Analog modulation: AM and FM

In short

Analog modulation puts a low-frequency message (speech, music, telemetry) onto a high-frequency carrier by varying the carrier's amplitude (AM) or its frequency/phase (FM/PM), so that it can be radiated by a practical antenna and share spectrum with other users. AM gives a simple receiver and narrow bandwidth but wastes power in the carrier and is vulnerable to noise; FM trades bandwidth for a large noise advantage and constant transmitted power. The topic is a guaranteed scorer in GATE EC/IN (power relations, Carson's rule, image frequency) and in university Analog Communication / Communication Systems papers, and it is standard interview ground for Indian employers such as AIR/Prasar Bharati, BEL, DRDO labs (LRDE, DEAL), ISRO, HAL, Airports Authority of India, Indian Railways (RRB JE/SSE, RDSO), Qualcomm, Analog Devices, Texas Instruments and Tejas Networks. It is also the physical basis of India's MW AM broadcast band, the 88-108 MHz FM band, VHF aviation radio and VHF/UHF two-way sets.

Explained

Why modulate, and the two families

• Reasons for modulation: practical antenna size (an efficient antenna is ~λ/4; at 1 kHz, λ = 300 km), frequency-division multiplexing of many users, reduced noise and interference, improved propagation, and matching the channel's passband.
• Baseband vs bandpass: the message m(t) occupies 0 to W Hz; modulation translates it to a band around the carrier fc, where fc >> W.
• Amplitude modulation: the carrier envelope follows m(t); information is in the amplitude. Linear (frequency-translation) modulation - the spectrum is simply shifted.
• Angle modulation: the carrier amplitude is constant and the angle carries the information - FM (frequency proportional to m(t)) and PM (phase proportional to m(t)). Non-linear: new sidebands are generated, so the spectrum is not a shifted copy.
• Classification of AM: DSB-FC (full carrier, ordinary AM), DSB-SC (suppressed carrier), SSB-SC (single sideband), VSB (vestigial sideband), and QAM (two DSB-SC signals in quadrature).

Amplitude modulation (DSB-FC): index, spectrum, power

• Signal: s(t) = [Ac + m(t)] cos ωc t. For a single tone m(t) = Am cos ωm t, s(t) = Ac[1 + μ cos ωm t] cos ωc t with modulation index μ = Am/Ac.
• Envelope: Amax = Ac(1 + μ), Amin = Ac(1 − μ). Hence μ = (Amax − Amin)/(Amax + Amin) - the standard oscilloscope/trapezoid measurement.
• μ < 1 → faithful envelope. μ = 1 → 100% modulation, Amin = 0. μ > 1 → overmodulation: the envelope reverses phase, causing envelope-detector distortion and splatter into adjacent channels. Broadcast transmitters use limiters to hold μ below 1.
• Spectrum: three lines - carrier at fc, USB at fc + fm, LSB at fc − fm, each sideband of amplitude μAc/2. Bandwidth BW = 2fm (generally 2W).
• Power: Pc = Ac²/2R; each sideband μ²Pc/4; Pt = Pc(1 + μ²/2).
• Transmission efficiency η = sideband power / total power = μ²/(2 + μ²). Even at μ = 1 this is only 33.3% - two-thirds of the power is in a carrier carrying no information.
• Multi-tone: effective index μt = √(μ1² + μ2² + ...); Pt = Pc(1 + μt²/2). Antenna current: It = Ic√(1 + μt²/2).
• Limitations: poor power efficiency, double the needed bandwidth, amplitude-sensitive (noise and fading corrupt the envelope directly), and the 9 kHz MW channel limits audio to about 4.5-5 kHz, which is why AM radio sounds dull.
• Advantages: very cheap receiver (a diode and an RC network), good coverage via ground-wave/sky-wave at MW/SW, no capture effect, and intelligibility is preserved in simultaneous transmissions - the reason aviation still uses AM.

