IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 15, NO. 9, SEPTEMBER 2005 549 Phase Error Resilience to I/Q Mismatch of a Simplified CPM Receiver Francisco A. Monteiro and António J. Rodrigues, Member, IEEE Abstract—A low complexity receiver which was devised in previous papers proved there to be quasioptimum over additive gaussian noise and also showed low power penalty with flat Rayleigh fading, i.e., with random phase, and despite the fact that one of three reduced complexity blocks of that receiver relies on symmetries on signal’s phase transitions. This letter analyzes the origin of that error resilience during the derivation of the metrics. Index Terms—Continuous phase modulation (CPM), phase ro- tation error, symmetry-based metrics derivation. I. INTRODUCTION C ONTINUOUS phase modulation (CPM) signals are insen- sitive to nonlinear radio frequency (RF) amplitude ampli- fication. Phase continuity allows good spectral performance and comprises a code gain. These properties motivated the wide- spread use of both minimum shift keying (MSK) and gaussian MSK. The use of other CPM schemes having both higher spec- tral and power efficiencies was restrained over time due to ex- cessive detection complexity [1]. However, complexity reduc- tion techniques made possible to employ CPM over digital radio relay links and it is also being considered for digital audio broad- casting [2]. CPM detection raises two problems: the optimum detector usually requires a very large bank of matched filters to obtain all phase transition metrics; afterwards, the number of phase states to track by a Viterbi algorithm may also be huge [1]. A quasioptimum low complexity CPM receiver was achieved by introducing three complexity reduction techniques on the op- timal receiver: i) replacement of the bank of matched filters by projections on a Walsh space [3], ii) sequence detection with the M-algorithm [4], and iii) a symmetry-based algorithm for the derivation of metrics from just 1/4 of them, which is to be further analyzed on this letter. This receiver has not only proved to be quasioptimum when assessed only with additive gaussian noise (AWGN) [5] but also demonstrated just about 3 dB power loss when assessed over Rayleigh fading with both amplitude Manuscript received February 15, 2005. This work was supported in part by the Portuguese Foundation for Science and Technology under both POSI and FEDER programs. The review of this letter was arranged by Guest Editors H. Nikookar and R. Prasad. F. A. Monteiro is with Telecommunications Institute, Lisbon 1049-001, Por- tugal and also with the Department of Information Sciences and Technologies, ISCTE, Lisbon 1049-001, Portugal (e-mail: frmo@lx.it.pt). A. J. Rodrigues is with Telecommunications Institute, Lisbon 1049-001, Por- tugal and also with the Department of Electrical and Computer Engineering, Instituto Superior Técnico, Technical University of Lisbon, Lisbon 1049-001, Portugal (e-mail: antonio.rodrigues@lx.it.pt). Digital Object Identifier 10.1109/LMWC.2005.855388 and phase additional impairments [6]. Hence, the geometric al- gorithm used to derive metrics proved robustness to phase rota- tions. This letter explains why and how that symmetry-depen- dent algorithm is resilient to phase rotations. II. CPM SIGNALS AND TESTING SCHEMES CPM signals are expressed by (1) The carrier frequency is , where , is the arbitrary initial phase, and is the energy per symbol, related with the bit energy by . Channel symbols are , forming the -ary sequence . Each symbol carries bits as a result of a natural mapping of the information bits. The information carried by channel symbols is keyed into signal’s phase by (2) A constant modulation index, , is considered, where and are integers with no common factors, so that the number of phase states is a finite one. Phase pulse shape, , af- fects phase transitions throughout symbols, however, its effect remains until the end of the sequence. is given by the fre- quency pulse . Making assures that the maximum phase transition is during a symbol time, . The most common pulses are LREC and GMSK [1]. LREC is given by , where for and zero elsewhere. The effect of phase rotations is initially studied on catastrophic -ary CPM schemes ( 1/2 and 2, i.e. MSK, and for 4, 8, and 16), taking advantage of their small number of states ( 4). Afterwards, the schemes 1REC 9/20 with and 8 are tested. These schemes are the best 4 and 8-ary 1REC schemes in joint power and spectral efficiencies whilst preserving a low number of states ( 40). Moreover, both schemes have minimum normalized squared Euclidean distances (MNSED) equal to their upper bound [1], [3]. III. METRICS DERIVATION For schemes with 0 4 only transitions metrics of transitions departing from states on the first quadrant must be calculated, and only for the in-phase branch; all the others can be derived from them. The phase states of those schemes can be distributed throughout the four quadrants in a symmetric 1531-1309/$20.00 © 2005 IEEE