Constellation Diagrams

Representing Digital Signal States

A constellation diagram represents the states of a digital modulation scheme. Each point corresponds to a defined combination of signal parameters and its assigned information. The transmitter selects the point assigned to the transmitted bits. In the common equal-prior additive white Gaussian noise (AWGN) model, maximum-likelihood detection chooses the nearest received point; other channel models or unequal priors use their specified decision regions.

Formally, it is a two-dimensional scatter plot in the complex plane, commonly called the I-Q plane. Each point represents a , which is a unique combination of amplitude and phase of the carrier wave.

The Building Blocks: The I-Q Plane

The diagram uses two perpendicular axes. In practice, they represent two carrier waves at the same frequency with a 90-degree phase difference. Varying the amplitudes of the two carriers selects a point in the constellation.

The I-axis (Horizontal): Represents the component. It corresponds to a carrier wave often represented by a cosine function ( $\cos(2\pi f_c t)$ ).
The Q-axis (Vertical): Represents the component. In the passband convention used below, it multiplies the negative-sine basis ( $-\sin(2\pi f_c t)$ ).

Every point on the diagram has coordinates (I, Q) that define the amplitudes of the cosine and negative-sine basis functions, respectively. Combining these waves produces one signal with an amplitude equal to its distance from the origin and a phase equal to its angle from the positive I-axis. The minus sign fixes the Q-axis convention; reversing it reflects Q and changes phase labels without changing symbol energy. The transmitted signal can be represented by the formula: $s(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t)$ .

Coordinates use a chosen reference scale. With unit-energy orthogonal basis functions, a normalized QPSK constellation can use the components $I,Q \in \{-1/\sqrt{2},+1/\sqrt{2}\}$ ; the symbol energy then satisfies $E_s=I^2+Q^2=1$ . With the unnormalized coordinates $\pm 1$ , the symbol energy is $E_s=2$ before applying the factor $1/\sqrt{2}$ . Comparisons should state which convention is used.

From Bits to Points: BPSK and QPSK

In constant-energy Phase-Shift Keying (PSK), only the signal phase changes while the amplitude remains constant. The ideal points lie on a circle centered at the origin.

BPSK (Binary Phase-Shift Keying)

BPSK has two states corresponding to binary data ('0' and '1'). Their phases are separated by 180°, so the modulation carries 1 bit per symbol ( $M=2^1=2$ ).

Point 1: Located on the positive I-axis (e.g., at (+A, 0)), represents a phase of 0° and could encode bit '1'.
Point 2: Located on the negative I-axis (e.g., at (-A, 0)), represents a phase of 180° and could encode bit '0'.

At a fixed symbol energy, the two BPSK points are maximally separated, so BPSK has strong noise tolerance. It carries one bit per symbol and, among the schemes listed here, has the lowest .

QPSK (Quadrature Phase-Shift Keying)

More points allow more bits per symbol. QPSK uses four points, allowing it to encode 2 bits per symbol ( $M=2^2=4$ ). This doubles the data rate of BPSK for the same symbol rate. The four phases are typically chosen to be 90° apart for maximum separation.

One possible bit-to-phase mapping is:

Point for '00': Phase 45°
Point for '01': Phase 135°
Point for '11': Phase 225°
Point for '10': Phase 315°

Advanced Constellations: QAM

QAM varies both phase and amplitude, which allows higher data rates. This is called Quadrature Amplitude Modulation (QAM). QAM constellations are typically arranged in a square grid.

For example, 16-QAM uses a 4x4 grid of 16 points. Since $16=2^4$ , each symbol encodes 4 bits of information. The points have different amplitudes (distances from the origin) and phases.

Higher-order schemes like 64-QAM (6 bits/symbol) and 256-QAM (8 bits/symbol) pack the points even more densely. They provide higher efficiency but require a higher signal-to-noise ratio for a given BER under the same channel, coding, and energy convention.

Why Constellations Matter: Noise, Errors, and Gray Coding

The layout of a constellation diagram affects system performance. The distance between points determines how much noise can occur before a decision boundary is crossed. BER comparisons require the same energy reference, channel model, coding, and detector.

Noise tolerance:Channel noise displaces the received point from its ideal location. A symbol error occurs when that displacement crosses into a neighboring symbol's decision region. Densely packed points, as in 64-QAM, are more susceptible to noise than widely spaced points, as in BPSK. For a fixed average symbol energy and receiver model, increasing the constellation order typically reduces the minimum distance and noise tolerance.
Gray coding:This bit-assignment strategy limits the impact of neighboring-symbol mistakes. In a Gray-coded constellation, adjacent symbols differ by only one bit. Under AWGN, when nearest-neighbor errors dominate, confusing a symbol with an adjacent symbol produces one bit error instead of several, improving the system's BER.