AM Demodulation

Getting the Message Back Out

Modulation was only half the job. The previous lesson put a message onto a carrier by making the carrier's amplitude trace the message; this lesson takes it back off. The process is called demodulation, or detection, and for standard AM it is startlingly cheap.

The reason it is cheap is the whole design of AM. The envelope of the transmitted wave is the message — not a code for it, not a transform of it, but a scaled and shifted copy of the original waveform. So a circuit that does nothing more than trace the outline of the received wave has already recovered the audio. There is no need to know the carrier's frequency precisely, and no need to know its phase at all.

That last point is worth marking, because it is what the rest of this lesson will charge you for. An envelope detector throws the carrier's phase away, and gets away with it. The moment the transmitter stops sending a carrier — which is exactly what DSB-SC and SSB do to save the two-thirds of power the previous lesson accounted for — the receiver has to reconstruct that phase, and the cheap circuit is no longer enough.

The Envelope Detector: Diode, Capacitor, Resistor

Despite its power inefficiency, standard AM has one compelling advantage: demodulation requires only a diode, a capacitor and a resistor. The three components are wired in the simplest arrangement imaginable — diode in series with the signal, capacitor and resistor in parallel across the output — and between them they do four things:

1. The diode rectifies the AM signal, passing the positive half-cycles and blocking the negative ones. What is left is a train of pulses whose heights follow the envelope.
2. The capacitor charges almost instantly to the peak of each carrier cycle, because the diode conducts hard on the rising edge and its resistance is very low.
3. Between carrier peaks the diode switches off, and the capacitor discharges through the resistor — slowly enough that it has barely drooped before the next peak arrives, but fast enough that it can follow the envelope downwards.
4. The voltage across the resistor is therefore the envelope — the recovered message, sitting on a DC offset contributed by the carrier, which a series capacitor removes.

Because the output tracks the peaks of the carrier, the recovered signal is a copy of the bracketed term in the AM equation of the previous lesson:

No carrier reference oscillator is needed. No phase synchronisation. A crystal radio — literally a coil, a variable capacitor, a diode and an earpiece — can receive AM broadcasts with no power supply at all, drawing the few microwatts it needs to move the earpiece diaphragm straight out of the radio wave. This simplicity was revolutionary in the 1920s and still explains AM's presence in aircraft communication, where a receiver that cannot fail in an interesting way is worth more than one that sounds good.

Choosing the Time Constant

Everything in that description hangs on one number: the product RC, the time constant of the capacitor and resistor. It is the only real design decision in the circuit, and it is squeezed from both sides.

Make RC too small and the capacitor discharges too far between carrier peaks. The output no longer joins the peaks in a smooth line; it saw-tooths between them, and that ripple is a copy of the carrier riding on your audio. In the limit, RC shrinks to nothing and the detector output is simply the rectified carrier — all ripple, no envelope.

Make RC too large and the opposite failure appears. The capacitor holds its charge so stubbornly that when the envelope falls, the output cannot fall with it. The diode stays reverse-biased for many cycles while the output coasts down its own exponential, and the recovered waveform shows straight diagonal runs where the envelope had curves. This is diagonal clipping — sometimes called negative-peak clipping — and it is a distortion, not a filtering artefact: information has been destroyed, and no later stage can put it back.

The design rule follows directly. RC must be long compared with one carrier period and short compared with one period of the highest message frequency:

Notice what that condition quietly requires: a carrier frequency much higher than the message bandwidth. Push f_c down towards f_m and the window closes from both ends until no value of RC works at all. Envelope detection is not a universal technique — it is a technique that depends on the frequency separation modulation created in the first place.

The upper limit can be made exact. To avoid diagonal clipping, the exponential discharge must never be slower than the steepest downward slope the envelope can produce, and that condition works out to:

That last observation is a second, independent reason to keep m below 1. The previous lesson ruled out m > 1 because the envelope goes negative and stops resembling the message. Here, m = 1 is ruled out for a different reason: the envelope reaches zero with a finite slope, and a capacitor discharging exponentially can never follow a straight line down to zero. Broadcasters therefore work at a peak modulation depth somewhat below 100%, and the two constraints happen to point the same way.

A worked example

Take a medium-wave station at f_c = 1 MHz carrying speech up to f_m = 5 kHz at a modulation depth of m = 0.8. The carrier period is 1 µs and the fastest message period is 200 µs, so the design window spans a factor of two hundred.

The clipping bound gives √(1 − 0.64) / (2π × 5000 × 0.8) = 0.6 / 25 133 = 23.9 µs. Choose RC = 20 µs and both requirements are satisfied at once: it is twenty carrier periods, so the droop between peaks is about 1/20 — a 5% ripple — and it is a tenth of the fastest message period, comfortably inside the 23.9 µs ceiling. The 5% ripple that survives is at 1 MHz, hundreds of times above the audio band, so the modest low-pass filter that follows the detector removes it without touching the speech.

