Analog Modulation¶

Exam-ready answer

Modulation is the process by which a baseband message controls a parameter of a high-frequency carrier \(c(t)=A_c\cos(2\pi f_ct+\phi_c)\). AM varies \(A_c\), FM varies instantaneous frequency and PM varies phase. The operation translates the message spectrum from near zero frequency to a selected passband while ideally preserving its information.

Modulation is necessary for: practical antennas, because \(\lambda=c/f\) and resonant dimensions are fractions of wavelength; efficient radiation and channel matching, because antennas, microwave links and other bandpass media work in assigned bands; frequency allocation and selective tuning, so a receiver can select one station; multiplexing, so independent messages occupy different carriers; and noise/interference planning, because carrier placement permits practical filtering and a modulation appropriate to the link.

For a \(3\,\text{kHz}\) message, \(\lambda=3\times10^8/(3\times10^3)=100\,\text{km}\), giving an impractical quarter-wave antenna of \(25\,\text{km}\). At a \(100\,\text{MHz}\) carrier, \(\lambda=3\,\text{m}\) and a quarter-wave is \(0.75\,\text{m}\). Separate carriers also form the nonoverlapping FDM channels shown; guard bands accommodate filters and prevent adjacent overlap.

Modulation does not by itself remove noise or guarantee greater range. It consumes bandwidth and circuit resources, and the best carrier depends on spectrum regulation, propagation, antenna gain, available power and receiver sensitivity. Applications include broadcast AM/FM, aeronautical radio, microwave/satellite links and cellular systems.

Practice target: 7-8 minutes; define modulation, calculate antenna size, and explain at least five independent reasons.

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In standard full-carrier amplitude modulation, the carrier amplitude varies linearly with the instantaneous message while carrier frequency and phase remain fixed. Let

With amplitude sensitivity \(k_a\), the modulation index is \(\mu=k_aA_m\) and

Using \(\cos A\cos B=\tfrac12[\cos(A+B)+\cos(A-B)]\),

Thus the spectrum contains a carrier at \(f_c\), upper sideband at \(f_c+f_m\) and lower sideband at \(f_c-f_m\). For a message limited to \(f_{m(max)}\) hertz, each sideband is a translated copy of the message and

From an oscilloscope envelope,

For \(0<\mu<1\) the envelope is faithful; at \(\mu=1\) it just reaches zero; for \(\mu>1\) it reverses/crosses zero and an envelope detector produces severe overmodulation distortion.

Across load \(R\) ohms, carrier power is \(P_c=A_c^2/(2R)\). Each sideband has amplitude \(\mu A_c/2\), hence

At the largest distortion-free single-tone value \(\mu=1\), \(P_T=1.5P_c\) and \(\eta_{max}=1/3=33.33\%\). The carrier consumes power but contains no unique message information, while USB and LSB carry duplicate information for a real message; this motivates DSB-SC and SSB. A low-level AM transmitter modulates before linear RF amplification, whereas high-level AM varies the final power stage. A diode-\(RC\) envelope detector is simple when \(\mu\leq1\); coherent detection is more linear but needs carrier synchronization.

As a check, if \(P_c=100\,\text{W}\) and \(\mu=0.6\), each sideband is \(9\,\text{W}\), total power is \(118\,\text{W}\) and efficiency is \(18/118=15.25\%\). Spectrally, a \(5\,\text{kHz}\) tone on \(1\,\text{MHz}\) produces lines at \(995\), \(1000\) and \(1005\,\text{kHz}\) with \(10\,\text{kHz}\) null span between sidebands.

Practice target: 16-18 minutes; derive the three spectral terms, draw the envelope/spectrum, and derive all four power relations.

Exam-ready answer

For single-tone standard AM, the carrier power is \(P_c\), each sideband power is \(\mu^2P_c/4\), total sideband power is \(\mu^2P_c/2\), and total transmitted power is

Given \(P_c=500\,\text{W}\) and \(\mu=0.8\), the upper-sideband power is

By spectral symmetry, the lower sideband contains the same power:

Therefore \(P_{SB}=80+80=160\,\text{W}\) and

Transmission efficiency is the information-bearing sideband power divided by total transmitted power:

The same result follows directly from \(\eta=\mu^2/(2+\mu^2)=0.64/2.64\). The power balance \(500+80+80=660\,\text{W}\) is the numerical check. Since \(\mu=0.8<1\), the signal is under-modulated and its envelope does not cross zero, so ideal envelope detection is possible. The carrier alone consumes \(500/660=75.76\%\) of total power without carrying unique message information, illustrating standard AM's low power efficiency. At \(\mu=1\), the theoretical maximum efficiency would still be only \(33.33\%\); increasing \(\mu\) above unity is not a valid efficiency remedy because it causes overmodulation and envelope-detector distortion.

