Maximum Multipath Diversity

• Maximum multipath diversity is the highest order of diversity in wireless systems, defined by the number of statistically independent fading paths available for signal processing.
• Techniques such as constellation rotation, space-time coding, and RIS optimization are employed to harness full diversity and significantly improve error performance at high SNR.
• Optimizing antenna geometry, channel modeling, and cooperative relay protocols are key design strategies to maximize multipath diversity and ensure reliable communication.

Maximum multipath diversity refers to the highest realizable order of diversity that can be harnessed by a wireless system in a multipath propagation environment. It is determined by the number of statistically independent fading branches (time, frequency, spatial, or pathwise) that can be coherently exploited by signaling, reception, and coding strategies. This parameter governs the asymptotic decay rate of error probability with increasing SNR and underpins optimal system design for communication reliability, radar sensing, and network robustness.

1. Mathematical Definition of Multipath Diversity Order

The diversity order $d_{\max}$ quantifies the exponential decay rate of the error (or outage) probability in the high-SNR regime: $$P_e(\mathrm{SNR}) \propto \left(\frac{1}{\mathrm{SNR}}\right)^{d_{\max}}, \qquad \mathrm{SNR} \rightarrow \infty.$$ It represents the number of independently faded signal branches combined at the receiver. In systems employing maximum-likelihood (ML) detection, the precise diversity order typically corresponds to the number of nonzero and linearly independent channel coefficients (resolvable multipath taps, spatial modes, or parallel paths), provided the signaling and modulation structures do not inadvertently destroy this degree of freedom (Li et al., 2022, Wang et al., 20 Jan 2026).

2. Fundamental Limits: Channel Models and System Architectures

2.1 Single-User, Frequency-Selective Channels

For a frequency-selective block-fading channel with $L$ independent resolvable taps modeled as independent fading gains (e.g., Nakagami-$m$), the maximum attainable multipath diversity in a single-antenna, cyclic-prefixed system is $d_{\max} = L$. RIS-empowered systems artificially increase $L$ by introducing delayed replicas via programmable reflection, attaining $d_{\max} = \sum_{r=0}^R L_r$ for $R$ non-overlapping RIS paths in addition to the direct path (Li et al., 2022).

2.2 Multi-Antenna Systems (MIMO)

For $N_T \times N_R$ Rayleigh-fading MIMO channels employing spatial multiplexing with $L$ selected transmit antennas, the maximal diversity is

$d_{\max} = (N_T - L + 1)(N_R - L + 1)$

when ZF or ZF-DF receivers and optimal antenna selection are used [0702138].

2.3 Multi-Hop and Cooperative Networks

In multi-hop relay topologies, maximum path diversity is achieved by identifying the maximal number of node-disjoint routes (paths) from source to destination. In $K$-parallel-path (KPP) networks, $d_{\max} = K$; if a direct link is present, $d_{\max} = K + 1$; and more generally, in layered and arbitrary directed-acyclic graphs, it equals the min-cut between source and sink (i.e., the maximal set of edge- or node-disjoint paths) (0802.1888, 0708.0386). The cut-set diversity bound governs the fundamental tradeoff with achievable multiplexing gain.

2.4 Spatial Diversity: Eigenvalue Perspective

The "diversity spectrum" is the set $\{\lambda_n\}$ of eigenvalues of the spatial autocorrelation kernel $R(x, x')$ of the fading field over a region or aperture. The effective spatial diversity (number of usable branches) can be summarized by

$N_{\mathrm{eff}} = \frac{1}{\sum_{n} \lambda_n^2}$

which equals the number of uncorrelated spatial modes present (Schulze, 2010).

3. Analytical Tools and Diversity-Order Verification

3.1 Pairwise Error Probability and Rank Criteria

Maximum multipath diversity is certified by showing that all nonzero distinct error events generate full-rank difference matrices post-channel. For general linearly precoded or modulated systems, random constellation rotations yield with probability one the full diversity order as long as the relevant "judgment matrices" $J_q$ for each symbol location satisfy $\operatorname{rank}(J_q) = L$ (Wang et al., 20 Jan 2026).

3.2 Spatial Karhunen–Loève Expansion

The Karhunen–Loève expansion of the spatial fading field quantifies the set of uncorrelated spatial branches as the eigenfunctions of $R(x, x')$. The diversity spectrum $\{\lambda_n\}$ allows explicit mapping from antenna geometry and power azimuth spectrum (PAS) to achievable spatial diversity. The effective number of branches is sharply bounded by the geometry, PAS, and aperture size (Schulze, 2010).

3.3 Network and Multihop Path Enumeration

For multi-hop or cooperative architectures, maximum diversity order equals the number of edge- or node-disjoint source-sink paths, provided each is activated independently (e.g., via pipelined amplify-and-forward protocols with orthogonal allocation) (0802.1888).

4. Practical Methods to Realize Maximum Multipath Diversity

4.1 Modulation and Precoding Strategies

Random per-symbol constellation rotation applied to linearly precoded CP-OFDM or other linear modulations guarantees almost surely full diversity order in both time-dispersive and doubly dispersive (delay–Doppler) channels. Diversity verification reduces to rank checks of $O(M)$ matrices of dimension $M \times L$ (Wang et al., 20 Jan 2026).

