Communication Systems · TOPIC LESSON
Follow a message from source to receiver and derive AM envelopes, sidebands and detection conditions.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. Check the current official syllabus separately; historical question frequency does not establish eligibility or predict future questions.
CONNECT THE CONCEPT TO THE EXAM
These selected examples are separate from the original practice below. Each identifies its exam, year and wording. They do not represent every paper or every selected year.
No source-verified exam questions have been published for this lesson yet. The worked examples and practice below are original teaching material, not past-exam questions.
01 / UNDERSTAND & APPLY
A basic communication chain has an information source, an input transducer when needed, a transmitter, a channel, a receiver and an output transducer or destination. A microphone converts sound into an electrical signal; a transmitter prepares that signal for the channel; a receiver selects and reconstructs the message; a loudspeaker can convert the recovered electrical signal back into sound.
The channel is the physical path, such as a guided cable, optical fibre or free-space radio link. Noise adds unwanted fluctuations, attenuation reduces signal strength and distortion changes the signal’s shape or relative frequency components. An amplifier can restore amplitude but generally amplifies noise present at its input too; it does not automatically recover lost information.
A signal’s bandwidth is the range of frequencies that must be conveyed with acceptable fidelity under the stated model. It is different from its highest carrier frequency and from its average power. A narrow high-frequency carrier can carry a low-frequency message only after some property of that carrier is varied to encode the message.
Analog information varies continuously in the chosen representation, while digital communication represents information with discrete symbols. Digital symbols still travel through physical analog waveforms and are subject to noise and bandwidth limits. This lesson emphasizes the elementary amplitude-modulation model, with brief comparisons to other methods rather than a claim to cover every modern coding system.
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02 / UNDERSTAND & APPLY
Modulation varies a property of a carrier according to a message. In amplitude modulation, the carrier amplitude changes; in frequency modulation, its instantaneous frequency changes; in phase modulation, its phase changes. The carrier provides a high-frequency oscillation on which slower message variation is encoded.
Antenna dimensions illustrate one reason for modulation. Efficient simple resonant antennas often have lengths comparable to a fraction of wavelength, such as a quarter wavelength for an idealized monopole with an appropriate ground reference. An audio-frequency electromagnetic wavelength is enormous, making direct low-frequency radiation inefficient for a small ordinary antenna. A higher carrier frequency permits a much smaller resonant structure.
Modulation also allows different messages to occupy distinct carrier bands so a receiver can select one without simply mixing every source together. This is a frequency-allocation and filtering idea. Modulation does not necessarily increase total radiated power or eliminate noise; it changes how information is represented in the spectrum.
The formula ℓ ≈ λ/4 is an illustrative antenna scale, not a universal minimum antenna length or a complete design rule. Real antennas can use other geometries, loading and matching, with efficiency and bandwidth tradeoffs. Use the ideal scale only when the question specifies that model.
ORIGINAL WORKED EXAMPLE
Using c = 3.0 × 10⁸ m s⁻¹, compare quarter-wavelength lengths at 1.0 kHz and 100 MHz.
Answer 75 km at 1.0 kHz; 0.75 m at 100 MHz in the stated model.
03 / UNDERSTAND & APPLY
Let a carrier be A_c cosω_ct and a single-tone message be represented by cosω_mt, with ω_c much greater than ω_m. Conventional full-carrier AM can be written s(t) = A_c[1 + μ cosω_mt]cosω_ct. The dimensionless modulation index μ measures the peak fractional carrier-amplitude variation in this normalized message convention.
For 0 ≤ μ ≤ 1, the nonnegative envelope ranges from A_min = A_c(1 − μ) to A_max = A_c(1 + μ). Therefore μ = (A_max − A_min)/(A_max + A_min). This formula uses envelope peak magnitudes under the ordinary undistorted full-carrier model, not arbitrary signed instantaneous samples of the rapid carrier.
At μ = 0 the carrier is unmodulated. At μ = 1 the minimum envelope just reaches zero. For μ > 1, the factor 1 + μcosω_mt changes sign, producing phase reversals and an envelope that no longer reproduces the message by simple positive-envelope following. This is overmodulation for the elementary envelope-detector receiver, although other modulation schemes can use different detection methods and conventions.
The message amplitude alone does not determine μ without the modulator’s scaling definition. In a common equation s = [A_c + A_m cosω_mt]cosω_ct, μ = A_m/A_c. More general modulators include a sensitivity coefficient. Always inspect the provided signal equation rather than divide two unrelated voltage readings.
