The learning of interpretable representations from raw data presents significant challenges for time series data like speech. In this work, we propose a relevance weighting scheme that allows the interpretation of the speech representations during the forward propagation of the model itself. The relevance weighting is achieved using a sub-network approach that performs the task of feature selection. A relevance sub-network, applied on the output of first layer of a convolutional neural network model operating on raw speech signals, acts as an acoustic filterbank (FB) layer with relevance weighting. A similar relevance sub-network applied on the second convolutional layer performs modulation filterbank learning with relevance weighting. The full acoustic model consisting of relevance sub-networks, convolutional layers and feed-forward layers is trained for a speech recognition task on noisy and reverberant speech in the Aurora-4, CHiME-3 and VOiCES datasets. The proposed representation learning framework is also applied for the task of sound classification in the UrbanSound8K dataset. A detailed analysis of the relevance weights learned by the model reveals that the relevance weights capture information regarding the underlying speech/audio content. In addition, speech recognition and sound classification experiments reveal that the incorporation of relevance weighting in the neural network architecture improves the performance significantly.