DSB-SC, SSB, VSB and QAM

• DSB-SC: s(t) = m(t)cos ωc t. Carrier removed → 100% of power in sidebands, BW = 2W, but needs coherent (synchronous) detection. Produced by a balanced modulator or ring (diode lattice) modulator.
• Coherent detector errors: a local phase error φ attenuates the output by cos φ (nulls at 90° - "quadrature null effect"); a frequency error Δf causes beating. Carrier recovery by pilot carrier, squaring loop or Costas loop.
• SSB-SC: one sideband only - BW = W, power = μ²Pc/4 for the same tone. Used in HF point-to-point, amateur radio, marine/military HF and old FDM carrier telephony (12-channel groups).
  • Filter method: generate DSB-SC, then a sharp crystal/ceramic filter removes one sideband; needs a spectral gap, so the message must have no energy below ~300 Hz (fine for speech, not for DC-coupled data).
  • Phase-shift (Hartley) method: two balanced modulators with 90° shifts in carrier and message (Hilbert transform); Weaver's third method avoids the wideband 90° network.
  • Limitations: complex transmitter and receiver, poor for music, frequency error shifts every tone (Donald-Duck effect).
• VSB: one full sideband plus a vestige of the other, with complementary filter roll-off - used for analogue TV video (System B, 625-line, 7 MHz channel: ~5 MHz video VSB + FM sound at +5.5 MHz). India's terrestrial analogue TV has largely been switched off in favour of digital (DVB-T2/DTH), so VSB is now mainly of exam interest.
• QAM: two messages on the same carrier in quadrature, m1(t)cos ωc t + m2(t)sin ωc t - bandwidth-efficient but needs exact coherent detection; the analogue ancestor of digital QAM and of the PAL/NTSC colour subcarrier.

Generating and detecting AM; the superheterodyne receiver

• Generation: square-law modulator (small-signal non-linear device + BPF), switching modulator (diode chopping), collector/plate-modulated class-C amplifier (high-level modulation, used in broadcast), and low-level modulation + linear class-A/B amplification (efficient only at low power).
• Envelope detector: series diode + RC. Design rule: 1/fc << RC << 1/W, specifically RC ≤ √(1 − μ²)/(μ ωm).
  • Diagonal clipping (RC too large - the capacitor cannot follow a falling envelope) and negative peak clipping (unequal DC and AC load resistances, worse at high μ) are the two classic distortions.
• Rectifier/square-law detector: simpler, but square-law detection introduces second-harmonic distortion; used only for small signals.
• Superheterodyne receiver: RF amplifier → mixer with local oscillator → fixed IF amplifier (usually 455 kHz for MW AM, 10.7 MHz for FM) → detector → audio amplifier. Fixed IF gives constant gain, selectivity and stability over the whole band.
• Image frequency: with high-side injection fLO = fs + fIF, the image is fsi = fs + 2 fIF, which the mixer translates to the same IF. Suppressed by RF pre-selection; image rejection ratio IRR = √(1 + Q²ρ²) with ρ = (fsi/fs − fs/fsi).
• Receiver performance terms: sensitivity (smallest input for a stated output/SNR), selectivity (ability to reject adjacent channels), fidelity (audio frequency response), double-spotting (a station heard at two dial points - poor image rejection), AGC (automatic gain control keeps output steady as signal strength varies), tracking (ganged capacitors keeping fLO − fs = fIF across the band, using a padder capacitor).

Frequency and phase modulation: deviation, index, spectrum, bandwidth

• Instantaneous frequency: fi(t) = fc + kf m(t), where kf is in Hz/volt. FM signal: s(t) = Ac cos[2πfc t + 2πkf ∫m(τ)dτ].
• PM signal: s(t) = Ac cos[2πfc t + kp m(t)]. FM is PM of the integral of m(t); PM is FM of the derivative of m(t) - so an FM modulator plus an input differentiator gives PM, and vice versa.
• Frequency deviation Δf = kf |m(t)|max (peak swing either side of fc). Modulation index β = Δf/fm (single tone); deviation ratio D = Δf/W for a general message.
• Spectrum: s(t) = Ac Σ Jn(β) cos[2π(fc + n fm)t] - infinitely many sidebands spaced fm apart, amplitudes given by Bessel functions of the first kind. Jn alternate in sign; J−n = (−1)ⁿJn. At β = 2.405 the carrier component vanishes (used to calibrate deviation).
• Power is constant: Pt = Ac²/2R regardless of modulation, since Σ Jn²(β) = 1. Modulation redistributes power from carrier to sidebands - no modulating power is added, so FM transmitters run class-C amplifiers at high efficiency.
• NBFM (β << 1, usually β < 0.3): only carrier and one pair of sidebands matter; BW ≈ 2fm, spectrum resembles AM but with one sideband phase-reversed. Used in two-way radio (12.5/25 kHz channels), police/railway sets, marine VHF.
• WBFM (β > 1): Carson's rule BW ≈ 2(Δf + fm) = 2fm(1 + β) - the same as counting sidebands up to n ≈ β + 1; the stricter "1% rule" (keep every |Jn| > 0.01, or read tabulated Jn) gives a somewhat wider figure at large β.
• Key behaviour: raising the message amplitude raises Δf and β (bandwidth grows); raising the message frequency at constant amplitude leaves Δf unchanged but lowers β (bandwidth grows only slightly). For PM, Δf is proportional to fm, so PM bandwidth grows with message frequency - one reason broadcast uses FM.

