Wideband spectrum sensing based on modulated wideband converter with nested array

Several spectrum sensing systems based on sub-Nyquist sampling have been extensively studied to deal with difﬁculties of traditional wideband spectrum sensing in cognitive radio (CR) networks. The modulated wideband converter (MWC) is an effective application and has drawn considerably more attention over the past few years. In this study, MWC with a nested array (NA) system is proposed to improve spectrum sensing performance, which is an array sensing system with sub-Nyquist sampling. The simulations show that the connection of co-array and MWC obtains the lower minimal system sampling rate than MWC. Second, the proposed system enables frequency and power spectrum estimation with sub-Nyquist sampling for more sources than sensors. Finally, our alternative spectrum sensing system outperforms other conventional methods in terms of sensing accuracy and design complexity.


INTRODUCTION
Spectrum resources are a kind of non-renewable resource. With the rapid growth of human communication services, limited spectrum resources are scarce. Improving the real-time utilisation of spectrum resources is expected to be an effective way to alleviate the shortage of spectrum resources [1]. Cognitive radio (CR) [2] is an effective solution for solving spectrum scarcity issue by exploiting its sparsity, which improves real-time utilisation of target bands by accurately monitoring the frequency bands already allocated to the primary users (PUs), capturing the spectral holes of the frequency band and finally providing the opportunity for the secondary users (SUs) spectrum access. The technology of spectrum sensing can be divided into narrowband and wideband sensing technologies. In the narrowband technology of spectrum sensing, energy detection [3] is one of the most common methods because of its low implementation cost. One weakness of this technology is that it cannot distinguish the user signals from the noise well, so it has poor performance at low signal-to-noise ratio (SNR) and is unable to identify unoccupied spectral opportunities over a wide frequency band. Therefore, it is necessary to apply CRs to high-frequency fields [4] to find unoccupied spectral bands quickly. To solve the obstacle, wideband spectrum sensing technology has attracted the attention of researchers.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. A few studies [5,6] focus on wideband spectrum sensing. In [5], a multiband joint detection algorithm based on Nyquist sampling is introduced, which can be used to detect spectrum opportunities on multiple frequency bands simultaneously. In [6], wavelet-based spectrum sensing has been proposed. Since it requires an analogue-to-digital converter (ADC) running at the Nyquist sampling rate, it also has high implementation and computational complexity. Therefore, many spectrum sensing applications deal with wideband signals leading to extremely high Nyquist rates. To overcome this obstacle, several sampling architectures and recovery algorithms of signals which are beneficial from the compressed sensing (CS) theory [7] have been proposed [8][9][10][11]. The emerged theory of CS is a technology with a sampling rate of less than twice, only a small number of sampled signals data can be perfectly reconstructed to obtain the original information, thereby breaking limitation of sampling. The methods in [8] used the multicoset sub-Nyquist sampling to reconstruct multiband signals at low rates. But it requires synchronisation between time shift elements and sufficient bandwidth of commercial ADCs in case of signal distortions in time-domain. In [9], Tropp et al. proposed the random demodulator that solves the problem of wideband spectrum sensing by generating a random matrix to compress and measure signals to obtain a small amount of important measurement data. The modulated wideband converter (MWC) sampling scheme [10,11] is able to implement with off-the-shelf ADCs, which can reconstruct blind and sparse signals at undersampling condition in frequency-domain. So far, sub-Nyquistbased wideband sensing technology has been widely studied in the academia and industry.
