From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals (0902.4291v3)

Abstract: Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then lowpass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, realtime performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.

• The paper introduces the modulated wideband converter (MWC) to enable efficient sub-Nyquist sampling of sparse wideband analog signals.
• The authors detail theoretical conditions for unique signal recovery by optimizing sampling rates and mixing function designs.
• The study compares MWC with conventional approaches, highlighting reduced digital processing loads and improved robustness in practical implementations.

Sub-Nyquist Sampling of Sparse Wideband Analog Signals: Theory and Practice

This paper by Moshe Mishali and Yonina C. Eldar provides a comprehensive analysis and a practical implementation strategy for sub-Nyquist sampling of sparse wideband analog signals. The focus is on designing a system capable of efficient hardware implementation while minimizing the computational load on digital processing. Key contributions include the development of the Modulated Wideband Converter (MWC) and innovative strategies for spectrum-blind recovery, which collectively address critical challenges in wideband signal acquisition.

Overview

The Modulated Wideband Converter (MWC)

The proposed MWC system leverages spread-spectrum techniques from communication theory, utilizing an analog mixing front-end that aliases the spectrum, bringing portions of each band into the baseband. This process involves:

1. Multiplying the input signal by a periodic waveform.
2. Truncating the signal spectrum using a lowpass filter.
3. Sampling the filtered output at a significantly lower rate than the Nyquist rate.

The MWC system facilitates the use of existing ADCs because it operates well below the Nyquist rate, avoiding the typical bandwidth limitations of current ADC technologies. The authors discuss specific design parameters, such as the period of the waveform, sampling rate, and the choice of mixing functions, to ensure optimal performance and uniqueness in signal recovery.

Key Theoretical Developments

Spectral Support and Unique Signal Recovery

The authors outline the conditions necessary for perfect signal recovery:

• The sampling rate of the system, denoted $f_s$, must meet $f_s \geq f_p$, where $f_p$ is derived from the periodic waveform.
• The number of sampling channels, $m$, must satisfy $m \geq 2N$ for spectrum-blind recovery, where $N$ is the number of bands in the input signal.

A critical aspect is the design of the mixing functions ($p_i(t)$). The paper demonstrates that by choosing these functions correctly, each band's spectrum can be projected uniquely, ensuring that the subsequent digital processing can accurately recover the original signal.

Practical Considerations

Parameter Selection and Sampling Strategies

Two configurations are discussed for parameter choice:

1. Option A: Where $f_s = f_p$, leading to a setup just above the theoretical minimum sampling rate.
2. Option B: Involving oversampling strategies where $f_s = qf_p$ (with $q$ integer), and fewer analog channels are required. This approach trades off digital processing complexity for reduced hardware requirements.

Robustness and Stability

The MWC incorporates mechanisms to handle real-world challenges like time-varying spectral support and noise. The system's robustness was empirically validated through simulations, demonstrating stable performance even with time-domain variability and quantization noise.

Comparative Analysis with Existing Technologies

The MWC system offers significant advantages over conventional multicoset sampling and the recently proposed random demodulator:

• Analog Flexibility: MWC's purely analog front-end mitigates the bandwidth limitations of digital processing, unlike the random demodulator, which faces computational infeasibility with truly analog signals.
• Low Digital Processing Load: The MWC reduces the digital processing requirements by focusing on the low-rate samples directly, avoiding the need for Nyquist rate interpolation, which is essential for multicoset sampling recovery.

Future Directions

The research opens pathways for further refinement in both theoretical and practical dimensions:

• Enhanced Sign Pattern Selection: Improving the choice of mixing functions through alternative randomness properties could enhance stability and performance.
• Adaptive Sampling Strategies: Ongoing research into adaptive schemes that respond dynamically to spectral changes could further optimize hardware resource use and processing time.

Implications

The practical implications of this work are multifaceted:

• Wideband Communications: Enabling efficient sampling and processing of wideband signals can enhance communication systems’ performance by reducing hardware costs and energy consumption.
• Digital Signal Processing: The advances in spectrum-blind recovery techniques can be leveraged for applications requiring real-time signal manipulation without the overhead of Nyquist rate sampling.

In conclusion, this paper bridges the gap between theoretical sub-Nyquist sampling methods and their practical implementation, offering a viable route to overcoming current ADC limitations. The MWC system stands out for its potential to revolutionize how sparse wideband analog signals are acquired and processed, making significant strides in the efficient use of hardware and computational resources.