4. Frequency Modulation: Trading Bandwidth for Noise Performance

We have only varied amplitude so far. What if we hold amplitude rigidly constant and vary the frequency of the carrier instead? The result is frequency modulation (FM), and it changes everything about the noise behavior.

4.1 Instantaneous frequency, the foundational idea

A carrier $A_c\cos(2\pi f_c t + \phi)$ has constant frequency $f_c$. Generalize: let the phase be a function of time, $\theta(t)$. The carrier becomes $A_c\cos\theta(t)$, and we define the instantaneous frequency as the rate of change of phase:

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

For a constant frequency carrier, $\theta(t) = 2\pi f_c t$ and the derivative gives back $f_c$. For FM, we want $f_i(t) = f_c + k_f m(t)$, so the instantaneous frequency tracks the message scaled by a sensitivity constant $k_f$ (in Hz/volt). Integrate to get the phase:

$\theta(t) = 2\pi f_c t + 2\pi k_f \int_0^t m(\tau)\,d\tau$

The FM signal is therefore:

$s_\text{FM}(t) = A_c\cos\!\bigg[2\pi f_c t + 2\pi k_f \int_0^t m(\tau)\,d\tau\bigg]$

Phase modulation (PM) uses the message directly as the phase deviation:

$s_\text{PM}(t) = A_c\cos\big[2\pi f_c t + k_p\,m(t)\big]$

PM and FM are duals. PM of $m(t)$ is identical to FM of $\dot{m}(t)$ (the derivative), and FM of $m(t)$ is identical to PM of $\int m(\tau)d\tau$. Practical broadcasting uses FM because audio is naturally bandlimited and integrating it does not blow up; the math is the same.

Flute-pitch-warble analogy. A flutist plays at constant volume but varies the pitch slightly to express phrasing (vibrato is exactly this: a small, periodic frequency deviation around the nominal pitch). Loudness is constant, pitch carries the expression. FM is identical: amplitude is constant, instantaneous frequency carries the message.

4.2 Single-tone FM and the modulation index

Take $m(t) = A_m\cos(2\pi f_m t)$. Then $\int m(\tau)d\tau = (A_m/2\pi f_m)\sin(2\pi f_m t)$, so:

$s(t) = A_c\cos\!\bigg[2\pi f_c t + \frac{k_f A_m}{f_m}\sin(2\pi f_m t)\bigg] = A_c\cos\big[2\pi f_c t + \beta\sin(2\pi f_m t)\big]$

where $\beta = k_f A_m / f_m = \Delta f / f_m$ is the FM modulation index, and $\Delta f = k_f A_m$ is the peak frequency deviation (the maximum amount the instantaneous frequency swings away from $f_c$ at the peak of the message).

This formula is deceptively compact. The phase deviation oscillates between $+\beta$ and $-\beta$ radians at the message rate. For broadcast FM the peak deviation is $\Delta f = 75$ kHz and the highest audio frequency is $W = 15$ kHz, giving a peak modulation index of $\beta = 5$. For narrowband FM (two-way radio, voice over an FM cellular system), $\Delta f$ is much smaller, often comparable to $f_m$, with $\beta < 1$.

4.3 The Bessel function spectrum

Now an unsettling fact. Even for a single-tone message, the FM spectrum is not three lines. It is infinite. We can prove this with a Fourier series expansion. The signal $A_c\cos[2\pi f_c t + \beta\sin(2\pi f_m t)]$ can be expanded using the identity:

$\cos[\beta\sin\theta] = J_0(\beta) + 2\sum_{n=1}^\infty J_{2n}(\beta)\cos(2n\theta)$ $\sin[\beta\sin\theta] = 2\sum_{n=0}^\infty J_{2n+1}(\beta)\sin((2n+1)\theta)$

where $J_n(\beta)$ is the Bessel function of the first kind, order $n$, evaluated at $\beta$. Expanding the cosine of a sum and substituting:

$s(t) = A_c\sum_{n=-\infty}^{\infty} J_n(\beta)\cos\big(2\pi(f_c + n f_m)t\big)$

The spectrum has a line at the carrier $f_c$ with amplitude $A_c J_0(\beta)$, plus an infinite number of sidebands at $f_c \pm f_m$, $f_c \pm 2 f_m$, $f_c \pm 3 f_m$, ... each with amplitude $A_c |J_n(\beta)|$. An FM signal with one audio tone produces an infinite series of sidebands.

The Bessel functions have a few useful properties:

• For small $\beta$ (narrowband FM, $\beta \ll 1$): $J_0(\beta) \approx 1$, $J_1(\beta) \approx \beta/2$, and $J_n(\beta) \approx 0$ for $n \geq 2$. The spectrum is essentially carrier + first sidebands, identical in shape to AM. The bandwidth is $\approx 2 f_m$.
• For large $\beta$: many Bessel functions are significant, but their amplitudes drop sharply once $|n| > \beta + 1$ or so. By the time $|n| = \beta + 2$, $J_n(\beta)$ is negligible.
• The total power is constant: $\sum_n J_n^2(\beta) = 1$ for all $\beta$. (FM amplitude is constant, so total power is fixed; redistributing among sidebands does not create more power.)

