3. Digital Modulation: Putting Bits on a Carrier

The physical channel does not accept bits. It accepts analog waveforms in some allowed band. To send bits, we map each $k$-bit chunk to one of $M = 2^k$ waveforms, send the waveform, and let the receiver decide which waveform was sent. The mapping and waveform set is the modulation. We are doing exactly what Chapter 7 did, except the message has only $M$ allowed values instead of being a continuous voltage.

3.1 ASK: amplitude shift keying

The simplest scheme. Bit 0: carrier off. Bit 1: carrier on. This is on-off keying (OOK), the original Morse-code-of-digital schemes.

$s_{\text{ASK}}(t) = a_n \cdot A_c \cos(2\pi f_c t), \quad a_n \in \{0, 1\}.$

ASK is everywhere small things must be cheap and battery is precious. Garage door openers, car key fobs, low-end RFID, IR remote controls, optical Ethernet at the simplest end. On a fibre, the laser blinks: on for 1, off for 0. ASK's weakness is sensitivity to amplitude noise. Anything that fades the channel's gain (rain, fluorescent light, an obstruction) gives an unpredictable threshold for "on" vs "off."

3.2 FSK: frequency shift keying

Two frequencies, one per bit.

$s_{\text{FSK}}(t) = A_c \cos(2\pi f_n t), \quad f_n \in \{f_0, f_1\}.$

FSK trades amplitude tolerance for spectrum. The amplitude is constant, so amplifier non-linearity does not matter. A simple frequency discriminator (or a pair of bandpass filters and envelope detectors) reads the bits. Bell 103 modems used 1070/1270 Hz on the originating end and 2025/2225 Hz on the answering end, giving 300 bps duplex over a phone line in 1962. Pagers and AIS marine transponders used FSK. Bluetooth Low Energy uses Gaussian FSK at 1 Mbps. LoRa is a chirped FSK descendant.

A coherent FSK receiver knows the carrier phase and uses two correlators (one per frequency). A non-coherent receiver uses two bandpass filters and envelope detectors; it is simpler but pays a roughly 0.5 to 1 dB penalty in SNR.

3.3 PSK: phase shift keying

Hold amplitude and frequency constant. Vary the phase. Two phases, $0$ and $\pi$, give binary phase shift keying (BPSK).

$s_{\text{BPSK}}(t) = A_c \cos(2\pi f_c t + \phi_n), \quad \phi_n \in \{0, \pi\}.$

That is the same as $\pm A_c \cos(2\pi f_c t)$, which makes BPSK identical to BPAM (binary pulse amplitude modulation) at the carrier; the two names refer to the same waveform. The constellation diagram, which we will lean on heavily, shows the two possible values plotted in the complex (in-phase, quadrature) plane.

        Q
        |
        |
 -A_c   .         . +A_c
 ───────●─────────●───────  I
        |
        |

Two points, one bit each. BPSK is the most robust digital modulation. It needs the lowest SNR for a given bit error rate, so it is what GPS chooses, what every deep-space probe chooses (Voyager, Mars rovers, Cassini), and what Wi-Fi falls back to at the edge of range. Pay attention to BPSK; its math underlies almost every other scheme.

3.4 DPSK: avoiding the carrier-recovery problem

Coherent BPSK detection requires the receiver to know the absolute carrier phase. That is hard when the channel rotates the phase unpredictably (fading, Doppler, oscillator drift). Differential PSK (DPSK) sidesteps the problem: encode bits as phase changes between consecutive symbols. Bit 0 = no change. Bit 1 = $\pi$ phase change. The receiver compares each symbol to the previous one and reads off the difference; absolute phase cancels out.

DPSK pays a penalty of about 1 dB in BER versus coherent BPSK, but the receiver is hugely simpler. Older satellite and modem standards (V.22, V.32 fallback modes) used DPSK for exactly this reason. Even today, when phase tracking is hard or expensive, DPSK is a sensible choice.

Drumbeat analogy. Coherent BPSK is like dancing on the downbeat, where you and the band have agreed in advance which beat is "down." DPSK is like dancing on the change from one beat to the next, where you and the band do not need to agree on the absolute phase, only on the transitions. If the band drifts in tempo a little, you still keep up.

3.5 QPSK: two bits per symbol

Why settle for two phases when you can have four? Quadrature phase-shift keying (QPSK) uses phases $\{0, \pi/2, \pi, 3\pi/2\}$, mapping two bits to each symbol. The constellation:

         Q
         |
   01    |    11
    ●    |    ●
         |
─────────+─────────  I
         |
    ●    |    ●
   00    |    10
         |

Two carriers in quadrature, $\cos(2\pi f_c t)$ and $\sin(2\pi f_c t)$. The first bit modulates the in-phase (I) carrier; the second bit modulates the quadrature (Q) carrier. The transmitted signal is

$s(t) = a_I \cos(2\pi f_c t) - a_Q \sin(2\pi f_c t),$

with each of $a_I, a_Q \in \{-1, +1\}$. A QPSK symbol is two BPSK symbols in quadrature. Crucially, at the same energy per bit, QPSK and BPSK have the same BER, because a QPSK demodulator decodes I and Q independently and each is a BPSK channel. QPSK doubles the bit rate at the same bandwidth and the same BER. That is why almost every digital radio worth caring about uses QPSK at minimum: GPS L1 C/A, every cellular control channel, satellite TV, every cable modem upstream, you name it.