In [12], the authors use CS to compress the sample vector into a smaller vector, and reconstruct the spectrum by minimising the l 1 norm. However, the multirate asynchronous sub-Nyquist sampling approaches relies on the noise power as prior knowledge. MWC does not have these disadvantages since it can recover signal without prior knowledge of signal or noise position. To address wireless channel fading due to the lack of space diversity exploitation in the process of spectrum sensing, the author proposes a multi-antenna system based on MWC called the multi-antenna generalised modulated converter (MAGMC) [13]. In [14], a blind sub-Nyquist spectrum sensing algorithm based on MWC sub-Nyquist sampling framework is proposed, referred to as the residual energy ratio-based detector (RERD). RERD no longer requires prior knowledge of the monitored spectrum and makes the way of performing spectrum sensing more autonomous. Unfortunately, there are still shortcomings with using MWC for wideband spectrum sensing. First, the main difficulty is choosing different periodic functions p n (t ) that alternates between the levels ±1 for each channel to let their Fourier coefficients fulfill CS requirements. Next, the sampler of MWC is one sensor composed of analogue processing channels so that all sensing channels are disrupted by the same noise, which causes the whole process of spectrum sensing to be difficult to stably accomplish in practical condition. In [15], Eldar et al. suggest a sub-Nyquist sampling scheme based on the MWC system with known or identical direction of arrivals (DOAs), and they also derived a system to solve joint spectrum sensing and DOA estimation is called CompreSsed CArrier and DOA Estimation (CaSCADE). Another method in [16] is proposed, which applies the co-prime array (CPA) to the MWC. The MWC-based CPA can detect more targets than traditional ULA-based MWC and reduce the effect of mutual coupling of ULA. Nevertheless, it is difficult to choose the optimal number of samplers in the condition of designing equations for a large number of elements since the number of CPAs must follow the prime principle. This problem also leads to the minimum sampling rate being very high for the sub-Nyquist sampling scheme. Another obstacle is that CPA has holes in the co-array so that the ULA part of the co-array is smaller than those of the nested array (NA).
Based on the receiver architecture MWC, this study considers a spectrum sensing system that uses MWC scheme based on an NA. First, the signals go through the NA antennas. Then the received signals are multiplied by the same periodic function, then filtered by a low-pass filter (LPF) and sampled at a low rate ADC below the Nyquist rate. NA configuration consists of contiguous virtual components (no holes) that can utilise the complete correlation information when signals are more than practical sampling sensors. One advantage of NA-based MWC is the simplicity of choosing the optimal number of samplers to attain the minimum system sampling rate. Another advantage is that each sampling sensor is corrupted by uncorrelated noises so that this configuration allows for noise averaging that increases signal to noise ratio (SNR).