Practice target: 7-8 minutes; state the formulas, show units at every result, and verify that carrier plus both sidebands equals total power.

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All four schemes translate a baseband message to a band around \(f_c\), but differ in retained carrier/sidebands, bandwidth, power and detector complexity.

Standard AM is \(s(t)=A_c[1+k_am(t)]\cos\omega_ct\). It transmits the full carrier and both sidebands, occupies \(2B_m\) hertz for message bandwidth \(B_m\), and permits a simple envelope detector if \(|k_am(t)|\leq1\). For a single tone, \(P_T=P_c(1+\mu^2/2)\) and maximum useful-sideband efficiency is only \(33.33\%\). Its simplicity suits MF broadcast and some aviation voice.

DSB-SC is \(s(t)=A_cm(t)\cos\omega_ct\). It suppresses the carrier but keeps both translated sidebands, so bandwidth remains \(2B_m\). All ideal transmitted power is information-bearing, but a product detector with a phase/frequency-synchronized carrier is compulsory. It is used as a modulation building block, in stereo subcarriers and in I/Q systems.

SSB-SC retains only USB or LSB. With Hilbert transform \(\hat m(t)\),

Its bandwidth is \(B_m\) and it removes carrier plus the redundant sideband, giving the best AM-family bandwidth/power economy. It is generated by DSB-SC plus a sharp filter, quadrature phase cancellation or Weaver processing. Product detection with a BFO/PLL is required; frequency error shifts recovered audio pitch. It is preferred for HF, marine and aeronautical long-distance voice.

VSB transmits one full sideband and a small vestige \(B_v\) of the other, usually with a residual/full carrier according to system design. Its approximate bandwidth is \(B_m+B_v\), between SSB and DSB. The vestige allows a realizable transition near the carrier when the message extends to DC; classical analog television video is the standard application.

For \(f_c=1\,\text{MHz}\) and \(B_m=5\,\text{kHz}\), AM and DSB-SC occupy \(995\)-\(1005\,\text{kHz}\), SSB occupies either \(1000\)-\(1005\,\text{kHz}\) or \(995\)-\(1000\,\text{kHz}\), and VSB is only slightly wider than \(5\,\text{kHz}\). An envelope detector fails on ordinary DSB-SC/SSB because their envelope does not preserve the signed message. Thus standard AM trades efficiency for simplicity, SSB maximizes economy, and VSB is a practical filtering compromise.

Practice target: 16-18 minutes; draw aligned spectra, then compare components, bandwidth, power, detection, applications and limitations.

Exam-ready answer

In double-sideband suppressed-carrier (DSB-SC) modulation, an ideal product modulator multiplies the message by the carrier:

For \(m(t)=A_m\cos\omega_mt\),

Only USB and LSB appear; no standalone term at \(f_c\) exists, and bandwidth is \(2f_{m(max)}\).

A balanced transistor multiplier or four-diode ring generates DSB-SC. Symmetry makes equal carrier-feedthrough terms cancel while cross-products add. Mismatch and transformer imbalance leave a practical residual carrier, so an output bandpass filter removes switching harmonics and unwanted products.

At the receiver, multiplication by \(2\cos(\omega_ct+\phi)\) gives

The LPF rejects the \(2f_c\) component and produces \(v_o(t)=A_cm(t)\cos\phi\). Thus phase error scales the output; at \(\phi=90^\circ\) it vanishes. A frequency error \(\Delta\omega\) produces \(m(t)\cos(\Delta\omega t+\phi)\), causing beats/fading, so a pilot, squaring loop or Costas loop restores carrier synchronization.

An envelope detector fails because the visible DSB-SC envelope is proportional to \(|m(t)|\), not signed \(m(t)\); each negative message excursion causes a \(180^\circ\) carrier-phase reversal that the envelope discards. Coherent detection preserves that sign and phase. DSB-SC saves carrier power but not bandwidth and is used as an SSB/IQ building block.

Practice target: 7-8 minutes; derive the two sidebands and the LPF output, then state phase/frequency-error effects and why envelope detection fails.

Exam-ready answer

Single-sideband suppressed-carrier (SSB-SC) transmits only the upper or lower sideband of a DSB-SC signal and ideally suppresses the carrier. For real message \(m(t)\) and its Hilbert transform \(\hat m(t)\),

The quadrature terms cancel one translated sideband and reinforce the other. If message bandwidth is \(B_m=f_{m(max)}\) hertz, then \(B_{SSB}=B_m\), half the bandwidth of AM/DSB-SC. Removing the carrier and one redundant sideband also gives the best power economy in the AM family.

Filter method: a balanced modulator first creates DSB-SC; a sharp crystal/mechanical bandpass filter passes USB or LSB. Generation is often at a fixed IF where narrow transition filters are practical, followed by mixing to the assigned RF. It becomes difficult when very low message frequencies make the two sidebands closely spaced.