FM generation, demodulation, pre-emphasis and practical effects

• Direct FM: a varactor (varicap) diode or reactance modulator shifts the frequency of an LC/crystal VCO. Large deviation easily, but poor carrier stability → needs AFC or a PLL with a crystal reference.
• Indirect (Armstrong) FM: integrate the message, make NBFM against a crystal carrier with a balanced modulator, then use frequency multipliers (which multiply both fc and Δf by n) and a mixer (which shifts fc but leaves Δf unchanged) to reach the wanted carrier and deviation. Excellent stability - the standard broadcast method.
• Demodulators:
  • Slope detector: off-tuned resonant circuit converts FM to AM, then an envelope detector; simple but non-linear.
  • Balanced slope detector: two off-tuned circuits, linear over a wider range.
  • Foster-Seeley phase discriminator: phase-shift transformer; linear and sensitive, but responds to amplitude noise → needs a preceding limiter.
  • Ratio detector: inherent amplitude limiting (output depends on the *ratio* of the two diode voltages), so no separate limiter; used in domestic receivers.
  • PLL demodulator: the loop filter output is proportional to the instantaneous frequency - linear, low distortion, works at low SNR; the IC standard today.
  • Quadrature detector and zero-crossing counter: used in ICs and in digital/SDR implementations.
• Amplitude limiter: clips amplitude variations before detection - this is where FM's noise immunity is realised.
• Pre-emphasis / de-emphasis: noise at the FM detector output rises with frequency (parabolic noise power spectrum), so high audio frequencies are boosted at the transmitter by a first-order network and cut by the matching network at the receiver, giving several dB of SNR improvement with no net frequency-response change. India and most of the world (ITU-R/CCIR practice) use a 50 µs time constant (corner ≈ 3.18 kHz); the Americas and South Korea use 75 µs (≈ 2.12 kHz).
• Capture effect: an FM receiver locks on to the stronger of two co-channel signals and suppresses the weaker; the capture ratio (1-2 dB in good receivers) is a selling point - AM instead produces heterodyne whistles.
• FM threshold: below about 10-12 dB carrier-to-noise ratio at the detector input the noise advantage collapses abruptly (clicks, then noise takeover). PLL and feedback demodulators give a few dB of threshold extension. (Quantitative SNR analysis belongs to the noise topic.)
• FM stereo multiplex: (L + R) at 0-15 kHz, a 19 kHz pilot, (L − R) as DSB-SC on a 38 kHz subcarrier, and RDS on 57 kHz - the composite baseband still deviating the carrier by at most 75 kHz.

