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Summary form only given. Sound textures, for instance a crackling fire, running water, or applause, constitute a large and largely neglected class of audio signals. Whereas tonal sounds have been effectively and flexibly modelled with sinusoids, aperiodic energy is usually modelled as white noise filtered to match the approximate spectrum of the original over 10-30 ms windows, which fails to provide a perceptually satisfying reproduction of many real-world noisy sound textures. We attribute this failure to the loss of short-term temporal structure, and we introduce a second modelling stage in which the time envelope of the residual from conventional linear predictive modelling is itself modelled with linear prediction in the spectral domain. This cascaded time- and frequency-domain linear prediction (CTFLP) leads to noise-excited resyntheses that have high perceptual fidelity. We perform a novel quantitative error analysis by measuring the proportional error within time-frequency cells across a range of timescales.