Frequency and phase modulation

Why it matters

Frequency modulation (FM) and phase modulation (PM) carry information in the angle of a constant-amplitude carrier. Because the envelope is constant, amplitude noise and non-linear power amplifiers do little harm, which is why FM is used in broadcast radio, two-way radio, telemetry links and many sensor transmitters. In instrumentation, frequency-output sensors and FM telemetry rely on exactly the same ideas.

Key ideas

Angle modulation. A carrier is written as s(t) = A_c·cos θ_i(t). The instantaneous frequency is the rate of change of the angle: f_i(t) = (1/2π)·dθ_i/dt. The amplitude A_c never changes, so the transmitted power is A_c²/2 (into 1 Ω) whatever the message.

Phase modulation. The phase deviates in direct proportion to the message: θ_i(t) = 2πf_c·t + k_p·m(t), where k_p is the phase sensitivity (rad/V). The peak phase deviation is Δφ = k_p·|m|max, and for a single tone this is the modulation index of PM.

Frequency modulation. The instantaneous frequency deviates in proportion to the message: f_i(t) = f_c + k_f·m(t), where k_f is the frequency sensitivity (Hz/V). The phase is then the integral of the message: θ_i(t) = 2πf_c·t + 2πk_f·∫m(τ)dτ. The peak frequency deviation is Δf = k_f·|m|max.

FM and PM are relatives. FM of m(t) equals PM of the integral of m(t); PM of m(t) equals FM of dm/dt. So an FM signal can be generated by integrating the message and feeding a phase modulator (the indirect, Armstrong method), and the two are indistinguishable for a single tone unless you vary the tone frequency.

Single-tone FM. For m(t) = A_m·cos(2πf_m·t), the FM signal is s(t) = A_c·cos(2πf_c·t + β·sin 2πf_m·t) with modulation index β = Δf/f_m (dimensionless, in radians of peak phase deviation). Note that for FM, Δf depends only on the message amplitude, so β falls as f_m rises. For PM, β = Δφ = k_p·A_m is independent of f_m, so the equivalent frequency deviation Δf = β·f_m rises with f_m.

Spectrum. The single-tone FM signal expands into a carrier and an infinite set of sidebands at f_c ± n·f_m with amplitudes A_c·J_n(β) (Bessel functions of the first kind). The carrier amplitude J_0(β) vanishes at β ≈ 2.405, which is used to calibrate deviation meters. Total power stays A_c²/2 because the Bessel terms satisfy ΣJ_n²(β) = 1.

Narrowband vs wideband. If β is small (below about 0.3), only the first pair of sidebands matters: narrowband FM (NBFM) occupies about 2f_m, like AM, but its sidebands are in phase quadrature with the carrier. When β is large (wideband FM, WBFM), the bandwidth is roughly 2Δf.

Carson's rule. About 98% of the power lies within B ≈ 2(Δf + f_m) = 2(β + 1)·f_m. For a general message use the peak deviation and the highest message frequency W: B ≈ 2(Δf + W); the ratio D = Δf/W is the deviation ratio.

Noise and trade-off. Above threshold, FM trades bandwidth for SNR: the figure of merit (output SNR relative to baseband) is 3β²/2 for a tone. Below the threshold carrier-to-noise ratio (about 10 dB at the discriminator input) the output breaks down sharply. FM output noise spectral density grows as f², which is why broadcast FM uses pre-emphasis at the transmitter and de-emphasis at the receiver; PM output noise is flat.

Generation and detection. Direct FM: a VCO or varactor-tuned oscillator. Indirect FM: NBFM via a phase modulator, then frequency multipliers to raise Δf (multiplying the carrier by n multiplies Δf and β by n). Detection: a slope or balanced discriminator, a Foster–Seeley or ratio detector, a zero-crossing counter, or a phase-locked loop (PLL). A limiter before the discriminator removes amplitude variations.

Formulas

f_i(t) = f_c + k_f·m(t) f_i instantaneous frequency (Hz), f_c carrier frequency (Hz), k_f frequency sensitivity (Hz/V), m(t) message (V).

Δf = k_f·|m(t)|max Δf peak frequency deviation (Hz).

θ_i(t) = 2πf_c·t + k_p·m(t) (PM), Δφ = k_p·|m(t)|max k_p phase sensitivity (rad/V), Δφ peak phase deviation (rad).

