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To develop an automated system to classify the severity of hypoxic-ischaemic encephalopathy injury (HIE) in neonates from the background electroencephalogram (EEG). By combining a quadratic time–frequency distribution (TFD) with a convolutional neural network, we develop a system that classifies 4 EEG grades of HIE. The network learns directly from the two-dimensional TFD through 3 independent layers with convolution in the time, frequency, and time–frequency directions. Computationally efficient algorithms make it feasible to transform each 5 min epoch to the time–frequency domain by controlling for oversampling to reduce both computation and computer memory. The system is developed on EEG recordings from 54 neonates. Then the system is validated on a large unseen dataset of 338 h of EEG recordings from 91 neonates obtained across multiple international centres.