2. Amplitude Modulation: The Original // Analog Communications // bhaswanth

The simplest scheme: take the carrier and let the message control its envelope. Loud audio means a tall envelope; quiet audio means a small one. AM was the only modulation in commercial use for the first thirty years of broadcasting, it is still alive on every shortwave band, and the math illustrates principles that recur in every other scheme.

2.1 The math, derived from intuition

Start with the carrier $c(t) = A_c \cos(2\pi f_c t)$ . We want the carrier's amplitude to follow the message $m(t)$ . Add the message to the carrier amplitude:

$s(t) = \big[A_c + m(t)\big]\cos(2\pi f_c t)$

If you plot this on a scope, you see the rapid carrier oscillation, with its envelope tracing out the shape $A_c + m(t)$ . The envelope rises when the message is positive, sags when the message is negative. The DC offset $A_c$ keeps the envelope strictly positive (assuming $|m(t)| < A_c$ ), so a simple peak detector can recover the audio.

Lightbulb-modulating-piano analogy. Imagine a stage lightbulb wired so that its brightness is controlled by an audio signal added to a baseline brightness. During a soft passage the bulb glows at the baseline. On a loud chord the bulb flares brighter. If you watched the filament with an ultra-fast camera, you would see the filament heating and cooling at audio rates. In AM, the carrier is the always-on baseline, and the message rides on top, brightening and dimming the envelope. Demodulation is just measuring the envelope.

2.2 Single-tone AM and the spectrum, from a trig identity

The cleanest case: a single tone for the message, $m(t) = A_m \cos(2\pi f_m t)$ . Substitute and let $\mu = A_m / A_c$ be the modulation index:

$s(t) = A_c \big[1 + \mu \cos(2\pi f_m t)\big]\cos(2\pi f_c t)$

Distribute, then apply the product-to-sum identity $\cos A \cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]$ :

$s(t) = A_c \cos(2\pi f_c t) \;+\; \frac{\mu A_c}{2}\cos\big(2\pi(f_c - f_m)t\big) \;+\; \frac{\mu A_c}{2}\cos\big(2\pi(f_c + f_m)t\big)$

Three sinusoids. The first is the original carrier at $f_c$ , untouched. The second is a new tone at $f_c - f_m$ called the lower sideband. The third is at $f_c + f_m$ , the upper sideband. The carrier stayed put; the message energy split into two copies, one shifted up by $f_m$ , one shifted down by $f_m$ . This is the famous "AM triangle":

plaintext

                    |
                    | carrier (height A_c)
              μA/2  |  μA/2
               |    |    |
      ─────────┼────┼────┼─────── f
            f_c-f_m f_c f_c+f_m

For a more general message $m(t)$ with bandwidth $W$ Hz, the multiplication by $\cos(2\pi f_c t)$ shifts the entire baseband spectrum up to $f_c$ and produces a mirror image below it. (This is exactly the frequency-shift property of the Fourier transform from Chapter 3, $m(t)\cos(2\pi f_c t) \leftrightarrow \tfrac{1}{2}[M(f - f_c) + M(f + f_c)]$ .) The transmitted spectrum has a delta-function carrier at $f_c$ flanked by two scaled copies of $M(f)$ , one occupying $[f_c, f_c + W]$ and the other occupying $[f_c - W, f_c]$ . The total bandwidth is $2W$ Hz, twice the message bandwidth. This 2x penalty is the price of being able to demodulate with a diode and a capacitor.

2.3 Modulation index and overmodulation

The index $\mu$ controls how deeply the carrier swings:

$\mu = 0$ : no modulation. The transmitter is just a carrier.
$0 < \mu < 1$ : under-modulation. The envelope, $A_c[1 + \mu\cos(2\pi f_m t)]$ , is always positive. A diode peak detector recovers the message cleanly.
$\mu = 1$ : 100% modulation. The envelope just kisses zero at the troughs.
$\mu > 1$ : overmodulation. The envelope dips below zero, and the carrier reverses phase during those intervals. A simple envelope detector reads $|s(t)|$ and produces a corrupted, half-rectified version of the message. The audio sounds distorted and harsh.

Broadcasters target $\mu \approx 0.85$ in normal operation, leaving headroom for transient peaks. They also use audio limiters and compressors to keep the loudest passages from pushing the carrier into overmodulation. The FCC monitors compliance, because an overmodulated AM signal sprays splatter into adjacent channels.

2.4 The power efficiency disaster

Here is the dirty secret of AM. For 100% modulation with a single tone, the average power in each part of the spectrum:

Carrier: $P_c = A_c^2 / 2$
Each sideband: $(\mu A_c / 2)^2 / 2 = \mu^2 A_c^2 / 8$
Total transmitted: $P_c + 2\cdot(\mu^2 A_c^2 / 8) = P_c (1 + \mu^2/2)$

For $\mu = 1$ , total power is $1.5 P_c$ . The information lives entirely in the sidebands (the carrier is a constant amplitude tone and tells you nothing about $m(t)$ ). So only $0.5 P_c$ out of $1.5 P_c$ is useful. Maximum efficiency: 33%. For typical $\mu = 0.85$ , efficiency drops to about 27%. A 50 kW AM station is burning roughly 35 kW heating its transmitter and antenna while only 15 kW carries information.

That is shocking, and it is the entire motivation for the next two schemes (DSB-SC, SSB). But before we leave AM, recognize what we got in exchange: the simplest possible receiver. Every transistor radio and crystal set in history exploited that simplicity.

