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Microwave backhaul links are characterized by high signal-to-noise ratios permitting spectrally-efficient transmission. The used signal constellation sizes and achievable data rates are typically limited by transceiver impairments, predominantly by phase noise from non-ideal carrier generation. In this paper, we propose a new method to improve the data rate over such microwave links. We make use of the fact that adjacent frequency channels are inactive in many deployment scenarios. We argue that additional data can be transmitted in the skirts of the spectral mask imposed on the transmission signal by regulation. To accomplish this task, we present a shaped wideband single-carrier transmission using non-Nyquist pulse shapes. In particular, we design spectrum-skirt filling (SSF) pulse shaping filters that follow the spectral mask response, and perform detection using an accordingly increased sampling frequency at the receiver. We evaluate the achievable information rates of the SSF-based transmission considering practical dispersive channels and non-ideal transmitter and receiver processing. To compensate for phase noise impairments, we derive carrier phase tracking and estimation techniques, and utilize them in tandem with nonlinear precoding which mitigates the intersymbol interference introduced by the non-Nyquist SSF shaping filter. Quantitative performance evaluations show that the proposed system design achieves higher data rates in a dispersive microwave propagation environment with respect to the conventional transmission with Nyquist pulse shaping.