Analog Modulation: AM, FM & PM

Modulation is the act of writing a slow, low-frequency message onto a fast carrier. A sinusoid has exactly three things you can vary — amplitude, frequency, phase — and analog modulation is the exhaustive enumeration of that list. Amplitude gives you AM and its bandwidth-saving relatives; frequency and phase turn out to be the same scheme seen from two sides of an integral, and together they buy noise immunity that AM cannot buy at any power level.

The two families fail differently, and that is the useful thing to carry away. AM degrades gracefully and linearly: noise on the channel appears as noise in the output, proportionally, all the way down. FM is dramatically better than AM above a carrier-to-noise ratio of roughly 10 dB and dramatically worse below it — there is a cliff, and the wider you make the FM signal to buy quieting, the higher up the cliff sits.

Why modulate at all #

Three independent reasons, any one of which would be sufficient.

Antennas are sized in wavelengths. A radiator only couples efficiently to free space when its dimensions are a meaningful fraction of $\lambda = c/f$; the quarter-wave monopole is the canonical yardstick. Run the numbers on a 3 kHz voice baseband and the argument ends immediately:

The 75 cm whip on a car is a quarter wave at 100 MHz. The 25 km version does not exist. Modulation is what lets a 3 kHz signal be radiated by an object a person can carry.

Spectrum is shared, and baseband is not shareable. Two stations both transmitting 0–15 kHz on the same medium are inseparable at the receiver, because no filter can distinguish them — they occupy identical frequencies. Translating each one to a distinct carrier makes them separable by a bandpass filter, which is precisely what multiplexing is. US AM broadcast stations sit on 10 kHz steps from 540 to 1700 kHz; ITU Regions 1 and 3 pack them on 9 kHz steps. FM broadcast channels are 200 kHz wide. Those grids exist because the modulator moved each station somewhere unique.

Where you put the signal decides what noise it meets. Atmospheric and man-made noise is heavily concentrated at low frequencies, and the countermeasures split by what they exploit — bandwidth, a correlated reference, or averaging — as the KPI signals course lays them out; propagation, antenna size and available bandwidth all change by orders of magnitude across the spectrum. Choosing a carrier is choosing a noise environment, a propagation mode, and a bandwidth budget in one decision.

flowchart LR
  M["baseband m(t)"] --> A["vary amplitude"]
  M --> G["vary angle"]
  A --> AM["AM: carrier plus 2 sidebands"]
  A --> DSB["DSB-SC: 2 sidebands"]
  A --> SSB["SSB: 1 sideband"]
  A --> VSB["VSB: 1 sideband plus vestige"]
  G --> PM["PM: phase follows m(t)"]
  G --> FM["FM: phase follows integral of m(t)"]

AM: the carrier that carries nothing #

Full-carrier amplitude modulation writes the message into the envelope of the carrier:

$$s(t) = A_c\,\bigl[\,1 + m\,x(t)\,\bigr]\cos(2\pi f_c t), \qquad |x(t)| \le 1$$

$m$ is the modulation index. Measured off an oscilloscope trace of the envelope,

$$m = \frac{A_{\max} - A_{\min}}{A_{\max} + A_{\min}}$$

Sideband structure #

Take a single tone $x(t) = \cos(2\pi f_m t)$ and expand the product:

$$s(t) = \underbrace{A_c\cos 2\pi f_c t}_{\text{carrier}} + \underbrace{\frac{mA_c}{2}\cos 2\pi (f_c - f_m) t}_{\text{LSB}} + \underbrace{\frac{mA_c}{2}\cos 2\pi (f_c + f_m) t}_{\text{USB}}$$

Three lines, always. A general message of bandwidth $W$ gives the carrier plus a mirrored pair of sidebands, so the occupied bandwidth is

$$B_{\text{AM}} = 2W$$

That factor of two is why a 10 kHz AM broadcast channel caps the audio at about 5 kHz — the entire "AM sounds muffled" experience is a direct consequence of double-sideband transmission inside a narrow allocation.

