Creating a Map for Digital Signals

A constellation diagram is a graphical way of representing the states of a digital modulation scheme. Think of it as a map where each specific location (a point) represents a unique piece of digital information. To send information, the transmitter simply sends a signal corresponding to a point on the map. The receiver's job is to look at the incoming signal and determine which point on the map it is closest to, thereby decoding the information.

This "map" is formally a two-dimensional scatter plot in the complex plane, often called the I-Q plane. Each point represents a symbol, which is a unique combination of amplitude and phase of the carrier wave.

The Building Blocks: The I-Q Plane

The map is built on two perpendicular axes. In practice, these axes represent two carrier waves of the same frequency that are 90 degrees out of phase with each other. By varying the amplitude of these two carriers, we can create any point on the map.

• The I-axis (Horizontal): Represents the In-phase 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 Quadrature component. It corresponds to a second carrier wave, represented by a sine function ($\sin(2\pi f_c t)$).

Every point on the diagram has coordinates (I, Q) that define the amplitude of the cosine and sine waves, respectively. The combination of these two waves results in a single signal with a specific amplitude (its distance from the origin) and a specific phase (its angle relative to the positive I-axis). The transmitted signal can be represented by the formula: $s(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t)$.

From Bits to Points: BPSK and QPSK

The simplest constellations are used in Phase-Shift Keying (PSK), where only the phase of the signal is changed, keeping the amplitude constant. All points lie on a circle centered at the origin.

BPSK (Binary Phase-Shift Keying)

This is the simplest modulation. Since we want to represent binary data ('0' and '1'), we need two distinct states. BPSK uses two phases, separated by 180°, to represent 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'.

Because the points are as far apart as possible, BPSK is very robust against noise but offers the lowest spectral efficiency.

QPSK (Quadrature Phase-Shift Keying)

To increase efficiency, we can use more points. 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.

A typical mapping would be:

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

Interactive Constellation Diagram

Advanced Constellations: QAM

To achieve even higher data rates, we can vary both the phase and the amplitude. 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, offering higher efficiency but requiring a better quality signal.

Why Constellations Matter: Noise, Errors, and Gray Coding

The layout of a constellation diagram is critical to a system's performance. The distance between points is a direct measure of the modulation's robustness to noise.

• Noise Immunity: Noise in the channel causes the received signal's point to be displaced from its ideal location. If the noise is strong enough to push the point closer to a neighboring symbol's decision region, a symbol error occurs. Densely packed points (like in 64-QAM) are more susceptible to noise than widely spaced points (like in BPSK). This is the fundamental trade-off: higher data rates for a lower noise tolerance.
• Gray Coding: This is an intelligent bit assignment strategy to minimize the impact of errors. In a Gray-coded constellation, adjacent symbols differ by only one bit. This means that the most likely error-mistaking a symbol for its immediate neighbor-will only result in a single bit error in the decoded data, not multiple bit errors. This significantly improves the overall Bit Error Rate (BER) of the system.