The paper proposes a novel model for time series displaying persistent stationary cycles, the fractional sinusoidal waveform process. The underlying idea is to allow the parameters that regulate the amplitude and phase to evolve according to fractional noise processes. Its advantages with respect to popular alternative specifications, such as the Gegenbauer process, are twofold: the autocovariance function is available in closed form, which opens the way to exact maximum likelihood estimation; secondly the model encompasses deterministic cycles, so that discrete spectra arise as a limiting case. A generalization of the process, featuring multiple components, an additive `red noise' component and exogenous variables, provides a model for climate time series with mixed spectra. Our illustrations deal with the change in amplitude and phase of the intra-annual component of carbon dioxide concentrations in Mauna Loa, and with the estimation and the quantification of the contribution of orbital cycles to the variability of paleoclimate time series.
 

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