $n$	$J_n(5)$	Timbre contribution
0	−0.1776	carrier (phase-inverted, reduced)
1	−0.3276	1st sideband
2	0.0466	2nd sideband
3	0.3648	3rd sideband (dominant)
4	0.3912	4th sideband (dominant)
5	0.2774	5th sideband
6	0.1310	6th sideband

Note that $J_0(5) < 0$ : the carrier component is phase-inverted. This “hollowing out” of the carrier and redistribution into high-order sidebands is what gives brass and bell timbres their inharmonic, cutting character.

Example 3 — DX7 Algorithm Complexity

With 6 operators and carrier $\omega_c$ modulated by a chain of depth $k$ , the instantaneous phase involves iterated FM at each level. A depth-2 chain (OP1 → OP2 → carrier output) produces:

y = \sin\!\bigl(\omega_c t + I_1 \sin(\omega_1 t + I_2 \sin(\omega_2 t))\bigr),

which expands to a double Bessel series $\sum_{n,m} J_n(I_1)J_m(I_2)\sin((\omega_c + n\omega_1 + nm\omega_2)t)$ . The number of significant partials grows multiplicatively, explaining why Algorithm 1 (depth-6 serial chain) produces extremely dense spectra from just six oscillators.

Musical Connection / 音乐联系

音乐联系

FM synthesis vs. subtractive synthesis exemplify two dual philosophies of timbre construction. In subtractive synthesis (

EP30

), one begins with a harmonically rich waveform (sawtooth, square) and removes partials with filters — a sculptor chipping away stone. FM synthesis does the opposite: a pure sine carrier gains sidebands as $I$ increases, building complexity from simplicity.

The DX7’s timbral vocabulary maps directly onto the Bessel function family:

Timbre	Typical $I$ range	Dominant Bessel orders
Sine / flute	0–0.5	$J_0$ only
Electric piano (Rhodes)	0.5–2	$J_0, J_1$
Clarinet / oboe	2–4	$J_0$ through $J_3$
Brass / horn attack	3–7	$J_2$ through $J_5$
Bell / gong / metal	5–12	spread across $J_0$ to $J_8$

Cross-episode thread. In

EP19 (Vibrato)

, we studied vibrato as FM synthesis with $I \approx 0.1$ and $\omega_m \approx 5$ Hz — a rate far below the audible spectrum, producing pitch fluctuation rather than timbre change. The DX7 uses exactly the same equation with $\omega_m$ in the audio range (50 Hz–several kHz) and $I$ up to 14. The mathematical structure is identical; only the parameter regime changes the perceptual result.

Feedback and edge-of-chaos timbres. The operator self-modulation in Theorem 34.4 connects to the broader theme of nonlinear dynamics from

EP30

. In the DX7, this is implemented with a one-sample-delayed feedback loop (with averaging to suppress instability), so the exact analytical model is an approximation — but the qualitative behavior matches: small $I$ gives pure tone, moderate $I$ gives sawtooth-like richness, large $I$ gives noise-like spectra used for breath, cymbal, and wind sounds.

Computational efficiency. FM synthesis requires only sine evaluations and a few multiplications per sample — far less than the filter banks needed for subtractive synthesis or the physical modeling waveguides of digital waveguide synthesis. This is why the DX7 could produce a six-operator polyphonic synthesizer in 1983 hardware: the Bessel sideband structure is “free” — it emerges automatically from the phase equation without any explicit sideband computation.

Limits and Open Questions / 局限性与开放问题

Carson’s rule is an approximation. The threshold $\epsilon$ at which a sideband is declared “negligible” is arbitrary; the true essential bandwidth depends on $I$ , $\epsilon$ , and the application context (broadcasting, audio synthesis, FM radio all use different standards). No sharp, parameter-independent bandwidth formula exists.
Feedback FM is not mathematically chaotic. The DX7’s self-modulation feedback uses a one-sample delay and averaging circuit, making it a discrete-time nonlinear map rather than the continuous-time system in Theorem 34.4. Whether the discrete system exhibits true period-doubling bifurcations in the sense of EP30 (versus mere spectral broadening from aliasing and quantization) remains an open engineering question.
Perceptual inversion problem. Given a target timbre (e.g., a recorded piano note), finding the FM parameters $(I, \omega_c, \omega_m, \text{algorithm})$ that best approximate it is an inverse problem with no closed-form solution. DX7 programmers famously found this “black magic” difficulty the instrument’s main usability barrier. Modern approaches use gradient descent on the Bessel series, but convergence to perceptually meaningful local minima is not guaranteed.
Inharmonicity and real instruments. The FM sideband model places sidebands at exact multiples $\omega_c \pm n\omega_m$ of the modulator frequency. Real instruments (pianos, bells) have inharmonic partials due to stiffness and geometry. The DX7 approximates these by choosing $\omega_c / \omega_m$ as a non-integer ratio, but this still produces exact rational relationships between partials, not the true irrational spacing of stiff-string inharmonicity.

Conjecture (Optimal FM Approximation)

For any target power spectrum

\{a_k^2\}_{k=1}^K

supported on

K

harmonics, the best two-operator FM approximation (minimizing mean-squared spectral error) satisfies

\omega_c/\omega_m \in \mathbb{Q}

and

I \leq 2K

. Falsification criteria: exhibit a target spectrum whose optimal FM approximation requires

I > 2K

, or show that the optimal solution uses

\omega_c/\omega_m \notin \mathbb{Q}

by constructing a counter-example via numerical optimization.

Academic References / 参考文献

Chowning, J. M. (1973). “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation.” Journal of the Audio Engineering Society, 21(7), 526–534. (The original paper introducing FM synthesis.)
Chowning, J. M., & Bristow, D. (1986). FM Theory and Applications: By Musicians for Musicians. Yamaha Music Foundation, Tokyo.
Abramowitz, M., & Stegun, I. A. (Eds.) (1964). Handbook of Mathematical Functions. National Bureau of Standards. Chapter 9: Bessel Functions.
Watson, G. N. (1922). A Treatise on the Theory of Bessel Functions. Cambridge University Press. (Comprehensive reference for Jacobi-Anger expansion and Bessel identities.)
Carson, J. R. (1922). “Notes on the Theory of Modulation.” Proceedings of the IRE, 10(1), 57–64. (Original statement of Carson’s bandwidth rule for FM radio.)
Oppenheim, A. V., & Schafer, R. W. (2010). Discrete-Time Signal Processing (3rd ed.). Pearson. Chapter 8: Frequency Analysis of Signals and Systems.
Roads, C. (1996). The Computer Music Tutorial. MIT Press. Chapter 6: Frequency Modulation Synthesis, pp. 224–289.
Schottstaedt, W. (1977). “The Simulation of Natural Instrument Tones Using Frequency Modulation with a Complex Modulating Wave.” Computer Music Journal, 1(4), 46–50.
Dodge, C., & Jerse, T. A. (1997). Computer Music: Synthesis, Composition, and Performance (2nd ed.). Schirmer. Chapter 4: Frequency Modulation.
Steiglitz, K. (1996). A Digital Signal Processing Primer. Addison-Wesley. Chapter 9: Modulation and Demodulation.