Efficient Sampling of Sparse Wideband Analog Signals

Periodic nonuniform sampling is a known method to sample spectrally sparse signals below the Nyquist rate. This strategy relies on the implicit assumption that the individual samplers are exposed to the entire frequency range. This assumption becomes impractical for wideband sparse signals. The current paper proposes an alternative sampling stage that does not require a full-band front end. Instead, signals are captured with an analog front end that consists of a bank of multipliers and lowpass filters whose cutoff is much lower than the Nyquist rate. The problem of recovering the original signal from the low-rate samples can be studied within the framework of compressive sampling. An appropriate parameter selection ensures that the samples uniquely determine the analog input. Moreover, the analog input can be stably reconstructed with digital algorithms. Numerical experiments support the theoretical analysis.

💡 Research Summary

The paper addresses the problem of sampling wideband analog signals that are spectrally sparse, i.e., signals whose Fourier support consists of a small number of narrow bands (multiband signals) spread over a large frequency range. Classical periodic non‑uniform sampling (also known as multi‑coset sampling) can theoretically reduce the average sampling rate to the information‑theoretic minimum, but it assumes that each ADC front‑end sees the full Nyquist bandwidth. In practice this is infeasible because commercial ADCs require an anti‑aliasing low‑pass filter with a cutoff at half the ADC’s maximum conversion rate; thus the front‑end must be able to handle the full Nyquist bandwidth even if the actual sampling pattern is sparse. Consequently, multi‑coset sampling is impractical for modern RF applications where the Nyquist rate can be orders of magnitude higher than the available ADC speed.

To overcome this limitation the authors propose a new front‑end architecture that replaces the high‑speed ADCs with a bank of low‑rate ADCs preceded by analog mixers (random demodulators) and low‑pass filters. The input signal x(t) is split into m parallel channels. In each channel i, x(t) is multiplied by a periodic mixing function p_i(t) that takes values ±1 on M equal sub‑intervals within a period T_p. The mixing functions are generated pseudo‑randomly and are independent across channels. After mixing, each channel passes through an ideal low‑pass filter h(t) with cutoff f_c = 1/(2T_s), where T_s is the sampling interval of the subsequent ADC. The filtered signal is then sampled at the low rate 1/T_s. By choosing T_s = T_p = T = M/f_Nyquist, the low‑pass filter captures exactly the baseband portion of the spectrum that contains the aliased copies of the original multiband signal.

In the frequency domain, multiplication by p_i(t) corresponds to a convolution with its Fourier series, which produces a weighted sum of shifted copies of X(f) (the Fourier transform of x(t)). The low‑pass filter selects the interval F_0 =