4.2 Space-Time Coding and Antenna Selection

In MIMO and relay networks, approximately universal space-time codes (such as NVD block codes for MIMO and cyclic division algebra codes for parallel/multihop structures) are shown to achieve the full cut-set DMT [0702138, (0708.0386, 0802.1888)]. Optimal antenna or path selection algorithms consistently orient the transmit/receive structure to maximize the minimum singular values driving diversity.

4.3 RIS and Delay Engineering

Engineering the reflection properties or delays in RIS or similar programmable metasurfaces can synthesize additional multipath taps that do not overlap, thereby increasing $d_{\max}$ beyond ambient physical scattering (Li et al., 2022, Zhang et al., 2024).

4.4 Antenna Geometry and Spatial Mode Expansion

Maximizing the spatial extent (in $\lambda$ units) and optimizing arrangement/deployment with respect to the measured PAS allows an aperture to support more uncorrelated spatial branches, up to the modal limit set by the spatial autocorrelation kernel (Schulze, 2010).

5. Limitations: Physical and Statistical Bounds

5.1 Physical Multipath Sparsity

In wideband (large $N$) noncoherent regimes, the number of independent degrees of freedom (DOF) or channel diversity bins $D$ grows sublinearly ($D \sim N^\alpha, \; 0 < \alpha < 1$) under sparse multipath, limiting the maximum attainable diversity no matter the modulation/coding strategy (0705.2848). Physical channel sparsity thus caps achievable diversity order.

5.2 Geometry and Angular Power Constraints

In spatial domains, PAS anisotropy or insufficient aperture size lead to rapid decay of eigenvalues $\lambda_n$. Beyond a critical aperture area or length (scaled by angular spread and wavelength), additional hardware yields diminishing benefit as eigenmodes become correlated (Schulze, 2010).

5.3 Cooperative and Multihop Networks

The min-cut bound—the number of edge- or node-disjoint source-sink paths—fundamentally limits achievable cooperative diversity, regardless of relaying protocol, while specific schemes may incur further losses unless scheduling and code design are carefully aligned to path diversity (0802.1888, 0708.0386).

6. Optimization and Design Guidelines

The following systematically maximize multipath diversity:

• Ensure all potential parallel paths (in time, frequency, spatial, or network sense) are resolved and do not overlap by judicious delay engineering, antenna placement, or RIS configuration.
• Apply randomized rotations or universal space-time codes to avoid modulation-induced diversity loss.
• In MIMO or relay networks, partition antennas/nodes to maximize the number and independence of activated branches. Exploit pipelined or orthogonal scheduling in multi-path networks.
• When deploying apertures or antenna arrays, tailor geometry (area, length, topology) and orientation to the PAS of the environment, maximizing $N_\mathrm{eff}$ or threshold eigenvalue count for given physical constraints (Schulze, 2010).
• In sparse or bandwidth-constrained regimes, jointly scale signaling duration and bandwidth along optimal trajectories to approach the diversity scaling limit imposed by physical sparsity ($D \sim N^\alpha$) (0705.2848).

7. Case Studies: RIS-Assisted Systems and Modern Schemes

In CPSC-RIS systems, configuring $R$ RIS super-elements with distinct cyclic delays adds $R$ non-overlapping virtual taps, thereby increasing $d_{\max} = \sum_{r=0}^R L_r$. Full diversity is validated by high-SNR BER slopes, and remains robust to moderate channel estimation error (Li et al., 2022). For RIS-assisted ISAC radar, the maximum spatial diversity (e.g. $D=2$ for direct plus RIS path) is achieved by jointly optimizing the transmit beamformer and RIS reflection phases using alternating MM and ADMM algorithms; each additional independent and controllable path (e.g. multiple RIS clusters) increases the diversity order by one (Zhang et al., 2024).

In multihop and cooperative relay networks, pipelined amplify-and-forward combined with universal cyclic division algebra codes attains the cut-set diversity $d_{\max}$ without a loss for half-duplex operation, provided the network supports that many node-disjoint paths (0802.1888).

• (Li et al., 2022): Channel Estimation and Multipath Diversity Reception for RIS-Empowered Broadband Wireless Systems Based on Cyclic-Prefixed Single-Carrier Transmission
• (Wang et al., 20 Jan 2026): Achieving Full Multipath Diversity by Random Constellation Rotation: a Theoretical Perspective
• (0705.2848): Non-Coherent Capacity and Reliability of Sparse Multipath Channels in the Wideband Regime
• [0702138]: On the Maximal Diversity Order of Spatial Multiplexing with Transmit Antenna Selection
• (0802.1888): Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design
• (Schulze, 2010): Diversity Spectra of Spatial Multipath Fading Processes
• (0708.0386): Diversity of MIMO Multihop Relay Channels
• (Zhang et al., 2024): Multipath Exploitation for Fluctuating Target Detection in RIS-Assisted ISAC Systems