A complete guide to amplitude modulation, frequency modulation, and phase modulation — with mathematical derivations, waveform diagrams, modulation index formulas, bandwidth calculations, and hands-on virtual lab experiments.
Audio signals like voice and music typically occupy frequencies between 20 Hz and 20 kHz. Transmitting these signals directly through a radio antenna would require a physically impractical antenna of several kilometres in length (antenna length ≈ λ/4, and at 10 kHz, λ = 30 km).
Modulation solves this problem by superimposing the low-frequency message signal onto a high-frequency carrier signal (e.g., 540 kHz for AM radio, 88–108 MHz for FM radio). The resulting modulated signal occupies a frequency band around the carrier that can be transmitted efficiently by a compact antenna.
1. Antenna size reduction — high carrier frequency means short, practical antenna. 2. Multiplexing — different stations use different carrier frequencies simultaneously without interference. 3. Noise reduction — FM shifts the signal away from low-frequency noise. 4. Propagation improvement — high-frequency waves propagate farther through the atmosphere and ionosphere.
The amplitude of the carrier changes in proportion to the message. Frequency and phase stay constant. Simple to implement; susceptible to noise.
The instantaneous frequency of the carrier varies with the message amplitude. Better noise immunity than AM. Used in high-fidelity radio broadcasting.
The instantaneous phase of the carrier varies with the message amplitude. Related to FM; used in digital satellite communication and GPS.
In amplitude modulation, the amplitude of the sinusoidal carrier signal is varied linearly with the instantaneous value of the message signal, while the carrier frequency and phase remain unchanged.
Let the carrier signal be c(t) = Ac cos(2πfct) and the message signal be m(t) = Am cos(2πfmt). The resulting AM signal is:
where μ = Am/Ac = modulation index | fc = carrier frequency | fm = message frequency
Figure 1: Amplitude Modulation — the message signal (amber) controls the envelope of the carrier (green), producing the AM output (red) whose amplitude varies at the message frequency. The dashed lines trace the upper and lower envelopes.
The modulation index (μ or m) quantifies how much the carrier amplitude varies relative to its unmodulated value. It can be measured directly from a CRO display:
Amax = maximum amplitude of AM envelope | Amin = minimum amplitude of AM envelope
| Condition | μ Value | Result |
|---|---|---|
| Under-modulation | 0 < μ < 1 | Normal AM; no distortion; Amin > 0 |
| 100% Modulation | μ = 1 | Ideal; maximum transmitted information; Amin = 0 |
| Over-modulation | μ > 1 | Distortion; envelope crosses zero; splatter sidebands appear |
When μ > 1, the carrier is overdriven and the envelope is clipped. The signal develops strong harmonic distortion and causes splatter — interference into adjacent channels — which violates telecommunications regulations. In practice, a limiter or AGC circuit prevents over-modulation.
Expanding the AM equation using the product-to-sum identity reveals three frequency components: the carrier at fc, the Upper Sideband (USB) at fc + fm, and the Lower Sideband (LSB) at fc − fm.
Figure 2: AM frequency spectrum — the carrier at fₒ is flanked by two sidebands, each at distance fₘ. Total bandwidth = 2fₘ. Both sidebands carry identical information; the carrier itself carries none.
Example: Voice signal 0–5 kHz → AM bandwidth = 10 kHz
Pc = Ac²/2R = carrier power | PLSB = PUSB = Pcμ²/4 | At μ=1: only 33% of Ptotal is in useful sidebands
Standard AM is inherently power-inefficient. At 100% modulation (μ = 1), only 33% of the total transmitted power is in the information-carrying sidebands. The remaining 67% is wasted in the carrier component. DSB-SC modulation eliminates this waste entirely.
Two important variants of AM eliminate redundant components to improve power efficiency and bandwidth utilization.
DSB-SC (Double Sideband Suppressed Carrier) is generated using a balanced modulator (ring modulator) that cancels the carrier term. Both sidebands are transmitted, so bandwidth remains 2fm, but 100% of the power is in the useful sidebands. The receiver must generate a synchronous carrier for demodulation (coherent detection).
SSB (Single Sideband) additionally removes one sideband using a sideband filter or phase-shift method, halving the bandwidth to fm. SSB is the most spectrally efficient analog voice modulation and is the standard for HF amateur and maritime radio.
| Type | Transmitted | Bandwidth | Power Efficiency | Demodulation |
|---|---|---|---|---|
| Standard AM | Carrier + 2 sidebands | 2fm | ≤ 33% | Simple envelope detector |
| DSB-SC | 2 sidebands only | 2fm | 100% | Synchronous (complex) |
| SSB | 1 sideband only | fm | 100% | Synchronous (complex) |
| VSB | Full sideband + vestige | ~1.25fm | High | Moderate |
In frequency modulation, the instantaneous frequency of the carrier is varied in proportion to the instantaneous amplitude of the message signal. The carrier amplitude remains constant at all times.
