Improved algorithm based on modulated wideband converter for multiband signal reconstruction
 Min Jia†^{1}Email author,
 Yao Shi^{1},
 Xuemai Gu^{1},
 Xue Wang^{1} and
 Zhisheng Yin^{1}
https://doi.org/10.1186/s136380160547y
© Jia et al. 2016
Received: 1 December 2015
Accepted: 4 February 2016
Published: 16 February 2016
Abstract
For the compressed sensing of multiband signals, modulated wideband converter (MWC) is used as the sampling system, and the signal is reconstructed by the simultaneous orthogonal matching pursuit algorithm (SOMP) and its derivative algorithms. In order to find matching atoms, we need to obtain the inner product between atoms in sensing matrix and columns in residual matrix. Next, several inner products corresponding to the same atom constitute an inner product vector. By calculating its 2norm, we can find the maximum value, whose corresponding atom is the matched atom. However, the inner product actually cannot reflect the relevancy between atoms and residual matrix very accurately, which may eventually lead to wrong results for a certain probability. The main idea of this paper is to change the inner product into the correlation coefficient, so that we can measure the relevancy between atoms and the residual better. Simulation results show that the improved algorithms can get higher probability of the signal reconstruction compared with the original algorithms in the condition of high signalnoise ratio (SNR). It also means that less samples were needed to reconstruct signals than traditional algorithms when the number of bands is unchanged. Since calculating correlation coefficient at each iteration will cost a lot of time, we also proposed a simplified algorithm, which can also improve reconstruction probability and reconstruction time is about the same as corresponding traditional algorithms.
Keywords
1 Introduction
Traditional compressed sensing theory is mainly used to process discrete and finitedimensional digital signal. However, it is expected to develop a technology which can sample continuous and infinite dimensional analog signal at subNyquist rate, so as to truly break the bandwidth limitation of existing ADC equipment and reconstruct original signal from the samples of baseband signal after lowpass filters and finally ease the pressure of hardware sampling. To address this problem, a variety of solutions have been proposed. Analog to information convertor (AIC) and Xampling for multiband analog signal [1–5] are two relatively mature technologies among them. Xampling uses modulated wideband converter [6] (MWC) to sample, whose results are infinite measurement vectors (IMV), and it cannot directly reconstruct signal using the traditional reconstruction algorithms. To solve this problem, we can reconstruct the original signal by turning IMV problem into multiple measurement vector (MMV) problem using continuous to finite (CTF) modular [7] under the premise of joint sparse [8].
The simultaneous orthogonal matching pursuit (SOMP) [9, 10], treated as a derivative algorithm of matching pursuit simultaneous (OMP) algorithm to solve MMV problem, is what used for reconstruction in CTF at present. Based on this method, many novel MMV algorithms have been proposed according to the derivative algorithms of OMP such as regularized orthogonal matching pursuit (ROMP) [11], stagewise orthogonal matching pursuit (StOMP) [12], compressive sampling matching pursuit (CoSaMP) [13], and subspace pursuit (SP) [14]. The main idea of these algorithms is firstly calculating the inner product between atoms in sensing matrix and columns in residual matrix. Then, several inner products corresponding to the same atom constitute an inner product vector. By calculating its 2norm. We can find the maximum value whose corresponding atom is the matched atom. In fact, the inner product cannot measure the relevancy between atoms and residual matrix very well and sometimes will eventually lead to errors.
The idea of this paper is to change the inner product into the correlation coefficient, so that we can measure the matching degree between atoms and the residual better. In order to check the performance of the changement, the SOMP algorithm and its derivative algorithms such as MMVROMP, MMVStOMP, MMVCoSaMP, and MMVSP, which can solve MMV problems, are chosen for comparison. The proposed algorithms are improved by changing the inner product into correlation coefficient as screening criterion of atoms, then compare the signal reconstruction probability of these methods with original algorithms. The results show that the improved algorithms have better reconstruction probability and there are less samples required.
2 Signal model of multiband
3 Modulated wideband converter
Multiband modulated wideband signal converter is shown in Fig. 2. There are m sampling channels, and the mixing function p _{ i }(t) is a pseudorandom sequence with T _{ p }period. Its value is {+1,−1}, and it has M pulses in each period whose interval is T _{ s }. The values of p _{ i }(t) in the first k intervals are denoted as α _{ ik }, h(t) is a lowpass filter whose cutoff frequency is 1/2T _{ s }, and the sampling frequency is 1/T _{ s }. The signal x(t) is transmitted through m channels at the same time, and it is multiplied with different pseudorandom sequences in each branch whose cycle is abiding and obey the same distribution. The samples y _{ i }[n],n=1,2,…,m can be obtained after lowpass filters and lowspeed sampling.
where f _{ p }=1/T _{ p }, F _{ s }=[−f _{ s }/2,f _{ s }/2], \({L_{0}} = \left [ {\frac {{f_{\textit {NYQ}}} + {f_{s}}}{2{f_{p}}}} \right ] 1\), and \({c_{\textit {il}}} = \frac {1}{{T_{p}}} \int _{0}^{T_{p}} {{p_{i}}(t){e^{j \frac {2\pi }{T_{p}} lt}}} dt\).
4 Improved simultaneous orthogonal matching pursuit algorithm

Input: M×L dimensional sensing matrix A, the number of sub frequency bands K, m×2K dimensional frame vector V, and residual threshold θ.

