上海理工大学学报  2020, Vol. 42 Issue (3): 240-246   PDF    
一种加权正交匹配追踪的盲多带信号重建方法
陈明夫, 渠刚荣, 石磊     
北京交通大学 理学院,北京 100044
摘要: 为研究多带信号的时域采样点盲重建该多带信号,将信号在适当大的包含其所有频带的频率区间上离散,信号频域重建归结为稀疏信号恢复问题。基于压缩感知恢复所需采样点少且其恢复稀疏信号要求观测矩阵的限制等距常数足够小,提出了一种改善观测矩阵的条件数,从而改善其限制等距常数的加权方法,以及相应的加权正交匹配追踪的盲多带信号重建方法,该方法对一般的稀疏信号恢复也适用。模拟中,对适当大的频率区间,取满足重建误差范围的适当小的离散间隔。模拟结果验证了对盲多带信号重建和一般的稀疏信号的恢复,提出的方法比直接用正交匹配追踪算法在相同条件下有更高的有效重建率。
关键词: 多带     压缩感知     条件数     限制等距常数    
A reweighted orthogonal matching pursuit method for blind multiband signal reconstruction
CHEN Mingfu, QU Gangrong, SHI Lei     
School of Science, Beijing Jiaotong University, Beijing 100044, China
Abstract: Blind multiband signal reconstruction is the reconstruction of the multiband signal from sampling points in the time domain. The signal was discretized over an appropriately large frequency range containing all of its bands, and then signal reconstruction in frequency domain boils down to sparse signal recovery. Based on the fact that fewer sampling points were needed for compressed sensing recovery and the RIC (restricted isometry constant) of the observation matrix was required to be small enough to recover the sparse signal, a reweighted method was proposed to improve the condition number of the observation matrix so as to improve its RIC, as well as a corresponding reweighted OMP (orthogonal matching pursuit) scheme for blind multiband signal reconstruction. The proposed method was also applicable to the recovery of general sparse signals. In simulations, for an appropriately large frequency range, an appropriately small discrete interval satisfying the reconstruction error range was taken. The simulation results verify that for blind multiband signal reconstruction and the recovery of general sparse signals, the reweighted OMP scheme has a higher efficient reconstruction rate than the OMP algorithm directly used under the same conditions.
Key words: multiband     compressed sensing     condition number     restricted isometry constant    

香农采样定理是现代通讯和信号处理的理论基础,它将模拟信号与离散表示法连接起来[1]。对于信号 $f(t) \in {L^2}(\mathbb{R})$ ,如果当 $\left| \omega \right| > \sigma > 0$ 时,其傅里叶变换 $F(\omega )$ 的值为0,那么信号 $f(t)$ 被称作 $\sigma $ -带限的。这里: $t$ 表示时间; ${L^2}(\mathbb{R})$ 表示实数域上的Lebesgue平方可积函数空间; $\omega $ 表示频率; $\sigma $ 表示一个常数。香农采样定理说明一个 $\sigma $ -带限的信号 $f(t)$ 能够从它所有的等步长 $h \leqslant {{\rm{1}}/{\left( {{\rm{2}}\sigma } \right)}}$ 的采样点准确重建[2]

本文考虑一类多带信号,其所有的频带都是连续的区间段,且组合一起分布在一个宽的频谱内。信号的重建是在时间区间 $[ - T,T]$