Signal denoising is a key preprocessing step for many applications, as the performance of a learning task is closely related to the quality of the input data. In this paper, we apply a signal processing based deep neural network architecture, a learnable extension of the wavelet packet transform. As main advantages, this model has few parameters, an intuitive initialization and strong learning capabilities. Moreover, we show that it is possible to easily modify the parameters of the model after the training step to tailor to different noise intensities. Two case studies are conducted to compare this model with the state of the art and commonly used denoising procedures. The first experiment uses standard signals to study denoising properties of the algorithms. The second experiment is a real application with the objective to remove audio background noises. We show that the learnable wavelet packet transform has the learning capabilities of deep learning methods while maintaining the robustness of standard signal processing approaches. More specifically, we demonstrate that our approach maintains excellent denoising performances on signal classes separate from those used during the training step. Moreover, the learnable wavelet packet transform was found to be robust when different noise intensities, noise varieties and artifacts are considered.