2.5 Generation of AM, two ways

Square-law modulator. A nonlinear device with a quadratic V-I characteristic ( $i \approx a_1 v + a_2 v^2$ ) is fed the sum of carrier and message. Squaring the sum produces cross products $2 a_2 m(t)\cdot c(t)$ , plus other terms (DC, $m^2$ , $c^2$ ) at distinct frequency bands. A bandpass filter centered at $f_c$ keeps only the AM term $\propto [\text{const} + m(t)]\cos(2\pi f_c t)$ and rejects the rest. This was the workhorse of vacuum-tube transmitters, where a pentode's input characteristic was inherently quadratic.

Switching modulator. Use a diode (or BJT switch) driven by a strong carrier. The carrier turns the diode hard on, hard off, on, off at the carrier rate, which is mathematically equivalent to multiplying the input signal by a square-wave switching function. The Fourier series of a square wave contains the fundamental $f_c$ and odd harmonics. Only the $f_c$ component, multiplied against the message, gives the desired AM after a bandpass filter. Switching modulators dominate solid-state AM transmitters because diodes are cheap, the switching function is well-defined, and the bandpass filter cleans up the rest.

In modern radio architectures, AM is more often generated by direct digital synthesis: a DSP forms $[A_c + m(t)]$ in the digital domain, then a DAC and an upconverter produce the final RF. The math is identical; only the implementation moved from pentodes to FPGAs.

2.6 Detection: the envelope detector

The reason AM lived for a century is that you can demodulate it with three components.

plaintext

   AM in ──[D]──*── audio out
                │
                C   R
                │   │
                ▼   ▼
               GND GND

The diode rectifies the AM signal, killing the negative half-cycles. The RC tank charges quickly through the diode on each carrier peak (small forward resistance) and discharges slowly through $R$ between peaks. With the time constants chosen correctly, the cap voltage tracks the peaks of the rectified RF, which is exactly the envelope of the original AM. The output is the audio, plus a DC offset (the carrier component, easily blocked by a series cap).

The RC time constant has to satisfy two constraints simultaneously:

Fast enough to follow the envelope: $RC \ll 1 / W$ , so the output can dip when the audio dips.
Slow enough to ignore the carrier ripple: $RC \gg 1 / f_c$ , so each carrier cycle does not push the output back to zero.

For broadcast AM with $f_c = 1$ MHz and $W = 5$ kHz: $1/f_c = 1\,\mu$ s and $1/W = 200\,\mu$ s. Pick $RC \approx 10\,\mu$ s and the detector is happy across the band.

Two failure modes are worth knowing:

Diagonal clipping. When $RC$ is too large, the cap cannot discharge fast enough to follow a rapidly falling envelope. The output cuts a straight diagonal line down instead of tracking the message. The audio sounds distorted on transients.

Negative peak clipping. Happens when the modulation index exceeds 1 (overmodulation), or when the load following the detector draws DC current that effectively raises the no-signal output above zero, clipping the negative envelope swings. The fix is light loading and AC coupling.

Almost every consumer AM radio from 1925 to today uses an envelope detector. It is dirt cheap, requires no oscillator, and works the moment you turn it on. The tradeoff is the wasted carrier power on the transmit side, which is somebody else's problem.

2.7 Synchronous (coherent) detection of AM

If you happen to have a perfect copy of the carrier locally available, you can multiply the AM signal by it and then low-pass filter:

$s(t)\cdot\cos(2\pi f_c t) = [A_c + m(t)]\cos^2(2\pi f_c t) = \tfrac{1}{2}[A_c + m(t)] + \tfrac{1}{2}[A_c + m(t)]\cos(4\pi f_c t)$

The first term is the message plus a DC offset; the second is a high-frequency component at $2 f_c$ that the LPF kills. This is synchronous (or coherent) detection. It is more complicated than envelope detection (you need a local oscillator phase-locked to the incoming carrier) but it does not suffer from envelope nonlinearities and works at any modulation index. It also extends naturally to schemes where there is no carrier to envelope-detect, which we are about to meet.

2.8 AM in the wild

Medium-wave broadcast (530 to 1700 kHz). Hundreds of high-power stations worldwide. Long range at night via skywave from ionospheric reflection.
Shortwave broadcast (3 to 30 MHz). International services (BBC World Service, Voice of America, Deutsche Welle in their day) used HF AM for transcontinental reach.
Aviation voice (118 to 137 MHz). This is the exception that proves the rule for choosing modulation. Aviation uses AM, not FM, even though FM is technically superior in noise. The reason is that AM lacks the capture effect: if two pilots key their microphones simultaneously, you hear both garbled but recognizable, and the controller knows there is a conflict. With FM, the stronger transmitter would win and the weaker (perhaps a distressed aircraft) would be silenced. Aviation also values graceful degradation: a weak AM signal still gives a faint but readable voice; a weak FM signal below threshold goes to noise and clicks. After a century of installed avionics, switching modulations is impossible in practice. So aviation stays AM.
CB radio (27 MHz). Mostly AM with some SSB.
Aviation transponders, distance-measuring equipment. Short pulses are AM-coded.
Air traffic control surveillance radar. Pulse modulation is a degenerate form of AM.

Hardware-security tie-in: replay attacks on simple AM control links. Old garage-door openers used a simple AM-OOK (on-off keying, the trivial limit of AM where $m(t)$ is binary) at 300 to 400 MHz, with a fixed 8 to 12-bit code that the receiver compared against a stored value. A $30 software-defined radio could capture the transmission once and play it back, opening the door. The fix was to replace the fixed code with a rolling code (HCS301, KeeLoq, etc.), but not before millions of vulnerable doors had been deployed. The attack is purely a Chapter 7 attack: identify the modulation, sample the envelope, replay the captured waveform.