AM vs FM, Indian standards and regulation

• MW (AM) broadcasting in India: 526.5-1606.5 kHz (ITU Region 3), 9 kHz channel spacing, audio limited to ~4.5-5 kHz, double-sideband full carrier; AIR/Prasar Bharati also runs SW HF bands and has converted several MW/SW transmitters to DRM digital broadcasting, with the analogue MW network being progressively rationalised.
• FM broadcasting in India: 88-108 MHz, maximum deviation ±75 kHz, a 100 kHz channel raster in ITU Regions 1/3 (200 kHz in the Americas); Indian city assignments are generally spaced more widely for filter and interference margin, the exact raster being fixed by WPC/MIB in each auction round. Private FM operates under MIB's Phase-III policy; community radio stations run low-power transmitters (of the order of 50-100 W ERP, ~10-15 km range) under the Community Radio Policy.
• Other Indian analogue uses: VHF civil aviation radio 118-136.975 MHz uses AM (DSB) with 25 kHz channels, plus AM-based VOR and ILS (AAI, airlines); marine VHF 156-174 MHz uses FM; VHF/UHF NBFM two-way radio for police, Railways, state transport and amateur operators; amateur HF uses SSB (ASOC licence from WPC).
• Regulation: spectrum assignment and licensing by the WPC Wing, Ministry of Communications (SACFA clearance for sites); tariff/policy recommendations by TRAI; broadcast content and licences by the Ministry of Information and Broadcasting. The legal base was the Indian Telegraph Act 1885 and Indian Wireless Telegraphy Act 1933; the Telecommunications Act 2023 replaces them and is being brought into force in stages, so check the current notified position before quoting it.
• Comparison in one line: AM is narrow and cheap but power-hungry and noise-prone; FM is wideband, power-efficient and noise-resistant above threshold; PM is mostly a building block (and the basis of digital PSK) rather than an end service.

Key points

• Modulation index (AM) - μ = Am/Ac = (Amax − Amin)/(Amax + Amin); must stay below 1 to avoid envelope distortion.
• AM bandwidth - 2W, independent of μ; FM bandwidth depends on both deviation and message frequency.
• AM power - Pt = Pc(1 + μ²/2); carrier power is unchanged by modulation, sidebands add.
• AM efficiency - η = μ²/(2 + μ²), maximum 33.3% at μ = 1; DSB-SC is 100%, SSB 100% with half the bandwidth.
• Multi-tone index - indices add in quadrature: μt = √(Σμi²).
• DSB-SC/SSB need coherent detection - phase error φ scales output by cos φ; 90° error gives zero.
• Envelope detector design - 1/fc << RC << 1/W; RC ≤ √(1 − μ²)/(μωm) avoids diagonal clipping.
• Superheterodyne IFs - 455 kHz (AM MW), 10.7 MHz (FM broadcast); image frequency = fs + 2fIF.
• Instantaneous frequency - fi = fc + kf m(t); FM amplitude and total power are constant.
• FM modulation index - β = Δf/fm = kf Am/fm; deviation ratio D = Δf/W.
• Carson's rule - BW ≈ 2(Δf + fm); for NBFM this collapses to 2fm.
• Bessel spectrum - amplitudes Ac Jn(β); Σ Jn² = 1; carrier null first at β = 2.405.
• FM vs PM - FM = PM of ∫m dt; in PM, Δf ∝ fm, so PM bandwidth worsens with higher audio frequencies.
• Multiplier vs mixer - a frequency multiplier scales fc and Δf by n; a mixer shifts fc and leaves Δf untouched.
• Pre-emphasis/de-emphasis - 50 µs in India (75 µs in the Americas) counters the rising noise spectrum at the FM detector.
• Capture effect and threshold - FM suppresses the weaker co-channel signal but loses its advantage below ~10 dB CNR.
• Indian bands - MW 526.5-1606.5 kHz with 9 kHz spacing; FM 88-108 MHz with ±75 kHz deviation; aviation VHF is AM.
• Ring/balanced modulator suppresses the carrier; a sharp filter or a 90° phase network then makes SSB.

Formulas and examples

Example 1 - AM power, efficiency and antenna current. An AM broadcast transmitter radiates 15 kW total at μ = 0.8 with one tone; the unmodulated antenna current is 8 A.

• Pc = Pt/(1 + μ²/2) = 15/(1 + 0.32) = 15/1.32 = 11.36 kW.
• Sideband power = 15 − 11.36 = 3.64 kW, i.e. 1.82 kW per sideband.
• η = 0.64/2.64 = 24.24% (check: 3.64/15 = 24.2% ✓).
• It = 8√1.32 = 8 × 1.1489 = 9.19 A.
• Now add a second tone so that μ1 = 0.6, μ2 = 0.5: μt = √(0.36 + 0.25) = 0.781; Pt = 11.36 × (1 + 0.61/2) = 11.36 × 1.305 = 14.83 kW.
• If instead only one sideband were sent (SSB) for the μ = 0.8 tone, the needed power would be μ²Pc/4 = 0.64 × 11.36/4 = 1.82 kW in half the bandwidth.

