基于稀疏AR建模信號去噪研究與應(yīng)用

第一作者宋歡歡女，碩士生，1989年4月生

通信作者葉慶衛(wèi)男，副教授，1970年4月生

基于稀疏AR建模信號去噪研究與應(yīng)用

宋歡歡，葉慶衛(wèi)，王曉東，周宇

(寧波大學(xué)信息科學(xué)與工程學(xué)院， 浙江寧波315211)

摘要：為去掉在不同環(huán)境、設(shè)備下所采集信號中的不同分布形態(tài)噪聲，引入稀疏優(yōu)化求解思路構(gòu)建新的去噪算法。設(shè)信號的AR模型系數(shù)是稀疏的，且噪聲對AR模型系數(shù)影響均衡分布，則可用采集的含噪聲信號構(gòu)建稀疏AR模型有效消除噪聲。用含噪聲信號構(gòu)建AR系數(shù)矩陣作為過完備稀疏基，通過多次重復(fù)隨機抽取方式獲得多個欠定方程組；利用稀疏優(yōu)化求解算法獲取AR模型稀疏系數(shù)；據(jù)稀疏系數(shù)平均值重構(gòu)信號。仿真實驗表明，信號含噪聲較大時該算法較經(jīng)典小波及中值濾波去噪效果更好。

關(guān)鍵詞：多頻信號；AR模型；稀疏表示；過完備稀疏基

收稿日期：2013-10-21修改稿收到日期：2014-04-03

中圖分類號:TP391.4文獻(xiàn)標(biāo)志碼：A

基金項目：國家科技支撐計劃項目(2011BAF14B04);江蘇省自然科學(xué)基金資助項目(BK2011504);江蘇高校優(yōu)勢學(xué)科建設(shè)工程資助項目(PAPD);江蘇省博士后基金資助項目(1201024B);國家博士后基金(2012M521008)

A new algorithm of signal de-noising based on sparse AR model

SONGHuan-huan,YEQing-wei,WANGXiao-dong,ZHOUYu(Information Science and Engineering College, Ningbo University, Ningbo 315211, China)

Abstract:A actual signal always contains noise due to different surroundings and collecting devices. And the noise has different forms. A signal de-noise algorithm is an important pre-processing tool. Here, a new de-noise algorithm was proposed based on sparse optimization. It was assumed that coefficients of signal’s AR model are sparse and the noise effect on coefficients of AR model has an even distribution. A sparse AR model was built for the signal with noise. The AR coefficient matrix was constructed with the noised signal, and the matrix was taken as the over-completed sparse basis. Several underdetermined equation sets were obtained by extracting randomly some rows several times from the over-completed sparse basis. Then, the sparse AR coefficients were solved with the sparse optimization algorithm. At last, the AR coefficients were averaged, and the de-noised signal was reconstructed with the averaged AR coefficients. The simulation results showed that the de-noising effect obtained with the proposed algorithm is better than those of the classical wavelet de-noising algorithm and the median filtering de-noising algorithm.

Key words:multi-frequency signal; AR model; sparse representation; over-completed sparse base

在實際實驗中信號分析只能針對采集的信號進(jìn)行研究。由于外界環(huán)境、檢測設(shè)備不完善造成所采信號含噪聲干擾，有用信息被噪聲淹沒，無法對信號準(zhǔn)確分析研究，故須先對信號進(jìn)行去噪預(yù)處理。小波去噪為常用的多頻信號去噪方法?；谛〔ㄗ儞Q的信號去噪算法有諸多閾值方法，如文獻(xiàn)[2]對閾值方法缺陷進(jìn)行闡述：在信號不連續(xù)區(qū)域會出現(xiàn)Gibbs現(xiàn)象；信噪比較大時去噪效果明顯下降；逐點處理小波系數(shù)時忽視其完整性；閾值選擇較關(guān)鍵，據(jù)信號不同，信噪比不同，閾值選擇隨之改變，閾值選擇不好會影響去噪效果，從而增加小波去噪實際操作難度。文獻(xiàn)[3-5]對小波去噪算法進(jìn)行改進(jìn)，提出的新小波閾值算法包含傳統(tǒng)硬、軟閾值方法優(yōu)點；并據(jù)傳統(tǒng)小波閾值算法將信號噪聲值域中的小波系數(shù)差別及權(quán)重加大以便確定閾值，取得更好的去噪效果，但小波算法的Gibbs現(xiàn)象仍然存在；基于稀疏表示的小波去噪引入隨機測量矩陣對小波系數(shù)進(jìn)行變換，并考慮小波系數(shù)的整體性，但仍要求據(jù)實驗經(jīng)驗選擇合適閾值及好的測量矩陣。

