Analog Modulation

Tip

Exam Focus

High-yield items: why we modulate (antenna size $\lambda=c/f$, allocation, multiplexing); the single-tone AM equation $s(t)=A_c[1+\mu\cos\omega_mt]\cos\omega_ct$ with $B=2f_m$, $P_T=P_c(1+\mu^2/2)$ and $\eta_{\max}=33.3\%$; the AM family (DSB-SC, SSB, VSB) and its spectra/detection; angle modulation with $\beta=\Delta f/f_m$, the Bessel spectrum and Carson’s rule $B\approx2(\Delta f+f_{m(\max)})$; FM generation (direct/Armstrong) and detectors (slope, Foster–Seeley, ratio, PLL); and the AM-vs-FM comparison.

Need for Modulation

Modulation is the controlled variation of a parameter of a high-frequency carrier $c(t)=A_c\cos(2\pi f_ct+\phi_c)$ by a lower-frequency message. Varying amplitude gives AM, frequency gives FM and phase gives PM.

Carrier parameter varied	Modulation
Amplitude $A_c$	AM
Frequency $f_c$	FM
Phase $\phi_c$	PM

Reasons for modulation.

1. Practical antenna size: efficient radiation needs a dimension $\sim\lambda/4$ with $\lambda=c/f$. A 3 kHz tone gives $\lambda=(3\times10^8)/(3\times10^3)=100\,km$, i.e. a $25\,km$ quarter-wave — impractical, so the message is translated up to RF.

2. Bandpass channel matching: antennas, microwave links, satellite transponders and AC-coupled circuits work only over assigned nonzero bands.

3. Frequency allocation: different stations use different carriers so a tuned receiver can select one.

4. Multiplexing: FDM places several messages on separate carriers in one medium.

5. Propagation choice: translation permits a band with suitable antenna gain, spectrum and propagation for the link.

6. Noise/interference planning: carrier placement enables filtering and a modulation with suitable immunity.

Frequency-division multiplexing: baseband channels are stacked in adjacent carrier bands $f_1\ldots f_4$ with guard bands between them.

Modulation does not by itself remove noise or guarantee greater range; the best carrier depends on regulation, propagation, antenna gain and power.

Standard Amplitude Modulation (AM)

In standard full-carrier AM the carrier amplitude varies linearly with the message, while carrier frequency and phase stay constant.

Single-Tone Derivation

With $m(t)=A_m\cos\omega_mt$ and $c(t)=A_c\cos\omega_ct$:

Caution

Key Formula — Standard AM signal

$$s(t)=A_c\,[1+\mu\cos\omega_mt]\cos\omega_ct,\qquad \mu=k_aA_m.$$

Only under the normalized sensitivity convention $k_a=1/A_c$ does $\mu=A_m/A_c$; that amplitude ratio is not the general relation. Expanding,

$$s(t)=A_c\cos\omega_ct +\tfrac{\mu A_c}{2}\cos(\omega_c+\omega_m)t +\tfrac{\mu A_c}{2}\cos(\omega_c-\omega_m)t.$$

The spectrum has a carrier at $f_c$ (amplitude $A_c$), a USB at $f_c+f_m$ and a LSB at $f_c-f_m$ (each amplitude $\mu A_c/2$).

Standard AM with $\mu=0.6$: time waveform and envelope (top), then its single-tone spectrum with carrier $A_c$ and sidebands $0.3A_c$ at $f_c\pm f_m$ (bottom).

Modulation Index

Caution

Key Formula — Index from the envelope

$$\mu=\frac{V_{\max}-V_{\min}}{V_{\max}+V_{\min}},\qquad \text{percentage modulation}=100\mu\%.$$

Condition	Index	Envelope
Under-modulation	$0<\mu<1$	Faithful, nonzero
Critical / 100%	$\mu=1$	Just reaches zero
Overmodulation	$\mu>1$	Crosses zero; diode detection distorts

For a general message the no-overmodulation limit is $\left\lvert k_am(t)\right\rvert\le1$.

Bandwidth and Power

If the message occupies $0\le f\le f_{m(\max)}$, the sidebands span $f_c\pm f_{m(\max)}$:

Caution

Key Formula — AM bandwidth and power relations

$$B_{AM}=2f_{m(\max)},\qquad P_c=\frac{A_c^2}{2R}.$$

Single-tone powers:

$$P_{USB}=P_{LSB}=\frac{\mu^2}{4}P_c,\quad P_{SB}=\frac{\mu^2}{2}P_c,\quad P_T=P_c\!\left(1+\frac{\mu^2}{2}\right).$$

Transmission efficiency:

$$\eta=\frac{P_{SB}}{P_T}=\frac{\mu^2}{2+\mu^2},\qquad \eta_{\max}\big|_{\mu=1}=\frac13=33.33\%.$$

The carrier carries most of the power but no unique information; for a real message both sidebands are duplicates.

