Within the deep learning paradigm, finite impulse response (FIR) filters are often used to encode audio signals, yielding flexible and adaptive feature representations. We show that a stabilization of FIR filterbanks with fixed filter lengths (convolutional layers with 1-D filters)leads to encoders that are optimally robust against noise and can be inverted with perfect reconstruction by their transposes. To maintain their flexibility as regular neural network layers, we implement the stabilization via a computationally efficient regularizing term in the objective function of the learning problem. In this way, the encoder keeps its expressive power and is optimally stable and noise-robust throughout the whole learning procedure. We show in a denoising task where noise is present in the input and in the encoder representation, that the proposed stabilization of the trainable filterbank encoder is decisive for increasing the signal-to-noise ratio of the denoised signals significantly compared to a model with a naively trained encoder.