Appearance
7.3 — Digital Modulation and Constellations
The message is now a stream of bits. The carrier still has only three things you can vary — amplitude, frequency, phase — so the schemes have the same three names with "shift keying" instead of "modulation".
What changes is that the receiver's job becomes a decision rather than a measurement. It does not have to reproduce a waveform; it only has to decide which of a finite set of symbols was sent. That single difference is why digital communication can be made essentially perfect over a channel that would ruin an analog signal.
This chapter carries the numbers of Volume I Chapter 1.3 over the physical channel of Chapter 7.1, using the same spectrum-shifting mechanism as Chapter 7.2. What changes is that the thing being sent is now one of a finite set of symbols, which is what makes perfect restoration possible.
1. The three basic schemes
ASK — amplitude shift keying
s(t)=\begin{cases}A\cos\omega_ct & \text{bit }1\\0 & \text{bit }0\end{cases}
On-off keying is the simplest form, and it is the oldest modulation there is — Morse code sent by switching a transmitter on and off.
Simple and poor. Amplitude is exactly what noise, fading and amplifier nonlinearity attack, so ASK is fragile. It survives in optical fibre, where the light is simply on or off and the channel does not fade, and in cheap remote controls.
FSK — frequency shift keying
s(t)=A\cos(\omega_1t) \text{ or } A\cos(\omega_2t)
Constant amplitude, so like FM it tolerates a nonlinear amplifier and resists amplitude noise.
Bandwidth by Carson's rule:
B\approx2(\Delta f+R_b)
Where it lives: early modems, Bluetooth (which uses a refined form called GFSK), and most low-power radio links, because a constant-amplitude signal lets the transmitter's amplifier run at full efficiency in saturation.
MSK — minimum shift keying — chooses the frequency separation to be exactly half the bit rate, which is the smallest spacing that keeps the two tones orthogonal, and arranges the phase to be continuous at every transition. Continuous phase means no sudden jumps, so the spectrum has far lower sidelobes and the signal does not splatter into neighbouring channels. GSM's GMSK adds a Gaussian filter to the data first, narrowing the spectrum further.
PSK — phase shift keying
s(t)=A\cos(\omega_ct+\phi_k)
Binary PSK uses two phases, 0° and 180°:
s(t)=\pm A\cos\omega_ct
Constant amplitude, and the two symbols are as far apart as they can possibly be — diametrically opposite. That maximum separation is why BPSK is the most noise-resistant of all the simple schemes.
QPSK uses four phases, 90° apart, carrying two bits per symbol. Same bandwidth as BPSK, twice the data rate, and — remarkably — the same bit error rate for the same energy per bit. Section 4 explains why.
2. The constellation diagram
Plot each symbol as a point in the complex plane: the horizontal axis is the in-phase component (I), the vertical is the quadrature component (Q), 90° out of phase.
s(t)=I\cos\omega_ct-Q\sin\omega_ct
Every digital modulation scheme is a set of points in that plane, and the picture tells you almost everything about the scheme's properties.
How to read a constellation:
- Number of points = symbols, so \log_2(\text{points}) bits per symbol.
- Distance between nearest neighbours = noise immunity. This is the quantity that decides the error rate.
- Distance from the origin = the power that symbol uses.
- In a measured constellation, the size of each cloud is the noise, and any rotation, squashing or smearing is a specific impairment — section 6.
3. QAM
Quadrature amplitude modulation varies amplitude and phase together, placing points on a rectangular grid.
| Scheme | Points | Bits/symbol | Needed SNR (for 10^{-6} BER) |
|---|---|---|---|
| BPSK | 2 | 1 | 10.5 dB |
| QPSK | 4 | 2 | 13.5 dB |
| 16-QAM | 16 | 4 | 20.5 dB |
| 64-QAM | 64 | 6 | 26.5 dB |
| 256-QAM | 256 | 8 | 32.5 dB |
| 1024-QAM | 1024 | 10 | 38.5 dB |
Read the pattern: every doubling of bits per symbol costs about 6 dB of SNR. That is not a coincidence — it is the same 6 dB per bit as the converter resolution formula of Chapter 3.5, and for the same underlying reason.
This table is the entire basis of adaptive modulation. A modern link measures its SNR continuously and picks the highest-order constellation the channel currently supports. When you walk away from a Wi-Fi access point, the throughput drops in steps — each step is the link dropping to a lower-order constellation.
