Digital Quadrature Amplitude Modulation: Optimal Detection in Receiver Noise

where p(t) is a (typically real valued) pulse shape, and the ck are QAM symbols drawn from a finite collection of complex numbers called a constellation based on input bits to be communicated over the channel. Some example QAM constellations include 4-QAM (a.k.a QPSK){±1± j}, 16-QAM {±a± jb} with a, b ∈ {1, 3}, 64-QAM {±a + jb} with a, b ∈ {1, 3, 5, 7}, and 256-QAM {±a ± jb} with a, b ∈ {1, 3, 5, 7, 9, 11, 13, 15}. These constellations are often also scaled to have unit power average power, e.g. for QPSK this scaling is {± 1 √ 2 ± 1 √ 2 }. The bits are mapped to constellation elements through one of a number of techniques, including Gray coding, Trellis coded modulation, and bit-interleaved coded modulation. Gray coding is the simplest, we will explain the other two when we explain forward error correction. For instance, under Gray coding bits are mapped to QPSK symbols through (0, 0) 7→ 1 √ 2 +j 1 √ 2 , (0, 1) 7→ 1 √ 2 − j 1 √ 2 , (1, 0) 7→ − 1 √ 2 + j 1 √ 2 , and (1, 1) 7→ − 1 √ 2 − j 1 √ 2 .