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This paper studies the differentially amplitude- and phase-encoded quadrature amplitude modulation (DAPE QAM) transmission for the amplify-and-forward multiple-relay system using the fixed-relay-gain mechanism over independent Rician and Nakagami-m fading links. In the multiple-relay system, the destination collects signals from the source and multiple relays in distinct phases. Operating over two successively received symbols, an equal-gain combining (EGC) receiver and a weighted-gain combining (WGC) receiver are developed to noncoherently combine received signals from direct and multiple-relay links and thereby achieve cooperative diversity. The EGC receiver operates without any link-side information, whereas the knowledge of average signal-to-noise ratios (SNRs) on source-relay and relay-destination links is required for realizing the WGC receiver. Based on Beaulieu's convergent series approach, efficient computation formulas of the bit error probability (BEP) upper bounds are derived for both receivers. Numerical and simulation results on 16- and 64-point signal constellations show that the WGC receiver exhibits better BEP performance than the EGC receiver when link SNR estimates are not significantly large in error. The WGC receiver for DAPE QAM is also shown to outperform, in terms of BEP, the conventional receiver for differentially detecting the differential phase-shift keying modulation signal with the same constellation size.