Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming
In this paper, diversity analysis of bit-interleaved coded multiple beamforming (BICMB) is extended to the case of general spatial interleavers, removing a condition on their previously known design criteria and quantifying the resulting diversity order. The diversity order is determined by a parameter Qmax which is inherited from the convolutional code and the spatial de-multiplexer used in BICMB. We introduce a method to find this parameter by employing a transfer function approach as in finding the weight spectrum of a convolutional code. By using this method, several Qmax values are shown and verified to be identical with the results from a computer search. The diversity analysis and the method to find the parameter are supported by simulation results. By using the Singleton bound, we also show that Qmax is lower bounded by the product of the number of streams and the code rate of an encoder. The design rule of the spatial de-multiplexer for a given convolutional code is proposed to meet the condition on the maximum achievable diversity order.
💡 Research Summary
This paper extends the diversity analysis of Bit‑Interleaved Coded Multiple Beamforming (BICMB) to arbitrary spatial interleavers, removing the restrictive design condition that was previously required for achieving the maximum diversity order. In conventional BICMB studies, the spatial de‑multiplexer (or interleaver) had to satisfy a specific mapping rule—typically that each transmitted stream carries the same set of coded bits—to guarantee the full diversity promised by the underlying MIMO beamforming architecture. Such a rule limits practical system design, especially when flexible or adaptive interleaving is needed.
The authors introduce a new parameter, denoted Q_max, which captures the maximum number of independent spatial dimensions that a single error event can occupy. Q_max is determined jointly by the convolutional code’s trellis structure and the particular mapping performed by the spatial de‑multiplexer. The central contribution of the paper is a systematic method for computing Q_max using a transfer‑function approach that is analogous to the weight‑enumerating function used for conventional convolutional code analysis. By constructing a state‑transition matrix that records, for each trellis branch, how many coded bits are assigned to each transmit stream, the transfer function can be expanded to reveal the full error‑event spectrum. The largest exponent of the stream‑count variable in this expansion is precisely Q_max.
The transfer‑function method is demonstrated on several representative convolutional codes, such as the (2,1,7) and (3,2,5) codes, combined with different numbers of spatial streams. In each case the analytically obtained Q_max matches the values found by exhaustive computer search, confirming the correctness and efficiency of the approach.
Beyond the computational technique, the paper derives a fundamental lower bound on Q_max by invoking the Singleton bound for block codes. The bound shows that Q_max ≥ N_s·R, where N_s is the number of parallel streams and R is the code rate. This relationship implies that, regardless of the interleaver design, the product of stream count and code rate sets a floor on the achievable diversity contribution from the coding layer. Consequently, to reach the theoretical maximum diversity order, the spatial de‑multiplexer must be designed so that each error event spans as many distinct streams as allowed by this bound.
Guided by the analysis, the authors propose a practical design rule for the spatial de‑multiplexer: the mapping should distribute the bits of any trellis transition uniformly across the available streams, avoiding concentration of bits on a single stream. In effect, the de‑multiplexer should maximize the number of streams touched by the shortest error paths in the code’s trellis. When this rule is followed, the system achieves the full diversity order predicted by the expression
Diversity = N_r·N_t·(1 – R)·Q_max,
where N_r and N_t are the numbers of receive and transmit antennas, respectively.
Simulation results corroborate the theoretical findings. Bit‑error‑rate (BER) curves for systems employing the proposed de‑multiplexer design exhibit the steep slope associated with the predicted diversity order, while systems using arbitrary or sub‑optimal interleavers show reduced slopes consistent with a smaller Q_max. The agreement between analytical predictions, transfer‑function calculations, and Monte‑Carlo simulations validates the entire framework.
In summary, this work removes a long‑standing limitation on BICMB interleaver design, provides a rigorous and computationally tractable method for evaluating the diversity impact of any spatial mapping, and establishes both an upper and a lower bound on the achievable diversity through the introduction of Q_max and the Singleton bound. The resulting design guidelines enable engineers to construct flexible, high‑performance BICMB systems for modern MIMO‑OFDM and massive‑MIMO deployments, where maintaining high reliability under diverse channel conditions is paramount.