The power split, and the 33% ceiling #

Into a unit resistance, the mean power of each component is half the square of its amplitude:

$$P_c = \frac{A_c^2}{2}, \qquad P_{\text{LSB}} = P_{\text{USB}} = \frac{1}{2}\left(\frac{mA_c}{2}\right)^{\!2} = \frac{m^2 A_c^2}{8}$$

$$P_t = P_c\left(1 + \frac{m^2}{2}\right)$$

Only the sidebands depend on $x(t)$, so only the sidebands carry information. The modulation efficiency is therefore

$$\eta = \frac{P_{\text{LSB}} + P_{\text{USB}}}{P_t} = \frac{m^2 A_c^2/4}{(A_c^2/4)(2+m^2)} = \boxed{\;\frac{m^2}{2+m^2}\;}$$

$\eta$ increases monotonically in $m$, and $m$ is capped at 1 by the distortion argument above. So:

$$\eta_{\max} = \eta\big|_{m=1} = \frac{1}{3} = 33.3\%$$

Envelope versus coherent detection #

That trade is the whole point of leaving the carrier in. With $m \le 1$ the envelope of $s(t)$ is $A_c[1 + m x(t)]$, so a diode followed by an RC low-pass recovers the message with no local oscillator, no phase reference and no frequency reference. A crystal set needs no power supply at all.

Coherent (synchronous) detection multiplies by a locally generated $\cos(2\pi f_c t + \phi)$ and low-passes. It works for every AM variant, needs a carrier-phase estimate, and is what you must build the moment you suppress the carrier.

Suppressing what you do not need #

Three refinements, each removing something the receiver could in principle reconstruct.

DSB-SC drops the carrier: $s(t) = A_c\,m(t)\cos 2\pi f_c t$. Now 100% of the power is in the sidebands, and $\eta = 1$. The bandwidth is unchanged at $2W$. The cost is that the envelope no longer tracks $m(t)$ — it tracks $|m(t)|$, and the carrier phase reverses every time $m(t)$ crosses zero — so envelope detection is out and you need a carrier-recovery loop (a Costas loop, or a squaring loop, or a transmitted pilot). A phase error $\phi$ in that loop attenuates the output by $\cos\phi$; a frequency error produces a beat that walks the output amplitude through zero.

FM stereo is DSB-SC hiding in plain sight: the L−R difference signal double-sideband-suppressed-carrier modulates a 38 kHz subcarrier, and a 19 kHz pilot — the subcarrier's exact half — is transmitted at 8–10% modulation so the receiver can regenerate the suppressed 38 kHz reference with the right phase.

SSB additionally discards one sideband. The two sidebands of a real message are mirror images and carry identical information, so throwing one away halves the bandwidth to $W$ with no information loss. Generation is either filtering (build the DSB-SC signal, then cut one side with a very steep filter — which needs a gap around DC in the message, fine for voice starting at 300 Hz) or phasing (a Hilbert-transform quadrature network, no steep filter required). SSB is why HF voice, both amateur and marine, sounds the way it does: it is the only way to fit useful traffic into a crowded 3 kHz-per-channel band.

VSB is the engineering compromise for messages that do have significant content near DC, where SSB's sharp filter is unbuildable. Transmit one sideband fully, plus a shaped remnant of the other, with the filter skirts arranged so the two overlapping halves add back to a flat response at the detector. Analog television is the canonical user: ITU-R System M puts a 4.2 MHz main sideband and a 0.75 MHz vestigial sideband inside a 6 MHz channel — nowhere near the 8.4 MHz that honest double-sideband would need, and nowhere near buildable as true SSB given video's DC-coupled brightness content.

Angle modulation: FM and PM are one mechanism #

Write any constant-envelope carrier as $s(t) = A_c\cos\theta(t)$ and define the instantaneous frequency as the derivative of the total phase:

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

Then the two schemes differ in exactly one place — whether the message enters the phase directly or through an integral:

$$\theta_{\text{PM}}(t) = 2\pi f_c t + k_p\, m(t)$$

$$\theta_{\text{FM}}(t) = 2\pi f_c t + 2\pi k_f\!\int_0^t\! m(\tau)\,d\tau$$

so $f_{i,\text{FM}}(t) = f_c + k_f\,m(t)$: the instantaneous frequency, not the phase, follows the message. The consequence is a hardware identity that Armstrong built his indirect FM transmitter on — an FM modulator is an integrator followed by a phase modulator, and a PM modulator is a differentiator followed by a frequency modulator:

flowchart LR
  m1["m(t)"] --> i1["integrate"]
  i1 --> pm1["phase modulator"]
  pm1 --> o1["FM output"]
  m2["m(t)"] --> d2["differentiate"]
  d2 --> fm2["frequency modulator"]
  fm2 --> o2["PM output"]

This matters because a crystal-stable phase modulator with a tiny deviation is easy to build and a stable wide-deviation direct FM oscillator is not; Armstrong's transmitter made narrowband PM and then multiplied the deviation up.

