基于全相位頻譜分析的正弦信號(hào)高精度參數(shù)估計(jì)方法

1、電力系統(tǒng)中,對(duì)電網(wǎng)電壓和電流基波參數(shù)的測(cè)量分析,通常采用傳統(tǒng)的離散傅里葉變換(DFT 進(jìn)行頻譜分析來(lái)實(shí)現(xiàn),當(dāng)信號(hào)的頻率不是DFT 頻率分辨率的整數(shù)倍的時(shí)候,即對(duì)信號(hào)非整周期采樣,會(huì)產(chǎn)生頻譜泄漏,使測(cè)得的幅值、頻率和相位偏離實(shí)際值,尤其相位測(cè)量誤差更大1-4?，F(xiàn)提出一種基于全相位頻譜分析的正弦信號(hào)高精度參數(shù)估計(jì)算法,在精度與實(shí)時(shí)性與算法復(fù)雜度上均優(yōu)于現(xiàn)有算法,軟件編寫(xiě)簡(jiǎn)單,在無(wú)噪情況下參數(shù)估計(jì)近似為無(wú)偏估計(jì),尤其對(duì)相位的估計(jì),誤差可達(dá)到0.001%。1全相位頻譜分析文獻(xiàn)5提出一種新型的頻譜估計(jì)算法,對(duì)傳統(tǒng)DFT 頻譜分析時(shí)數(shù)據(jù)的截?cái)喾绞竭M(jìn)行了改進(jìn),可以很大程度地減小頻譜泄漏。若將N 階中心對(duì)稱窗

2、和N 階矩形窗卷積產(chǎn)生的一個(gè)2N -1階窗作為窗函數(shù),則是全相位單窗頻譜分析,若將N 階中心對(duì)稱窗和自身卷積產(chǎn)生的一個(gè)2N-1階窗作為窗函數(shù),則是全相位雙窗頻譜分析,其框圖如圖1所示。首先,推導(dǎo)對(duì)具有單一頻率f 0的單頻信號(hào)進(jìn)行全相位頻譜分析得到的幅度。設(shè)單頻信號(hào):x=e j 2f 0f sn其中,f 0為信號(hào)頻率,f s 為采樣頻率。對(duì)于時(shí)間序列中的一點(diǎn)x (N ,存在也只存在N 個(gè)包含該點(diǎn)的N 維向量:X 0=x (N x (N+1x (2N-1T X 1=x (N-1x (N x (2N-2TX N-1=x (1x (2x (N T將每個(gè)向量進(jìn)行循環(huán)移位,把樣本點(diǎn)x (N 移到首位,則得

3、到另外的N 個(gè)N 維向量:X 0=x (N x (N+1x (2N-1T X 1=x (N x (N+1x (N-1TX N-1=x (N x (1x (N-1T對(duì)準(zhǔn)x (N 相加得到全相位數(shù)據(jù)向量:X A P =1NNx (N (N-1x (N+1+x (1x (2N-1+(N-1x (N-1T基于全相位頻譜分析的正弦信號(hào)高精度參數(shù)估計(jì)方法黃曉紅1,王兆華2(1.河北理工大學(xué)信息學(xué)院,河北唐山063000;2.天津大學(xué)電子信息工程學(xué)院,天津300072摘要:對(duì)傳統(tǒng)頻譜分析的輸入數(shù)據(jù)截?cái)喾绞竭M(jìn)行了改進(jìn),提出了全相位頻譜分析,理論證明全相位頻譜分析幅度譜是傳統(tǒng)FFT 頻譜分析幅度譜的平方,可以很

4、大程度上減小頻譜泄露,同時(shí),給出了其實(shí)現(xiàn)框圖。另外,全相位頻譜分析方法具有良好的相位分析性能,不受頻偏影響,在信號(hào)是非整數(shù)倍頻率采樣情況下,由該方法分析的相位與信號(hào)真實(shí)相位誤差極小,無(wú)需校正。提出基于全相位相位差法估計(jì)正弦信號(hào)幅值、頻率和相位的新算法,對(duì)傳統(tǒng)相位差法其序列的取法進(jìn)行了改進(jìn),對(duì)單頻余弦信號(hào)進(jìn)行非整周期采樣,分別用全相位相位差法和離散頻譜綜合相位差校正法對(duì)信號(hào)的頻率、幅值和相位進(jìn)行了校正。仿真結(jié)果證明該算法參數(shù)估計(jì)精度高于現(xiàn)有算法,具有較好的實(shí)用價(jià)值,在無(wú)噪情況下參數(shù)估計(jì)近似為無(wú)偏估計(jì),尤其對(duì)相位的估計(jì),誤差達(dá)到0.001%。關(guān)鍵詞:全相位頻譜分析;參數(shù)估計(jì);相位差;校正中圖分類號(hào)

5、:TN911文獻(xiàn)標(biāo)識(shí)碼:A文章編號(hào):1006-6047(200807-0054-03收稿日期:2007-04-06;修回日期:2007-12-05電力自動(dòng)化設(shè)備Electric Power Automation EquipmentVol.28No.7Jul.2008第28卷第7期2008年7月圖1傳統(tǒng)頻譜分析與全相位頻譜分析框圖Fig.1Block diagrams of traditional and apFFT spectrum analysisx (n Z -1Z -1Z -1N 階對(duì)稱窗W N 階FFT (a N 階傳統(tǒng)FFT 頻譜分析框圖(b N 階全相位頻譜分析框圖Z -1Z -1

