Full Rate L2-Orthogonal Space-Time CPM for Three Antennas

To combine the power efficiency of Continuous Phase Modulation (CPM) with enhanced performance in fading environments, some authors have suggested to use CPM in combination with Space-Time Codes (STC). Recently, we have proposed a CPM ST-coding scheme based on L2-orthogonality for two transmitting antennas. In this paper we extend this approach to the three antennas case. We analytically derive a family of coding schemes which we call Parallel Code (PC). This code family has full rate and we prove that the proposed coding scheme achieves full diversity as confirmed by accompanying simulations. We detail an example of the proposed ST codes that can be interpreted as a conventional CPM scheme with different alphabet sets for the different transmit antennas which results in a simplified implementation. Thanks to L2-orthogonality, the decoding complexity, usually exponentially proportional to the number of transmitting antennas, is reduced to linear complexity.

💡 Research Summary

The paper addresses the long‑standing challenge of combining the power‑efficiency of continuous‑phase modulation (CPM) with the spatial diversity benefits of multiple‑input multiple‑output (MIMO) systems when more than two transmit antennas are used. While earlier works have successfully merged CPM with space‑time coding (STC) for two antennas, extending these schemes to three antennas typically incurs a loss in transmission rate or an exponential increase in decoding complexity. The authors propose a novel “Parallel Code” (PC) family that achieves full transmission rate, full diversity, and linear decoding complexity for three‑antenna CPM systems by exploiting L₂‑norm orthogonality rather than conventional matrix orthogonal designs.

The system model treats the transmitted signals as blocks of CPM waveforms, each block spanning three symbol intervals (3 T). The instantaneous phase of each antenna’s signal consists of a common initial phase, a data‑dependent term (modulation index h, data symbols d(l,i), and phase‑smoothing function q(t)), and a correction factor cₘ,ᵣ(t) that is introduced to enforce L₂ orthogonality across antennas. The orthogonality condition is expressed as an integral of the product of the transmitted signal matrix with its Hermitian transpose over a block, which must equal a scaled identity matrix. By imposing parallel mapping (identical data symbols on all antennas) and a specific repetitive mapping in time, the authors reduce the orthogonality condition to a simple phase‑difference equation. Solving this yields two admissible phase‑difference pairs: (2π/3, 2π/3) and (4π/3, 4π/3). Consequently, the difference between the phase‑memory terms ξₘ(3l+r) of any two antennas must be either 1/3 or 2/3 of a symbol interval.

Two concrete realizations of the correction factor are presented. The “linear PC” (LinPC) uses a piecewise‑linear function that jumps from 0 to ±1/3 over each symbol interval, while the “offset PC” (OffPC) embeds the correction into the CPM data term using a sum of the smoothing function q(t). In the OffPC case the three antennas can be interpreted as conventional CPM signals with three distinct alphabet sets, shifted by ±2/3 h. This viewpoint clarifies why the introduced frequency offsets are modest and do not degrade performance.

To prove full diversity, the authors construct the signal difference matrix Cₛ and apply the pairwise error probability framework. They show that Cₛ is full‑rank because any non‑zero linear combination of the antenna difference vectors would require a degree‑(Lₜ−1) polynomial to have more zeros than its degree, which is impossible given the structure of the correction factors. Hence, the code achieves the maximal diversity order equal to the number of transmit antennas (three).

Decoding complexity is dramatically reduced by the L₂ orthogonality: cross‑correlation terms between antennas vanish, allowing the Euclidean distance metric to be computed separately for each antenna and each time slot. The trellis therefore contains only 3 M branches per state (instead of M·Lₜ), while the number of states remains the same as in conventional CPM (p M γ − 1). This yields a linear‑in‑Lₜ decoding complexity, making practical MLSE feasible.

Simulation results over Rayleigh fading channels confirm the theoretical claims. Using both binary and quaternary CPM, the LinPC and OffPC schemes outperform uncoded CPM by roughly 2–3 dB at a BER of 10⁻⁴, while preserving a full transmission rate (one symbol per channel use). The modest frequency offsets introduced by the correction factors do not cause noticeable degradation, validating the practicality of the approach.

In summary, the paper introduces a robust, full‑rate, full‑diversity space‑time CPM scheme for three transmit antennas based on L₂ orthogonality. The Parallel Code family offers a compelling solution for power‑constrained wireless systems—such as satellite links, low‑power IoT, and next‑generation cellular uplinks—where CPM’s constant‑envelope property is desirable, yet spatial diversity is needed. The methodology also opens the door for extensions to more antennas and other CPM configurations, representing a significant step forward in the design of efficient, high‑performance MIMO CPM systems.