Wireless 101
M05 · L03
Module 5 · Lesson 3

A Picture of Every Symbol

We have drawn modulation as waveforms in time. There is a second map — the one engineers actually read — that shows each symbol as a single point. It turns noise and reliability into geometry.

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Wireless 101
M05 · L03
The map

The I–Q Plane

In-phase & Quadrature
s(t) = I\cos(2\pi f_c t) - Q\sin(2\pi f_c t)

Two numbers per symbol: I (a cosine) and Q (a sine). Plot them as a point — distance from origin is amplitude, angle is phase. The set of allowed points is the constellation.

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Wireless 101
M05 · L03
Two dots, then four

BPSK & QPSK

BPSK is two points on the horizontal axis, chosen by sign — one bit. QPSK adds the vertical axis: one point per quadrant, two bits per symbol. The picture makes the phase difference literally an angle.

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Wireless 101
M05 · L03
Packing more in

8-PSK to 256-QAM

Keep points on a circle → 8-PSK. Fill a grid → QAM. Each step doubles the points and adds a bit: b = log₂M.

16-QAM
4 bits
64-QAM
6 bits
256-QAM
8 bits
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Wireless 101
M05 · L03
Who gets the vote

Decision Regions

The channel moves the received point off its ideal spot. The receiver decodes to the nearest constellation point. Perpendicular bisectors between neighbours carve the plane into one territory per symbol — quadrants for QPSK, a grid of squares for QAM.

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Wireless 101
M05 · L03
Noise is a cloud

Crossing the Line

Symbol error rides on d₋ₐₓₖ / σ
P_e \;\sim\; Q\!\left(\dfrac{d_{\min}}{2\sigma}\right)

Noise smears each point into a Gaussian blob. An error happens only when a sample crosses into a neighbour’s region — so what matters is the spacing dmin against the noise σ.

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Wireless 101
M05 · L03
Try it

Add some noise

Scheme QPSK
Noise SER 0%
Interactive: add noise to any constellation and plot its BER waterfall — AI-generated
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Wireless 101
M05 · L03
The density trade

Closer Dots, Cleaner Channel

At a fixed transmit power, every constellation fits in the same circle — so more points sit closer together. Each extra 2 bits/symbol shrinks dmin enough to need about 6 dB more SNR for the same error rate. It is why systems drop to a smaller constellation when the link gets noisy (adaptive modulation, Module 6).

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Wireless 101
M05 · L03
Symbol → bit

One Slip, One Bit

When noise pushes a symbol across a boundary, it lands on a neighbour. Gray coding (next lesson) labels neighbours to differ by exactly one bit — so the common error costs a single bit, not a whole symbol. Slipping to a distant point is exponentially rarer.

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Wireless 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

10 / 11
Wireless 101
M05 · L03
Recap

What you learned

  • Every symbol is a point (I, Q): distance = amplitude, angle = phase
  • Bits/symbol = log₂M; PSK sits on a circle, QAM fills a grid
  • Decode to the nearest point; noise clouds crossing a boundary cause errors
  • Reliability rides on dmin/σ; denser needs ~6 dB more SNR per +2 bits
Up next in Module 5
QAM →
All lessons Read article
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