Analog Modulation and Demodulation

Introduction and Context

A communication system transfers information from a source to a destination through a physical channel. In many systems the information signal is a low-frequency or baseband waveform, whereas the channel is designed to operate efficiently over a different frequency range. Modulation is the controlled variation of a parameter of a carrier wave according to the information signal. Demodulation is the recovery of the information from the received modulated waveform.

Analog modulation remains important despite the dominance of digital communication. It is used in commercial broadcasting, legacy telemetry, aircraft communication, instrumentation, radio navigation, and as a conceptual foundation for digital modulation, sampling, frequency conversion, phase-locked loops, and software-defined radio.

The central design problem is a trade-off among bandwidth, transmitted power, noise performance, circuit complexity, linearity, and regulatory constraints. Amplitude modulation is generally simple but power-inefficient. Frequency and phase modulation require more bandwidth, but offer improved immunity to amplitude noise and can provide better signal quality under suitable conditions.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Explain the purpose and mathematical structure of analog modulation.
2. Derive spectra, bandwidth, power, and efficiency for principal AM systems.
3. Distinguish conventional AM, DSB-SC, SSB, and VSB.
4. Explain FM and PM, including their bandwidth and noise behavior.
5. Select suitable demodulators and evaluate their limitations.
6. Analyse practical receiver architectures, synchronization requirements, and regulatory issues.
7. Compare analog modulation methods in an engineering design context.

1. Signals, Carriers, and Modulation

Let the message or information signal be m(t), with highest significant frequency W hertz. A sinusoidal carrier may be written as

where A_c is carrier amplitude in volts, f_c is carrier frequency in hertz, and φ_c is carrier phase in radians. In a bandpass communication system, f_c is normally much greater than W.

A modulator maps m(t) into a waveform centred around f_c. A receiver generally performs the following functions: band selection, amplification, frequency translation if necessary, demodulation, filtering, and audio or data recovery. The recovered waveform is not necessarily identical to m(t); amplitude scaling, delay, distortion, and noise are unavoidable practical considerations.

For a real low-pass signal M(f), modulation translates its spectral content to positive and negative frequencies around the carrier. The translated copies are called sidebands. The spectral representation is particularly useful because bandwidth and interference are frequency-domain properties.

2. Conventional Amplitude Modulation

2.1 AM waveform

In conventional or double-sideband large-carrier AM, the carrier amplitude varies in proportion to the message:

Here m_n(t) is a normalized message satisfying |m_n(t)| ≤ 1, and μ is the modulation index. For a single-tone message m_n(t) = cos(2πf_m t), 0 ≤ μ ≤ 1 is required for envelope detection without envelope inversion.

For a general message m(t), an alternative form is

where k_a has units inverse to those of m(t). The no-overmodulation condition is |k_a m(t)| ≤ 1 for every t. If the condition is violated, the envelope crosses zero and a simple envelope detector produces severe distortion.

Using the identity cos α cos β = [cos(α + β) + cos(α − β)]/2, a single-tone AM signal becomes

Thus AM contains a carrier at f_c and two sidebands at f_c + f_m and f_c − f_m. For a message of bandwidth W, the occupied transmission bandwidth is

provided the sidebands are transmitted without intentional filtering.

2.2 AM power and efficiency

For a resistive load R, the carrier power is

for the peak carrier amplitude A_c. With a single-tone modulation index μ, each sideband has power μ²P_c/4, so total sideband power is μ²P_c/2. Therefore,

and the useful sideband power efficiency is

The maximum efficiency under ordinary envelope detection, μ = 1, is 1/3, or 33.3%. The remaining power is in the carrier, which contains no message information but supports simple demodulation.

For a general multicomponent message, power depends on the mean-square value of the normalized message. If m_n(t) has mean-square value P_m and zero mean, the sideband-to-carrier power ratio is μ²P_m, and the total power is P_c(1 + μ²P_m). The single-tone result follows from P_m = 1/2.

2.3 AM generation and detection

AM can be generated by a multiplier, a nonlinear device followed by filtering, or a controlled-gain amplifier. The transmitter must maintain carrier frequency stability and prevent overmodulation.

An envelope detector consists of a diode, capacitor, and load resistor. Its time constants should satisfy approximately

The first inequality allows the capacitor to follow carrier peaks while suppressing carrier ripple; the second allows it to follow the message envelope. These are design inequalities rather than exact universal rules. Excessively large RC causes diagonal clipping, whereas excessively small RC causes carrier-frequency ripple.

