Efficient Algorithms for Calculating Euclidean Distance Spectra of Muliti-User Continuous Phase Modulation Systems Li Bing State Key Lab of ISN Xidian University Xi’an, Shaanxi, 710071, P.R.China Email: libingprc@gmail.com Tor Aulin Computer Science and Engineering Chalmers University of Technology SE-412 96 G¨ oteborg, Sweden. Email: aulin@chalmers.se Baoming Bai State Key Lab of ISN Xidian University Xi’an, Shaanxi, 710071, P.R.China Email: bmbai@mail.xidian.edu.cn Abstract—The performance of maximum likelihood detection of a coded modulation scheme over an additive white Gaussian noise channel can be predicted precisely by its Euclidean distance spectrum. In this paper, three algorithms are developed to generate the Euclidean distance spectra. The first algorithm is an exhaustive tree-search algorithm which is simplified by exploiting the correspondence between a code tree and the trellis. This algorithm is then modified to incorporate the A* algorithm and results in two heuristic algorithms. Moreover, the technique of trellis minimization is further developed for the multi user case and employed to reduce the complexity and memory requirements. Numerical results confirm that the proposed algorithms can generate the distance spectra of all systems with lower complexity relative to previous methods and are particularly suited for multi user CPM systems. I. I NTRODUCTION Continuous Phase Modulation (CPM) is a class of modula- tion schemes, which is widely used in wireless communication systems due to its good power-efficiency and bandwidth- efficiency [1]. It is generally recognized that the performance of max- imum likelihood (ML) detection of a modulation scheme over the additive white Gaussian noise (AWGN) channel can be predicted by its Euclidean distance spectrum (EDS) [6]. Existing algorithms mainly focus on deriving closed-form expression [5], [6], [7] of EDS using transfer matrix or transfer diagram. However, as the number of encoder states becomes high, the calculations quickly become cumbersome even using some newly developed techniques [10], [11]. On the other hand, instead of generating closed-form expression, numerical algorithms are designed to calculate the EDS using exhaustive search for a given length of observation. Unfortunately, a considerable amount of memory is required to enumerate and identify the error events being examined [8]. In this paper, three algorithms are proposed to calculate the EDS for multi-user CPM (MUCPM) systems. The first algorithm is an exhaustive tree-search algorithm which is sim- plified by exploiting the correspondence between a code tree and the trellis. This algorithm is then modified to incorporate the A* algorithm [12] and results in two simplified heuristic algorithms. Moreover, the concept of trellis minimization [11] is generalized to the MUCPM systems and employed by the proposed algorithms to reduce the search effort. This paper is organized as follows. Section II presents the systems model. Section III describes the algorithms and the technique of trellis minimization. Numerical results and performance comparison are given in Section IV. Sections V concludes the paper. II. SYSTEM MODEL Shown in Figure 1 is an MSK-based multiple-access type MUCPM system proposed in [9], which consists of K users                                               Figure 1. System Model sharing a common channel. The kth user maps independently its information sequence a k to a CPM signal waveform s k (t), which reads s k (t)=  2E k T k exp  j  2πh +∞  i=0 a i k q k (t - nT )  (1) where T k = T , E k , q k (t) and h are the symbol duration, average signal energy, phase response and modulation index 2012 IEEE International Symposium on Information Theory Proceedings 978-1-4673-2579-0/12/$31.00 ©2012 IEEE 2391