Iterative Equalization of CPM With Unitary Approximate Message Passing

TL;DR

The paper targets reliable CPM detection in frequency-selective fading by introducing a factor-graph–based iterative receiver that combines a $UAMP$-based equalizer with an FFT/CP unitary transform and a BP-based demodulator/decoder. The approach explicitly decouples the densely connected equalizer from the discrete-variable CPM demodulator/decoder, enabling efficient inference via a three-part pipeline and a carefully designed message-passing schedule. Empirical results show that the proposed receiver delivers substantial BER improvements (approximately 1.5–1.7 dB at BER $10^{-5}$) over MMSE/SIC-based turbo equalizers, with fast convergence and comparable computational complexity. This work demonstrates that unitary transforms and message-passing on factor graphs can effectively tackle nonlinear CPM detection in realistic channel conditions, offering practical gains for high-efficiency wireless systems.

Abstract

Continuous phase modulation (CPM) has extensive applications in wireless communications due to its high spectral and power efficiency. However, its nonlinear characteristics pose significant challenges for detection in frequency selective fading channels. This paper proposes an iterative receiver tailored for the detection of CPM signals over frequency selective fading channels. This design leverages the factor graph framework to integrate equalization, demodulation, and decoding functions. The equalizer employs the unitary approximate message passing (UAMP) algorithm, while the unitary transformation is implemented using the fast Fourier transform (FFT) with the aid of a cyclic prefix (CP), thereby achieving low computational complexity while with high performance. For CPM demodulation and channel decoding, with belief propagation (BP), we design a message passing-based maximum a posteriori (MAP) algorithm, and the message exchange between the demodulator, decoder and equalizer is elaborated. With proper message passing schedules, the receiver can achieve fast convergence. Simulation results show that compared with existing turbo receivers, the proposed receiver delivers significant performance enhancement with low computational complexity.

Figures (6)

• Figure 1: Data block structure of $\boldsymbol{\alpha}^{(j)}$ in the case of cyclic prefix.
• Figure 2: Factor graph representation of (23).
• Figure 3: BER performance of the receiver with various equalizers. The blue and orange curves represent the TU-6 and Proakis’ C channels, respectively.
• Figure 4: BER performance of the receiver with various numbers of iterations. The blue and orange curves represent the TU-6 channels at $4$dB and Proakis’ C channel at $6.5$dB, respectively.
• Figure 5: Convergence performance of the receiver with various numbers of inner iterations under the TU-6 channel.