Carson’s Rule

 

Carson’s Rule

For an FM signal, the approximate bandwidth is

$$\boxed{BW \approx 2(\Delta f+f_m)}$$

where:

• $\Delta f$ = maximum frequency deviation
• $f_m$ = highest frequency in the modulating/message signal

It gives a practical estimate of the bandwidth containing roughly 98% of the FM signal power. 

Where does the derivative come in?

An FM signal can be written as

$$s(t)=A_c\cos\left(2\pi f_ct+\phi(t)\right)$$

The instantaneous phase is

$$\theta(t)=2\pi f_ct+\phi(t)$$

and instantaneous frequency is obtained by differentiating phase:

$$\boxed{f_i(t)=\frac{1}{2\pi}\frac{d\theta(t)}{dt}}$$

Simple example

Suppose an FM transmitter has:

$$\Delta f=75\,kHz$$

and the highest modulating frequency is

$$f_m=15\,kHz.$$

Then

$$BW\approx2(75+15)$$

$$\boxed{BW\approx180\,kHz}$$

So you'd allocate roughly 180 kHz of channel bandwidth.

Practical uses

Carson's Rule is useful when designing or checking:

• FM radio channels — determining how much spectrum an FM station needs.
• Wireless communication systems — estimating whether an FM signal will fit within an allocated channel.
• Transmitters/receivers — choosing appropriate RF filters and intermediate-frequency bandwidths.
• Spectrum planning — preventing neighboring FM channels from interfering with each other.
• Exam problems — quickly calculating FM bandwidth without calculating every individual sideband.

The mathematical foundation comes from FM phase modulation + Fourier series/Bessel functions.

1. Start with an FM signal

For a sinusoidal message,

$$m(t)=A_m\cos(2\pi f_m t)$$

the FM signal is

$$s(t)=A_c\cos\left(2\pi f_ct+\beta\sin(2\pi f_mt)\right)$$

where

$$\boxed{\beta=\frac{\Delta f}{f_m}}$$

is the FM modulation index.

So the entire problem becomes:

How much spectrum does
$\cos(\omega_ct+\beta\sin\omega_mt)$ occupy?

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2. The surprising part: FM creates infinitely many sidebands

Using the Jacobi–Anger/Bessel expansion,

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

Therefore,

$$s(t)$$

contains frequencies

$$f_c,\quad f_c\pm f_m,\quad f_c\pm2f_m,\quad f_c\pm3f_m,\ldots$$

with amplitudes determined by

$$J_0(\beta),J_1(\beta),J_2(\beta),\ldots$$

So mathematically, FM technically has infinite bandwidth.

That's the first important point.

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3. Then where does Carson's rule come from?

We need to decide:

How many of those sidebands are significant enough to keep?

The significant sidebands extend roughly to an order

$$n\sim\beta+1$$

because the Bessel coefficients $J_n(\beta)$ become very small when $n$ gets sufficiently larger than $\beta$.

Each sideband is separated from the carrier by $n f_m$.

So approximately,

$$f_{\text{max deviation}} \approx \beta f_m+f_m$$

But

$$\beta f_m=\Delta f$$

because

$$\beta=\frac{\Delta f}{f_m}.$$

Therefore,

$$f_{\text{max deviation}} \approx \Delta f+f_m.$$

The spectrum extends approximately this far on both sides of the carrier:

$$f_c-(\Delta f+f_m)$$

to

$$f_c+(\Delta f+f_m).$$

Hence

$$\boxed{BW\approx2(\Delta f+f_m)}$$

That's the mathematical origin of Carson's rule.

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4. But there's an even deeper way to see it

Remember that instantaneous frequency is

$$f_i(t)=\frac{1}{2\pi}\frac{d\theta(t)}{dt}.$$

For

$$\theta(t)=2\pi f_ct+\beta\sin(2\pi f_mt),$$

differentiate:

$$f_i(t) = f_c+\beta f_m\cos(2\pi f_mt).$$

Since

$$\beta f_m=\Delta f,$$

we get

$$\boxed{f_i(t)=f_c+\Delta f\cos(2\pi f_mt)}.$$

Thus the carrier's frequency swings between

$$f_c-\Delta f$$

and

$$f_c+\Delta f.$$

So $\Delta f$ tells us the main frequency excursion.

But the message itself changes at a rate $f_m$. That changing frequency produces the additional spectral spread of roughly $f_m$ on each side.

That's why the combination is

$$\Delta f+f_m.$$

And because there are two sides:

$$\boxed{2(\Delta f+f_m)}.$$

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5. Why isn't it exactly true?

This is important.

Carson's rule is not an exact mathematical bandwidth formula.

FM has infinitely many Bessel sidebands:

$$f_c\pm nf_m,\qquad n=1,2,3,\ldots$$

Carson's rule is an engineering approximation based on retaining the portion containing approximately 98% of the signal power.