SIGNAL AND ARRAY MODEL
Consider K uncorrelated far-field narrow-band complex signals{x i (t )} K i=1 impinging on the array with unknown disjoint frequencies. Each signal x i (t ) is modulated by a carrier fre- (1) We denote by f Nyq the Nyquist rate of u(t ). The Fourier transform of u(t ) is given by where X i ( f ) is the Fourier transform of x i (t ). The Fourier transform of u(t ) is zero for every f ∉  .

NA BASED ON MWC
Consider a two-level NA consisting of two ULAs, which has N 1 and N 2 elements. The distance between elements of two ULAs is d and d 2 = (N 1 + 1)d , respectively. The sets of sensors locations that can be defined by The definition of is the half-wavelength of the signals whose frequency is the highest. Therefore, there are N 1 + N 2 = N sensors in total. We choose the number of sub-array elements to be N 1 = N 2 = N ∕2 for convenient discussion and assume that N is even in this work. Similar arguments can be applied for the odd case as well. The structure of NA composed of N 1 = N 2 = 3 is illustrated in Figure   The received signal from each sensor is mixed to the baseband through a periodic mixing function p(t ) whose period is T p = 1∕ f p and then filtered by a LPF with cut-off frequency f s ∕2. Finally, it is sampled at a rate f s ≥ f p below the Nyquist rate. To derive the relation about sample sequences from the nth sensor, we define where n ( ) = dn c cos( ) is the relative time advance of the nth sensor from the direction with respect to the first one at the origin. is the known or identical DOA and c denotes the speed of light.
Due to the narrow-band assumption, the approximation in Equation (3) can be written as The Fourier transform of the received signal u n (t ) is given by where X i ( f ) is the spectrum of the nth baseband signal before modulation. Then the next step is mixing the received signal with a periodic mixing function Therefore, the analogue multiplicationz n (t ) is given bỹ We denote the Fourier transform ofz n (t ) bỹ Therefore, the sum in Equation (7) contains at most ⌈ where L 0 is chosen as the smallest integer such that the sum contains all non-zero contributions. The exact value of L 0 is calculated by We whereX i ( f ) is the output signal at baseband after modulation.
X i ( f ) is a cyclic shifted and scaled version of X i ( f ) in the interval  p as shown in Figure 3. Next, z n (t ) is sampled at rate f s to obtain the sample sequences, which is the inverse Fourier transform of Z n ( f ). Equation (11) can be written in a matrix form as where is expressed as In the time domain, Equation (12) can be written as By taking noise into account, Equation (14) is modified as where A represents the array manifold matrix and the noiseñ[v] is assumed to be temporally and spatially white, and uncorrelated from each sensor. In this study, we choose Additive White Gaussian Noise (AWGN) which follows a normal distribution with a mean of 0 and a variance of 2 n .

Carrier frequency recovery with known DOA
The covariance matrix of z[v] is given by where } is also a covariance matrix. The meaning of (•) H is conjugate transpose. Next, the equivalent covariance matrixR zz of ULA (N ) matching the property of NA (N 1 ,N 2 ) will be derived. We start by vectoring R zz to get the following vector: Here, rxx is vector form of Rxx, ⃗ I n = [e T 1 , … , e T N ] T and ⊙ denotes Khatri-Rao product. Comparing Equations (15) with (13), the vector of r zz behaves like z[v] and the distinct rows of A * ⊙ A are the manifold of NA (N 1 ,N 2 ). Following Equation [17], we can work with the difference co-array of NA whose sensor locations are given by the distinct values S N = {nd, n = −M, … , M, M = N 2 (N 1 + 1) − 1} instead of from the original NA.
Since the property from NA is a virtual and filled ULA, A * ⊙ A is like a Vandermonde matrix with N 2 ∕2 + N − 1 distinct rows. Hence, its rank is K ≤ N 2 ∕2 + N − 1.
To obtain the matrix of signals corresponding to this property, first, we need to construct a new matrixĀ of size (N 2 ∕2 + N − 1) × K from A * ⊙ Awhere we have removed the repeated rows and sorted them so that the row corresponds to the sensor location ℝ (N 2 ∕2+N −1)×1 . This is equivalent to removing the corresponding rows from the vector r zz and sorting them to get a new vectorrr where e ′ ∈ ℝ (N 2 ∕2+N −1)×1 is an equivalent vector which is also sorted and replaced of repeated rows. The new steering matrix Each array sensor separation of our sensing system is defined from −(N 2 ∕4 + N ∕2 − 1)d to (N 2 ∕4 + N ∕2 − 1)d . Then this filled ULA is divided into N 2 ∕4 + N ∕4 + N ∕2 overlapping sub-arrays, each with N 2 ∕4 + N ∕4 + N ∕2 elements. Consequently, the sensor position of ith sub-array is expressed as { We can denote ther in ith sub-arraȳ whereĀ i consists of the row from (N 2 ∕4 + N ∕2 − i + 1)th to (N 2 ∕ + N − i )th ofĀ. Consequently,Ā i can be written as where diagonal matrix Taking the average ofR i , we can get a spatially smoothed matrix In addition,R zz can be expressed as Therefore, where the form ofR zz is same as the signal received by a longer virtual and filled ULA of the structure NA (N 1 ,N 2 ). The definition of is the diagonal matrix with covariance elements from K received signals. The proof and theorem about Equation (24) follow in Equation [18]. Finally, we can applyR zz to estimate carrier frequencies.
DecomposingR zz using the singular value decomposition (SVD), we can writeR where U is a left singular vector with size ofR zz , ∑ is a diagonal matrix with the values of elements which are permutated from large to small and V is a right singular vector.
To extract signal subspace, we get U z as We define that U 1 consists of first N 2 ∕4 + N ∕2 − 2 rows and U 2 consists of last N 2 ∕4 + N ∕2 − 2.
Then, the carrier frequencies can be estimated aŝ where ∠(•) denotes the corresponding angle of its argument and eig(•) denotes that eigen-decompose argument to get eigenvectors composed of eigenvalues.

