Pattern recognition from audio signals is an active research topic encompassing audio tagging, acoustic scene classification, music classification, and other areas. Spectrogram and mel-frequency cepstral coefficients (MFCC) are among the most commonly used features for audio signal analysis and classification. Recently, deep convolutional neural networks (CNN) have been successfully used for audio classification problems using spectrogram-based 2D features. In this paper, we present SpectNet, an integrated front-end layer that extracts spectrogram features within a CNN architecture that can be used for audio pattern recognition tasks. The front-end layer utilizes learnable gammatone filters that are initialized using mel-scale filters. The proposed layer outputs a 2D spectrogram image which can be fed into a 2D CNN for classification. The parameters of the entire network, including the front-end filterbank, can be updated via back-propagation. This training scheme allows for fine-tuning the spectrogram-image features according to the target audio dataset. The proposed method is evaluated in two different audio signal classification tasks: heart sound anomaly detection and acoustic scene classification. The proposed method shows a significant 1.02\% improvement in MACC for the heart sound classification task and 2.11\% improvement in accuracy for the acoustic scene classification task compared to the classical spectrogram image features. The source code of our experiments can be found at \url{https://github.com/mHealthBuet/SpectNet}