Maximum Likelihood Detection (MLD)

MLD performs a brute-force search over the entire set of possible transmitted symbol vectors. For a system with N_t transmit antennas and a modulation order M, the detector evaluates M^(N_t) candidate vectors.

• Computes the Euclidean distance ||y - Hx||² for every candidate x
• Selects the vector x̂ that minimizes this distance
• Guarantees the minimum error probability when all vectors are equally likely
• Complexity grows exponentially with the number of antennas and constellation size

Under the assumption of additive white Gaussian noise (AWGN), MLD is equivalent to minimum distance detection. The detector maximizes the likelihood function p(y|x, H), which reduces to finding the symbol vector closest to the received signal in the complex signal space.

• Achieves the full diversity order of N_r in a N_t × N_r MIMO system
• Provides a lower bound on the Bit Error Rate (BER) for all practical detectors
• Serves as the gold standard for benchmarking sub-optimal algorithms like MMSE and ZF
• Optimality holds only when noise is Gaussian and symbols are equiprobable

The primary drawback of MLD is its prohibitive computational cost for practical systems. The search space explodes combinatorially, making real-time implementation impossible for high-order MIMO configurations.

• For 4x4 MIMO with 64-QAM: 16.7 million candidate vectors per detection
• For 8x8 MIMO with 256-QAM: over 1.8 × 10¹⁹ candidates
• Latency scales linearly with the number of candidates, violating real-time constraints
• Drives the need for sphere decoding and other complexity-reduction techniques

In modern coded communication systems, MLD can be extended to produce soft-decision outputs in the form of Log-Likelihood Ratios (LLRs). This variant, often called Soft MLD, provides reliability information for each bit to the channel decoder.

• Computes the LLR for each bit by comparing the best candidate where the bit is '0' versus '1'
• Requires maintaining a list of candidate vectors rather than just the single best
• Enables near-Shannon-limit performance when paired with turbo or LDPC codes
• Further increases computational load due to dual-hypothesis evaluation per bit

Sphere Decoding (SD) is a prominent technique that achieves ML performance while drastically reducing average complexity. It restricts the search to candidate vectors lying within a hypersphere of radius r centered at the received signal point.

• Performs a depth-first tree search through the lattice of possible symbol vectors
• Prunes branches whose partial Euclidean distance already exceeds the current best radius
• Achieves ML-optimal performance with significantly fewer evaluations on average
• Worst-case complexity remains exponential, making it sensitive to channel condition number

MLD provides a fundamental performance advantage over linear receivers like Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE), especially in poorly conditioned channels.

• ZF Receiver: Inverts the channel matrix but suffers from noise enhancement at low SNR
• MMSE Receiver: Balances interference suppression and noise, but still sub-optimal
• MLD avoids noise enhancement entirely by jointly detecting all streams
• The performance gap widens as the channel condition number increases or the spatial correlation grows

The condition number of the MIMO channel matrix dictates how much linear detectors degrade relative to MLD. A high condition number indicates a poorly conditioned channel where ZF and MMSE suffer severe noise enhancement.

• MLD is immune to the condition number's effect on noise amplification
• The performance gap between MLD and linear receivers grows with spatial correlation
• Antenna design and placement directly impact the condition number