Deviation and index #

For a single tone $m(t) = A_m\cos 2\pi f_m t$:

$$\Delta f = k_f A_m \quad\text{(peak frequency deviation, Hz)}, \qquad \beta_{\text{FM}} = \frac{\Delta f}{f_m}$$

$$s_{\text{FM}}(t) = A_c\cos\!\bigl(2\pi f_c t + \beta\sin 2\pi f_m t\bigr)$$

$\beta$ is the peak phase excursion in radians, which makes the FM/PM difference sharp:

FM broadcast is defined at $\Delta f = \pm 75$ kHz for 100% modulation with a 15 kHz audio ceiling, so $\beta = 5$ at the top of the audio band and $\beta = 375$ at 200 Hz. Land-mobile FM, narrowbanded to 12.5 kHz channels, lives near $\beta \approx 1$ — deep enough to get the limiter's benefit, shallow enough to fit.

The spectrum is Bessel, the bandwidth is Carson's #

Angle modulation is nonlinear in the message: you cannot superpose spectra, and the sideband structure of a single tone is already infinite. The Jacobi–Anger expansion gives it exactly:

$$s_{\text{FM}}(t) = A_c\sum_{n=-\infty}^{\infty} J_n(\beta)\,\cos\bigl(2\pi (f_c + n f_m)\,t\bigr)$$

Sidebands at every multiple of $f_m$, with amplitude $J_n(\beta)$ — the $n$-th order Bessel function of the first kind. Since $\sum_n J_n^2(\beta) = 1$, total power is $A_c^2/2$ regardless of $\beta$: modulating an FM transmitter harder does not change its power, it only redistributes it outward. This is the deepest structural difference from AM, where modulation adds power.

Values computed from $J_n(\beta)$; entries below $10^{-4}$ shown as —. Note the sign changes: sidebands are not merely attenuated copies, and $J_0$ goes negative between its zeros.

Carrier nulls #

$J_0(\beta) = 0$ at $\beta = 2.40483,\ 5.52008,\ 8.65373,\ 11.79153,\ 14.93092,\ \dots$ — the zeros of the zeroth-order Bessel function. At those exact indices the carrier line disappears entirely and all the power sits in the sidebands.

Carson's rule, stated precisely #

The spectrum is infinite, so "the bandwidth" must be a truncation rule. Carson's, from his 1922 paper:

$$\boxed{\;B_T = 2\bigl(\Delta f + W\bigr) = 2\Delta f\left(1 + \frac{1}{\beta}\right) = 2W\,(\beta + 1)\;}$$

where $\Delta f$ is the peak frequency deviation and $W$ the highest message frequency. The claim attached to it is quantitative: approximately 98% of the modulated signal's power falls inside $B_T$. Verifying that against the Bessel expansion — summing $J_0^2 + 2\sum_{n=1}^{\lfloor\beta+1\rfloor} J_n^2$ — gives:

The rule holds — the worst case in this range is 98.2% (at $\beta = 5.52$) — but notice the last column. Carson's bandwidth is consistently narrower than the span of sidebands that are individually still visible on an analyser. At $\beta = 5$, Carson says 6 sideband pairs and 8 pairs exceed 1% of the unmodulated carrier.

FM broadcast checks out exactly: $B_T = 2(75 + 15) = 180$ kHz inside a 200 kHz channel, leaving 10 kHz of guard on each side.

Where FM's noise advantage comes from #

Two distinct mechanisms, and they compound.