A synchronous detector multiplies the received AM signal by a locally generated carrier and low-pass filters the result. It is more complex but can demodulate suppressed-carrier signals and can be more tolerant of certain envelope distortions.

3. Suppressed-Carrier and Reduced-Bandwidth AM

3.1 Double-sideband suppressed-carrier AM

DSB-SC removes the carrier and transmits only the two sidebands:

For a single tone, the two sidebands each have amplitude A_c/2 when m_n(t) has unit amplitude. The bandwidth remains 2W, but all transmitted power is associated with the message. A coherent detector is required because the envelope no longer represents the message uniquely.

The local oscillator at the receiver must have the correct frequency and phase. A phase error θ scales the recovered signal by cos θ and introduces a quadrature component in more general signals. A 90° phase error can ideally suppress coherent recovery entirely.

3.2 Single-sideband modulation

Single-sideband (SSB) transmits only one sideband and suppresses or greatly reduces the carrier. Its bandwidth is

For equal peak message and ideal filtering, SSB requires approximately half the bandwidth of DSB. It also avoids transmitting redundant spectral information. SSB generation commonly uses a sharp crystal or mechanical filter, or a phasing method based on a Hilbert-transform network.

The upper-sideband and lower-sideband forms are not interchangeable in all systems. The receiver must use the correct carrier reinsertion frequency and phase. Frequency error produces pitch or spectral displacement in voice communication, so oscillator stability is important.

3.3 Vestigial-sideband modulation

VSB transmits one complete sideband and a vestige of the other. It is useful when the message contains significant low-frequency content and a sharply selective SSB filter would be difficult to realize. Analog television historically used VSB because it provided a practical compromise between bandwidth and filter complexity. VSB receivers require an appropriate Nyquist slope or equalization to avoid amplitude distortion near the carrier.

3.4 Spectrum illustration

Figure: Conceptual line-spectrum comparison of conventional AM, DSB-SC, and SSB for a single-tone message.

4. Angle Modulation: FM and PM

In angle modulation, the carrier amplitude remains constant while its instantaneous angle varies. The general signal is

where θ_i(t) is instantaneous phase. The instantaneous frequency is

4.1 Frequency modulation

For FM,

where k_f is the frequency sensitivity in hertz per unit of message amplitude. The instantaneous frequency deviation is Δf(t) = k_fm(t). For a single-tone signal m(t) = A_m cos(2πf_m t), the peak deviation is Δf = |k_fA_m| and the modulation index is

FM has infinitely many mathematical sidebands, although only a finite number are practically significant. Carson’s rule estimates the transmission bandwidth as

where Δf is the peak frequency deviation and W is the highest significant message frequency. Carson’s rule is an engineering approximation, commonly associated with retaining about 98% of the signal power for typical spectra; it is not an exact law for every waveform or required power criterion.

For a single tone, the exact spectral components occur at f_c ± nf_m, with amplitudes proportional to Bessel functions J_n(β). The carrier component has amplitude A_cJ_0(β), and the nth pair has amplitude A_cJ_n(β). At certain β values J_0(β) is zero, but the transmitted FM signal still contains power in sidebands.

4.2 Phase modulation

For PM,

where k_p is phase sensitivity in radians per unit of message amplitude. For a sinusoidal message of amplitude A_m, the phase modulation index is β_PM = |k_pA_m|. The corresponding instantaneous frequency deviation is proportional to the derivative of the message:

Consequently FM and PM are closely related. FM can be produced by integrating the message and applying PM; PM can be produced by differentiating the message and applying FM. In practical systems, pre-emphasis and de-emphasis networks modify this relationship and improve noise performance.

4.3 Narrowband and wideband angle modulation

For β ≪ 1, FM or PM can be approximated as a carrier plus small quadrature sidebands. This is narrowband FM/PM and occupies approximately 2W. When β is large, many sidebands become significant and the bandwidth increases. Wideband FM provides a substantial improvement in noise performance at the cost of bandwidth and transmitter linearity requirements.

5. Demodulation of FM and PM

An FM discriminator converts frequency variation into amplitude variation. A practical discriminator may use a tuned-slope network, a balanced discriminator, or a ratio detector. Since slope detectors are sensitive to amplitude noise, a limiter is usually placed before the discriminator to remove amplitude fluctuations.