Note

Example — AM sideband power and efficiency (5 marks)

Carrier power $P_c=500\,W$, $\mu=0.8$.

$$P_{USB}=P_{LSB}=\frac{0.8^2}{4}(500)=80\,W,$$$$P_T=500\!\left(1+\frac{0.8^2}{2}\right)=660\,W,\qquad \eta=\frac{160}{660}\times100=24.24\%.$$

Practice: quote the three formulas first, then substitute — most marks are for the working, not the final number.

Generation/detection overview. Low-level AM modulates at low power then linearly amplifies; high-level AM modulates the final PA. An envelope detector (diode–$RC$) suffices for $\mu\le1$; a synchronous product detector gives better linearity and noise performance.

Double Sideband Suppressed Carrier (DSB-SC)

DSB-SC transmits both sidebands but suppresses the carrier:

Caution

Key Formula — DSB-SC signal

$$s_{DSB}(t)=A_c\,m(t)\cos\omega_ct =\frac{A_cA_m}{2}\big[\cos(\omega_c+\omega_m)t+\cos(\omega_c-\omega_m)t\big].$$

Bandwidth stays $2f_{m(\max)}$; all transmitted power is in the information-bearing sidebands; coherent detection is required.

Balanced modulator: two AM modulators fed by $+m(t)$ and $-m(t)$ share a common carrier; subtracting the outputs cancels the carrier and leaves the product $\propto m(t)\cos\omega_ct$.

Practical product modulators use balanced transistor or diode multipliers so symmetry cancels carrier feedthrough. A double-balanced diode-ring modulator suppresses both carrier and message feedthrough ideally, leaving the desired sum-and-difference products that form DSB-SC.

Coherent (product) detector: multiply the received signal by a synchronized local carrier and low-pass filter to recover $m(t)$.

Multiplying by $2\cos(\omega_ct+\phi)$ and low-pass filtering gives $A_cm(t)\cos\phi$: an envelope detector fails because the DSB-SC envelope is $\propto\left\lvert m(t)\right\rvert$ and loses the sign of $m(t)$. Phase error costs $\cos\phi$; frequency error causes beating.

Single Sideband Suppressed Carrier (SSB-SC)

SSB transmits only one sideband (usually carrier-suppressed). With Hilbert transform $\hat m(t)$:

Caution

Key Formula — SSB signals and bandwidth

$$s_{USB}(t)=m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct,$$$$s_{LSB}(t)=m(t)\cos\omega_ct+\hat m(t)\sin\omega_ct,\qquad B_{SSB}=f_{m(\max)}.$$

SSB halves the bandwidth of AM/DSB-SC and removes the carrier plus one redundant sideband, giving the best power/bandwidth efficiency in the AM family. For a single tone under the corresponding amplitude normalization, 100% AM sends $P_c+2(P_c/4)=1.5P_c$, whereas one suppressed-carrier SSB component is $P_c/4$. This is an apparent $83.3\%$ transmitter-power saving, but it is not a universal SSB ratio: the comparison must hold message normalization, output amplitude and transmitter reference conditions fixed.

SSB generation: (a) filter method — balanced modulator followed by a sharp sideband BPF; (b) phasing method — two balanced modulators with $90^\circ$ shifts of message and carrier summed so one sideband cancels.

In the filter method, DSB-SC is generated at a convenient IF so a sharp crystal or mechanical filter can select one sideband before conversion to the final RF. The phase-shift method cancels one sideband by adding or subtracting the outputs of quadrature message/carrier paths. The Weaver method instead uses two quadrature mixing stages with low-pass filters and is convenient in DSP and IC implementations.

Detection: reinsert a carrier with a BFO/PLL and product-detect; carrier-frequency error shifts all audio components (unnatural pitch). A small transmitted pilot can assist carrier-frequency and phase synchronization. Advantages: best bandwidth/power efficiency, narrower receiver noise bandwidth, ideal for HF voice and marine/aeronautical links. Limitations: complex generation/filtering, needs an accurate frequency reference, and simple envelope detection is unavailable.

Vestigial Sideband (VSB)

VSB transmits one full sideband plus a small vestige of the other, often with a residual carrier. It suits messages that extend near DC where an abrupt SSB filter is impractical.

Caution

Key Formula — VSB bandwidth

$$B_{VSB}\approx f_{m(\max)}+f_v,$$

where $f_v$ is the vestigial width.

Classic use: analog television video — saves bandwidth versus DSB while preserving low video frequencies.

AM-Family Comparison

Sideband occupancy: AM keeps carrier $+$ both sidebands; DSB-SC removes the carrier (dashed) but keeps both sidebands; SSB keeps one sideband only.

