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

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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 h∑ γ i q (t − iTs ) . i =0

NS

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I. INTRODUCTION Continuous phase modulation (CPM) signals have constant amplitude and so they are a good solution for systems requiring insensitivity to non-linear amplitude amplification. Their phase continuity allows good spectral performance and implies a code gain due to the inherent memory effect. These properties have motivated the common use of GMSK (gaussian minimum shift keying), which is a simple member of the CPM family, in widespread use systems such as GSM/DCS, PCS, DECT, CT2 and Bluetooth. The use of others CPM schemes more spectrally efficient and better power efficient was restrained owing to excessive detection complexity [1]. The number of analogue matched filters (or correlators) needed is often unbearable for practical implementation. The number of phase states to be detected can very large as well. Conception of simple receivers is nowadays a main concern within CPM research. This paper shows that Walsh functions can generate a space where it is possible to find signals close to the original CPM signals. Digital signal processing (DSP) allows fast matrix calculus using both received and stored signals. II. CPM FORMATTING AND PERFORMANCE Every CPM signals can be expressed in the form:

A constant modulation index, h=p/q, is considered, where p and q are integers with no common factors. (h is rational in order to have a finite number of phase states.) Phase transition pulse shape, q(t), affects phase transitions shape during L symbols. However, its effect remains until the end of the transmitted sequence. q(t) is t defined by the frequency pulse g(t): q (t ) = ∫ g (τ )dτ . The ∞ −∞ normalisation q(t) = ∫ g(τ ) dτ = 1/2 is applied so that the 0 maximum phase transition during a symbol time, Ts, is h⋅(M-1)⋅π. Different frequency pulses define different CPM families. The most common are: LREC, LRC (L is the variable mentioned above) and GMSK [1,2]. LREC is defined by g(t)=rect[t/(LTs)]/2, where rect(t)=1 for −1/2