This paper presents a circuit-algorithm co-design framework for learnable analog front-end (AFE) in audio signal classification. Designing AFE and backend classifiers separately is a common practice but non-ideal, as shown in this paper. Instead, this paper proposes a joint optimization of the backend classifier with the AFE's transfer function to achieve system-level optimum. More specifically, the transfer function parameters of an analog bandpass filter (BPF) bank are tuned in a signal-to-noise ratio (SNR)-aware training loop for the classifier. Using a co-design loss function LBPF, this work shows superior optimization of both the filter bank and the classifier. Implemented in open-source SKY130 130nm CMOS process, the optimized design achieved 90.5%-94.2% accuracy for 10-keyword classification task across a wide range of input signal SNR from 5 dB to 20 dB, with only 22k classifier parameters. Compared to conventional approach, the proposed audio AFE achieves 8.7% and 12.9% reduction in power and capacitor area respectively.