1564 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009 Spectrally-Ef ﬁcient Continuous Phase Modulations Alan Barbieri, Dario Fertonani, Student Member, IEEE, and Giulio Colavolpe, Member, IEEE Abstract—We investigate the spectral efﬁciency of continuous phase modulations (CPMs). To this end, we need an effective bandwidth deﬁnition for a CPM signal, whose power spectral density has in principle an inﬁnite support. The deﬁnition we adopt is based on the spacing between adjacent carriers in a frequency division multiplexed CPM system. We consider the inter-channel interference, which depends on the channel spacing, and we evaluate the spectral efﬁciency achievable by a single-user receiver in the considered multi-channel scenario. We then optimize the channel spacing with the aim of maximizing the spectral efﬁciency, showing that impressive improvements with respect to the spectral efﬁciencies reported in the literature and obtained by heuristic approaches can be achieved. Index Terms—Continuous phase modulation, interchannel in- terference, multiuser channels, information rate, spectral efﬁ- ciency. I. I NTRODUCTION C ONTINUOUS phase modulations (CPMs) form a class of constant envelope signaling, widely studied thanks to their appealing properties [1]. In particular, besides the robustness to non-linearities stemming from the constant en- velope, some CPM formats are claimed to exhibit an excellent power and spectral efﬁciency [1], [2]. The spectral efﬁciency is an important quality ﬁgure for a modulation format, since it quantiﬁes how many information bits per second can be loaded per unity of the available spectrum. To evaluate it, a suitable bandwidth deﬁnition is required. In fact, although the power of a CPM signal is typically concentrated in a small portion of the spectrum [3], the power spectral density (PSD) has rigorously an inﬁnite support. The most commonly employed deﬁnition is that based on the power concentration, which deﬁnes the signal bandwidth as the bandwidth that contains a given fraction (this fraction being a parameter) of the overall power [4]–[6]. A different deﬁnition is used in [7], [8], which is based on the Carson’s rule and exhibits a nice mathematical tractability. We also notice that some papers, as [9], investigated the information rate achievable by CPMs Manuscript received May 20, 2008; revised August 18, 2008; accepted October 23, 2008. The associate editor coordinating the review of this paper and approving it for publication was D. Huang. A. Barbieri and D. Fertonani are with Scuola Superiore Sant’Anna, Via G. Moruzzi 1, I-56124 Pisa (e-mail: a.barbieri@sssup.it; dario.fertonani@gmail.com). G. Colavolpe is with Università di Parma, Dipartimento di Ingegneria dell’Informazione, Viale G. P. Usberti, 181A, I-43100 Parma, Italy (e-mail: giulio@unipr.it). The paper has been presented in part at the IEEE International Symposium on Information Theory (ISIT’08), Tornto, Canada, July 2008, and at the 10th International Workshop on Signal Processing for Space Communications (SPSC’08), Rhodes Island, Greece, Oct. 2008. This work is funded by the European Space Agency, ESA-ESTEC, Noord- wijk, The Netherlands, under contract no. 19370/05/NL/JD. Digital Object Identiﬁer 10.1109/TWC.2009.080679 without accounting for the bandwidth and thus without giving any insight on the spectral efﬁciency. A critical drawback of the above mentioned bandwidth deﬁnitions is that they are not prone to describe the impact of the power leakage outside the deﬁned bandwidth in terms of interference caused to adjacent signals. In other words, the values of spectral efﬁciency they predict are nearly useless in a real multi-user scenario. As a matter of fact, from a communication system perspective, the ﬁnal aim of any bandwidth deﬁnition is to evaluate the portion of the available spectrum taken up by a signal, so as to realize how far the adjacent channels should be placed from the signal under analysis when a frequency division multiplexed (FDM) system is considered. Based on this consideration, we consider an FDM-CPM transmission over a channel impaired by additive white Gaussian noise (AWGN), where all sub-channels are equally spaced and adopt the same CPM format, so that the effective bandwidth of a CPM signal is implicitly deﬁned as the separation between two adjacent channels. Then, we evaluate the ultimate information rate achievable when single- user detection is employed, that is, when each of the sub- channels is processed by a dedicated receiver that neglects the inter-channel interference (ICI)—we will use the terms “channels” and “users” interchangeably. For the considered scenario, which turns out to be an instance of mismatched decoding [10], the information rate can be evaluated by means of the simulation-based algorithm described in [11], as a function of the spacing between the channels (i.e., the effective bandwidth). Hence, a method for evaluating the ultimate achievable spectral efﬁciency results, which improves the existing methods since it accounts for the presence of ICI. By means of the proposed framework, we show that, for any given CPM format, the achievable spectral efﬁciency is signiﬁcantly larger than that resulting from the conventional approaches, in which a ﬁxed, non optimized spacing is con- sidered. In particular, it turns out convenient to consider fre- quency spacings between the channels much lower than those deriving from the above mentioned bandwidth deﬁnitions, which can provide a better tradeoff between degradation of the information rate due to the ICI and usage of the available spec- trum. Moreover, we show that the optimal spacing depends on the signal-to-noise ratio (SNR), that is like to say that the effective bandwidth of a CPM signal does depend on the SNR. Hence, for a given CPM format, the frequency spacing must be selected during design according to the operating SNR. The framework proposed in this paper can be also used for selecting, based on information-theoretic arguments, a set of CPM formats suitable for a particular application, similarly to what has been done in [6] and [12] (see also [13]), where anyway the effect of ICI has been neglected—a comparison 1536-1276/09$25.00 c  2009 IEEE