A phase-locked loop (PLL) is a powerful coherent FM demodulator. A PLL includes a phase detector, loop filter, and voltage-controlled oscillator (VCO). When locked, the VCO control voltage tracks the input frequency deviation and therefore represents the message after suitable scaling. The loop must be designed with adequate capture range, lock range, noise bandwidth, and stability.

A PLL is not automatically superior in every situation. A narrow loop suppresses noise but may fail to track rapid modulation; a wide loop tracks modulation better but admits more noise. Loop-filter design is therefore a bandwidth and dynamic-response compromise.

PM may be demodulated with a phase detector or PLL. A practical phase detector compares the received carrier phase with a reference. Carrier recovery is essential for suppressed-carrier systems and must be robust against frequency offset and noise.

6. Noise Performance and Threshold Effects

Additive white Gaussian noise (AWGN) is a common analytical model. It does not describe every radio channel, but it provides a useful baseline. AM envelope detection converts amplitude noise directly into output noise. FM limiting removes amplitude noise before frequency discrimination, so FM can achieve improved output signal-to-noise ratio above a threshold signal level.

For an ideal FM system with sinusoidal modulation and sufficiently high carrier-to-noise ratio, the output signal-to-noise ratio is approximately

where β is the FM modulation index and the relationship assumes a particular normalized noise model, ideal limiting and discrimination, and operation above threshold. It should not be applied indiscriminately to arbitrary message spectra or low input levels.

FM exhibits a threshold effect: below a critical input signal-to-noise ratio, the output quality deteriorates rapidly rather than gradually. Impulse noise, multipath propagation, oscillator phase noise, and nonlinear distortion can also limit performance. Pre-emphasis boosts high-frequency message components before transmission; de-emphasis attenuates them at the receiver, reducing high-frequency noise while approximately restoring the original spectrum.

7. Transmitter and Receiver Architectures

A superheterodyne receiver converts the selected RF signal to a fixed intermediate frequency (IF). A typical chain is antenna → RF filter and low-noise amplifier → mixer → IF filter and amplifier → detector → baseband amplifier. Fixed IF filtering provides predictable selectivity and gain.

Important practical measures include:

• Automatic gain control (AGC): maintains an appropriate detector input level, especially for AM.
• Automatic frequency control (AFC): reduces frequency offset in some receivers.
• Limiter: removes amplitude variation before FM detection.
• Image-rejection filtering: prevents an undesired frequency from mixing to the same IF.
• Local-oscillator stability: reduces tuning error and demodulation distortion.
• Linearity: prevents intermodulation and adjacent-channel interference.

A direct-conversion receiver mixes the RF signal directly to baseband. It can be compact and integrated, but DC offsets, oscillator leakage, flicker noise, and I/Q imbalance require careful design. Modern software-defined radios digitize an intermediate-frequency or baseband signal and implement filtering and demodulation numerically, although analog front-end linearity and sampling constraints remain essential.

8. Worked Engineering Examples

Example 1: Conventional AM power and bandwidth

A 1 MHz carrier has peak amplitude 10 V and modulates a 5 kHz sinusoid with μ = 0.8. The load is 50 Ω.

Carrier power:

Total transmitted power:

Each sideband has power μ²P_c/4 = 0.16 W, and the two sidebands together carry 0.32 W. The bandwidth is B_AM = 2f_m = 10 kHz. The efficiency is 0.32/1.32 ≈ 24.2%.

Example 2: FM bandwidth

An FM transmitter has maximum deviation Δf = 75 kHz and message bandwidth W = 15 kHz. Carson’s estimate is

This is an estimate based on the stated deviation and message bandwidth. If the message amplitude or highest frequency changes, the occupied bandwidth also changes.

Example 3: SSB carrier accuracy

Suppose a suppressed-carrier voice receiver uses a 1.000 MHz carrier reinsertion oscillator. If the oscillator is high by 100 Hz, every recovered voice spectral component is displaced by approximately 100 Hz. This can produce a noticeable pitch error in voice, demonstrating why frequency accuracy is more critical in SSB than in ordinary envelope-detected AM.

9. Applied Case: Selecting a Modulation Method for a Remote Instrument Link

Background. An illustrative industrial system must transmit a 0–5 kHz analog sensor signal over a licensed narrowband radio link. The receiver is located in an electrically noisy plant. Available options are conventional AM, SSB, and FM.

