Random Multiplexing

TL;DR

Random Multiplexing introduces a unitary random transform that decouples the multiplexing operation from the physical channel, yielding an equivalent dense, input-isotropic channel that lies in the universality class $\\mathscr{U}$. This enables AMP-type detectors, notably the cross-domain MAMP (CD-MAMP), to achieve replica MAP BER optimality and replica-constrained capacity across arbitrary norm-bounded, spectrally convergent channel matrices, with reduced complexity by exploiting time-domain sparsity. The work develops RT-domain power allocation and a principled coding framework that attains capacity-like performance in practice, supported by state evolution analyses and numerical results showing sizable BER/BLER and rate gains over traditional schemes such as OFDM, OTFS, and AFDM in realistic high-mobility settings. The framework also offers opportunities to improve spectral efficiency, adapt to various modulations, and integrate AI for finite-length design, broadening applicability to future 6G deployments and beyond.

Abstract

As wireless communication applications evolve from traditional multipath environments to high-mobility scenarios like unmanned aerial vehicles, multiplexing techniques have advanced accordingly. Traditional single-carrier frequency-domain equalization (SC-FDE) and orthogonal frequency-division multiplexing (OFDM) have given way to emerging orthogonal time-frequency space (OTFS) and affine frequency-division multiplexing (AFDM). These approaches exploit specific channel structures to diagonalize or sparsify the effective channel, thereby enabling low-complexity detection. However, their reliance on these structures significantly limits their robustness in dynamic, real-world environments. To address these challenges, this paper studies a random multiplexing technique that is decoupled from the physical channels, enabling its application to arbitrary norm-bounded and spectrally convergent channel matrices. Random multiplexing achieves statistical fading-channel ergodicity for transmitted signals by constructing an equivalent input-isotropic channel matrix in the random transform domain. It guarantees the asymptotic replica MAP bit-error rate (BER) optimality of AMP-type detectors for linear systems with arbitrary norm-bounded, spectrally convergent channel matrices and signaling configurations, under the unique fixed point assumption. A low-complexity cross-domain memory AMP (CD-MAMP) detector is considered, leveraging the sparsity of the time-domain channel and the randomness of the equivalent channel. Optimal power allocations are derived to minimize the replica MAP BER and maximize the replica constrained capacity of random multiplexing systems. The optimal coding principle and replica constrained-capacity optimality of CD-MAMP detector are investigated for random multiplexing systems. Additionally, the versatility of random multiplexing in diverse wireless applications is explored.

Figures (17)

• Figure 1: A comparison of the equivalent channel matrices $\bf{H}_{{\rm{output}}\times{\rm{input}}}^{\rm{eqv}}=\bf{\Xi }^{\rm{H}}\bf{H}\bf{\Xi}$ of OFDMtse2005fundamentals ($\bf{\Xi}=\bf{F}^{\rm{H}}$) in static multipath channels, and OTFS ($\bf{\Xi}={\bf{F}}^{\rm{H}}\otimes \bf{I}$)OTFS1, AFDM ($\bf{\Xi}=\bf{\Lambda}_{c_1}^{\mathrm{H}}\bf{F}^{\mathrm{H}}_{{N}}\ \bf{\Lambda}_{c_2}^{\mathrm{H}}$) AFDM and the random multiplexing (RM) in Definition \ref{['Def:RUP']} in time-varying doubly-selective channels, where $\bf{H}$ denotes the time-domain channels, $\bf{\Xi}$ the multiplexing matrix, and $\bf{\Lambda}_{c_i}\triangleq\mathrm{diag}(e^{-j2\pi c_in^2}, n=0, \cdots\!, N-1)$, $i=1,2$.
• Figure 2: The linear system with random multiplexing, where the mapper corresponds to the constellation constraint $\bf{s}\overset{\rm i.i.d.}{\sim} P_{s}$, and $\bf{\Xi}$ denotes the RT matrix that is independent of the signal vector $\bf{s}$, the measurement matrix $\bf{A}$, and the channel noise $\bf{n}$.
• Figure 3: Visualization and BER comparison under ML detection for 2D BPSK linear systems with orthogonal and unitary multiplexing.
• Figure 4: BER comparisons for 1) orthogonal multiplexing $\bm{y} = \bm{\Sigma}_A \bm{x} + \bm{n}$ using an element-wise MMSE detector, 2) unitary multiplexing $\bm{y} = \bm{\Sigma}_A\bm{\Xi} \bm{x} + \bm{n}$ using ML and AMP-type detectors. The $\bm{\Sigma}_A$ and $\bm{\Xi} = \bm{V}_A^{\rm H}$ are obtained from the SVD on $10^5$ normalized IID Gaussian matrices $\bm{A}$. For each $\bm{A}$, the BER is averaged over $10^4$ Monte Carlo simulations for all detectors. The system dimension is set to $N \in \{ 8, 32, 512\}$.
• Figure 5: The CD-MAMP framework for the linear system with random multiplexing, where $t$ denotes the iteration index, $\bf{\Xi}$ the RT, $\bf{\Xi}^{\rm H}$ the IRT, and DEM the demodulation.

Theorems & Definitions (37)

• Definition 1: Spectrally Convergent Matrix
• Lemma 1: Replica MMSE and MAP BER
• Lemma 2: Scalar I-MMSEGuo2005
• Lemma 3: Replica Constrained CapacityBarbier2018ISITKabashima2006Tulino2013
• Definition 2: Universal Multiplexing
• Lemma 4: IID MatricesRishabh2024
• Definition 3: Random Multiplexing
• Theorem 1: Permutation-Invariant Matrices
• Corollary 1
• Lemma 5: Haar-Distributed MatricesRishabh2024