The instantaneous frequency is: fi(t) = fc + kf · m(t), where kf is the frequency sensitivity (Hz/V) and Δf = kf · Am is the peak frequency deviation.
β = Δf / fm = FM modulation index | Δf = kf Am = peak frequency deviation
Figure 3: Frequency Modulation — where the message is at its positive peak, the FM carrier oscillates faster (higher frequency); where the message is at its negative peak, the FM carrier oscillates slower. Carrier amplitude stays constant throughout.
β < 1 → Narrowband FM (NBFM) — two-way radios | β >> 1 → Wideband FM (WBFM) — commercial radio (β ≈ 5)
The exact bandwidth requires an infinite Bessel function series. Carson's rule provides a practical approximation capturing ~98% of the total signal power:
Commercial FM radio: Δf = 75 kHz, fm = 15 kHz → β = 5 → BW = 2(75+15) = 180 kHz
NBFM (β < 1): BW ≈ 2fm — same as AM. Used in aviation and emergency communication where spectrum is scarce. WBFM (β > 1): BW = 2(Δf + fm). Used in FM broadcasting (88–108 MHz) for high audio fidelity. Each FM channel occupies 200 kHz, spaced every 100 kHz with guard bands. Wideband FM provides ~25 dB better signal-to-noise ratio than AM at the same received power (above the FM threshold).
In phase modulation, the instantaneous phase of the carrier is varied in proportion to the instantaneous value of the message signal, while frequency and amplitude remain nominally constant.
kp = phase sensitivity (rad/V) | mp = kp Am = maximum phase deviation (radians)
Figure 4: Phase Modulation — the carrier frequency stays constant but the phase shifts abruptly at each message transition (red dashed lines). For a sinusoidal message, the phase shift is continuous and the PM signal superficially resembles FM.
FM and PM are closely related forms of angle modulation. An FM modulator preceded by an integrator produces PM; a PM modulator preceded by a differentiator produces FM. The key practical difference is their spectral response to message frequency:
| Property | FM | PM |
|---|---|---|
| Modulation index depends on… | Message amplitude AND frequency (β = Δf/fm) | Message amplitude only (mp = kpAm) |
| Instantaneous variable | Frequency fi(t) | Phase θ(t) |
| Deviation at high fm | β decreases (pre-emphasis needed) | mp constant |
| Application | FM radio, VHF/UHF voice | GPS, satellite links, BPSK/QPSK |
| Parameter | AM | FM | PM |
|---|---|---|---|
| Varied parameter | Amplitude | Instantaneous frequency | Instantaneous phase |
| Constant parameters | f, φ | A, φ | A, f |
| Modulation index | μ = Am/Ac (0 to 1) | β = Δf/fm (0 to ∞) | mp = kpAm |
| Bandwidth | 2fm | 2(Δf + fm) — Carson's rule | Similar to FM |
| Noise immunity | Poor | Excellent | Excellent |
| Power efficiency | Poor (carrier wastage) | Good (constant amplitude) | Good (constant amplitude) |
| Circuit complexity | Simple | Moderate | Moderate–Complex |
| Transmitter range | Long (MW/SW, ionosphere) | Shorter (VHF, line-of-sight) | Shorter |
| Audio fidelity | Lower (~5 kHz audio BW) | Higher (15 kHz audio BW) | High |
| Demodulation | Envelope detector (simple) | Limiter + discriminator / PLL | Coherent detector / PLL |
| Applications | AM radio 535–1705 kHz, aircraft comms | FM radio 88–108 MHz, TV audio | GPS, satellite, BPSK/QPSK modems |
| Quantity | AM | FM |
|---|---|---|
| Modulation index | μ = Am/Ac | β = Δf/fm |
| Bandwidth | BW = 2fm | BW = 2(Δf + fm) |
| Total power | Pt = Pc(1 + μ²/2) | Pt = Ac²/2R (constant) |
| Max efficiency | 33% (at μ=1) | 100% (constant amplitude) |
Practice the concepts covered in this article with these interactive lab experiments. Each includes a step-by-step virtual simulation with circuit diagrams, observation tables, and analysis. Click any experiment card to launch the demo.
Generate an AM signal using a multiplier/modulator IC and observe the waveform on a CRO. Measure modulation index from the trapezoid pattern.
Try DemoConnect the AM signal and message signal to the X and Y inputs of a CRO to obtain the trapezoid pattern and calculate μ from its dimensions.
Try DemoGenerate an FM signal using the ICL8038 function generator IC and demodulate it using a PLL (CD4046). Measure frequency deviation on a spectrum analyser.
Try DemoBuild a double balanced ring modulator using four diodes to generate DSB-SC. Verify suppression of the carrier component (> 40 dB) on the spectrum analyser.