Why 1024-QAM needs 38.5 dB. With 1024 points in the same power envelope, adjacent points are extremely close, so tiny amounts of noise, phase error or amplifier nonlinearity cause errors. It is used only on very clean channels — cable television, short-range Wi-Fi 6 at close quarters, and microwave backhaul with a clear line of sight.
4. Bit error rate
The measure of digital link quality:
\text{BER}=\frac{\text{bits received wrongly}}{\text{bits sent}}
For BPSK in additive white Gaussian noise:
P_b=Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right)
where E_b is energy per bit, N_0 is noise power spectral density, and Q is the tail probability of the standard Gaussian:
Q(x)=\frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2}dt
Read the formula in words: the probability of error is the probability that noise pushes the received point across the halfway line between the two constellation points.
E_b/N_0 — pronounced "ee-bee over en-nought" — is the fundamental measure, and it differs from SNR:
\frac{E_b}{N_0}=\text{SNR}\cdot\frac{B}{R_b}
SNR depends on your bandwidth choice; E_b/N_0 does not. That makes it the right quantity for comparing schemes that use different bandwidths.
The curve is extraordinarily steep:
| E_b/N_0 | BER for BPSK |
|---|---|
| 0 dB | 7.9\times10^{-2} |
| 4 dB | 1.3\times10^{-2} |
| 8 dB | 1.9\times10^{-4} |
| 10 dB | 3.9\times10^{-6} |
| 12 dB | 9\times10^{-9} |
Two more decibels between 10^{-4} and 10^{-6}, and two more again to 10^{-9}. That steepness is the digital cliff: a link that works fine can fail completely with a 3 dB change in conditions.
Why QPSK matches BPSK
QPSK carries two bits per symbol yet has the same bit error rate as BPSK at the same E_b/N_0. That looks like something for nothing, and it is not.
The explanation: QPSK is two independent BPSK signals, one on the I axis and one on Q, and they do not interfere because sine and cosine are orthogonal. Each carries one bit and each sees the same noise as a BPSK link would. The symbol energy is doubled because there are two bits in it, so the energy per bit is unchanged.
So QPSK doubles the data rate in the same bandwidth for free. This is why QPSK, not BPSK, is the baseline for almost every practical system, and BPSK survives only where the channel is so poor that even QPSK cannot be decoded — deep space, and the control channels of cellular systems.
Beyond QPSK the free lunch ends. 16-QAM's points are closer together, so it genuinely needs more energy per bit.
5. Getting closer to the limit
Three techniques, all essential in any modern system.
Pulse shaping
A rectangular pulse has a sinc spectrum (Chapter 4.4) with sidelobes falling off slowly, so it splatters energy into adjacent channels. Shape the pulse instead.
The raised cosine filter has zero at every other symbol instant, so consecutive symbols do not interfere — the Nyquist criterion for zero intersymbol interference. Its bandwidth is
B=\frac{R_s(1+\alpha)}{2}
with \alpha the roll-off factor, between 0 and 1.
\alpha=0 gives the minimum possible bandwidth and an impulse response that decays very slowly, making the system intolerant of timing error. \alpha=1 doubles the bandwidth and is very forgiving. Practical systems use 0.2 to 0.35, and that number appears in every standard.
The filtering is split between transmitter and receiver — each applies a root raised cosine, and the product is the full raised cosine. That arrangement makes the receiver's filter a matched filter (Chapter 4.3), which is optimal for detection in noise.
Error correction coding
Add redundancy so that errors can be corrected rather than merely detected.
\text{code rate } r = \frac{\text{information bits}}{\text{total bits}}
Coding gain is how much less E_b/N_0 you need for the same error rate. Typical values:
| Code | Gain | Where |
|---|---|---|
| Convolutional, r=1/2, Viterbi | 5 dB | older systems |
| Reed-Solomon | 6 dB | CDs, DVDs, deep space |
| Turbo | 9–10 dB | 3G, deep space |
| LDPC | 10–11 dB | Wi-Fi, 5G, DVB-S2 |
A 10 dB coding gain is a factor of ten in transmitter power, or a factor of three in range. It is the cheapest decibel in any system, since it costs computation rather than watts, and computation has been getting cheaper for fifty years while transmitter power has not.
Turbo codes, discovered in 1993, and LDPC codes, invented in 1962 and forgotten until 1996, both come within about 0.5 dB of the Shannon limit of Chapter 7.5 — a limit that had stood essentially unapproached for forty-five years.
Equalisation
A channel with multipath spreads each symbol into the next, which is intersymbol interference. An equaliser applies the inverse channel response.
