Minimum Distance Decoding

The decision rule computes the squared Euclidean distance between the received noisy vector r and every candidate constellation point s_i in the IQ plane. The receiver selects the symbol that minimizes ||r - s_i||². This is equivalent to maximizing the correlation between the received signal and the reference symbols when all constellation points have equal energy. The metric inherently accounts for both amplitude and phase differences, making it the optimal detector for AWGN where noise components are independent and identically distributed Gaussian random variables.

The IQ plane is partitioned into Voronoi regions—convex polygons surrounding each constellation point. Each region contains all received signal locations closer to its associated constellation point than to any other. The boundaries between regions are the perpendicular bisectors of lines connecting neighboring constellation points. For a regular QAM constellation, these boundaries form a rectangular grid. For PSK, they form angular sectors radiating from the origin. The shape of these decision regions directly determines the symbol error probability.

In an AWGN channel with equal prior symbol probabilities, minimum distance decoding is identical to maximum likelihood (ML) detection. The likelihood function for each candidate symbol is a Gaussian distribution centered at the constellation point. Maximizing this likelihood is equivalent to minimizing the Euclidean distance because the log-likelihood simplifies to -||r - s_i||² / (2σ²). This equivalence establishes minimum distance decoding as the theoretically optimal detection strategy, achieving the lowest possible symbol error rate for the given constellation and channel conditions.

The decision boundary between two adjacent constellation points s_a and s_b is the line where ||r - s_a||² = ||r - s_b||². Expanding this equality reveals it is the perpendicular bisector of the line segment connecting s_a and s_b. For a binary phase shift keying (BPSK) constellation with points at +√E and -√E on the I-axis, the decision boundary is the Q-axis (I=0). For quadrature phase shift keying (QPSK), boundaries are the ±45° diagonal lines through the origin, creating four quadrant decision regions.

The probability of symbol error for minimum distance decoding is determined by the minimum Euclidean distance d_min between any two constellation points. For an M-ary PSK constellation with energy E_s, the symbol error rate is approximately 2Q(√(2E_s/N_0) · sin(π/M)) at high SNR, where Q(·) is the Gaussian tail probability. For square M-QAM, the error rate scales as 4(1 - 1/√M)Q(√(3E_s/((M-1)N_0))). The error performance improves exponentially with increasing d_min, motivating constellation designs that maximize this parameter under power constraints.

A brute-force minimum distance decoder computes distances to all M constellation points, requiring O(M) operations per symbol. For high-order modulations like 4096-QAM, this becomes computationally expensive. Practical implementations exploit constellation regularity: - Rectangular QAM: Independent I and Q decisions reduce complexity to O(√M) - PSK: Phase-only comparison eliminates amplitude calculations - Lattice-based decoding: Uses closest-point search algorithms for structured constellations These optimizations preserve optimality while enabling real-time operation in high-throughput receivers.

The convex polygon surrounding a constellation point that defines its exclusive decision area. Every location within a Voronoi region is closer to its associated constellation point than to any other. Minimum distance decoding is geometrically equivalent to identifying which Voronoi region contains the received signal vector. The boundaries between regions are the perpendicular bisectors of lines connecting neighboring constellation points.

A geometric threshold in the IQ plane that partitions the signal space into distinct decision regions. For minimum distance decoding, these boundaries are straight lines equidistant from two adjacent constellation points. A received symbol falling exactly on a boundary is equally likely to belong to either constellation point, representing the point of maximum ambiguity. The shape and placement of these boundaries directly determine the symbol error probability.

The straight-line distance between two points in the complex plane, calculated as sqrt((I1-I2)^2 + (Q1-Q2)^2). Minimum distance decoding uses this metric because, in additive white Gaussian noise (AWGN), the likelihood of a received point given a transmitted symbol decays exponentially with the squared Euclidean distance. This makes the nearest constellation point the maximum likelihood estimate under AWGN assumptions.

A decision-theoretic framework that selects the hypothesis (constellation point) that maximizes the probability of observing the received signal. In AWGN channels, the likelihood function is a Gaussian centered on each constellation point. Because the noise variance is identical for all hypotheses, maximizing likelihood is mathematically equivalent to minimizing Euclidean distance. Minimum distance decoding is therefore the optimal detection strategy for AWGN.

A quantitative metric measuring the Euclidean distance between the ideal reference constellation point and the actual received signal point. EVM is the magnitude of the error vector at the instant of symbol decision. High EVM values indicate that received symbols are far from their ideal locations, increasing the probability that minimum distance decoding will select an incorrect constellation point. EVM directly quantifies modulation fidelity.

A bit-to-symbol mapping scheme where adjacent constellation points differ by only a single bit. When minimum distance decoding makes an error due to noise, the most likely mistake is selecting a neighboring constellation point. Gray coding ensures that these most probable symbol errors result in only one bit error per symbol, minimizing the overall bit error rate (BER) for a given symbol error rate (SER).