Feature	AM	DSB-SC	SSB-SC	VSB
Carrier	Full	Suppressed	Suppressed/pilot	Residual/full
Sidebands	Both	Both	One	One $+$ vestige
Bandwidth	$2f_m$	$2f_m$	$f_m$	$f_m+f_v$
Efficiency	Low	Better	Best	Intermediate
Detector	Env./coherent	Coherent	Coherent/BFO	Env./coherent
Complexity	Lowest	Medium	Highest	Medium/high
Application	Broadcast	Subcarrier	HF voice	Analog TV

Comparison of the amplitude-modulation family.

Angle Modulation

Here the carrier amplitude is constant and information changes the instantaneous phase $s(t)=A_c\cos\theta_i(t)$:

Caution

Key Formula — Instantaneous frequency

$$\omega_i(t)=\frac{\mathrm{d}\theta_i(t)}{\mathrm{d}t},\qquad f_i(t)=\frac{1}{2\pi}\frac{\mathrm{d}\theta_i(t)}{\mathrm{d}t}.$$

Angle modulation includes FM and PM.

Angle-modulation waveforms: message $m(t)$ (top), FM whose frequency deviation tracks $m(t)$ (middle) and PM whose phase deviation tracks $m(t)$ (bottom); both have a constant envelope.

Note

Key Point

Constant envelope $\Rightarrow$ constant average power $P_T=A_c^2/2R$; efficient saturated PAs and amplitude limiting can be used. Total FM power is fixed but split among a carrier and Bessel sideband pairs — it is not all in the sidebands. Angle modulation generally occupies more bandwidth than standard AM, especially at large modulation index.

Frequency Modulation (FM)

The instantaneous frequency deviation $f_i(t)-f_c$, not the total instantaneous frequency, is proportional to the message:

Caution

Key Formula — FM signal

$$f_i(t)=f_c+k_fm(t),\qquad s_{FM}(t)=A_c\cos\!\left[2\pi f_ct+2\pi k_f\!\int_{-\infty}^{t}\!m(\tau)\,d\tau\right].$$

Single-Tone FM

For $m(t)=A_m\cos2\pi f_mt$:

Caution

Key Formula — Deviation and index

$$s_{FM}(t)=A_c\cos[2\pi f_ct+\beta\sin2\pi f_mt],$$$$\Delta f=k_fA_m\quad\text{(peak deviation)},\qquad \beta=\frac{\Delta f}{f_m}\quad\text{(modulation index)}.$$

For fixed $A_m,k_f$, $\Delta f$ is independent of $f_m$ while $\beta$ falls as $f_m$ rises.

Type	Index	Spectrum / usage
NBFM	$\beta\ll1$	Carrier $+$ first sideband pair; $B\approx2f_m$
WBFM	$\beta>1$	Many sideband pairs; broadcast/hi-fi

These are descriptive regimes rather than a sharp physical boundary: NBFM means $\beta$ is small enough for the first-pair approximation, while increasing $\beta$ progressively makes more sideband pairs significant.

FM Spectrum and Carson’s Rule

Caution

Key Formula — Bessel expansion

$$s_{FM}(t)=A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t], \qquad \sum_{n=-\infty}^{\infty}J_n^2(\beta)=1.$$

Lines occur at $f_c\pm nf_m$; the carrier amplitude is $A_cJ_0(\beta)$ (which can vanish for certain $\beta$), and the $n$th sideband pair has magnitude $A_c\left\lvert J_n(\beta)\right\rvert$. There are infinitely many theoretical sidebands, but high-order terms are negligible.

FM line spectrum: Bessel-weighted sidebands at $f_c\pm nf_m$ with the significant-sideband span given by Carson’s rule.

Caution

Key Formula — Carson’s rule

$$B_{FM}\approx2(\Delta f+f_{m(\max)}).$$

For a single-tone message, where $f_{m(\max)}=f_m$ and $\beta=\Delta f/f_m$, this becomes

$$B_{FM}\approx2f_m(\beta+1).$$

Carson’s rule is an engineering occupied-bandwidth approximation commonly interpreted as containing roughly 98% of single-tone FM power. Exact occupied bandwidth depends on the message spectrum, deviation and chosen power criterion.

Note

Example — FM bandwidth by Carson’s rule (5 marks)

Broadcast FM with $\Delta f=75\,kHz$, $f_{m(\max)}=15\,kHz$:

$$\beta_{\max}=\frac{75}{15}=5,\qquad B\approx2(75+15)=180\,kHz.$$

Practice: always state whether the index asked is $\beta$ or the peak deviation — they are related by $\beta=\Delta f/f_m$.

