Limits for Cpm Functions

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Limits for CPM Signals Representation by Walsh Functions
Francisco A. Monteiro1, António J. Rodrigues1, 2
2

Instituto de Telecomunicações and ISCTE Instituto de Telecomunicações and Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal Tel: +351 218418484; Fax: +351 218418472; E-mail: frmo@lx.it.pt

1

Abstract — This paper explores the feasible limits for complexity reduction of a very simple front-end block for the calculus of phase transition metrics on a continuous phase modulation (CPM) receiver. A quasi-optimum receiver of very low complexity is attained by splitting the function of the optimum receiver bank filters in two blocks: calculus of projections coefficients on a low dimensional space of Walsh functions followed by simple matrix calculus. A sequence detection algorithm follows this block. The presented approach enables the reduction of the matched filters or correlators to just two integrators, regardless of the CPM scheme. Research on the reduction limits of the space dimension is conducted using catastrophic M-ary CPM schemes, taking advantage of their very low number of phase states. Performance of 1REC h=1/2 16-ary scheme is for the fist time presented. A rule is defined concerning the number of Walsh functions that must be used. That outcome proves to be valid for two CPM schemes of high power gain. The receiver is tested under additive white gaussian noise (AWGN).

The carrier frequency is fc, where ωc=2πfc, ϕ0 is the arbitrary initial phase and Es is the energy per symbol, related with the bit energy by Es=log2(M)⋅Eb. Channel symbols are γi∈{±1, ±3, ⋅⋅⋅, ±(M-1)}, forming the M-ary sequence γ . Each symbol γi carries log2(M) bits as a result of a natural mapping of the information bits stream α. The information carried by NS channel symbols is keyed in signal’s phase:

ϕ (t , γ ) = 2π…...

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