Constellation Diagram
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A constellation diagram is a representation of a signal modulated by a digital
modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...
scheme such as
quadrature amplitude modulation Quadrature amplitude modulation (QAM) is the name of a family of digital modulation methods and a related family of analog modulation methods widely used in modern telecommunications to transmit information. It conveys two analog message signa ...
or
phase-shift keying Phase-shift keying (PSK) is a digital modulation process which conveys data by changing (modulating) the phase of a constant frequency carrier wave. The modulation is accomplished by varying the sine and cosine inputs at a precise time. I ...
. It displays the signal as a two-dimensional ''xy''-plane scatter diagram in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
at
symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...
sampling instants. In a manner similar to that of a phasor diagram, the angle of a point, measured counterclockwise from the horizontal axis, represents the
phase shift In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
of the carrier wave from a reference phase; the distance of a point from the origin represents a measure of the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
or power of the signal. It could be considered a
heat map A heat map (or heatmap) is a 2-dimensional data visualization technique that represents the magnitude of individual values within a dataset as a color. The variation in color may be by hue or intensity. In some applications such as crime analy ...
of I/Q data. In a
digital modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...
system, information is transmitted as a series of samples, each occupying a uniform time slot. During each sample, the carrier wave has a constant
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
, which is restricted to one of a finite number of values. So each sample encodes one of a finite number of "symbols", which in turn represent one or more
binary digit Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical o ...
s (bits) of information. Each symbol is encoded as a different combination of amplitude and phase of the carrier, so each symbol is represented by a point on the constellation diagram, called a ''constellation point''. The constellation diagram shows all the possible symbols that can be transmitted by the system as a collection of points. In a
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
or phase modulated signal, the signal amplitude is constant, so the points lie on a circle around the origin. The carrier representing each symbol can be created by adding together different amounts of a
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
wave representing the "I" or ''in-phase'' carrier, and a
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
wave, shifted by 90° from the I carrier called the "Q" or ''quadrature'' carrier. Thus each symbol can be represented by a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, and the constellation diagram can be regarded as a
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, with the horizontal real
axis An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
representing the I component and the vertical imaginary
axis An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
representing the Q component. A coherent detector is able to independently demodulate these carriers. This principle of using two independently modulated carriers is the foundation of
quadrature modulation Quadrature may refer to: Mathematics * Quadrature (geometry), drawing a square with the same area as a given plane figure (''squaring'') or computing that area ** Quadrature of the circle ** ''Quadrature of the Parabola'' ** Quadrature of the ...
. In pure
phase modulation Phase modulation (PM) is a signal modulation method for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal f ...
, the phase of the modulating symbol is the phase of the carrier itself and this is the best representation of the modulated signal. A 'signal space diagram' is an ideal constellation diagram showing the correct position of the point representing each symbol. After passing through a
communication channel A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for infor ...
, due to
electronic noise In electronics, noise is an unwanted disturbance in an electrical signal. Noise generated by electronic devices varies greatly as it is produced by several different effects. In particular, noise is inherent in physics and central to thermod ...
or
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
added to the signal, the amplitude and phase received by the demodulator may differ from the correct value for the symbol. When plotted on a constellation diagram the point representing that received sample will be offset from the correct position for that symbol. An electronic test instrument called a vector signal analyzer can display the constellation diagram of a digital signal by sampling the signal and plotting each received symbol as a point. The result is a 'ball' or 'cloud' of points surrounding each symbol position. Measured constellation diagrams can be used to recognize the type of interference and distortion in a signal.


Interpretation

The number of constellation points in a diagram gives the size of the "alphabet" of symbols that can be transmitted by each sample, and so determines the number of bits transmitted per sample. It is usually a power of 2. A diagram with four points, for example, represents a modulation scheme that can separately encode all 4 combinations of two bits: 00, 01, 10, and 11, and so can transmit two bits per symbol. Thus in general a modulation with N constellation points transmits \log_2 N bits per symbol. After passing through the communication channel the signal is decoded by a
demodulator Demodulation is the process of extracting the original information-bearing signal from a carrier wave. A demodulator is an electronic circuit (or computer program in a software-defined radio) that is used to recover the information content from ...
. The function of the demodulator is to classify each sample as a symbol. The set of sample values which the demodulator classifies as a given symbol can be represented by a region in the plane drawn around each constellation point. If noise causes the point representing a sample to stray into the region representing another symbol, the demodulator will misidentify that sample as the other symbol, resulting in a symbol error. Most demodulators choose, as its estimate of what was actually transmitted, the constellation point which is closest (in a
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
sense) to that of the received sample; this is called
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
detection. On the constellation diagram these detection regions can be easily represented by dividing the plane by lines equidistant from each adjacent pair of points. One half the distance between each pair of neighboring points is the amplitude of additive noise or distortion required to cause one of the points to be misidentified as the other, and thus cause a symbol error. Therefore, the further the points are separated from one another, the greater the noise immunity of the modulation. Practical modulation systems are designed to maximize the minimum noise needed to cause a symbol error; on the constellation diagram this means that the distance between each pair of adjacent points is equal. The received signal quality can be analyzed by displaying the constellation diagram of the signal at the receiver on a vector signal analyzer. Some types of distortion show up as characteristic patterns on the diagram: *
Gaussian noise Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
causes the samples to land in a random ball about each constellation point * Non-coherent single frequency interference shows as samples making circles about each constellation point *
Phase noise In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). Generally speaking, radio-frequency enginee ...
shows as constellation points spreading into arcs centered on the origin * Amplifier compression causes the corner points to move towards the center A constellation diagram visualises phenomena similar to those an
eye pattern In telecommunications, an eye pattern, also known as an eye diagram, is an oscilloscope display in which a digital signal from a receiver is repetitively sampled and applied to the vertical input (''y-axis''), while the data rate is used to tri ...
does for one-dimensional signals. The
eye pattern In telecommunications, an eye pattern, also known as an eye diagram, is an oscilloscope display in which a digital signal from a receiver is repetitively sampled and applied to the vertical input (''y-axis''), while the data rate is used to tri ...
can be used to see timing
jitter In electronics and telecommunications, jitter is the deviation from true periodicity of a presumably periodic signal, often in relation to a reference clock signal. In clock recovery applications it is called timing jitter. Jitter is a signifi ...
in one dimension of modulation.


See also

* Error vector magnitude * Eye diagram * Modulation error ratio *
Quadrature amplitude modulation Quadrature amplitude modulation (QAM) is the name of a family of digital modulation methods and a related family of analog modulation methods widely used in modern telecommunications to transmit information. It conveys two analog message signa ...


References

{{Reflist Quantized radio modulation modes Diagrams