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Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems^{ †}

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## Abstract

**:**

## 1. Introduction

- We propose SNSS, which can determine the occupancy status of the spectrum from multiple sub-sampled data. Taking advantage of CRT, SNSS not only simplifies the system architecture but also reduces the computation amount.
- We make comprehensive experiments to characterize the effect on the accuracy of undersampled sensing of sampling rate, bandwidth resolution and the SNR of the original signal.
- We design an adaptive policy ASNSS that can determine the optimal sampling rate and bandwidth resolution when the spectrum occupancy or the strength of the existing signals is changed. It is shown that, with the adaptive policy, far better performance is achieved for the sub-Nyquist spectrum sensing.

## 2. Related Work

#### 2.1. Nyquist Wideband Sensing

#### 2.2. Sub-Nyquist Wideband Sensing

## 3. Problem Definition

**w**(n) stands for the noise and n the time. As for spectrum sensing, we need to determine whether the observation

**y**was gotten under the hypothesis ${H}_{0}$ or ${H}_{1}$. To handle this, firstly, we should calculate a statistic

**Y**from the sampled data. Then, we need to compare it with a threshold $\lambda $

## 4. Basic Idea

#### 4.1. SNSS: System Architecture

**Theorem**

**1.**

#### 4.2. SNSS: Algorithm

- Step1
- Initialize all the variables.Initialize all the variables including signal bandwidth (BW), spectrum occupancy ($\rho $), channel number (N) and spectrum occupancy vector. Then, calculate the number of sampling branches and their corresponding sampling rates.
- Step2
- For all the branches, sample the signal using its corresponding sampling rates.All the rates are set as what is discussed in Section 4.
- Step3
- Calculate the FFT transformation of the undersampled data.
- Step4
- Energy detection and Frequency reconstruction.Conduct energy detection directly on the undersampled data and reconstruct the frequencies from the result. The DETECT function calculate the aliased frequency band energy and output the aliased channel occupation sequence. The REPROJECT algorithm is to get the possible occupation status from the aliased sequences using the CRT theorem.
- Step5
- Information fusion.Make a fusion of all the information from each branch. As shown in Figure 4, when we get all the reproject sequences from all the channels, we can easily get the final results by making a union operation on all the sequences.

Algorithm 1: SNSS Algorithm |

Data: Input: Signal Bandwidth $WB$, Spectrum Occupancy $\rho $, Channels N Data: Output: Spectrum Occupancy Vector(SOV) $\mathbf{F}$ 1 Initialize SOV $\mathbf{F}\leftarrow (0,0,0,\cdots ,0)$; 2 Initialize the number of sampling branches: $\gamma \leftarrow \eta \rho +\theta $; 3 their value are defined in Equation (13); 4 Initialize sampling rates: ${\mathbf{F}}_{\mathbf{s}}\leftarrow ({f}_{1},{f}_{2},{f}_{3},\cdots ,{f}_{\gamma})$; |

#### 4.3. SNSS: Example

## 5. Further Study

#### 5.1. The Impact of Sampling Rate

#### 5.2. The Impact of Bandwidth Resolution

#### 5.3. The Impact of the SNR of the Original Signal

## 6. ASNSS: Adaptive Algorithm for SNSS

#### 6.1. Strength of PU’s Signal Changes

- Detection. We set a threshold of the detected SNR of the signal, $SN{R}_{TH}$, when the SNR of the sampled data is lower than $SN{R}_{TH}$, i.e., $SN{R}_{DT}<SN{R}_{TH}$, the adaptive process is triggered.
- Adaptive policy. To improve the detected SNR, it is feasible to adopt a higher sampling rate and use more points in FFT.