Signal power spectrum recovery
In order to enable perfect reconstruction of the power spectrum of transmissions r X ( f ), we choose Equations (12) not (13) to be the recovery setting. Therefore, from Equation (9) we hold that where RX ( f ) Then, the next processing steps are similar to the set of equations such as Equations (15), (16) and (19). We vectorise R Z ( f ) and then remove the redundancies and obtain the corresponding virtual filled ULA model The steering matrix A defined in Equation (17) can be constructed only if the carrier frequency f i is recovered. SinceĀ is constructed, the power spectrum ofX ( f ) then gets by inverting the steering matrix,rX where i is the power spectrum of ith transmissionX i ( f ). Note that the steering matrixĀ is a full rank matrix with size of (N 2 ∕2 + N − 1) × K since the number of source signals should be less than the number of physical and virtual sensors.
Next, we need to derive the relationship betweenX i ( f ) and X i ( f ) so that the relationship between [rX ( f )] i and [r X ( f )] i can be found and then finish signal power spectrum recovery.
In the interval  p , the sampled outputX i ( f ) is given bỹ If a l ≠ 0, −L 0 ≤ l ≤ L 0 and f ′ ⊂  ⊂  p , we can writẽ where a l ′ = ⌊ f p ⌋ and l ′ is the index of one of the two f p -bins. ⌊·⌋ means to return the nearest integer towards negative infinity.
From Equation (34), [rX ( f ′ )] i can be expressed as Finally, after a change of variables, we have The power spectrum r X ( f ′ ) can be reconstructed perfectly from sub-Nyquist samples if f s ≥ f p ≥ B.

Comparison with other schemes related to MWC
Although NA-, CPA- [16], ULA-based system [15] and MWC [10,11] all can sample the multiband signals below the Nyquist rate, they have different properties. First, the sampler of MWC is one sensor that consists of N 1 + N 2 channels to finish analogue processing. However, the NA-and ULA-based system use N 1 + N 2 sensors and each sensor works like one channel. Accordingly, all the MWC channels are impacted by the same antenna noise but array-based sampling system has uncorrelated antenna noise. Then, a practical obstacle of the MWC is choosing realisable periodic functions {p n (t )} in hardware condition. The NA-, CPA-and ULA-based system allow all sensors to use the same mixing function p(t ), making the task easier.
The NA-based MWC composed of N = N 1 + N 2 physical sensors can provide O(N 2 ) degrees of freedom while CPA only can provide O(N 1 N 2 ) in same physical sensors condition. Therefore, the maximum detectable number of sources of our proposed system is more than CPA-based MWC. In other words, NA-based system is better at detecting more signal sources than physical sensors. Since NA-based system has more detection elements by constructing a longer-filled ULA through sensing in the condition of same physical sensors, it has better robustness to noise than CPA-and ULA-based system.