Example 2 - index from an oscilloscope, and detector design. An AM envelope measured across 50 Ω shows Amax = 12 V, Amin = 3 V, with fm = 5 kHz, fc = 1 MHz.

• μ = (12 − 3)/(12 + 3) = 9/15 = 0.6; Ac = (12 + 3)/2 = 7.5 V, Am = 4.5 V.
• Pc = 7.5²/(2 × 50) = 56.25/100 = 0.5625 W; Pt = 0.5625 × (1 + 0.18) = 0.664 W; BW = 10 kHz.
• Detector: RC ≤ √(1 − 0.36)/(0.6 × 2π × 5000) = 0.8/18,850 = 42.4 µs. With C = 10 nF, R ≤ 4.24 kΩ; and RC must stay >> 1/fc = 1 µs, so RC ≈ 20-40 µs is a good choice.

Example 3 - FM deviation, index and bandwidth. fc = 100 MHz, kf = 10 kHz/V, Ac = 10 V across 50 Ω, m(t) = 5 cos(2π × 5000 t).

• Δf = 10 kHz/V × 5 V = 50 kHz; β = 50/5 = 10.
• Carson: BT = 2(50 + 5) = 110 kHz; sideband-count rule n ≈ β + 1 = 11 → 2 × 11 × 5 = 110 kHz (agrees).
• Transmitted power = 10²/(2 × 50) = 1 W, the same modulated or unmodulated.
• Double the message frequency to 10 kHz: Δf stays 50 kHz, β = 5, BT = 2(50 + 10) = 120 kHz.
• Double the message amplitude to 10 V at 5 kHz: Δf = 100 kHz, β = 20, BT = 2(100 + 5) = 210 kHz - and this would exceed the ±75 kHz broadcast limit.

Example 4 - Armstrong indirect FM transmitter. NBFM is generated at f1 = 200 kHz with Δf1 = 25 Hz; the output must be 96 MHz with Δf = 75 kHz.

• Required deviation multiplication = 75,000/25 = 3000; split as n1 = 60 and n2 = 50.
• After ×60: carrier 200 kHz × 60 = 12 MHz, Δf = 25 × 60 = 1.5 kHz.
• The final stage ×50 must start from 96/50 = 1.92 MHz, so mix down with fLO = 12 − 1.92 = 10.08 MHz (the mixer does not change Δf).
• After ×50: carrier = 1.92 × 50 = 96 MHz, Δf = 1.5 × 50 = 75 kHz ✓.

Example 5 - image frequency and rejection. An MW receiver with fIF = 455 kHz is tuned to 600 kHz; the pre-selector has Q = 100.

• fLO = 600 + 455 = 1055 kHz; image fsi = 600 + 910 = 1510 kHz.
• ρ = 1510/600 − 600/1510 = 2.5167 − 0.3974 = 2.1193; Qρ = 211.93 → IRR = √(1 + 44,914) ≈ 211.9 → 46.5 dB.
• The same receiver tuned to 1500 kHz has an image at 2410 kHz: ρ = 1.6067 − 0.6224 = 0.9843, IRR ≈ 98.4 → 39.9 dB - image rejection worsens at the high end of the band.
• An FM set at 100 MHz with fIF = 10.7 MHz has fLO = 110.7 MHz and image = 121.4 MHz, far outside 88-108 MHz, so image interference is negligible.

Where it is used

• Broadcasting: AIR/Prasar Bharati MW and SW AM transmitters and the nationwide 88-108 MHz FM network, plus private FM and community radio - roles in transmitter maintenance, studio-to-transmitter links and RF measurement (deviation, modulation depth, spurious emissions).
• Aviation and navigation: AAI's VHF air-ground AM radios, VOR and ILS localiser/glide path (AM-based), marker beacons; defence communication sets at BEL, DRDO (DEAL, LRDE) and HAL avionics.
• Public safety and transport two-way radio: NBFM VHF/UHF sets for police, fire, Indian Railways train control, state road transport and port operations; marine VHF (FM) and HF SSB for coastal and merchant shipping.
• Test, design and semiconductor roles: RF/analog design and applications engineers at Analog Devices, Texas Instruments, Qualcomm, Skyworks India centres working on VCOs, PLLs, mixers, limiters and FM/AM demodulator IP; spectrum-analyser and modulation-analyser work in EMI/EMC and type-approval labs.
• Instrumentation and sensing: FM telemetry and FM/FSK data links, AM-based lock-in and carrier amplifiers in strain and bridge measurement, varactor-tuned oscillators, FM-CW radar level gauges and automotive sensors.
• Legacy and systems context: SSB-FDM carrier telephony, VSB analogue TV (now digital in India), FM subcarrier systems such as RDS, and FM's role as the conceptual base for FSK, PLL synthesisers and frequency-agile SDR front ends.