There are zeros of $J_0(\beta)$ at $\beta \approx 2.405, 5.520, 8.654, \ldots$ At those exact modulation indices, the carrier component vanishes entirely! This is sometimes used in calibration: drive the modulator with a known tone at increasing amplitude, watch on a spectrum analyzer for the carrier to disappear, and you know your peak deviation precisely.

4.4 Carson's rule

The full Bessel-function bandwidth is infinite, but the practical bandwidth (containing 98 to 99% of the power) is given by Carson's rule:

$B_\text{FM} \approx 2(\Delta f + W) = 2 f_m(\beta + 1)$

The rule says the bandwidth is twice the sum of peak deviation and message bandwidth. It is an empirical rule of thumb derived by inspecting Bessel coefficients and asking "where do the sidebands drop below 1% of the carrier amplitude," and it works astonishingly well across all practical $\beta$.

For broadcast FM ($\Delta f = 75$ kHz, $W = 15$ kHz): $B \approx 180$ kHz. The FCC allocates 200 kHz channels, leaving a 10 kHz guard band on each side. For narrowband FM in two-way radio ($\Delta f \approx 5$ kHz, voice bandwidth 3 kHz): $B \approx 16$ kHz, fitting comfortably inside 25 kHz channel allocations.

Anticipating confusion: why is FM wide-band even for a narrow message? A naive intuition says: if I only swing the carrier by 75 kHz, the signal must occupy 75 kHz of spectrum. That intuition is wrong. Even tiny phase modulation on a high-frequency carrier produces sidebands at every integer multiple of the modulation rate. The reason is that the phase and amplitude of the carrier together encode the modulation, and a sinusoidal phase wiggle is mathematically equivalent to a sum of many fixed-phase tones at different frequencies (the Bessel expansion). The message frequency $f_m$ sets the spacing of sidebands; the modulation index $\beta$ sets how many of them have nontrivial amplitude.

4.5 Why FM beats AM in noise: trading bandwidth for SNR

A noisy AM channel translates into amplitude noise on the recovered audio. A noisy FM channel translates into phase noise, but the FM receiver has a limiter stage right before the demodulator that hard-clips the input to constant amplitude. Any amplitude noise (which is most of the additive Gaussian noise from a thermal channel) is killed by the limiter before it ever reaches the demodulator. Only phase noise gets through, and even that is partly suppressed by the differentiating action of the demodulator.

The SNR improvement of WBFM over AM, well above threshold:

$\frac{\text{SNR}_o}{\text{SNR}_i} \approx 3\beta^2(\beta + 1)$

For $\beta = 5$ (broadcast FM): the output SNR is about 450x or 26.5 dB better than the input SNR. That is the entire reason FM sounds dramatically cleaner than AM at the same transmitter power, and why it is the broadcast modulation of choice for high fidelity. Wide-band FM trades spectrum (more bandwidth) for SNR (less audible noise). This is the classic Shannon-theorem tradeoff playing out in an analog setting, decades before Shannon formalized it.

There is a catch, the threshold effect. The SNR improvement holds only above an input SNR of roughly 10 dB. Below threshold, the limiter occasionally tracks a noise spike instead of a signal cycle, the demodulator outputs a spurious "click", and the audio falls off a cliff. This is why FM "fades to noise" abruptly when you drive out of range, while AM gradually loses fidelity. We will analyze the threshold more in section 6.

4.6 The capture effect

Two FM transmitters on the same frequency. The stronger one wins, and the weaker is suppressed entirely, even if the SNR ratio is only a few dB. The reason: the limiter clips to whichever carrier dominates instantaneously, and the FM demodulator follows that dominant phase. Compare to AM, where the two signals add and you hear both garbled together.

The capture effect is great for broadcasting (you do not get two stations bleeding through at once) but bad for cooperative voice (aviation, where simultaneous transmissions need to be detectable as collisions). It is also a hardware-security relevant phenomenon: an attacker with a slightly stronger transmitter on a victim's FM frequency completely silences the legitimate signal, a kind of trivial denial-of-service jamming that AM does not exhibit at the same effectiveness.

4.7 Pre-emphasis and de-emphasis

The noise out of an FM demodulator has a triangular power spectrum, rising as $f^2$ (parabolic in PSD; linear in amplitude). The noise is worst at high audio frequencies, exactly where most of the message is least present (audio energy is concentrated at low frequencies, by the inverse-frequency tilt typical of natural sound). To equalize, FM systems boost high audio frequencies before transmission (pre-emphasis) with a first-order high-pass shelf, then attenuate them at the receiver (de-emphasis) with a matching low-pass shelf. The boost-cut cancels for the message, but the noise added in the channel is only de-emphasized, suppressing the high-frequency hiss substantially.

Standardized time constants: 75 microseconds in the US (corner around 2.1 kHz), 50 microseconds in Europe (corner around 3.2 kHz). Pre-emphasis applies in broadcast FM and analog cellular (NMT, AMPS). It does not apply in narrowband FM (two-way radio) because the SNR benefit there is small relative to the implementation cost.