Simulation results
In the NA-based system, an NA with N 1 = 3 and N 2 = 3 is deployed. Therefore, the number of physical sensors is N 1 + wheref i and f i are the estimated frequency and actual frequency of ith transmission, respectively. A total of 500 Monte Carlo simulations are used in each scenario. The SNR is from -10 to 10 dB with step size 2 dB. The simulation result is depicted in Figure 4. Since the number of sources is less than that of physical sensors, all methods can detect all the sources and successfully finish spectrum sensing. It can be observed that the proposed system outperforms the ULA-based system because the NA can produce an equivalent virtual and filled ULA that has much larger aperture than the ULA-based scheme and has reduced mutual coupling compared to ULAs. In addition, the sensing accuracy of MWC is constrained by the mixing functions p n (t ) and worse than the remaining methods. NA-based MWC system does not have such limitation and is better than other methods under different SNR conditions except noncompressed method. With increasing SNR, the RMSE of our proposed system is gradually becoming more closely related to the non-compressed method, especially in high SNR regimes.
In the second scenario, we test the RMSE performance of our sensing system and ULA-based system under different number of signals conditions. The formula for RMSE is defined in Equation (37). The proposed system can identify signals up to N 2 ∕4 + N ∕2 − 1 = 11 when there are six physical sensors. The practical sampling channels of the CPA-based MWC are eight in same physical sensors condition since the degrees of freedom of NA are more than CPA in some conditions [21]. In the ULA-based system, the minimal number of sampling channels is K + 1 when there are K uncorrelated signal sources [15]. The SNR is 0 dB. The number of signals K is between 1 and 11. Figure 5 shows that the proposed system is superior to other systems and also has a good performance of RMSE when there are more signals than the physical sensors. This is because the NA-based system has larger apertures than other sensing systems when they have the same number of physical sensors, as shown in Figure 6. In the NA-based MWC system, the optimal number of virtual apertures is N 2 ∕4 + N ∕2 − 1. Figure 6, the number of virtual apertures of our method is larger than that of others methods with the augment of the number of physical sensors N . Accordingly, this property allows our system to accommodate more signal sources and has better robustness to noise.
In the third scenario, the success probability of NA-based system is shown in Figure 7. The success probability is defined as the number of Monte Carlo simulation with RMSE <5%, divided by the number of simulations. In this example, we test the probability of success under different numbers of signals. The setup of simulation is same as most parts of the second scenario. Consider a multiband signal model containing K uncor- related signal sources, the number of which is between 1 and 11. The number of physical sensors in this test is 6. The number of Monte Carlo simulation is 500 as before. In Figure 7, the success probability decreased significantly when K >9, indicating that virtual units from co-array are not as stable as physical sensors in this condition. Therefore, the number of sources no longer needs to highly approximate maximal detectable number of signal sources if signals are needed perfect reconstruction.
In the last scenario, minimal sampling rate of different sensing systems including NA-, CPA-and ULA-based system is taken into consideration. We define the minimum sampling rate f ms by f ms = N * × f s , where N * is the number of sampling channels and f s is sampling rate of each channel. As mentioned above, the only limitation of f s is f s ≥ f p ≥ B. In the ULAbased system, the minimum sampling rate is f ms = (K − 1) × f s , referencing to the second scenario. In our derived system, the minimum sampling rate can be obtained by finding the is the optimal solution of gcd(N 1, N 2) = 1 and N 1 × N 2 +N 1 > K referring to [21,22]. We choose the number of signals K ∈ [10, 100] to complete the experiment. The optimal numberN op can be calculated through the mathematic formulas shown above. The simulation result is depicted in Figure 8. It is evident that our proposed system has a lower minimum sampling rate, especially K , which is a large number.

CONCLUSION
In this study, a NA-based MWC array system is considered for wideband spectrum sensing using sub-Nyquist sampling to enhance the sensing performance. The basic procedure of the proposed system involves array sensors sampling of signals, followed by carrier frequency estimation and power spectrum detection. The key point of the proposed wideband spectrum sensing system is the NA processing. The sampler of our proposed system is an NA that can construct a longer virtual array and acquire a larger virtual aperture to detect more signal sources than sensors and achieve better sensing performance. Comparing with other methods, our proposed configuration can reduce the minimal sampling rate of system and complexity to design and implement. Simulation results confirm that the proposed system can achieve better sensing performance at a lower sampling rate with fewer sensors.