β = Δf/f_m (single-tone FM; for PM β = Δφ) β modulation index (rad, dimensionless), f_m tone frequency (Hz).

s(t) = A_c·Σ J_n(β)·cos[2π(f_c + n·f_m)t], n from −∞ to +∞ A_c carrier amplitude (V), J_n Bessel function of order n.

B ≈ 2(Δf + W) = 2(D + 1)·W (Carson's rule) B transmission bandwidth (Hz), W highest message frequency (Hz), D = Δf/W deviation ratio.

P = A_c²/2 (average power into 1 Ω, independent of modulation)

Worked examples

Example 1 (standard). An FM broadcast station has peak deviation Δf = 75 kHz and the highest audio frequency is W = 15 kHz. Find the deviation ratio and the Carson bandwidth.

1. D = Δf/W = 75/15 = 5.
2. B = 2(Δf + W) = 2(75 + 15) kHz.
3. B = 2 × 90 kHz = 180 kHz. Answer: D = 5, B ≈ 180 kHz. (Channels are spaced 200 kHz apart, leaving a guard band.)

Example 2 (GATE level). The message m(t) = 2·cos(2π·2000·t) V is applied (a) to an FM modulator with k_f = 5 kHz/V and (b) to a PM modulator with k_p = 2 rad/V. Find the Carson bandwidth in each case, and then repeat with the tone frequency doubled to 4 kHz (same amplitude).

1. FM: Δf = k_f·A_m = 5 kHz/V × 2 V = 10 kHz; β = Δf/f_m = 10/2 = 5; B = 2(10 + 2) kHz = 24 kHz.
2. PM: β = Δφ = k_p·A_m = 2 × 2 = 4 rad; equivalent Δf = β·f_m = 4 × 2 kHz = 8 kHz; B = 2(8 + 2) kHz = 20 kHz.
3. FM at f_m = 4 kHz: Δf stays 10 kHz, β = 10/4 = 2.5; B = 2(10 + 4) = 28 kHz.
4. PM at f_m = 4 kHz: β stays 4 rad, Δf = 4 × 4 = 16 kHz; B = 2(16 + 4) = 40 kHz. Answer: FM 24 kHz → 28 kHz; PM 20 kHz → 40 kHz. Doubling f_m barely changes FM bandwidth but doubles PM bandwidth — the cleanest way to tell the two apart.

Example 3 (indirect FM). An NBFM stage produces Δf = 25 Hz at a carrier of 200 kHz. How much frequency multiplication gives Δf = 75 kHz?

1. Multiplication factor n = 75 000/25 = 3000.
2. The carrier would become 3000 × 200 kHz = 600 MHz, so a mixer is used part-way to bring the carrier back down; mixing shifts the carrier but does not change Δf. Answer: n = 3000.

Common mistakes

• Using β = Δf/f_m for PM. For PM the index is the peak phase deviation itself, k_p·A_m, in radians.
• Thinking a mixer changes the deviation. Only frequency multiplication scales Δf and β; mixing only translates the carrier.
• Writing cos(2π·f_i(t)·t) for the FM signal. The phase is the integral of 2πf_i(t), not f_i(t) multiplied by t.
• Forgetting the factor 2 in Carson's rule, or using f_c in it. The carrier frequency never enters the bandwidth.
• Assuming FM power rises with modulation. The total power is fixed at A_c²/2; modulation only redistributes it among carrier and sidebands.
• Treating NBFM as identical to AM. The bandwidth is the same (2f_m), but the NBFM sideband phases differ, so an envelope detector cannot recover it.

For GATE IN

• Numericals on Δf, β and Carson bandwidth, often with a given s(t) expression from which you read β and f_m.
• Distinguishing FM from PM when the tone frequency or amplitude changes.
• Frequency multiplier and mixer chains in Armstrong transmitters: track f_c and Δf separately.
• Instantaneous frequency of a given angle expression by differentiation.
• Bessel-function sideband questions: carrier null at β ≈ 2.405, power in a given set of sidebands.

Quick check

1. s(t) = 10·cos(2π·10⁸·t + 4·sin 2π·10³·t). What are β and Δf?
2. What is the average power of the signal in Q1 into 1 Ω?
3. In PM, what happens to bandwidth when the tone frequency doubles?
4. Which circuit block raises the frequency deviation in an Armstrong transmitter? Answers: 1. β = 4, Δf = 4 kHz. 2. 50 W. 3. It roughly doubles (β constant, Δf = β·f_m doubles). 4. The frequency multiplier.