Unfolding networks have shown promising results in the Compressed Sensing (CS) field. Yet, the investigation of their generalization ability is still in its infancy. In this paper, we perform a generalization analysis of a state-of-the-art ADMM-based unfolding network, which jointly learns a decoder for CS and a sparsifying redundant analysis operator. To this end, we first impose a structural constraint on the learnable sparsifier, which parametrizes the network's hypothesis class. For the latter, we estimate its Rademacher complexity. With this estimate in hand, we deliver generalization error bounds -- which scale like the square root of the number of layers -- for the examined network. Finally, the validity of our theory is assessed and numerical comparisons to a state-of-the-art unfolding network are made, on synthetic and real-world datasets. Our experimental results demonstrate that our proposed framework complies with our theoretical findings and outperforms the baseline, consistently for all datasets.