Phase-shift method: one branch uses \(m(t)\cos\omega_ct\) and the other uses \(\hat m(t)\sin\omega_ct\). Accurate \(90^\circ\) message/carrier phase shifts and equal amplitudes allow addition/subtraction to cancel one sideband. Broadband phase/amplitude accuracy is its limitation. The Weaver method instead uses two quadrature mixing stages and low-pass filters and is convenient in DSP/IC systems.

Detection uses a product detector and reinserted carrier from a BFO, PLL or pilot. Mixing the selected sideband with \(2\cos[(\omega_c+\Delta\omega)t+\phi]\) translates it to baseband. Phase error changes recovered amplitude/phase, while carrier-frequency error \(\Delta f\) shifts every recovered audio component by that amount; speech then sounds unnaturally pitched. An ordinary envelope detector cannot recover suppressed-carrier SSB faithfully.

Advantages are half AM bandwidth, lower transmitter power, less received noise because of narrower filtering, and reduced selective-fading distortion. Limitations are generation/filter complexity, need for linear amplification, accurate tuning/carrier reinsertion and poor fidelity when the BFO is offset. SSB is used for HF long-distance, aeronautical, marine and amateur voice. For a \(300\)-\(3000\,\text{Hz}\) voice band on \(10\,\text{MHz}\) USB, the occupied RF is \(10.0003\)-\(10.0030\,\text{MHz}\), only \(2.7\,\text{kHz}\); DSB would require twice that span. This numerical placement also shows why a receiver-frequency error of even tens of hertz is audible.

Practice target: 16-18 minutes; write the Hilbert expressions, draw both generation methods, and cover detection error, advantages, limitations and applications.

Exam-ready answer

In angle modulation, carrier amplitude is constant and information changes instantaneous angle:

In frequency modulation (FM), instantaneous frequency deviation is proportional to the message:

where \(k_f\) is in hertz per message unit. Integrating angular frequency gives

For \(m(t)=A_m\cos2\pi f_mt\), peak deviation is \(\Delta f=k_fA_m\) hertz and

In phase modulation (PM), phase deviation is directly proportional to the message:

where \(k_p\) is radians per message unit. For the same sinusoid, \(\beta_p=k_pA_m=\Delta\phi_{max}\) radians and

The waveform shows constant envelope in both cases. In FM, cycles compress or spread according to message amplitude; in PM, greatest instantaneous frequency departure occurs where message slope is greatest. For fixed \(A_m\), FM peak deviation is independent of \(f_m\) while \(\beta\) decreases as \(f_m\) rises. PM phase index is independent of \(f_m\), but its frequency deviation rises in proportion to \(f_m\). FM can therefore be generated by integrating \(m(t)\) before a PM modulator; PM can be generated by differentiating \(m(t)\) before an FM modulator.

Both allow efficient nonlinear RF power amplification and amplitude limiting, but generally require more bandwidth than AM. Practical FM uses transmitter pre-emphasis to boost high audio frequencies and complementary receiver de-emphasis to restore response while suppressing high-frequency demodulated noise. FM is detected by a discriminator or PLL; PM uses a phase detector/PLL, or an FM discriminator followed by integration.

For FM with \(\Delta f=50\,\text{kHz}\) and \(f_m=10\,\text{kHz}\), \(\beta=5\). For PM with \(\beta_p=2\) radians at the same \(f_m\), \(\Delta f=20\,\text{kHz}\). Total ideal angle-modulated power is constant, \(P_T=A_c^2/(2R)\), but it is distributed among carrier and many Bessel sidebands. Capture and threshold effects limit weak-signal FM, and oscillator/phase noise limits both schemes.

Practice target: 16-18 minutes; derive both expressions from instantaneous angle, define indices with units, then compare deviation behavior and generation/detection.

Exam-ready answer

For a sinusoidal message, an FM signal is

where \(\Delta f\) is peak frequency deviation in hertz and \(f_m\) is modulating frequency in hertz. Narrowband FM (NBFM) has \(\beta\ll1\); using the small-angle approximation, only the carrier and first sideband pair are important and bandwidth is roughly \(2f_m\). Wideband FM (WBFM) has \(\beta>1\) and several significant sideband pairs, giving better potential noise performance at the cost of bandwidth. These are engineering descriptions rather than an abrupt physical boundary.

The exact single-tone Bessel expansion is

Therefore spectral lines occur at \(f_c\pm nf_m\) for \(n=0,1,2,\ldots\); carrier amplitude is \(A_cJ_0(\beta)\) and each \(n\)th sideband amplitude follows \(A_cJ_n(\beta)\) with sign representing phase. The theoretical spectrum is infinite, but high-order terms become negligible. At zeros of \(J_0\), the spectral carrier component vanishes even though the transmitted waveform still has constant amplitude. The identity \(\sum_{n=-\infty}^{\infty}J_n^2(\beta)=1\) proves that total power remains \(A_c^2/(2R)\) while being redistributed among lines.