The limiter throws amplitude away. An FM signal has constant envelope by construction, so any amplitude variation at the receiver is known to be noise or fading. A hard limiter ahead of the discriminator clips the signal to a constant amplitude and discards it. Half of the noise — the in-phase component — is simply deleted. AM cannot do this, because in AM the amplitude is the message. This is also why FM tolerates nonlinear (saturated, efficient) power amplifiers while AM demands linear ones.

Differentiating phase shapes the residual noise. The discriminator recovers $d\theta/dt$, and differentiation multiplies the noise spectrum by $f^2$. The output noise power spectral density is therefore parabolic, rising toward the top of the baseband, while the signal is spread across the band by the modulator with no such tilt. Wider deviation spreads the signal further without changing the noise floor, so output SNR climbs with $\beta$.

Comparing schemes needs a common reference. Take the figure of merit $\gamma$ — output SNR divided by the SNR a baseband system would achieve with the same received power $P_R$ and the same message bandwidth, $P_R/(N_0 W)$:

For single-tone modulation the FM entry is $\gamma_{\text{FM}} = \tfrac{3}{2}\beta^2$, and the AM entry is $\gamma_{\text{AM}} = m^2/(2+m^2)$ — numerically the same expression as the modulation efficiency, which is not a coincidence: the carrier contributes power and no information, so it degrades output SNR by exactly the fraction of power it wastes. Referred instead to the carrier-to-noise ratio measured in the IF passband $B_T = 2(\beta+1)W$, the FM result reads

$$\frac{\text{SNR}_{\text{out}}}{\text{CNR}} = 3\beta^2(\beta + 1)$$

which at $\beta = 5$ is a factor of 450, or 26.5 dB of demodulation gain.

Pre-emphasis and de-emphasis #

The parabolic output noise means the top octave of the audio band arrives far noisier than the bottom, while typical programme material has less energy up there — a double loss. The fix is to boost high frequencies before modulating and cut them by the exact inverse after demodulating, so the message comes out flat and the noise comes out tilted down.

The networks are single-pole and specified by a time constant $\tau$, with the corner at $f_0 = 1/(2\pi\tau)$:

The improvement follows from integrating the parabolic noise density with and without the de-emphasis response $|H(f)|^2 = 1/(1 + (f/f_0)^2)$. With $r = W/f_0$:

$$I = \frac{\displaystyle\int_{-W}^{W} f^2\,df}{\displaystyle\int_{-W}^{W} \frac{f^2}{1 + (f/f_0)^2}\,df} = \frac{r^3}{3\bigl(r - \arctan r\bigr)}$$

For broadcast FM, $W = 15$ kHz:

13 dB from two resistors and two capacitors, which is why the technique survived every generation of receiver electronics.

Two cliffs FM falls off #

Capture effect. Because the limiter tracks zero crossings and zero crossings are dominated by whichever signal is larger, an FM receiver presented with two co-channel signals will lock to the stronger one and suppress the weaker almost entirely. The capture ratio — the power ratio at which the receiver cleanly rejects the weaker signal — is a published tuner specification and is typically a few dB. That is qualitatively different from AM, where two co-channel signals simply sum and you hear both plus a heterodyne whistle.

Capture is a feature for broadcast (you get one clean station rather than two muddy ones) and a hazard for shared-channel operation, which is exactly why the VHF airband is still AM: if two aircraft transmit at once, a controller hears both plus a heterodyne squeal and knows to ask for a repeat, whereas FM would silently deliver only the stronger one and lose the other transmission entirely. Aviation kept AM through the move to 8.33 kHz channel spacing for that reason among others.

Threshold effect. All of the SNR analysis above assumes the noise is small enough that it merely perturbs the carrier's phase. Once noise amplitude occasionally exceeds carrier amplitude, the resultant phasor encircles the origin and the discriminator emits a $2\pi$ phase jump — an impulse, heard as a click. Below the threshold, click rate rises rapidly and output SNR collapses far faster than input CNR falls.

The comparison, end to end #

PM is the odd one out in deployment: as a standalone analog scheme it is rare, because its bandwidth grows with modulating frequency and it needs an absolute phase reference at the receiver. Its descendants are everywhere — every PSK constellation in digital modulation is phase modulation with a finite symbol alphabet, which removes the phase-reference problem by making phase discrete and recoverable.

Real allocations, with the numbers #

Sharp edges #