An analog-to-information converter (AIC) is an emerging method for compressive sampling that overcomes the limitations imposed by the Nyquist rate. The classical AIC employs a pseudo-random (PN) sequence at the Nyquist conversion rate to mix the input signal, then down-samples it by spreading the sparse signal across the entire frequency band, enabling the extraction of low-frequency information. However, in wireless communications applications, signals are often bandpass. The excessively high PN conversion rate leads to excessive sampling redundancy and significant non-ideal effects, degrading the quality of the reconstructed signal. To address these issues, this paper proposes a new design method for ultra-low-rate PN sequences based on bandpass sampling theory. This method uses the actual bandwidth of the input signal rather than its maximum frequency to determine the PN sequence rate. Based on this method, we constructed a novel bandpass random demodulation (RD) AIC architecture that employs a Sigma Delta ADC as its core component. By decreasing the PN sequence switching rate, the architecture effectively minimizes non-ideal effects associated with high-speed PN switching, ensuring the integrity of bandpass signal information while significantly improving the compression ratio and the reconstruction signal-to-noise ratio (RSNR). Experimental results show that for an input signal with a frequency range of 780~790 kHz, the proposed bandpass RD AIC can lower the PN sequence conversion rate to 52 kHz. This advancement achieves a sampling compression ratio of 30 and an average RSNR of 68 dB. Compared with the latest RD Sigma Delta AIC design, the proposed architecture improves the sampling compression ratio by 7 times and enhances the RSNR by 146 dB.