多頻信號中的噪聲主要分為加性噪聲與乘性噪聲。加性噪聲主要有脈沖噪聲及白噪聲，而乘性噪聲會改變信號的振動幅值，較復(fù)雜。本文實驗仿真主要對含白噪聲的多頻信號去噪。先用采集的帶噪聲信號構(gòu)建自適應(yīng)過完備稀疏基，隨機抽取其m行構(gòu)造矩陣，其中m小于過完備稀疏基列數(shù)；再對信號進(jìn)行稀疏表示，因過完備稀疏基含信號的結(jié)構(gòu)特征，而噪聲信號無此特征，則噪聲信號映射到過完備稀疏基的參數(shù)非常小，用l1算法計算可自動清除較小噪聲參數(shù)；再用所求稀疏系數(shù)與過完備稀疏字典相乘獲得去噪后信號。此時所得信號并非完整信號，其前部分并未得到去噪處理。本文實驗仿真用雙向稀疏去噪完成對整個信號的去噪。因需處理的信號符合AR模型，將信號倒置，所得新信號亦符合AR模型。據(jù)此，本文對倒置后所得信號去噪處理，與原始采集信號去噪所得信號合并進(jìn)結(jié)果。本文定義此操作為雙向稀疏去噪，通過與小波去噪、中值濾波去噪算法比較，證明該算法去噪效果較好。

1基于稀疏AR建模的信號去噪基本原理

1.1基于AR模型的過完備稀疏基構(gòu)造

設(shè)信號符合AR模型，即

(1)

由式(1)看出，AR模型具有較強的時間相關(guān)性，即第k時刻信號xk只與其前p時刻信號xk-l(l=1,2,…,p)有關(guān)，與其它時刻信號、激勵無關(guān)。第k時刻信號xk可以由前p個時刻信號線性表示。對某信號采樣n點，因該信號符合AR模型，得方程組為

(2)

將式(2)寫成矩陣形式，即

(3)

(4)

式中：Z定義為過完備稀疏基，且要求n-p>p。

1.2稀疏表示

(5)

式中：i1，i2，…，im為行號。令

式(5)為欠定方程組，有無窮個解，而所需解為最稀疏解，則方程組可轉(zhuǎn)換為

(6)

式中：ε為實數(shù)。

(7)

1.3重構(gòu)去噪

(8)

(9)

圖1為基于稀疏AR建模信號去噪算法流程圖及算法具體步驟。

圖1　基于稀疏AR建模信號去噪算法流程圖 Fig.1 The flow chart ofsignal de-noising algorithm based on sparse AR model

2實驗仿真

原始信噪比為

(10)

復(fù)原后信噪比為

(11)

小波去噪復(fù)原后信噪比為

(12)

中值濾波去噪復(fù)原后信噪比為

(13)

仿真采用?1內(nèi)點法求解稀疏系數(shù)，其中參數(shù)λ=0.1，精度為0.001。設(shè)仿真信號為多頻信號，即

x0(t)=sin(200πt+0.1)+3cos(260πt+0.9)+

0.7sin(340πt+1.5)+1.2cos(140πt)

(13)

(a) SNR0=9.54dB,SNR1=18.89dB(b) SNR0=9.54dB,SNR2=12.86dB(c) SNR0=9.54dB,SNR3=10.47dB圖2 不同算法去噪復(fù)原信號對比Fig.2Thechartofcomparesignalswhichrecoveryfromonesignalusingtwodifferentde-noisingalgorithms

圖3　兩種算法對不同信噪比的信號去噪比較圖 Fig.3 Comparision chart of different SNR signal’s de-noising use two algorithms

3實際應(yīng)用

用本文算法對實際采集的振動信號進(jìn)行去噪。工程數(shù)據(jù)采自寧波某斜拉索大橋。大橋全長67m，由102根直徑0.15m拉索構(gòu)成支撐系統(tǒng)；用WS-ZHT2振動設(shè)備及雙傳感器采集振動信號。雙傳感器安裝于拉索及梁端鉸支處，以便能有效感應(yīng)索-梁耦合的拉索振動，見圖4。

圖4　傳感器安裝圖 Fig.4 The picture of sensor installation

圖4　本文算法對實采信號去噪前后對比 Fig.4 The figure of compare original signal to de-noising signal using the algorithm the paper proposed

4結(jié)論

(2)本文算法基于信號自身的結(jié)構(gòu)特點進(jìn)行去噪處理，在實驗仿真、實際應(yīng)用中對帶較大噪聲符合AR模型信號的去噪效果均較好。參數(shù)L、m據(jù)采集的信號數(shù)據(jù)及經(jīng)驗人為設(shè)定，L、m值不同去噪效果不同。

(3)本文中過完備字典D由隨機抽取完整字典矩陣Z的m行構(gòu)成(m

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