In digital modulation schemes, data symbols are transmitted by modulating the amplitude and/or phase of a carrier wave having a certain frequency. For example, a data symbol typically represents an M-bit fragment of data, resulting in N=2M possible symbols. The set of N possible symbols are mapped to a set of N respective fixed complex numbers, which are referred to as constellation points and may be represented in the complex plane in the form of a constellation diagram. In order to transmit a given symbol, a complex carrier wave is multiplied by the value of the constellation point corresponding to the symbol, thereby modulating the amplitude and phase of the carrier by amounts corresponding respectively to the amplitude and phase of the constellation point.
Various constellations designs are used in various modulation schemes, including N-Quadrature Amplitude Modulation (QAM) in which the constellation comprises a square lattice of N regularly-spaced constellation points, and N-Phase Shift Keying (PSK) in which the constellation comprises a circular lattice of N regularly-spaced constellation points. Various other constellation designs are also known.
In order to measure the performance of a given constellation or between different constellations, various metrics may be used.
For example, capacity is a measure of the maximum rate of information that can be reliably transmitted over a communications channel. The maximum theoretical capacity of a channel is given by a well-known formula derived by Shannon. The Coded Modulation (CM) capacity is the maximum capacity achievable using a fixed non-uniform constellation without any coding constraints. The Bit Interleaved Coded Modulation (BICM) capacity is the maximum capacity achievable using a certain binary Forward Error Correction (FEC) scheme and fixed non-uniform constellation.
In addition, when comparing two systems, the difference in Signal-to-Nose Ratio (SNR) required achieving the same Bit Error Rate (BER) may be referred to as the SNR gain.
In contrast to uniform constellations, a non-uniform constellation is a constellation in which the constellation points are not regularly spaced. One advantage of using a non-uniform constellation is that performance may be increased, for example for SNR values below a certain value. For example, the BICM capacity may be increased by using a non-uniform constellation, when compared to an equivalent uniform constellation. Using a non-uniform constellation may also achieve a SNR gain over an equivalent uniform constellation.
A constellation may be characterised by one or more parameters, for example specifying the spacing between constellation points. Since constellation points of a uniform constellation are regularly spaced, the number of parameters needed to characterise a uniform constellation is typically equal to 1. For example, for a QAM type constellation, the constellation is characterised by the (constant) lattice spacing. For a PSK type constellation, the constellation is characterised by the (constant) distance of each constellation point from the origin. On the other hand, since the spacing between constellation points in a non-uniform constellation varies, the number of parameters needed to characterise a non-uniform constellation is relatively high. The number of parameters increases as the order of the constellation (i.e. the number of constellation points) increases.
One problem with designing a non-uniform constellation is that a relatively high number of parameters need to be searched to find the optimum constellation. This problem is increased in the case of constellations of higher order. In the case of high-order constellations (e.g. constellations comprising more than 1024 constellation points), an exhaustive search across all parameters may be unfeasible.
Therefore, what is desired is a technique for designing non-uniform constellations, and in particular, for designing non-uniform constellations for optimising performance (e.g. capacity and SNR performance). What is also desired is a technique for designing non uniform constellations using an algorithm having a relatively low complexity and relatively high computational efficiency.