The Diode Is Not an Ideal Switch

The description above treats the diode as a perfect one-way valve. Real diodes are not, and the discrepancy shows up exactly where the signal is weakest.

A diode needs a forward voltage before it conducts usefully — roughly 0.6 V for silicon, 0.2 to 0.3 V for germanium. If the peak of the received AM signal is comparable with that threshold, the diode does not switch; it curves. In that region its current is approximately proportional to the square of the applied voltage, so the detector output contains a term in the square of the envelope rather than the envelope itself, and squaring the sum of a carrier and a message generates harmonics and intermodulation products that were never transmitted.

This is why the distortion of an envelope detector is worst at low modulation depth and low signal level, which is the reverse of most people's intuition. A strong, deeply modulated signal keeps the diode firmly in its linear switching region for most of each cycle; a weak or lightly modulated one leaves it loitering in the curved region near the origin. It is also why crystal sets used germanium point-contact diodes rather than silicon: a lower turn-on voltage moves the linear region closer to zero, so a weaker station still gets detected honestly. A practical receiver solves the same problem by putting 60 dB or more of gain ahead of the detector, so that the signal arriving at the diode is always large.

When the Carrier Is Suppressed: Synchronous Detection

Now spend the saving. The previous lesson showed that at m = 1 two-thirds of an AM transmitter's power sits in a carrier that conveys nothing, and that DSB-SC removes it. What does that do to the receiver?

It breaks the envelope detector completely. With the carrier gone the transmitted signal is A_c·x(t)·cos(2πf_ct), and its envelope is |x(t)| — the magnitude of the message, not the message. Every time x(t) crosses zero the carrier's phase jumps by 180°, and an envelope detector, which by construction cannot see phase, reports the negative half of the waveform as though it were positive. Feed music through it and you get a full-wave-rectified caricature, rich in harmonics that were never sent.

The cure is to stop discarding phase. A synchronous — or coherent — detector multiplies the received signal by a locally generated cosine at the same frequency and the same phase as the carrier that was suppressed, then low-pass filters the product:

The identity behind it is the same one the previous lesson used to find the sidebands, run backwards: multiplying by a cosine shifts a spectrum both up and down in frequency, so the pair of sidebands sitting either side of f_c is translated back down onto baseband where they add, and simultaneously up to 2f_c where the filter throws them away. Coherent detection is not a different idea from modulation — it is modulation applied a second time.

The Price of Coherence

"The same frequency and the same phase" is easy to write and expensive to build. Suppose the local oscillator is at the right frequency but its phase is off by φ. Then the recovered signal comes out multiplied by cos φ:

A frequency error is worse than a static phase error, because φ then grows without limit and cos φ sweeps through zero repeatedly: the output fades in and out at the difference frequency. On a voice channel a small offset makes speech sound like a cartoon character, which is the familiar mis-tuning artefact of single-sideband radio. Envelope detection has no equivalent failure, because it never had a phase to get wrong.

So the receiver must derive the missing carrier from the signal itself. Three approaches are standard: transmit a low-power pilot carrier — a few percent of the power the full carrier would have needed — and lock a narrow filter or a phase-locked loop to it; recover the phase from the sidebands directly with a Costas loop, which uses two multipliers in quadrature to sense which way the phase is drifting; or, in SSB voice work, simply give the operator a tuning control and let a trained ear do the phase-locking. The point is not which one you pick but that all of them cost something. That is the trade this module has been circling: the carrier is a payment made by the transmitter so that the receiver can be stupid, and suppressing it moves the bill to the other end of the link.

Where the Detector Sits: The Superheterodyne Receiver

A crystal set puts the detector straight across the antenna's tuned circuit, and pays for that simplicity twice over: it is deaf to weak stations and it cannot separate two stations that are close together, because a single tuned circuit is a poor filter. Every AM radio built since the 1930s uses the superheterodyne arrangement instead, and the envelope detector is one block near the end of it.

The mixer translates whichever station you tune to a single fixed intermediate frequency, conventionally 455 kHz for medium-wave AM. That one trick is what makes the rest of the receiver possible: the filter and the amplifier never have to be re-tuned, so they can be as sharp and as high-gain as the designer likes. Sixty to eighty decibels of that gain sits ahead of the detector, which is what keeps the diode out of its square-law region and the distortion low.

The DC component of the detector output — the term the coupling capacitor blocks on its way to the audio amplifier — is not wasted. It is proportional to the carrier strength, so it is fed backwards as automatic gain control, reducing the IF gain when a strong station is tuned. This is the reason stations of wildly different strengths come out of the loudspeaker at roughly the same volume, and it is a small piece of engineering elegance: the unwanted by-product of demodulation turns out to be exactly the measurement the receiver needed.