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ORIGINAL WORKED EXAMPLE
A conventional AM waveform has maximum envelope amplitude 12 V and minimum envelope amplitude 4 V. Find carrier amplitude and modulation index.
Answer Carrier amplitude 8 V; μ = 0.50.
04 / UNDERSTAND & APPLY
Expand the AM expression with cos a cos b = [cos(a + b) + cos(a − b)]/2. The result contains A_c cosω_ct plus two terms of amplitude μA_c/2 at angular frequencies ω_c + ω_m and ω_c − ω_m. Thus a single-tone conventional AM signal has a carrier and upper and lower sidebands.
The sidebands carry the changes that encode the message. The carrier remains useful for simple envelope detection but carries no varying message by itself in this model. Merely adding an audio waveform to a carrier without the required multiplicative or nonlinear operation does not produce the same AM sideband structure.
For a baseband message containing frequencies from near zero to a highest frequency B_m, ordinary double-sideband AM occupies the interval approximately f_c − B_m to f_c + B_m, giving transmission bandwidth 2B_m. The two sidebands each contain a shifted version of the message spectrum. Single-sideband and suppressed-carrier systems use different transmission/detection arrangements.
For a pure tone, the three ideal spectral components are lines, but the frequency span between sidebands is still 2f_m. Practical bandwidth also depends on filtering and definitions of acceptable spectral leakage. A finite-duration signal cannot have infinitely sharp eternal spectral lines, so the ideal line picture is a teaching model.
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ORIGINAL WORKED EXAMPLE
A signal is s(t) = 10[1 + 0.60 cos(2π × 2000t)]cos(2π × 500000t) V. Find carrier and sideband frequencies, sideband peak amplitudes and sideband-to-sideband span.
Answer Carrier 500 kHz; sidebands 498 and 502 kHz, each 3.0 V peak; span 4 kHz.
05 / UNDERSTAND & APPLY
For a conventional single-tone AM voltage across a resistance R, with A_c a peak voltage, carrier power is P_c = A_c²/(2R). Each sideband has peak amplitude μA_c/2 and power μ²P_c/4. Summing the three time-averaged components gives total power P_total = P_c(1 + μ²/2).
Total sideband power is μ²P_c/2. The fraction of total power in the sidebands is μ²/(2 + μ²). At μ = 1, that fraction is one third for this full-carrier single-tone model. This is a modulation-power efficiency measure, not the electrical efficiency of a transmitter’s power supply or amplifier.
Increasing modulation index raises sideband power while leaving the unmodulated carrier term unchanged in the given equation. Real transmitters have amplitude and power limits, so the ideal expression cannot be extended indefinitely without distortion. A multi-tone or general message requires its mean-square modulation value rather than blindly substituting one tone’s μ into every formula.
ORIGINAL WORKED EXAMPLE
A single-tone AM transmitter has carrier power 100 W and modulation index 0.80. Find total sideband power, total power and the sideband fraction.
Answer 32 W in sidebands; 132 W total; approximately 24% sideband fraction.
06 / UNDERSTAND & APPLY
A nonlinear device can generate frequency mixing because its output includes products of input signals. For a simple square-law model y = ax + bx², applying a carrier plus message produces terms including their product. A suitable band-pass filter around the carrier selects the carrier and nearby sum/difference components needed for AM while rejecting unwanted low-frequency and harmonic terms.
At a receiver, tuning selects the desired frequency band, amplification raises signal level, and demodulation extracts the message. A diode envelope detector rectifies the rapid oscillation and uses a capacitor/resistor network to follow its slowly varying amplitude. The capacitor charges near peaks and discharges between them; the output includes an offset that can be removed by suitable coupling.
A basic time-scale condition is carrier period much less than RC, while RC is short compared with the message-variation time scale. Too small RC leaves excessive carrier ripple; too large RC cannot follow a falling envelope and produces distortion. For deep modulation, detailed slope conditions can be stricter than this simple separation-of-scales rule.
An ordinary envelope detector requires a suitably non-overmodulated full-carrier input and adequate signal amplitude relative to diode behavior. A suppressed-carrier signal generally needs coherent detection rather than this simple method. Noise, diode drops and loading affect practical recovery even when the ideal signal algebra is correct.
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ORIGINAL WORKED EXAMPLE
A teaching AM system has carrier frequency 1.0 MHz and message frequency 1.0 kHz at moderate modulation depth. Compare RC values of 0.10 μs, 20 μs and 10 ms for a simple envelope detector.
Answer 20 μs is the plausible time scale among the supplied options.
