An improved method of closed-loop synchronization of a radio receiver with the phase of a carrier signal modulated by Gaussian minimum-shift keying (GMSK) has been proposed. Synchronization of the receiver with the phase of the carrier signal ("carrier synchronization" for short) is necessary for coherent detection of the data modulation. The method could improve the performances of digital wireless communication systems - particularly European cellular systems, wherein GMSK is the standard form of modulation but efficient means of carrier synchronization for coherent detection have thus far been lacking.

In GMSK, continuous-phase frequency-modulation pulses are used to convey digital data. The specific pulse shape is such that each pulse can last longer than one baud interval or bit period, *T _{b}*. The pulse duration is given by

*LT*, where

_{b}*L*is an integer that is typically chosen to equal 4. The overlapping of pulses when

*L*>

*1*gives rise to additional inter-symbol interference (ISI) - beyond the ISI attributable to the memory associated with continuity of phase. In older GMSK carrier-synchronization methods, ISI is not taken into account; consequently, GMSK carrier-synchronization systems designed according to those methods perform suboptimally. In the proposed method, ISI is taken into account, making it possible to approach optimum performance.

The present method is based on a combination of (1) maximum* a posteriori *(MAP) estimation of digital modulation containing ISI and (2) the Laurent amplitude-modulation pulse (AMP) representation of continuous-phase modulation conveying digital data. In the Laurent AMP representation, a GMSK signal is described in terms of a superposition of 2^{L}^{-1} amplitude-and-phase-modulated pulse streams, some containing pulses that extend beyond *T _{b}*. Thus, effects of ISI are included.

In the typical case of *L* = 4, the Laurent AMP representation contains 8 pulse trains. However, two of the pulse trains (for which the pulse durations are 5*T _{b}* and 3

*T*, respectively) contain most of the signal energy (the fraction of signal energy in the other six pulse trains is only 2.63 × 10

_{b}^{-5}). Therefore, the signal can be approximated closely by the 5

*T*and 3

_{b}*T*pulse trains. This is advantageous because the design of a closed carrier-synchronization loop based on these two pulse trains only can be simpler than a design based on all eight pulse trains.

_{b}The loop design is derived from a combination of (1) the foregoing two-pulse-train representation and (2) an equation for an error signal as a function of the estimated carrier phase for the GMSK signal observed (along with noise) during a given number of baud intervals. The zero-error condition is an estimated carrier phase equal to the open-loop MAP phase estimate. One would close the loop by updating the phase estimate in the effort to null the error signal.

It turns out that the right side of the equation for the error signal as a function of the estimated carrier phase can be decomposed into two components, each corresponding to one of the two pulse streams in the approximate AMP representation. Thus, a closed-loop GMSK carrier synchronizer could be constructed as a superposition of two loops, each contributing one of the components of the error signal (see figure).

At the time of reporting the information for this article, the method had been tested in some computational simulations, with promising results; simulated carrier-phase synchronizers designed according to the proposed method exhibited excellent performance. Moreover, inasmuch as the second pulse stream contains significantly less energy than the first one does, it might be possible to reduce the complexity of the basic synchronizer design by use of a single-pulse-stream AMP representation of GMSK.

*This work was done by Marvin K. Simon of Caltech for NASA's *Jet Propulsion Laboratory*.*

*In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to*

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*Refer to NPO-20482*