Constellation_diagram

# Constellation diagram A **constellation diagram** is a two-dimensional plot of a digitally modulated carrier's possible symbol states, drawn on the complex in-phase/quadrature (I/Q) plane at the instant each symbol is sampled. Because [[Signal_modulation|digital modulation]] schemes such as [[Phase-shift_keying|phase-shift keying]] and [[Amplitude-shift_keying|amplitude-shift keying]], and combinations of the two, restrict a symbol's carrier to one of a finite set of amplitude-and-phase combinations, every such scheme has an ideal constellation: a fixed, finite set of points that real, noisy samples plotted on the same axes will scatter around rather than land on exactly. Read as a [[Cartesian_coordinate_system|Cartesian]] plot, a point's distance from the origin is a measure of the symbol's amplitude and its angle from the horizontal axis is the carrier's phase shift from a reference; read as data, the spread of real samples around each ideal point is a direct picture of the noise and distortion the [[Communication_channel|channel]] has added. The pattern the decoded samples fall into is closely related to the timing display used downstream of the decision circuit, the [[Eye_pattern|eye pattern]]. Test equipment such as a vector signal analyzer recovers the carrier's phase and amplitude from a live radio signal and plots exactly this picture, so that a constellation diagram is as much a bench instrument's everyday display as it is a textbook figure. This article carries the Constellation diagram sketch, which draws a live scatter of QPSK, 8-PSK, 16-QAM or 64-QAM symbols corrupted by adjustable noise and lets the reader watch the resulting bit-error rate. ## Interpretation Every symbol interval, a digital transmitter sets the carrier to one of M allowed combinations of amplitude and phase, and a constellation diagram marks each of those M combinations as an ideal point in the I/Q plane: the in-phase component on the horizontal axis, the quadrature component, ninety degrees out of phase with it, on the vertical axis. Together the pair behaves as the real and imaginary parts of a single complex number per symbol, so the whole modulated signal reduces, at the sampling instants, to a sequence of points rather than a continuous waveform. Because each symbol carries log2(M) bits, denser constellations pack more bits into every symbol: a 4-point constellation such as QPSK carries two bits per symbol, a 16-point constellation carries four, and a 64-point constellation carries six.[^ell133] Adjacent points are almost always assigned bit patterns that differ in only one bit, a Gray-coded labeling that makes the most likely mistake, confusing a symbol with its nearest neighbor, cost a single bit rather than several.[^ell133] A symbol's distance from the origin measures its amplitude, and squaring that distance gives a measure of its instantaneous power; the average of that quantity over all M symbols is the mean symbol energy, related to the energy carried by a single bit through the number of bits each symbol encodes.[^ell142] That relationship, together with the symbol rate and the noise power in the receiver's bandwidth, sets the working figure of merit for every scheme plotted this way, the ratio of energy per bit to noise power spectral density, usually read directly off the sketch's own control.[^ell142] An ideal transmitter would place every sample exactly on its symbol's ideal point, but a real channel adds noise that a receiver cannot separate from the signal, so each ideal point becomes the center of a cloud of received samples whose spread is set by the noise power in each of the two axes.[^ell145] A receiver decides which symbol was actually sent by choosing the ideal point closest to the sample it received, which is the optimal rule when the noise added by the channel is Gaussian and independent on the two axes; the boundary between two neighboring decision regions is exactly the line of points equally likely under either symbol. When the noise clouds are small relative to the spacing between ideal points, decisions are almost always correct; as the noise grows, or as a denser constellation crowds the ideal points closer together at the same average power, the clouds begin to overlap the decision boundaries and symbol errors appear. This is why the closed-form error probability of coherent phase-shift keying is written directly in terms of the Gaussian tail function, and why doubling a constellation's point count for the same average power always costs several decibels of extra energy per bit for the same error rate: uncoded four-point phase-shift keying needs on the order of eleven decibels of energy per bit over noise density for a bit-error rate of one in a million, and the denser amplitude-and-phase-shift and quadrature-amplitude schemes need several decibels more for that same target.