Try DemoTrace the signal through a superheterodyne AM receiver: RF amplifier, mixer, IF filter at 455 kHz, envelope detector, and audio amplifier stage.
Try DemoUse a virtual spectrum analyser to measure the FM signal bandwidth, frequency deviation Δf, and verify Carson's rule for different modulation indices.
Try DemoAnalog modulation is the process of varying a property (amplitude, frequency, or phase) of a high-frequency carrier signal in proportion to a low-frequency message signal. It is needed because low-frequency audio and video signals (20 Hz–15 kHz) cannot be transmitted over long distances directly — the required antenna would be kilometres long. Modulation shifts the signal to a high carrier frequency, dramatically reducing the antenna size and allowing multiple stations to share the radio spectrum simultaneously without interference.
The modulation index μ = Am/Ac (message amplitude / carrier amplitude). At μ = 1 (100% modulation), the envelope just touches zero — the maximum useful value. When μ > 1, the envelope is clipped and the signal develops intermodulation products that spill into adjacent channels (called splatter), violating spectrum regulations. In practice, AGC circuits and limiters prevent over-modulation. The modulation index can also be measured from a CRO display as μ = (Amax − Amin) / (Amax + Amin).
Most natural and man-made electrical noise (lightning, ignition, power lines) manifests as amplitude variations. An FM receiver employs a limiter circuit before the discriminator that clips all amplitude variations, stripping away the noise. Since the information in FM is encoded in frequency (not amplitude), the limiter removes only the noise, not the information. This is the FM noise capture advantage, and it gives wideband FM a signal-to-noise improvement of 3β²(β+1) over AM at the same received power — for commercial FM with β = 5, this is approximately 25 dB better than AM.
Carson's rule states that the approximate bandwidth of an FM signal is BW ≈ 2(Δf + fm), capturing about 98% of the total FM signal power. For commercial FM radio: Δf = 75 kHz, fm = 15 kHz → BW = 2(75+15) = 180 kHz. Each channel is allocated 200 kHz (with 10 kHz guard bands each side). For narrowband FM police/taxi radios: Δf = 5 kHz, fm = 3 kHz → BW = 16 kHz, channelised in 12.5 or 25 kHz slots. The exact Bessel function approach gives a slightly narrower bandwidth in practice.
Both DSB-SC and SSB suppress the carrier to achieve 100% power efficiency. DSB-SC transmits both sidebands and occupies the same bandwidth (2fm) as standard AM. SSB transmits only one sideband (USB or LSB), halving the bandwidth to fm — twice as spectrally efficient as DSB-SC. The trade-off is that SSB requires more precise filter design (suppressing the unwanted sideband by 40–60 dB) and a more accurate synchronous carrier at the receiver. SSB is the standard for HF amateur and maritime radio where the spectrum is congested.
FM has a capture effect: when two signals are on the same frequency, the FM receiver locks onto the stronger one and completely suppresses the weaker one. In aviation, if two pilots transmit simultaneously (a "double"), FM would cause complete loss of both transmissions — a serious safety hazard. With AM, both signals add together and some intelligibility is preserved. Aviation also uses AM (118–137 MHz) because the historic installed base of global AM infrastructure is enormous, and the AM receiver can hear signals from multiple ground stations that overlap in frequency.
The FM threshold is the minimum carrier-to-noise ratio (CNR) below which FM demodulation performance degrades sharply — the noise advantage suddenly collapses and output SNR falls below that of equivalent AM. For wideband FM the threshold is typically around 10–12 dB CNR. Above threshold, FM outperforms AM by 3β²(β+1) — for commercial FM with β = 5, about 25 dB advantage. The threshold effect is why FM is unsuitable for very weak signals (deep-space probe downlinks), where SSB or phase-coherent schemes with lower threshold are preferred.
Angle modulation is the collective term for FM and PM, where the instantaneous angle (phase or frequency) of the carrier is varied by the message. It is a nonlinear modulation scheme — the spectrum of the modulated signal is not simply a shifted version of the message spectrum (unlike AM/DSB/SSB, which are linear). This nonlinearity means angle modulation generates infinite sidebands (described by Bessel functions) and is why FM requires more bandwidth than AM for the same message, yet delivers better noise performance above the threshold.
Several methods exist: (1) Slope detector — uses the sloped response of a tuned circuit to convert frequency to amplitude, then envelope-detects. Simple but non-linear. (2) Foster-Seeley discriminator — balanced transformer circuit; good linearity. (3) Ratio detector — modified Foster-Seeley with built-in AM rejection; used in older FM radios. (4) Phase-Locked Loop (PLL) — the most common modern approach; the VCO tracks the instantaneous frequency and the error voltage is the demodulated output. High linearity, good noise rejection. (5) Quadrature detector — used in FM ICs (e.g. LM3361, CA3089); cheap and simple for consumer electronics.