A common labelling is Gray coding: assign the bits to the constellation so that adjacent points differ in only one bit. In the picture above, the bottom-left is 00, top-left 01, top-right 11, bottom-right 10. A symbol error to a neighbour gives only one bit error. Gray coding cuts the BER by a factor of $\log_2 M$ at high SNR.

3.6 OQPSK and MSK

A drawback of QPSK is that on a 180° phase change, the I and Q signals both flip simultaneously, sending the carrier momentarily through zero amplitude. Real amplifiers, especially saturated ones in satellites and handsets, hate that. Offset QPSK (OQPSK) delays the Q stream by half a symbol so I and Q never transition together; the maximum phase change per chip is 90°. The amplitude envelope stays much closer to constant, so satellite power amplifiers can run at saturation with less spectral regrowth.

Minimum shift keying (MSK) goes further: a kind of FSK where the frequency deviation is exactly half the symbol rate, equivalent to a continuous-phase OQPSK with half-sinusoidal pulse shaping. MSK has a continuous phase, so it is even friendlier to non-linear amplifiers. GMSK (Gaussian MSK) smooths the phase further with a Gaussian pulse-shaping filter and was the modulation of GSM cellular for two decades. It is also the modulation in classic Bluetooth Basic Rate. GSM at 270.8 kbps over 200 kHz channels using GMSK was one of the great engineering hits of the 1990s.

3.7 M-PSK and M-QAM: more bits per symbol

We can keep going. Eight phases, equally spaced, gives 8-PSK, three bits per symbol. Sixteen phases gives 16-PSK. The constellation is a circle of $M$ points. Each new bit halves the angular spacing between adjacent points, which means the receiver must distinguish smaller phase differences in the presence of noise. 8-PSK was the basis of EDGE cellular (the "2.75 G" upgrade to GSM).

Beyond 8-PSK, restricting symbols to a single circle wastes the magnitude axis. Quadrature Amplitude Modulation (QAM) uses both phase and amplitude. The constellation is a 2D grid of points. 16-QAM has four bits per symbol, 64-QAM has six, 256-QAM has eight, and Wi-Fi 6 and 5G allow up to 1024-QAM (10 bits per symbol) on excellent channels.

                    Q
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
   ─────────────────+───────────────── I
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
    ●   ●   ●   ●   |   ●   ●   ●   ●
                    |
                            (8x8 grid = 64-QAM)

Airport analogy. Picture a constellation diagram as a map of airports on a flat plane. The transmitter is sending you to one of those airports per symbol. Noise is wind that pushes you off course. The bigger the map (higher $M$), the closer the airports, and the more accurately you have to fly to land at the right one. BPSK is two airports on opposite sides of a continent: easy. 1024-QAM is 1024 airports packed into a city: you need a precise navigation system.

The progression looks linear in dB on the table but it is a steep one. Doubling $M$ costs about 3 dB. Modulation choice is therefore an SNR-driven decision. A modern adaptive radio constantly measures the channel, picks the densest constellation that hits its target BER, and shifts as conditions change. Walk away from your access point, watch the signal-strength indicator drop, and you can almost feel the constellation collapsing through 256, 64, 16, QPSK, BPSK as you go.

3.8 OFDM: many parallel QAM tones

A modern wideband channel is not flat. Multipath fading carves nulls in the spectrum, often dozens of dB deep, while leaving other frequencies pristine. The classical fix is a complicated equalizer in the time domain. The modern fix is orthogonal frequency-division multiplexing (OFDM): split the wide band into hundreds or thousands of narrow subcarriers, each less than 100 kHz wide. Each subcarrier sees a flat channel and can be modulated with its own QAM. Subcarriers that hit deep fades carry low-rate constellations or no data; subcarriers in good frequencies carry up to 1024-QAM.

OFDM is implemented elegantly with the FFT (Chapter 17). The transmitter takes a block of QAM symbols, runs them through an inverse FFT to get a time-domain block, prepends a cyclic prefix, and sends. The receiver does the inverse. Wi-Fi 4/5/6, LTE, 5G NR, DVB-T, DAB, DOCSIS 3.1, and powerline networking all use OFDM. It is the dominant wideband modulation of the modern era. We give it more attention in Chapter 23.