[^ell152][^ell163] Because a constellation diagram is built from samples taken once per symbol, it says nothing about how the carrier moves between those instants; that motion, and the timing margin left for a receiver to decide correctly, is what the related [[Eye_pattern|eye pattern]] display shows instead, built from the same recovered symbol clock. A constellation diagram is nonetheless the more direct picture of the modulation itself, and engineers reduce that picture to a single number, the error vector magnitude, defined as the root-mean-square distance of the measured points from their ideal positions, expressed as a fraction (or a [[Decibel|decibel]] figure) of the ideal amplitude. A well-behaved digital transmitter typically holds its error vector magnitude to a few percent; a receiver working from a diagram whose points have drifted far larger than that will see a rising bit-error rate even before any single decision region is visibly crossed. The shape of a real constellation is also a diagnostic on its own. Phase noise in the local oscillator smears every point into a short arc around its ideal position rather than a round cloud; an imbalance between the in-phase and quadrature amplifier gains stretches the whole pattern into an ellipse; a direct-current offset in either channel shifts the entire constellation away from the origin; and gain compression in a transmitter's power amplifier pulls the outermost points of a dense constellation inward, toward the center, more than it pulls the inner ones. Reading a constellation diagram is, in this sense, reading the [[Distortion|distortion]] a system has picked up as much as it is reading the data the system is carrying. The layout of the ideal points is itself a design choice, not a fixed law. Phase-shift keying places every point on a single circle, so that every symbol has the same amplitude and only the phase varies; a constant-envelope signal like this tolerates a transmitter power amplifier driven into its nonlinear region far better than a signal whose amplitude must be reproduced faithfully, which is why phase-shift keying and its close relatives dominate links with cheap or heavily driven amplifiers. Quadrature amplitude modulation instead spaces its points on a square or near-square grid, trading that constant envelope for a more efficient packing of points into the available power budget, so that for the same peak power a grid layout can carry more bits per symbol than a ring of the same size, at the price of demanding a more linear amplifier and a receiver able to resolve smaller gaps between neighboring points. ## Microsims The Constellation diagram sketch draws a live [[Scatter_plot|scatter]] of received symbols for a selectable modulation scheme, QPSK, 8-PSK, 16-QAM or 64-QAM, each ideal point marked as a glowing halo. A slider sets the [[Signal-to-noise_ratio|signal-to-noise ratio]] from −5 to 30 decibels, controlling how far each new sample lands from its ideal point; a second slider sets how quickly new symbols arrive, and a third sets how long old points linger before fading, so that a slow fade turns the scatter into a rough density map of where symbols actually land, not unlike a [[Heat_map|heat map]] of the same data. The sketch's own HUD reports the chosen scheme and its point count, the number of samples drawn so far, the per-axis noise standard deviation, and a running bit-error-rate tally built from a nearest-point receiver, next to the additive-noise model and signal-to-noise definition the sketch is computing from. *Try:* Hold the noise level fixed and switch the modulation from QPSK to 64-QAM: the ideal points crowd closer together on the same scale, the noise clouds start to overlap their neighbors, and the sketch's live bit-error-rate tally climbs even though nothing about the noise itself has changed. A three.js companion sim renders this same noise cloud in more depth, relating the energy-per-bit-to-noise ratio, phase error and error-vector magnitude to the constellation's closed-form bit-error rate.  **Microsim — three.js (Wikitube framework), pending deploy:** *Constellation diagram: symbols as a noise cloud* will play here once `https://wikitube-3d-microsims.netlify.app/signal/Constellation_diagram.html` is live.  *Built from `MICROSIM_GUIDE/specs/sims/Constellation_diagram.json`; part of the [[Signal_processing]] set ([[PORTAL_Signal_Processing]]).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Constellation_diagram) : [Wikitube](https://en.wikitube.io/wiki/Constellation_diagram) Skeleton mirrored at revision 1279051979. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Phase-shift_keying]] - [[Amplitude-shift_keying]] - [[Eye_pattern]] - [[Communication_channel]] - [[Signal_modulation]] - [[Scatter_plot]] ## References The definition of the I/Q plane, the additive-Gaussian-noise decision rule and the general relationship between constellation size and required signal-to-noise ratio are standard results in digital communication theory and are not separately footnoted here, per Wikitube style guide §6.1. The worked figures below are cited to their page in the source textbook. [^ell133]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, pp. 133-134 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell142]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, pp. 142-143 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell145]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, pp. 145-151 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell152]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, pp. 152-154 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell163]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, p. 163 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC.