Spectral reconstruction of signals from periodic nonuniform subsampling based on a Nyquist folding scheme
 Kaili Jiang^{1}Email authorView ORCID ID profile,
 Jun Zhu^{1} and
 Bin Tang^{1}
DOI: 10.1186/s136340170458z
© The Author(s). 2017
Received: 30 June 2016
Accepted: 17 February 2017
Published: 23 February 2017
Abstract
Periodic nonuniform sampling occurs in many applications, and the Nyquist folding receiver (NYFR) is an efficient, low complexity, and broadband spectrum sensing architecture. In this paper, we first derive that the radio frequency (RF) sample clock function of NYFR is periodic nonuniform. Then, the classical results of periodic nonuniform sampling are applied to NYFR. We extend the spectral reconstruction algorithm of time series decomposed model to the subsampling case by using the spectrum characteristics of NYFR. The subsampling case is common for broadband spectrum surveillance. Finally, we take example for a LFM signal under large bandwidth to verify the proposed algorithm and compare the spectral reconstruction algorithm with orthogonal matching pursuit (OMP) algorithm.
Keywords
Nyquist folding receiver Periodic nonuniform subsampling Spectral reconstruction1 Introduction
Under the condition of modern information warfare, reconnaissance receiver faces the gradually complex electromagnetic environment; accompanied by diversification of electromagnetic radiation sources and coexistence of jamming and antijamming. The features of received signals are wide timefrequencyspace domain, waveform complexity, and large dynamic range. So to speak, the problem is receiving and dealing with the wideband signals. In recent years, with the rapid development of radar technology, the range of the frequency spectrum is from 5 MHz to 95 GHz and enlarges gradually [1]. The existing reconnaissance receiver cannot match the coverage of radar because of the limited sampling rate and precision of analogtodigital converter (ADC) [2]. Therefore, how to solve this problem becomes a focus.
Reconnaissance receiver as a channelized receiver [3, 4], in general, is based on the Nyquist theorem for design of the data acquisition of wideband signals [5]. And Nyquist rate is only a necessary but not sufficient condition for signals recovered accurately [6]. For another, nonuniform sampling exists extensively in the practical system of nonideal and compressed sensing (CS) theory as a typical example of nonuniform sampling. The research in analogtoinformation (A2I) conversion is still limited in prototype and numerical simulation [7]. And there are some requirements for the sparse characteristic of the received signals based on CS [8–11].
Periodic nonuniform sampling introduces enough nonuniform to differentiate the frequency band of the received signals, whose randomness of sampling is between uniform sampling and random sampling. JENQ presents the detailed Fourier spectrum and digital spectrum of periodic nonuniformly sampled signals by a time series decomposed model [12], and its spectral reconstruction algorithm under the Nyquist theorem described in the reference [13]. Similarly, the fractional Fourier spectrum of periodic nonuniformly sampled signals and the fractional spectral reconstruction are discussed by Ran Tao [14, 15], for linear frequency modulation (LFM) signals. However, the spectral reconstruction of periodic nonuniform subsampling based on Fourier or fractional Fourier has not been reported by far.
2 Periodic nonuniform sampling
where f _{ s } is the average sampling frequency, and θ(t) is the phase modulation function.
where f _{ θ } is the frequency of the sinusoid phase modulation function.
2.1 Periodicity
Besides, considering an extreme case, if f _{ s } and f _{ θ } are coprime, which will introduce much randomization, whose randomness of sampling is between uniform sampling and random sampling. Then, the influence of aliasing will be suppressed, and the original information of inputs will be more complete at the cost of increased algorithm complexity.
The focus of this paper is not on how to set the parameter of NYFR more suitable. However, it is given a further understanding of NYFR architecture based on periodic nonuniform sampling.
2.2 Nonuniformity
3 Spectral reconstruction of NYFR
where m ∈ {0, 1, 2, …, M − 1} is the index of sample time in one period, l ∈ {0, 1, …, ceil(N/M) − 1} is the index of period, and ceil(⋅) denotes round up. And t _{ m } = mT − r _{ m } T as shown in reference [12].
Eq. (11) means that Ã(l) is the Fourier transform of the sinusoid modulation function. To simplify Ã(l), we use Eq. (7) as shown in reference [16]. In the equation, p(t) is a pulse model, and k represents the index of Nyquist zone (NZ) from zero to κ, where κ denotes the number of NZ by NYFR covered. So k can be obtained from l, that is to say k = ⌊(l + M/2)/M⌋ ∈ Z and ⌊ ⋅ ⌋ denotes floor.
when − M/2 + 1 ≤ l < M/2, the index of NZ is k = ⌊(l + M/2)/M⌋ = 0; likewise, when M/2 + 1 ≤ l < 3M/2 corresponds to k = ⌊(l + M/2)/M⌋ = 1, et al. And the analysis object turns from a point into a zone.
It is noted that the matrix A is column orthogonality, and then, the matrix A ^{− 1} exists certainly. However, we need to reevaluate the matrix for each different index value of NZ. Finally, by choosing different value of ω _{0}, we can get enough uniformly sampled points of the original signal spectrum. And by scanning k _{ H } from zero to κ, we can reconstruct the spectrum of NYFR.
4 Simulation results and discussion
The simulation settings table
Average sampling frequency  f _{ s }  1 GHz 
Sinusoid modulation frequency  f _{ θ }  10 MHz 
Simulation points  N  1000 points 
Amplitude of LFM  A _{0}  1 
Initial phase of LFM  φ _{0}  0 
Initial frequency of LFM  f _{0}  3.52 GHz 
Bandwidth of LFM  B _{0}  0.95 GHz 
Modulation rate of LFM  k _{0}  9.5e6 GHz 
 (a)
Fourier spectrum of uniform sampling
 (b)
Digital spectrum of periodic nonuniform subsampling
 (c)
Spectral reconstruction
5 Conclusions
NYFR is an efficient A2I conversion model, and its spectral reconstruction can use the traditional CS recovery algorithms. However, if the signal is not sparse in frequency domain as shown in simulation, the existing CS algorithms as OMP cannot reconstruct the received signal accurately. In this paper, we first derive that the RF sample clock function of NYFR is periodic nonuniform. Then, the classical results of periodic nonuniform sampling are applied to NYFR. We extend the spectral reconstruction algorithm of time series decomposed model to the subsampling case by using the spectrum characteristics of NYFR. And finally, we take an example for a LFM signal under large bandwidth to verify the proposed algorithm and compare the spectral reconstruction algorithm with OMP algorithm. But for the influence of noise, the parameter estimation of wideband LFM signals will be more difficult. In the future work, we will study the fractional spectrum reconstruction of periodic nonuniform subsampling and their applications.
Abbreviations
 A2I:

AnalogytoInformation
 ADC:

Analog to digital converter
 CS:

Compressive sensing
 LCM:

Least common multiple
 LFM:

Linear frequency modulation
 NYFR:

Nyquist folding receiver
 NZ:

Nyquist zone
 OMP:

Orthogonal matching pursuit
 RF:

Radio frequency
 RIP:

Restricted isometry property
 SFM:

Sinusoid frequency modulation
Declarations
Acknowledgements
The authors thank the National Hightech R&D Program of China and the National Natural Science Foundation of China for their supports for the research work. The authors also thank the reviewers for their suggestions and corrections to the original manuscript.
Funding
This work was supported by the 863 Project (2015AA8098088B&2015AA7031093B) and the National Natural Science Foundation of China (61571088).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
KJ is the first author and corresponding author of this paper. Her main contributions include (1) the basic idea, (2) the derivation of equations, (3) computer simulations, and (4) writing of this paper. JZ is the second author whose main contribute includes analyzing the basic idea and checking simulations. BT is the third author and his main contribute includes refining the whole paper. All authors read and approved the final manuscript.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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