Problem. The link must fit a constrained channel, tolerate amplitude interference, and remain maintainable by technicians familiar with conventional RF equipment.

Analysis. Conventional AM requires approximately 10 kHz bandwidth and places a large fraction of power in an uninformative carrier. It is simple to detect but vulnerable to amplitude noise. SSB requires approximately 5 kHz bandwidth and is power-efficient, but carrier recovery and filter alignment increase complexity. FM rejects amplitude noise after limiting and is operationally robust, but Carson’s bandwidth depends on the selected deviation. For example, Δf = 10 kHz and W = 5 kHz gives approximately 30 kHz, possibly exceeding the allocation.

Approach. If the regulator permits only a 6 kHz channel, SSB is the technically strongest choice, provided the system can maintain carrier accuracy and provide calibration. If a 25–30 kHz channel is available, narrowband FM may be preferable because of its noise resilience and simpler operational diagnosis. In either case, input anti-aliasing or reconstruction is not relevant to a purely analog link, but sensor conditioning, shielding, grounding, and surge protection are essential.

Outcome. The decision is not based on modulation efficiency alone. It combines bandwidth allocation, interference environment, receiver complexity, maintenance capability, oscillator stability, and licensing constraints.

Learning points. Modulation selection is a systems-engineering decision. The nominal bandwidth, power, and noise formulas must be interpreted together with channel regulation and implementation risks.

10. Limitations, Regulation, and Contemporary Practice

Analog systems are vulnerable to component drift, nonlinear distortion, adjacent-channel interference, and gradual quality degradation. They may also require high-quality inductors, filters, and stable oscillators. Digital systems often provide stronger error control and flexible processing, but analog modulation remains embedded in RF front ends and is still valuable where low latency, graceful degradation, or legacy compatibility matters.

Regulatory authorities specify carrier frequencies, occupied bandwidth, maximum power, out-of-band emissions, frequency deviation, and electromagnetic compatibility limits. Engineers must not infer legal compliance solely from a theoretical bandwidth equation. Actual occupied bandwidth depends on filtering, transient behavior, modulation statistics, and nonlinearities. Power must be measured using the regulator’s specified detector and bandwidth.

Best practice includes maintaining spectral masks, using calibrated test equipment, documenting modulation index and deviation, checking spurious emissions, providing fail-safe power levels, and protecting communications from intentional or accidental interference. In safety-critical systems, analog demodulator failure modes should be analysed rather than assuming that a loss of signal produces a safe output.

Chapter Summary

Analog modulation translates a baseband signal to a suitable carrier-frequency range, while demodulation recovers the information. Conventional AM transmits a carrier and two sidebands, occupies twice the message bandwidth, and is simple to envelope-detect but power-inefficient. DSB-SC removes the carrier and requires coherent detection. SSB transmits one sideband, halves the ideal bandwidth, and demands accurate carrier reinsertion. VSB provides a practical compromise between bandwidth and filter complexity.

FM and PM vary carrier angle rather than amplitude. Their constant-envelope property enables limiting and improved resistance to amplitude noise, but angle modulation generally requires more bandwidth and can exhibit a threshold effect. Carson’s rule is a useful FM bandwidth estimate, not an exact universal result. PLLs, discriminators, synchronous detectors, envelope detectors, filters, AGC, and frequency-conversion stages form the practical foundation of analog receivers.

Engineering selection must consider bandwidth, power, noise, synchronization, complexity, regulation, and maintenance. These principles also underpin modern software-defined radios and digital communication systems.

Key Terms

Modulation: Controlled variation of a carrier parameter by an information signal.

Demodulation: Recovery of information from a modulated waveform.

Carrier: Periodic high-frequency waveform used to transport information.

Baseband: Original information-frequency range before modulation.

Modulation index: Dimensionless measure of modulation depth or deviation.

Sideband: Spectral component displaced from the carrier by message frequencies.

Conventional AM: AM with carrier and both sidebands transmitted.

DSB-SC: Double-sideband suppressed-carrier modulation.

SSB: Single-sideband modulation transmitting only one sideband.

VSB: Vestigial-sideband modulation transmitting one full and one partial sideband.

Envelope detector: Noncoherent AM detector following the signal envelope.

Synchronous detector: Detector using a phase- and frequency-related carrier.