07 / UNDERSTAND & APPLY
Ground-wave propagation follows the Earth’s surface with attenuation depending on frequency, ground conductivity and geometry. Sky-wave propagation uses the ionosphere’s frequency-dependent refracting effect to return some radio waves toward Earth. It is not a perfect mirror that reflects every electromagnetic frequency equally, and its conditions vary with time and environment.
Space-wave communication includes approximately line-of-sight direct and reflected paths through the lower atmosphere. At sufficiently high frequencies, simple links are often limited by horizon geometry, obstacles and propagation losses rather than reliable ionospheric return. Satellites and repeaters can extend coverage by providing another path, but they do not remove delay or signal-to-noise constraints.
For a spherical Earth of radius R and antenna height h much smaller than R, the geometric distance to the tangent horizon is approximately √(2Rh). For two antennas, a simple direct-path range estimate is √(2Rh_t) + √(2Rh_r). This ignores atmospheric refraction, terrain, diffraction and required clearance, so it is a model estimate rather than guaranteed communication coverage.
Optical fibres guide electromagnetic waves using their refractive-index structure, often explained initially through total internal reflection for suitable rays. Attenuation, dispersion and coupling losses still matter. A communication channel is chosen using bandwidth, loss, distance and environment, not just whether its carrier has the highest available frequency.
ORIGINAL WORKED EXAMPLE
Use an ideal spherical Earth of radius 6.4 × 10⁶ m. Two antennas have heights 20 m and 45 m. Estimate their combined geometric line-of-sight horizon range, ignoring atmospheric refraction and obstacles.
Answer Approximately 40 km.
08 / UNDERSTAND & APPLY
Identify the modulation convention, whether amplitudes are peak or rms, and whether a question concerns a message frequency, carrier frequency or occupied bandwidth. Keep propagation models distinct from modulation algebra.
ORIGINAL PRACTICE · 1
It converts the source information into a suitable physical signal representation.
A microphone, for example, converts sound variation into an electrical signal.
ORIGINAL PRACTICE · 2
No.
An ordinary amplifier raises both signal and noise present at its input.
ORIGINAL PRACTICE · 3
150 m.
λ = c/f = 3 × 10⁸/(2 × 10⁶).
ORIGINAL PRACTICE · 4
2/3, approximately 0.667.
μ=(10−2)/(10+2)=8/12.
ORIGINAL PRACTICE · 5
The minimum envelope reaches zero.
A_min=A_c(1−μ), at the boundary of the simple non-overmodulated model.
ORIGINAL PRACTICE · 6
The modulation factor changes sign and the detected positive envelope no longer follows the intended message directly.
This introduces phase reversals and envelope distortion in the stated full-carrier model.
ORIGINAL PRACTICE · 7
995 kHz and 1005 kHz.
They are f_c−f_m and f_c+f_m.
ORIGINAL PRACTICE · 8
8 kHz.
Both shifted sidebands together span 2B_m.
ORIGINAL PRACTICE · 9
2 V.
Each sideband amplitude is μA_c/2.
ORIGINAL PRACTICE · 10
90 W.
P_total=P_c(1+μ²/2)=60(1.5).
ORIGINAL PRACTICE · 11
No.
It is the sideband fraction of the ideal RF signal power, not power-supply conversion efficiency.
ORIGINAL PRACTICE · 12
No.
It should smooth rapid carrier oscillations while still tracking slower envelope changes.
ORIGINAL PRACTICE · 13
As √h.
d≈√(2Rh), so quadrupling height doubles the geometric horizon distance.
ORIGINAL PRACTICE · 14
No.
Its response depends on frequency and ionospheric conditions; many high-frequency links use space-wave paths instead.
09 / UNDERSTAND & APPLY
Begin with the information-bearing quantity and the transmitter/receiver chain. In an AM calculation, read the modulation index from the specified equation, then expand the product to obtain sidebands. Use envelope extrema only when the conventional positive-envelope assumptions hold. Keep peak amplitudes distinct from average powers.
For detection, check time-scale separation and whether the receiver type can recover the chosen modulation. For propagation, state the physical route and the approximations behind any range formula. Do not infer noise immunity, bandwidth or coverage from carrier frequency alone, and do not treat these foundational models as a statement of current exam syllabus inclusion or real-world radio authorization.
KEEP THE BIG IDEAS
READ WITH CONTEXT
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Factual cross-check only. Explanations, numerical examples and practice are independently authored.
Factual cross-check only. Explanations, numerical examples and practice are independently authored.
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