Output: 2K×1 dimensional support set S.

Initialization: support set S=∅, residual matrix R=V.

Iteration: Repeat the following steps until the residual is less than the threshold or the number of iteration reach K.

(1) P=A ^{ T } R.

(2) d _{ k }=∥P _{ k }∥_{2},k=1,2,…,L, P _{ k } is kth column of matrix P.

(3) \({z_{\text {k}}} = \frac {d_{k}}{{\left \ {\textbf {A}_{k}} \right \}_{2}}\), k=1,2,…,L, z={z _{1},z _{2}…,z _{ L }}, A _{ k } is the kth column of matrix A.

(4) Find the largest item in the vector z and add its corresponding index k into the support set S, and the symmetric index value L+1−k is also added to the support set V. It is generally considered that the traditional signal is real signal, and its support set is symmetric.

(5) Construct the matrix A _{ S } corresponding to the columns in matrix A and support set S.

(6) Evaluate \(\hat {\mathbf {U}}\), \(\hat {\mathbf {U}} = \mathbf {A}_{\textbf {S}}^{\mathbf {\dag }} \textbf {V} = {\left (\textbf {A}_{\textbf {S}}^{T}{\textbf {A}_{\textbf {S}}}\right)^{1}} \textbf {A}_{\textbf {S}}^{T}\textbf {V}\).

(7) Update signal residuals \(\textbf {R} = \textbf {V}  {\textbf {A}_{\textbf {S}}}\hat {\mathbf {U}}\).
It also excludes the effect of the norm of atoms, and the mean value of atom is zero. Although both the inner product and the correlation coefficient can reflect the relevancy, but in some cases, for the same signal and sensing matrix, the atoms they selected are different, just as the following example: There are two atoms X=(1,2,3), Y=(2,2,3), signal Z=(4,5,6), after normalizing them, we can get X∗,Y∗, and Z∗. If we use the inner product as screening criterion of atoms, the inner product of X∗ and Z∗ is 0.975 and the inner product of Y∗ and Z∗ is 0.995.The most matched atoms should be Y∗. However, when the correlation coefficient is used as screening criterion of atoms, the correlation coefficient of X∗ and Z∗ is 1 and the correlation coefficient of Y∗ and Z∗ is 0.866. The most matched atoms should be X∗. The results are different using two criterions. Then which one is better? To solve this problem, we need to analyze the process of signal reconstruction.
if and only if when s=q, the equality holds. Thus, using correlation coefficient as screening criterion of atoms is more accurate.
Since the sensing matrix A and the residual matrix R are multidimensional, we cannot calculate the correlation coefficient between them directly, so we need to calculate the correlation coefficient between atoms in matrix A and columns in residual matrix R, and then squaring the correlation coefficients and sum the correlation coefficients corresponding to the same atom; finally, extract the root of the sum and find the largest one, whose corresponding atom is matched atom. This improvement will only change the first step of the SOMP algorithm into \({P_{k}} = \sum \limits _{i = 1}^{m} {r_{{\textbf {A}_{k}}{\textbf {R}_{i}}}}\), other steps remain unchanged.
5 Simulation results
6 Conclusions
In order to improve the reconstruction probability of MWC sampling system, we improve the SOMP algorithm and its derivative algorithms in this paper by changing the inner product into correlation coefficient as screening criterion of atoms. The simulation experiments show that the improved algorithm can increase the reconstruction probability, especially for MMVICoSaMP algorithm and MMVISP algorithm, the improvement is very significant. For SOMP, MMVROMP, and MMVStOMP algorithms, the improvement can also increase the reconstruction probability to a limited extent. It also means that less samples were needed to reconstruct signals than traditional algorithms when the number of bands is unchanged. Because of the improvement of correlation coefficient, the complexity of the algorithms increase, so the reconstruction time correspondingly increase and the algorithm is not suitable for the system with high realtime requirements. For systems that require high accuracy, the improved algorithms can be used. Since calculating correlation coefficient at each iteration will cost a lot of time, we also proposed a simplified algorithm, which can also improve reconstruction probability and reconstruction time is about the same as traditional algorithms. In addition, relatively small SNR has an adverse effect on reconstruction probability of improved algorithms, so the improved algorithms are more suitable for low noise channels.
Declarations
Acknowledgements
This work was supported in part by National Natural Science Foundation of China under Grants No. 61201143 and No. 91438205.
Open Access This 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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