4.8 Generation of FM, two architectures

Direct FM. The varactor (a reverse-biased diode whose junction capacitance varies with voltage) sits inside an LC oscillator's tank circuit. Apply $m(t)$ to the varactor, the capacitance varies in proportion, the oscillation frequency varies with capacitance (per $f = 1/(2\pi\sqrt{LC})$), and you have FM directly. Simple, but the absolute frequency depends on component tolerances and temperature, which drift unacceptably for broadcast. AFC (automatic frequency control) or a phase-locked loop holds the center frequency steady.

Indirect (Armstrong) FM. Generate narrowband FM at a low carrier frequency, where the modulation index is small enough that all the math is well-behaved (a single-tone message produces essentially carrier + first sidebands, like AM). Then frequency-multiply the resulting signal by a chain of nonlinear stages and bandpass filters: a multiplier of factor $N$ multiplies both the carrier and the deviation by $N$. By the time the chain finishes, the modest deviation at the start has grown into the broadcast 75 kHz, and the carrier is at its assigned 88 to 108 MHz. The advantage is that the original oscillator is a stable low-frequency crystal, so the carrier is rock-steady. Edwin Armstrong invented this in the 1930s, and it dominated commercial FM transmission for decades.

4.9 Detection of FM, four architectures

All FM demodulators ultimately convert frequency variations to amplitude variations, then envelope-detect.

Slope detector. Use the side of an LC tank's bandpass response (where the response amplitude varies linearly with frequency near the half-power point) to convert frequency to amplitude. Crude, asymmetric, distorts on large deviations. Mostly historical, but useful as the seed of intuition for what a discriminator does.

Balanced slope detector. Use two LC tanks, tuned to slightly above and slightly below $f_c$. Subtract the two envelope-detected outputs. The asymmetry of the two responses combines into a linear discriminator characteristic over a wider range of deviation.

Foster-Seeley discriminator. A coupled-resonator circuit (a tuned primary, a center-tapped tuned secondary, and a phase-shift network) whose output is the difference of two diode-rectified signals. The phase relationship between primary and secondary varies linearly with frequency around resonance, producing a clean, linear discriminator characteristic over $\pm$50 kHz or more. The Foster-Seeley was the gold standard in 1950s vacuum-tube hi-fi tuners (Marantz, McIntosh, Scott), praised for its low distortion when properly aligned.

Ratio detector. A close cousin of Foster-Seeley with a stabilizing capacitor across the output that holds the total amplitude constant, suppressing AM noise in addition to the limiter. Slightly worse linearity than Foster-Seeley but better AM rejection. Dominated 1960s-70s consumer FM tuners and TV sound demodulators.

PLL detector. A phase-locked loop with its VCO running at $f_c$. The PLL forces the VCO to track the input signal's instantaneous phase. The control voltage driving the VCO is exactly the demodulated audio: when the input frequency rises, the PLL pushes its VCO frequency up by raising the control voltage, which is the message. Modern integrated FM receivers (CXA1622M, TDA7088, the Si473x series) use PLL-based detection because it is cheap, settles in software, and handles wide deviation ranges without alignment.

In modern receivers, especially software-defined ones, FM demodulation is a one-line DSP operation: compute the analytic signal (Hilbert transform), unwrap the phase, differentiate. The output is the message. RTL-SDR dongles do exactly this in real time on a $30 device.

4.10 FM applications

• Broadcast FM (88 to 108 MHz). The high-fidelity standard since the 1940s. Stereo (L+R baseband, 19 kHz pilot, 38 kHz DSB-SC L-R subcarrier) was added in 1961 in a way that is backward-compatible with mono receivers. RDS (Radio Data System) adds another subcarrier at 57 kHz with low-rate digital station info, song titles, traffic alerts.
• Analog TV sound (1948 to 2009 in the US). TV audio was FM-modulated on a separate subcarrier within the analog TV channel; that is why old TVs continued to carry sound clearly even when the picture was snowy.
• Two-way radio (police, fire, taxi, marine VHF). Narrowband FM dominates because of capture effect (one strong transmitter at a time) and the limiter's noise rejection.
• First-generation cellular (AMPS, 1983 to 2008 in the US). Voice was narrowband FM on 30 kHz channels.
• FM telemetry, SCA subcarriers, baby monitors. Niche but ubiquitous.

Hardware-security tie-in: SDR demodulation of TEMPEST emissions. Modern computer CRT monitors radiated their video signal as parasitic AM-like emissions at clock harmonics; modern LCD monitors leak similar signals through pixel-clock harmonics, sometimes at MHz to low GHz. An attacker with a directional antenna and an SDR can sample these emissions and reconstruct the displayed image. The demodulation involves both AM and FM-like processing: clock harmonics are amplitude-modulated by the data being shown, but they are also slightly frequency-modulated by jitter on the clock, and a clever FM-style demodulator can sometimes pull out the data more cleanly than envelope detection alone. This is the modern descendant of the original van Eck phreaking demonstrated in 1985.