Modulation FundamentalsModule 2

Modulation is the process of imprinting an information signal onto a high-frequency carrier wave by varying one of its three properties: amplitude, frequency, or phase. Without it, radio communication as we know it would be impossible.

The clearest reason is antenna physics. An efficient radiator must be a significant fraction of a wavelength — typically λ/4. Voice frequencies sit between 300 Hz and 3 kHz; the wavelength of 3 kHz in air is 100 km, demanding a 25 km antenna. Push that audio onto a 1 GHz carrier and the wavelength shrinks to 30 cm — a practical 7.5 cm antenna. A factor-of-300,000 problem becomes a pocket-sized one.

Modulation also enables frequency-division multiplexing: hundreds of FM stations and millions of cellphones share the spectrum because each is shifted to its own carrier band. A third benefit is noise rejection — by moving the information away from low-frequency 1/f noise and into a band the receiver can filter tightly, the link tolerates more interference.

The first commercial radios were AM transmitters in the early 1900s. By the 1920s, voice broadcasting on the medium-wave band was transforming public life. Edwin Armstrong demonstrated wide-band FM in 1933, trading bandwidth for noise immunity — a trade modern digital schemes still make in different forms. Every cellphone, WiFi router, GPS receiver, and satellite link descends from those two ideas.

In AM, the amplitude of the carrier is varied in proportion to the instantaneous message. The carrier frequency and phase stay constant; only its envelope changes.

Here x(t) is the normalised message (|x| ≤ 1) and m is the modulation index. With m = 0 the transmitter sends a pure carrier carrying no information; with m = 1 the envelope just touches zero on the negative peaks. Push m above 1 and the envelope inverts — a phenomenon called over-modulation that produces severe distortion at an envelope detector and splatters energy into adjacent channels.

In the frequency domain, multiplying a single tone of frequency f_m by the carrier creates two new components — the upper sideband at f_c+ f_m and the lower sideband at f_c − f_m. The total occupied bandwidth is therefore 2 · f_m,max. Twice the message bandwidth — exactly double what is theoretically needed, because both sidebands carry the same information.

FM varies the instantaneous frequency of the carrier in proportion to the message. Amplitude stays constant — the wave is a fixed-height sinusoid whose cycles bunch up or spread out as the message swings positive or negative.

The frequency deviation Δf is how far the instantaneous frequency swings from the carrier at peak message amplitude. The dimensionless ratio β = Δf / f_m is the FM modulation index. A small β (≪ 1) is narrowband FM, used by two-way radios; a large β is wideband FM, the variety that produces broadcast FM's famous immunity to amplitude noise.

The exact spectrum of FM is described by Bessel functions and is theoretically infinite in extent. In practice, almost all of the power is captured by Carson's rule:

FM broadcast in the 88–108 MHz band uses a 200 kHz channel and a ±75 kHz peak deviation, so β ≈ 5 for typical 15 kHz audio. By spending bandwidth lavishly, FM trades it for an SNR improvement of 3β² · (β + 1) over AM — the physical reason FM hi-fi sounds dramatically cleaner than AM at the same transmit power.

In PM, the carrier's phase tracks the message directly: a positive message offsets the carrier ahead in phase, a negative message holds it behind. Amplitude stays constant, just like FM.

Phase and frequency are tightly coupled — frequency is just the time derivative of phase. As a consequence, phase-modulating x(t) is equivalent to frequency-modulating its derivative dx/dt, and vice versa. Most analog radios that you'd loosely call "FM" are actually PM transmitters with a 6 dB/octave equaliser applied to the message.

Pure analog PM is rare; its real importance is as the foundation of every digital scheme that follows. BPSK, QPSK, 8-PSK, and the phase-half of QAM are all discrete-state PM — instead of letting the phase wander continuously, the modulator snaps it to a small set of allowed values, one per symbol.

Any narrowband bandpass signal can be written as the sum of two orthogonal carriers — one a cosine, the other a sine — each scaled by its own slowly-varying baseband signal:

I ("in-phase") and Q ("quadrature") are the two real-valued baseband channels. Because the cosine and sine carriers are mathematically orthogonal, the two channels do not interfere — a single hardware mixer pair can send any waveform you can describe as a complex baseband.

The complex envelope I + jQ is what software-defined radios actually stream over USB, and what every modern transmitter pushes into its DAC. AM is a pure-I signal with zero Q. FM is a constant-magnitude signal whose phase I + jQ swings smoothly. Digital schemes quantise the complex envelope to a finite set of values — the constellation.

A constellation diagram plots each allowed (I, Q) point at the moment of decision. The receiver samples the incoming complex envelope once per symbol and decides which point it was supposed to be. Each symbol carries log₂(M) bits, where M is the constellation size.