An engineering estimate containing about \(98\%\) of power is Carson's rule:

for a single highest-frequency tone. It is approximate; exact occupied bandwidth depends on message spectrum and the specified power mask. For broadcast values \(\Delta f=75\,\text{kHz}\) and \(f_{m(max)}=15\,\text{kHz}\), \(\beta_{max}=5\) and

A nominal \(200\,\text{kHz}\) channel therefore leaves allowance for filters/guarding. By comparison, an AM signal carrying \(15\,\text{kHz}\) audio needs only \(30\,\text{kHz}\). WBFM uses this extra spectrum to gain noise immunity above threshold; a limiter removes amplitude noise, and pre-emphasis/de-emphasis improves high-frequency audio SNR. Below the FM threshold, however, output noise rises abruptly, and the capture effect favors the stronger co-channel signal. These limitations mean wider bandwidth is not a free improvement and must be supported by channel allocation and adequate input carrier-to-noise ratio.

Practice target: 16-18 minutes; define both FM classes, label Bessel lines and amplitudes, prove constant power, and complete one Carson calculation.

Exam-ready answer

An FM generator must produce \(f_i(t)=f_c+k_fm(t)\), and an FM detector must convert deviation \(f_i-f_c\) back to a proportional voltage.

Direct FM: the message controls a VCO, varactor-tuned LC oscillator or reactance modulator, directly changing resonant frequency. It is simple and can provide large deviation, but modulating the oscillator can impair center-frequency stability. A slow PLL may stabilize the average carrier while allowing message-rate deviation.

Armstrong indirect FM: a crystal oscillator supplies a stable carrier; the message is integrated and applied to a phase modulator to create narrowband FM. Frequency multipliers increase both carrier frequency and deviation, and mixers place the result in the assigned band without changing deviation. Its frequency stability is excellent, but it needs more stages and careful multiplication/mixing.

Every detector has an S-shaped transfer near \(f_c\), with zero output at center and opposite polarities above/below it.

• A slope detector places \(f_c\) on a tuned-circuit slope so FM becomes AM, then envelope-detects it. It is simple but nonlinear, narrow-range and amplitude-noise sensitive; a limiter is required. A balanced pair improves symmetry.
• A Foster-Seeley discriminator uses a double-tuned transformer to convert frequency deviation into phase/diode-voltage imbalance. It has high output and very good linearity but responds to amplitude, so a preceding limiter is essential.
• A ratio detector uses related transformer/diode connections and a large capacitor to hold the diode-voltage sum approximately constant. The output ratio rejects AM variation inherently, usually avoiding a separate limiter, though output and linearity may be lower than a well-aligned Foster-Seeley circuit.
• A PLL detector makes its VCO follow the incoming FM. Within lock and loop bandwidth, control voltage is

where \(K_{VCO}\) is in hertz per volt. It offers excellent linearity, amplitude immunity and IC integration without a precision transformer.

The detector must cover peak deviation without leaving its linear/lock range, and its bandwidth must pass the highest message frequency while rejecting noise. A de-emphasis network follows detection in broadcast FM. If \(K_{VCO}=100\,\text{kHz/V}\) and input deviation is \(75\,\text{kHz}\), the PLL control deviation is about \(0.75\,\text{V}\), verifying the proportional conversion. Slope and Foster-Seeley circuits need limiting; ratio and PLL methods provide more inherent AM rejection. Direct generation favors simplicity/large deviation, whereas Armstrong favors carrier accuracy.

Practice target: 17-19 minutes; draw both generators, sketch an S-curve, and compare the four detectors by principle, limiter need, linearity and limitation.

Exam-ready answer

In AM, carrier amplitude follows the message:

whereas in FM, instantaneous frequency follows it:

For \(f_{m(max)}=15\,\text{kHz}\), AM needs \(30\,\text{kHz}\). Broadcast FM with \(\Delta f=75\,\text{kHz}\) needs about \(180\,\text{kHz}\) by Carson's rule, so FM trades spectrum and circuit complexity for better noise immunity/fidelity. AM is power-inefficient because its carrier contains no unique information, but reception is inexpensive. FM uses constant transmitter power and supports amplitude limiting and pre/de-emphasis, yet below its threshold quality collapses rapidly. Neither is universally superior: AM suits bandwidth- and cost-limited links, while FM suits links where bandwidth is available and noise performance is more important.

Practice target: 7-8 minutes; state both defining equations and compare at least seven features including bandwidth and special distortions.