How To Draw A Constellation Diagram

Part 1 looked at the eye diagram, a simple yet powerful visual tool which reveals many specifics about the quality of a recovered bit stream. Part 2 looks at the constellation diagram, another powerful visual tool which is used to show the signal space of complex encoding schemes and relationship among the symbols.

Q: Why do we need to show the "signal space"? What is that, anyway?

A: Let's step back for a moment. While the original signals being encoded and transmitted were undoubtedly binary (two-valued), the actual encoding and modulation scheme is more complicated. Various types of multilevel encoding are used so that each transmitted symbol can convey 4, 8, 16, or more distinct multibit digital values.

In practice, this is usually done using some form of quadrature encoding, where two carriers are modulated, Figure 1. These carriers are 90° out of phase with each other, and are called in-phase (I) and quadrature (Q) signals. Each of these quadrature signals is then amplitude or phase modulated to encode the bits, and the two are combined and sent as one signal. Since they are in quadrature, the two signals do not "interfere" with each other (if the difference is exactly 90°).

Fig 1: In a commonly used quadrature-modulation scheme, a single carrier is 90° phase-shifted (sine versus cosine function), both signals are modulated, and then the two streams are recombined for transmission. (Source: Auburn University)

For example, if the I and Q signals are grouped into three-bit units, and each shifted to one of two values, then the combined signal can represent 2³ = 8 distinct values with a single symbol, Figure 2. For example, in quadrature amplitude modulation (QAM) encoding, multiple amplitudes and phases are used to represent 16, 32, 64, or even 256 states/symbol, with each state defined by a specific amplitude and phase pairing; other encoding schemes such as phase-shift keying (PSK) vary only the I and Q phases, but not their amplitudes.

Fig 2: Each multilevel symbol represents three encoded bits, and is used to modulate the I and Q carriers. (Source: Auburn University

Q: Why go to all this effort? Why not just use simple binary encoding?

A: It's a matter of bandwidth use and spectrum-use efficiency. Binary signals obviously have the greatest noise immunity, as the two signal states are separated as widely as possible for the given voltages available to represent a 1 or a 0. However, binary signals make poor use of the available bandwidth and spectrum, as they only encode one bit/symbol.

At the other end of the efficiency/noise immunity range are analog signals, which make the most efficient use of bandwidth but have no noise immunity: an analog signal of a given value with added noise is indistinguishable from any analog signal of the same combined value. Multilevel coding between these two extremes offers tradeoff in spectrum-use efficiency (bits encoded/symbol) and noise immunity. (Note that analog signals can be thought of as infinite-level encoding).

Q: What does the constellation diagram do?

A: The constellation diagram is a two-dimensional plot demodulated data signals along the I and Q axes, Figure 3. Depending on the data encoding used, these can look like m × m symbol encoding graphs (such as 2 × 2, 3 × 3, 4 × 4, 8 × 8, and higher-number), representing 2^m bits/symbol. If the signals are received with no noise, the graph points of each symbol will be perfect dots at each code value.

Fig 3: This constellation diagram of an idealized situation with m = 4; there are thus 16 possible states, each representing 4 bits. (Source: Forum for Electronics/EDABoard.com)

However, the reality is that noise, distortion, channel imperfections, and other factors will result in less-than-perfect symbols, Figure 4. The "cloud" around each point represents uncertainty that the decoder must deal with when it attempts to take the received symbol and decide what bit pattern it actually represents. As the cloud (and thus uncertainty) increases, the distance between the symbols decreases and the decoder may decode incorrectly. In severe cases, the clouds begin to overlap and merge, and the decoding will have an increasing number of errors.

Fig 4: Imperfections in the channel and nose cause the idealized points of the constellation diagram to spread, which reduces noise margin as the "spread" zones get closed to adjacent ones. (Source: ResearchGate GmbH)

Q: What else can the constellation diagram show?

A: Besides showing the symbol space, noise situation, and noise margin among symbols, it can also reveal some aspects of the noise and imperfections. If the noise is truly random, the cloud around each symbol point will be random as well. However, if there is a mismatch between the I and Q channels (imperfect 90° difference. for example), possibly due to channel or circuit imperfections, or other reasons, the cloud will not be random but will lean towards one axis more than the other. Also, if there are non-linearity or saturation issues with the I and Q channels, the constellation diagram will not be evenly spaced, or it may be tilted, Figure 5.

Fig 5: The constellation diagram can also show channel distortion, non-linearities, and many other problems; the tilting here indicates that there are phase differences (uneven path delays) between the I and Q signal paths. (Source: ResearchGate GmbH)

Like the eye diagram, the constellation diagram is a real-time visual representation. As such, it can easily show changes in system performance as the channel changes, as new distortions occur, or as changes are made in the system to improve performance. Despite its initially qualitative nature, it can offer strong indications as to the nature of any problems of shortcomings of the demodulation and decoding process for multi-level signals and symbols of digital communication systems.

References

Auburn University, "Signal Constellations"
RF Wireless World, "Difference between constellation Diagram vs Eye Diagram"
Questel, "Quadrature Amplitude Modulation (QAM) Constellation"