The catch, from Chapter 4.2: where the channel has a deep null, the inverse needs enormous gain, which amplifies noise. So a zero-forcing equaliser that perfectly inverts the channel can perform worse than one that does not.
The MMSE equaliser minimises the total of residual interference plus amplified noise, rather than eliminating interference alone. It is always at least as good and usually much better, and it is what real receivers use.
Adaptive equalisers track a changing channel, using the LMS algorithm of Chapter 5.4 — the same three-line update, doing yet another job.
6. Reading a measured constellation
The single most useful diagnostic in digital radio, because each impairment has a distinct visual signature.
| What you see | Cause |
|---|---|
| Fuzzy round clouds | thermal noise — low SNR |
| Whole constellation rotated | carrier phase offset |
| Constellation slowly spinning | carrier frequency offset |
| Points smeared into arcs | phase noise in an oscillator |
| Outer points compressed inward | amplifier compression |
| Rectangular grid squashed | I/Q gain imbalance |
| Grid sheared into a rhombus | I/Q phase imbalance |
| Points doubled or ghosted | multipath |
The measure is error vector magnitude:
\text{EVM}=\frac{\text{rms error vector}}{\text{rms reference vector}}\times100\%
Requirements by scheme: QPSK tolerates about 17% EVM; 64-QAM needs under 5%; 1024-QAM needs under 2%. That tightening requirement is why high-order QAM demands expensive, linear transmitters — the constellation is so dense that ordinary amplifier compression destroys it.
7. Synchronisation
The receiver must know three things it is not told, and getting any of them wrong makes the data unrecoverable.
Carrier frequency and phase. Even a small frequency error rotates the constellation continuously. A Costas loop — a phase-locked loop variant that works on suppressed-carrier signals — recovers both.
Symbol timing. Sampling at the wrong instant within each symbol reduces the margin and eventually causes errors. The eye diagram — overlaying many symbol periods on one plot — shows this directly. The "eye" opening's width is the timing tolerance and its height is the noise tolerance, and a closing eye is the first visible sign of trouble.
Frame timing. Knowing where a packet starts. Done with a known preamble whose autocorrelation has a sharp peak (Chapter 4.3), so the receiver correlates continuously and fires when it finds the pattern.
8. Spectral efficiency
\eta=\frac{R_b}{B} \ \text{bits per second per hertz}
| Scheme | Efficiency |
|---|---|
| BPSK | 1 |
| QPSK | 2 |
| 16-QAM | 4 |
| 64-QAM | 6 |
| 256-QAM | 8 |
In practice these are reduced by pulse-shaping roll-off, guard bands, coding overhead and pilot symbols — a real 64-QAM system with rate-3/4 coding and 20% overhead delivers about 3.6 bits/s/Hz rather than 6.
And there is a hard ceiling. Chapter 7.5 gives it: for any SNR there is a maximum spectral efficiency no scheme can exceed, and modern codes sit within a fraction of a decibel of it. The era of large gains from better modulation is over, which is why the last twenty years of progress have come instead from using more bandwidth, more antennas and smaller cells.
Chapter 7.4 is about sharing: how many users occupy the same medium without destroying each other, and the trick that made Wi-Fi and 4G possible.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Digital modulation
\eta=\frac{R_b}{B}, \qquad R_b=R_s\log_2M
Bit error rate for BPSK in AWGN:
P_b=Q\left(\sqrt{\frac{2E_b}{N_0}}\right), \qquad Q(x)=\frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2}dt
QPSK has the same P_b as BPSK, because it is two orthogonal BPSK signals sharing the symbol energy between two bits.
For M-QAM:
P_s\approx4\left(1-\frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3E_s}{(M-1)N_0}}\right)
\frac{E_b}{N_0}=\text{SNR}\cdot\frac{B}{R_b}
SNR needed rises about 6 dB per extra bit per symbol — the same 6 dB per bit as converter resolution, for the same reason.
Raised cosine bandwidth:
B=\frac{R_s(1+\alpha)}{2}
\alpha=0.2 to 0.35 in practice; \alpha=0 is the theoretical minimum and intolerant of timing error.
\text{EVM}=\frac{\text{rms error vector}}{\text{rms reference}}\times100\%
QPSK tolerates ~17%; 64-QAM needs under 5%; 1024-QAM under 2%.
What the next chapter fixes
One signal in one channel is the easy case. Chapter 7.4 fits many conversations into the same piece of spectrum at once, by separating them in frequency, in time, or by code — and ends at the scheme that made 4G and Wi-Fi fast, which turns the multipath echo that ruins other systems into something harmless.