ASK is the digital cousin of AM — instead of a continuous message, the amplitude is switched between a small set of discrete levels, one per symbol. The simplest case is OOK (on-off keying): a 1 turns the carrier on, a 0 turns it off. One bit per symbol, no phase information at all.

OOK is the workhorse of cheap, simple links: garage door openers, keyless car fobs, the classic 433 MHz remotes, and the IR carriers behind every TV remote. RFID readers in the 125 kHz / 13.56 MHz bands also use a form of ASK. The hardware is trivial — a transistor and a resonant tank are enough to transmit, an envelope detector is enough to receive.

M-ary ASK with multiple amplitude levels is rarely used on its own because it's fragile in two ways: it's vulnerable to additive noise (which directly distorts amplitude), and it demands a strictly linear power amplifier. Combine it with phase modulation, however, and the result is QAM — which solves both problems at once.

FSK encodes bits by switching between two (or M) discrete carrier frequencies. 2-FSK sends a 0 at f₁ and a 1 at f₂; the receiver simply decides which of the two frequencies has more energy. Like FM, it is a constant-envelope signal — the amplitude never changes — which lets the transmitter use a highly efficient, non-linear class-C power amplifier.

The two frequencies must be far enough apart that the receiver can tell them apart. For optimum (orthogonal) detection, the separation Δf must be at least equal to the symbol rate. Tighter spacing creates inter-symbol interference; wider spacing wastes bandwidth.

MSK (minimum shift keying) is the special case where Δf is exactly half the symbol rate — the minimum orthogonal spacing — and the phase is forced to be continuous across symbol boundaries. Continuous phase means no spectral splatter from sudden discontinuities. GMSK goes further by passing the data through a Gaussian filter before the modulator, smoothing the phase transitions and shrinking the spectral footprint.

PSK encodes bits in the carrier's phase, not its amplitude or frequency. The two cleanest forms are BPSK (two phases, 180° apart, 1 bit per symbol) and QPSK (four phases, 90° apart, 2 bits per symbol).

BPSK is the most robust digital scheme there is. With only two constellation points sitting as far apart as physically possible, every other scheme requires more energy per bit to reach the same error rate. It is the modulation of choice when the link budget is brutal: GPS L1 C/A, every deep-space probe from Voyager onward, and the BPSK pilot that anchors LTE.

QPSK packs two bits into each symbol and — perhaps surprisingly — achieves identical BER to BPSK at the same E_b/N₀. The trick is that QPSK is essentially two orthogonal BPSK signals, one on the I channel and one on the Q channel. You get twice the data rate in the same bandwidth at no SNR cost.

Both schemes use Gray coding: adjacent constellation points differ by only one bit. When noise nudges a symbol into the wrong decision region, it almost always lands on an immediate neighbour — so a symbol error becomes a single bit error rather than two.

QAM combines amplitude and phase modulation on the I and Q axes independently. 16-QAM arranges 16 points on a 4×4 grid (4 bits per symbol). 64-QAM uses an 8×8 grid (6 bits), and 256-QAM and 1024-QAM continue the pattern. Each rectangular layer doubles in side length, packing four times the points and two more bits per symbol.

The cost is SNR. Squeezing more points into the same constellation area means the points are closer together, so the receiver has less margin against noise. As a rule of thumb, every doubling of the bits-per-symbol roughly demands 6 dB more SNR for the same error rate. WiFi and LTE keep a ladder of schemes — BPSK, QPSK, 16-QAM, 64-QAM, 256-QAM, and (in WiFi 6) 1024-QAM — and adapt on the fly: each symbol's modulation is chosen for the channel quality measured an instant earlier.

All modern QAM links lean on two more tricks not visible in the constellation: pulse shaping (root-raised-cosine filters) to pack symbols tightly without inter-symbol interference, and forward error correction (LDPC, turbo codes) to claw back the SNR margin that high-order QAM gives up.

Every modulation scheme is a point on the same trade-off curve: how many bits per second per Hertz of bandwidth can you push through a channel of a given SNR before the bit-error rate becomes unacceptable? Claude Shannon answered this in 1948 with the channel-capacity theorem — and his answer is an absolute upper bound that no scheme, however clever, can beat.

C/B is the spectral efficiency in bits per second per Hz, and SNR is the linear signal-to-noise ratio at the receiver input. The curve below plots Shannon's limit (in green) and overlays the SNR each common scheme needs for a textbook BER of 10⁻⁶ over an AWGN channel. Every dot sits below the limit.

Three lessons fall out of the chart:

• Doubling the bits per symbol roughly costs 6 dB of extra SNR. This is why high-order QAM is reserved for short-range, clean links.
• BPSK and QPSK sit at the same Eb/N0 because QPSK is two independent BPSK signals — you get twice the rate in the same bandwidth without paying any noise penalty.
• The right modulation depends entirely on the channel. Deep-space probes use BPSK because the SNR is brutal; cable modems use 4096-QAM because the SNR is plentiful.