#### 6.2. Spectrum Occupancy Change

- Detection. We use the uncertain level to trigger the adaptive process. When the number of devices is too low, there will be some channels whose status cannot be determined uniquely.
- Adaptive policy. Let E be the number of the uncertain channels and $\theta $ be a predefined threshold, if $E>\theta $, which means that the status of a lot of channels cannot be determined. We set $\theta $ to be 0.1 in our simulations. At this time, it is necessary to increase the bandwidth resolution. Otherwise, the set of devices is not sufficient to monitor the spectrum. At this time, we try to decrease the resolution to obtain a more accurate view of the spectrum usage.

#### 6.3. ASNSS System

#### 6.4. ASNSS Algorithm

Algorithm 2: Adaptive Algorithm for SNSS |

Data: Input: Signal Bandwidth $WB$, Spectrum Occupancy $\rho $, Channel Number N Data: Output: Spectrum Occupancy Vector $\mathbf{F}$ 1 ${\mathbf{F}}_{{\mathbf{s}}_{max}}\leftarrow ({f}_{{1}_{max}},{f}_{{2}_{max}},{f}_{{3}_{max}},\cdots ,{f}_{{\gamma}_{max}})$; 2 Set the maximum sampling rate of each device; 3 ${\mathbf{F}}_{{\mathbf{s}}_{min}}\leftarrow ({f}_{{1}_{min}},{f}_{{2}_{min}},{f}_{{3}_{min}},\cdots ,{f}_{{\gamma}_{min}})$; 4 Set the minimum sampling rate of each device; 5 $\mathbf{FFTWINDOW}\leftarrow (FF{T}_{1},FF{T}_{2},FF{T}_{3},\cdots ,FF{T}_{\gamma})$; 6 Initialize the window size of FFT transformation; 7 $\alpha \leftarrow f(SN{R}_{Sub},SN{R}_{Ny})$; 8 β is the sampling rate adjustment factor, $\alpha \in (0,1)$; 9 INITIALIZE $\beta $; 10 INITIALIZE $Occupanc{y}_{low},Occupanc{y}_{high}$; 11 $\mathbf{F}\leftarrow (0,0,0,\cdots ,0)$; 12 Initialize Spectrum Occupancy Vector; 13 $\gamma \leftarrow \eta \rho +\theta $; 14 Initialize the number of sampling Branches, their value are defined in Equation (13); 15 ${\mathbf{F}}_{\mathbf{s}}\leftarrow ({f}_{1},{f}_{2},{f}_{3},\cdots ,{f}_{\gamma})$; 16 Initialize sampling rates; 17 INITIALIZE $SN{R}_{low}$, $SN{R}_{high}$; 18 When $SNR<SN{R}_{low}$, increase the sampling rates; or when $SNR>SN{R}_{high}$ decrease the sampling rates; 19 $Occupancy\leftarrow \rho $; |

## 7. Performance Evaluation

#### 7.1. SNSS Performance

#### 7.2. ASNSS Performance

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Illustration of the basic idea of sub-Nyquist sensing. (

**a**) Sampling result with ${f}_{s}=2$; (

**b**) Sampling result with ${f}_{s}=3$.

**Figure 5.**The relationship between the sampling rate and the detected signal energy. The sampling rates are normalized with respect to the Nyquist rate.

**Figure 6.**The signal SNR for different sampling rates. The vertical dash line stands for the Nyquist rate and the horizonal one represents the SNR detected under Nyquist sampling rate.

**Figure 7.**The impact of bandwidth resolution. The x-axis stands for the actual bandwidth resolution.

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**MDPI and ACS Style**

Lu, Y.; Lv, S.; Wang, X.
Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems. *Symmetry* **2019**, *11*, 342.
https://doi.org/10.3390/sym11030342

**AMA Style**

Lu Y, Lv S, Wang X.
Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems. *Symmetry*. 2019; 11(3):342.
https://doi.org/10.3390/sym11030342

**Chicago/Turabian Style**

Lu, Yong, Shaohe Lv, and Xiaodong Wang.
2019. "Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems" *Symmetry* 11, no. 3: 342.
https://doi.org/10.3390/sym11030342