Modulo sampling is a promising technology to preserve amplitude information that exceeds the observable range of analog-to-digital converters during the digitization of analog signals. Since conventional methods typically reconstruct the original signal by estimating the differences of the residual signal and computing their cumulative sum, each estimation error inevitably propagates through subsequent time samples. In this paper, to eliminate this error-propagation problem, we propose an algorithm that reconstructs the residual signal directly. The proposed method takes advantage of the high-frequency characteristics of the modulo samples and the sparsity of both the residual signal and its difference. Simulation results show that the proposed method reconstructs the original signal more accurately than a conventional method based on the differences of the residual signal.
S AMPLING plays a fundamental role in modern signal processing. An analog-to-digital converter (ADC) for sampling is typically characterized by two primary performance metrics: the sampling period and the observable amplitude range. When the maximum frequency of the input signal exceeds half of the sampling frequency, aliasing distortion arises during reconstruction [1], [2]. Furthermore, when the signal amplitude exceeds the observable range, the signal is clipped and accurate recovery of the original signal becomes generally impossible. Signal clipping can cause serious problems in a wide range of practical applications [3], [4]. To prevent such distortions, a higher sampling rate or a wider dynamic range is required, which in turn leads to increased power consumption. To balance energy efficiency and reconstruction accuracy, it is desirable to use an ADC with a lower sampling rate and a narrower dynamic range without causing signal clipping.
To avoid clipping in an ADC with a small observable amplitude range, modulo sampling has been proposed [5]. In modulo sampling, a folding operation is performed by a modulo operator before measurement so that the amplitude of the resulting folded signal lies within [-λ, λ) (λ > 0). The folded signal is then sampled by an ADC that can This work will be submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.
This work was supported in part by The Telecommunications Advancement Foundation and Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant JP24K17277.
Haruka Kobayashi is with the Graduate School of Engineering Science, The University of Osaka, 560-8531, Osaka, Japan.
Ryo Hayakawa is with the Institute of Engineering, Tokyo University of Agriculture and Technology, 184-8588, Tokyo, Japan. measure only the range [-λ, λ). Fig. 1 illustrates the concept of modulo sampling. By comparing the original signal f with the folded signal f λ obtained from modulo sampling, we can see that modulo sampling prevents clipping and preserves the waveform in part even when the amplitude exceeds the observable range. However, since we sample the folded signal in modulo sampling, it is necessary to reconstruct the original unfolded signal from these samples. Various methods have been proposed to reconstruct the original signal from the samples obtained by modulo sampling [5]- [16]. A hardware-oriented approach [9] uses a reset count map to record the number of foldings together with the folded signal. This approach, however, requires complex electronic circuitry as well as additional power and memory. As a reconstruction method without the reset count map, an algorithm using high-order differences of the signal has been studied [5], [10]. Subsequently, a prediction-based algorithm has been proposed, demonstrating that in the absence of noise, perfect recovery is possible for finite energy signals at any sampling rate above the Nyquist rate [11]. More recently, a more noise-robust method called beyond bandwidth residual reconstruction (B 2 R 2 ), which focuses on the high-frequency band of the folded signal, has been proposed [12]. Building on this approach, least absolute shrinkage and selection operator-B 2 R 2 (LASSO-B 2 R 2 ) has been introduced to further improve performance [13]. This method exploits the sparsity of the first-order difference of the residual signal, which represents the number of wraps and is defined as the difference between the signal obtained by modulo sampling and the original signal. Although this approach can reconstruct the signal efficiently, reconstruction methods based on the difference of the residual signal suffer from the problem that a reconstruction error at one time index propagates to all subsequent points.
In this study, we propose an optimization problem for directly reconstructing the residual signal in modulo sampling. In the proposed residual signal-based optimization problem, we first introduce a regularization term to exploit the sparsity arXiv:2602.16418v1 [eess.SP] 18 Feb 2026 of the first-order difference of the residual signal as in conventional methods. In addition, we newly focus on the sparsity of the residual signal itself and introduce a regularization term that promotes this property as well. To solve the optimization problem efficiently, we derive an algorithm based on the alternating direction method of multipliers (ADMM) [17], [18]. Simulation results show that the proposed method achieves a lower normalized mean squared error (NMSE) than the conventional LASSO-B 2 R 2 .
Throughout this paper, R, C, and Z denote the sets of all real numbers, complex numbers, and integers, respectively. For a
. The operators ⌈•⌉ and ⌊•⌋ denote the ceiling and floor functions, respectively.
In typical sampling, the signal is clipped when the amplitude of the input signal exceeds the observable amplitude range of the ADC. The clipped
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