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Damage detection and the classification of carbon fiber-reinforced composites using non-destructive testing (NDT) techniques are of great importance. This paper applies an acoustic emission (AE) technique to obtain AE data from three tensile damage tests determining fiber breakage, matrix cracking, and delamination. This article proposes a deep learning approach that combines a state-of-the-art deep learning technique for time series classification: the InceptionTime model with acoustic emission data for damage classification in composite materials. Raw AE time series and frequency-domain sequence data are used as the input for the InceptionTime network, and both obtain very high classification performances, achieving high accuracy scores of about 99%. The InceptionTime network produces better training, validation, and test accuracy with the raw AE time series data than it does with the frequency-domain sequence data. Simultaneously, the InceptionTime model network shows its potential in dealing with data imbalances.