Interview questions

• Q: Why can an AM transmitter never exceed 33.3% efficiency? A: Efficiency is μ²/(2 + μ²), which reaches 1/3 only at μ = 1, because the carrier always carries two-thirds of the power and carries no information.
• Q: Does AM bandwidth change with modulation index? A: No - the sidebands stay at fc ± fm, so bandwidth is 2W; only the power split changes. (Overmodulation does splatter extra energy, which is a distortion effect, not true bandwidth.)
• Q: Why is FM transmitted power independent of modulation? A: The envelope is constant, and the Bessel coefficients satisfy Σ Jn²(β) = 1, so modulation only redistributes a fixed total power among carrier and sidebands.
• Q: What happens to Δf and β if the message frequency is doubled at constant amplitude? A: Δf is unchanged (it depends on amplitude), β halves, and Carson bandwidth rises slightly, by 2Δfm.
• Q: Difference between a frequency multiplier and a mixer in an FM transmitter? A: The multiplier multiplies carrier frequency and deviation by the same n; the mixer translates the carrier without changing the deviation - Armstrong's method uses both to set fc and Δf independently.
• Q: Why can't an envelope detector be used for DSB-SC or SSB? A: Without a large carrier the envelope is not proportional to m(t) (for DSB-SC it is |m(t)|), so coherent detection with a locally regenerated carrier is required.
• Q: What is the image frequency and how do you suppress it? A: A second input frequency fs + 2fIF that the mixer also translates to the IF; it is suppressed by selective RF pre-selection, a higher IF, or double conversion.
• Q: Why is pre-emphasis used in FM broadcasting, and what value does India use? A: Detected noise power rises as f², so high audio frequencies are boosted at the transmitter and cut at the receiver; India follows the 50 µs time constant (~3.18 kHz corner), while the Americas use 75 µs.
• Q: What is the capture effect and why does AM not show it? A: An FM limiter-discriminator locks on to the stronger of two co-channel signals and suppresses the weaker; AM detection is linear in amplitude, so both signals add and produce heterodyne whistles.
• Q: Why does civil aviation still use AM rather than FM? A: AM has no capture effect, so simultaneous transmissions are heard as overlapping speech (and the pilot knows of the clash), channels are narrow, and the equipment base is standardised worldwide.
• Q: What is FM threshold, and how is it extended? A: Below about 10-12 dB CNR the noise advantage disappears abruptly; PLL and feedback (FMFB) demodulators give a few dB of threshold extension.
• Q: How would you measure deviation on the bench without a modulation analyser? A: Use the Bessel null method - increase the modulating amplitude until the carrier line on a spectrum analyser vanishes (β = 2.405), then Δf = 2.405 × fm.

Common mistakes

• Treating the carrier power as varying with μ - Pc is fixed; it is the total power Pc(1 + μ²/2) that rises.
• Adding multi-tone modulation indices arithmetically instead of in quadrature (μt = √Σμi²).
• Applying Carson's rule to NBFM without noticing it correctly reduces to 2fm, or using Δf as a peak-to-peak swing when it is the peak deviation from fc.
• Confusing β with μ, and assuming β must be less than 1 - FM broadcast runs β ≈ 5 at 15 kHz and far higher at low audio frequencies.
• Forgetting that a frequency multiplier multiplies the deviation too, and that a mixer does not - the classic Armstrong numerical error.
• Choosing an envelope-detector RC that violates RC ≤ √(1 − μ²)/(μωm), then reporting the resulting diagonal clipping as "circuit noise".
• Using low-side LO injection in image calculations without saying so, or quoting the image as fs − 2fIF when high-side injection was assumed.

Related topics

Signals and systems · Fourier, Laplace and Z-transforms · Noise in communication systems · Sampling and digital modulation · Information theory and coding · Satellite communication