So far we have described modulation by drawing waveforms in time. There is a second picture — far more useful once the schemes get dense — that shows every symbol as a single point. It is called the constellation diagram, and it turns questions about noise, bit rate, and reliability into simple geometry: how many points, how far apart, and how big is the noise. This lesson builds that picture and uses it to see exactly how noise causes bit errors.
Any passband signal can be written as the sum of two carriers a quarter-cycle apart: a cosine (the in-phase, or I, component) and a sine (the quadrature, or Q, component). Because a sine and a cosine at the same frequency are orthogonal — a receiver can recover each without interference from the other — the pair (I, Q) completely describes the symbol:
This single picture unifies everything from the last lesson. Keying the amplitude moves a point toward or away from the origin; keying the phase rotates it around the origin; keying both at once — QAM — places points anywhere on the plane. ASK, PSK, and QAM are not different mechanisms; they are different arrangements of dots.
BPSK is two points on the horizontal axis, at +1 and −1 — one bit, chosen by sign. QPSK adds the vertical axis: four points, one in each quadrant, two bits per symbol. Push further and the points multiply:
| Scheme | Points M | Bits / symbol | Arrangement |
|---|---|---|---|
| BPSK | 2 | 1 | 2 points on a line |
| QPSK | 4 | 2 | 4 points on a circle (a square) |
| 8-PSK | 8 | 3 | 8 points on a circle |
| 16-QAM | 16 | 4 | 4 × 4 grid |
| 64-QAM | 64 | 6 | 8 × 8 grid |
| 256-QAM | 256 | 8 | 16 × 16 grid |
The receiver never gets an exact point back — the channel has moved it. Its job is to decide which of the M constellation points was most likely sent, and for equal-power symbols in Gaussian noise the answer is simply the nearest one. Draw the perpendicular bisector between every pair of neighbouring points and the plane divides into decision regions: one territory per symbol. For QPSK these are the four quadrants; for 16-QAM they are a grid of squares. As long as the received point lands in the right region, the symbol is decoded correctly.
Thermal noise (Module 9) adds an independent random nudge to both I and Q. On the constellation this smears each ideal point into a fuzzy cloud — a two-dimensional Gaussian blob. Most of the time the received point stays near its origin and decodes correctly. But if the noise is large enough to push it across a decision boundary, the receiver picks a neighbour and a symbol error occurs.
Whether that happens is governed by one number: the minimum distance dmin between neighbouring points, measured against the noise standard deviation σ. The probability of a symbol error is dominated by the closest neighbours:
This is why a symbol error is usually a single-bit error. Constellations are labelled with Gray coding (next lesson) so that neighbouring points differ by exactly one bit — when noise pushes a symbol into the adjacent region, only one of its bits flips. Slipping to a far-away point, which would corrupt many bits, is exponentially less likely.
The density trade, made visible. For a fixed average transmit power, every constellation must fit inside the same circle — so packing in more points forces them closer together, shrinking dmin. Going from QPSK to 16-QAM to 64-QAM doubles then triples the bits per symbol, but each jump of two bits shrinks the spacing enough to need roughly 6 dB more SNR to hold the same error rate. That is the constellation-diagram version of the Module 5 refrain: more bits per symbol always costs noise margin. It is exactly why real systems switch to a smaller constellation when the channel degrades (adaptive modulation, Module 6).