FM Generation

FM generation: (a) direct — message drives a VCO/reactance modulator (large deviation, less stable centre); (b) Armstrong indirect — integrate, phase-modulate a crystal carrier, then frequency-multiply (very stable).

FM Detection

Families of FM detector, all converting frequency deviation to voltage.

• Slope: tuned slope converts FM$\to$AM; simplest, poor linearity, needs a limiter.

• Foster–Seeley: transformer phase discriminator; excellent linearity but requires a limiter.

• Ratio: diode-voltage ratio gives inherent AM rejection; usually no separate limiter.

• PLL: VCO control voltage tracks instantaneous frequency; excellent IC method.

• Quadrature: frequency-dependent phase shift then phase detection.

Tuned-transformer discriminators: (a) Foster–Seeley needs a preceding limiter; (b) the ratio detector has inherent amplitude rejection via a large stabilising capacitor.

A PLL provides a feedback-based alternative by converting the VCO tracking voltage into the recovered message.

PLL FM demodulator: the loop drives the VCO to track the input frequency, so the loop-filter voltage is the recovered message.

Noise improvement. FM allows amplitude limiting and better output SNR above threshold. Because demodulated high-frequency noise rises, broadcast FM uses pre-emphasis (boost highs before Tx) and de-emphasis (complementary Rx cut). FM also shows the capture effect (stronger co-channel signal dominates) and a threshold effect (SNR collapses below a critical CNR).

Phase Modulation (PM)

Phase deviation is proportional to the message:

Caution

Key Formula — PM signal

$$s_{PM}(t)=A_c\cos[2\pi f_ct+k_pm(t)],\qquad \beta_p=k_pA_m=\Delta\phi_{\max}\ \text{(rad)}.$$

For $m(t)=A_m\cos2\pi f_mt$,

$$s_{PM}(t)=A_c\cos[2\pi f_ct+\beta_p\cos2\pi f_mt].$$

Generally, $f_i(t)=f_c+\frac{k_p}{2\pi}\frac{\mathrm{d}m(t)}{\mathrm{d}t}$; for this tone, $\Delta f=\beta_pf_m$, so PM deviation grows with $f_m$.

For a general message, peak PM frequency deviation depends on the maximum message derivative, not merely its maximum amplitude. Applying Carson’s approximation to the single tone above gives $B_{PM}\approx2(\Delta f+f_m)=2f_m(\beta_p+1)$.

FM–PM relationship: FM from a PM modulator = integrate $m(t)$ first; PM from an FM modulator = differentiate $m(t)$ first. Detection: phase detector, PLL or Costas loop, or an FM discriminator followed by an integrator.

Comparisons

Feature	FM	PM
Controlled quantity	Instantaneous frequency	Instantaneous phase
Phase term	$\propto\int m(t)\,dt$	$\propto m(t)$
Single-tone index	$\beta=\Delta f/f_m$	$\beta_p=k_pA_m$
Fixed $A_m$	$\Delta f$ independent of $f_m$	$\Delta f=\beta_pf_m$
Generation	Integrator $+$ PM	Differentiator $+$ FM
Detector	Discriminator/PLL	Phase det. or discrim. $+$ integrator

FM versus PM.

Feature	AM	FM
Varied parameter	Amplitude	Frequency
Envelope	Carries info; varies	Ideally constant
Bandwidth	$2f_{m(\max)}$	$2(\Delta f+f_{m(\max)})$
Noise immunity	Lower	Better above threshold
Tx power	Varies with $\mu$; carrier wasteful	Constant, Bessel-distributed
RF PA	Must preserve envelope	Efficient nonlinear PA
Receiver	Simple envelope detector	Limiter $+$ discriminator/PLL
Special effects	Overmodulation	Capture, threshold
Typical use	MF broadcast, aviation	VHF hi-fi, telemetry

AM versus FM.

Note

Key Point — Revision — Analog Modulation

• Standard AM: $A_c(1+\mu\cos\omega_mt)\cos\omega_ct$, $B=2f_m$.

• AM power: each SB $=\mu^2P_c/4$, $P_T=P_c(1+\mu^2/2)$, $\eta_{\max}=33.3\%$.

• DSB-SC suppresses the carrier, keeps both sidebands, needs coherent detection.

• SSB uses bandwidth $f_m$ and needs accurate carrier reinsertion; VSB $=$ one SB $+$ vestige (analog TV).

• FM index $\beta=\Delta f/f_m$; PM index $\beta_p=k_pA_m$.

• FM has infinitely many Bessel sidebands with constant total power.

• Carson’s rule: $B\approx2(\Delta f+f_{m(\max)})$.

• Foster–Seeley needs a limiter; ratio and PLL detectors reject amplitude inherently.

• FM from PM $\to$ integrator; PM from FM $\to$ differentiator.