Frequency modulation: Modulation in which instantaneous frequency varies with the message.

Phase modulation: Modulation in which instantaneous phase varies with the message.

Frequency deviation: Maximum departure of instantaneous frequency from carrier frequency.

Carson’s rule: Approximate FM bandwidth relation B ≈ 2(Δf + W).

PLL: Feedback system that locks oscillator phase to an input signal.

Pre-emphasis: Boosting selected message frequencies before transmission.

Threshold effect: Abrupt degradation of angle-modulation quality below adequate input SNR.

Occupied bandwidth: Frequency range containing the specified proportion of transmitted power.

Review Questions

Short Answer Questions

1. Why is modulation required in a bandpass communication system?
2. State the no-overmodulation condition for conventional AM.
3. Why does DSB-SC require coherent detection?
4. Define FM modulation index for a single-tone message.
5. What is the purpose of a limiter in an FM receiver?

Long Answer Questions

1. Derive the spectrum, bandwidth, power, and efficiency of single-tone conventional AM.
2. Compare conventional AM, DSB-SC, SSB, and VSB in terms of bandwidth, power, receiver complexity, and applications.
3. Explain the operation and design trade-offs of an FM PLL demodulator.
4. Discuss the relationship between FM and PM and derive the instantaneous-frequency expression for PM.
5. Explain threshold effects, pre-emphasis, de-emphasis, and the role of noise in angle modulation.

Analytical / Application Questions

1. A 50 Ω AM transmitter has carrier peak amplitude 20 V and modulation index 0.6. Calculate carrier power, total power for a single-tone message, sideband power, and efficiency.
2. An FM link carries a signal whose highest frequency is 12 kHz with peak deviation 40 kHz. Estimate its bandwidth using Carson’s rule and discuss what additional information is needed for regulatory compliance.
3. Select AM, SSB, or FM for a narrowband industrial link with strong amplitude interference. Justify the decision using bandwidth, synchronization, noise, and maintenance considerations.

Multiple Choice Questions with Answers

Question 1

For a single-tone conventional AM signal with modulation index μ, the total transmitted power is

A. P_c(1 + μ)

B. P_c(1 + μ²/2)

C. P_c(1 + 2μ²)

D. P_cμ²/2

Correct Answer: B. P_c(1 + μ²/2)

Question 2

The ideal bandwidth of an AM signal carrying a message of bandwidth W is

A. W/2

B. W

C. 2W

D. 4W

Correct Answer: C. 2W

Question 3

At μ = 1, the maximum theoretical sideband power efficiency of conventional AM is

A. 25%

B. 33.3%

C. 50%

D. 100%

Correct Answer: B. 33.3%

Question 4

Which detector is normally suitable for conventional AM when there is no overmodulation?

A. Envelope detector

B. Frequency discriminator only

C. Limiter alone

D. Differentiator only

Correct Answer: A. Envelope detector

Question 5

Compared with DSB-SC, ideal SSB transmission

A. doubles bandwidth

B. uses the same bandwidth and twice the power

C. halves bandwidth by transmitting one sideband

D. requires no carrier synchronization

Correct Answer: C. halves bandwidth by transmitting one sideband

Question 6

For FM with peak deviation Δf and message bandwidth W, Carson’s approximate bandwidth is

A. Δf + W

B. 2(Δf + W)

C. 2Δf − W

D. W/Δf

Correct Answer: B. 2(Δf + W)

Question 7

For a sinusoidal FM signal, the modulation index is

A. f_m/Δf

B. Δf/f_m

C. Δf + f_m

D. f_c/Δf

Correct Answer: B. Δf/f_m

Question 8

The main purpose of a limiter before an FM discriminator is to

A. increase carrier amplitude indefinitely

B. remove amplitude variations caused by noise

C. recover the suppressed carrier

D. double the message bandwidth

Correct Answer: B. remove amplitude variations caused by noise

Question 9

A PM demodulator must account for the fact that instantaneous frequency depends on

A. the integral of message amplitude only

B. the derivative of message amplitude

C. carrier power only

D. load resistance only

Correct Answer: B. the derivative of message amplitude

Question 10

An AM transmitter has P_c = 2 W and μ = 0.5 for a single-tone message. Its total power is

A. 2.125 W

B. 2.25 W

C. 2.5 W

D. 3 W

Correct Answer: A. 2.125 W