6、N 階對(duì)稱窗W 1與N 階對(duì)稱窗W 2卷積N 階FFTx (n Z -1+Z -1+根據(jù)DFT 移位性質(zhì),X i (i =0,1,N-1的離散傅里葉變換X i (k 和X i (i =0,1,N-1的離散傅里葉變換X i (k 之間有很明確的關(guān)系:X i (k =X i (k e j2kiN(1全相位頻譜是由X i (k 之和組成,所以有X A P (k =1N !i=0N -1X i(k =1N !i=0N-1X i (k e j 2ki N =1N !i=0N-1!n=0N -1e j2f 0f s(N-i+n e -j 2N kn e j 2ki N =1Ne j2f 0f sN !i=

7、0N -1e -j2f 0f s-k N "#i !n=0N -1e j2f 0f s-kN "$n=1N e j 2f 0f sN sin 2N (f 0/f s -k /N sin 2(f 0/f s -k /N (2由式(2可見(jiàn)全相位頻譜分析幅度譜為1Nsin N (f 0/f s -k /N sin (f 0/f s -k /N 2其為傳統(tǒng)DFT 頻譜分析幅度譜的平方,這對(duì)減小頻譜泄漏很有益。全相位頻譜分析另一個(gè)重要的特點(diǎn)是其相位不用校正,恒定不變,不受頻偏影響,即當(dāng)信號(hào)未做整周期截?cái)鄷r(shí),全相位頻譜分析仍能求出信號(hào)的真實(shí)相位,得到的相位與信號(hào)的真實(shí)相位誤差極小,近似

8、相等,這是傳統(tǒng)DFT 頻譜分析所不具備的,也是用全相位頻譜分析進(jìn)行參數(shù)校正的基礎(chǔ)?，F(xiàn)以余弦信號(hào)cos (1.2×2t /6+100/180為例進(jìn)行N=6階全相位頻譜分析,分析全相位方法求相位誤差小的原因。11個(gè)取樣信號(hào)為:-0.1736,-0.9903,-0.4384,0.7193,0.8829,-0.1736,-0.9903,-0.4384,0.7193,0.8829,-0.1736。全相位輸入信號(hào)由6組N=6的取樣信號(hào)組成,第1組是由11個(gè)取樣信號(hào)的最后6個(gè)組成,第2組由向左移1位的6個(gè)數(shù)組成,但-0.1736要循環(huán)移到首位,其他類同。信號(hào)排列如表1所示。這6組N=6的取樣信號(hào)的

9、相位如表2所示。因?yàn)轭l率是1.2Hz ,從每組第2個(gè)相位可見(jiàn),6個(gè)相位中3個(gè)偏離100°增大,3個(gè)偏離100°減小。全相位輸入數(shù)據(jù)是上面6組信號(hào)之和的平均,相位互相抵消,使得與原始信號(hào)的相位差為零,所以全相位頻譜分析后求出的相位即為信號(hào)真實(shí)相位。實(shí)驗(yàn)結(jié)果表明:全相位無(wú)窗頻譜分析時(shí),求整數(shù)倍頻率相位準(zhǔn)確,如果求偏離整數(shù)倍頻率相位時(shí),全相位加kaiser (N ,9.5雙窗誤差最小。此例使用全相位kaiser (N ,9.5雙窗求出的信號(hào)相位如下:180.0000°,100.0069°,100.0004°180.0000°,259.999

10、6°,259.9931°與信號(hào)真實(shí)的相位誤差僅為0.69%,可認(rèn)為相等。從具體的數(shù)據(jù)分析,可清楚地看出全相位頻譜分析具有良好的相位分析性能,在信號(hào)是非整數(shù)倍頻率情況下相位不變。而傳統(tǒng)譜分析方法相位會(huì)發(fā)生很大誤差。用計(jì)算機(jī)生成如下信號(hào),采樣頻率128Hz ,作譜點(diǎn)數(shù)為128的分析,頻率分辨率為1Hz ,有s 1(t =cos (9×2t /N+50/180+cos (19.1×2t /N+50/180+cos (29.2×2t /N+50/180+cos (39.3×2t /N+50/180+cos (49.4×2t /N+5

11、0/180全相位頻譜分析采用hanning (N 雙窗,傳統(tǒng)DFT 頻譜分析采用hanning (N 窗,分析結(jié)果如圖2(a (b 所示?？梢?jiàn)全相位雙窗頻譜分析的各頻率成分信號(hào)的相位均為信號(hào)真實(shí)相位50°,而傳統(tǒng)加窗分析的信號(hào)相位除了第一個(gè)沒(méi)有偏離整數(shù)倍頻率信號(hào)的相位為50°,其余頻率成分信號(hào)的相位分析均與真實(shí)相位有很大誤差,且頻率偏離整數(shù)倍越遠(yuǎn),相位分析結(jié)果誤差越大。若將信號(hào)相位發(fā)生變化,即產(chǎn)生信號(hào):s 2(t =cos (9×2t /N+10/180+cos (19.1×2t /N +30/180+cos (29.2×2t /N +50&#

12、215;/180+cos (39.3×2t /N+70/180+cos (49.4×2t /N+90/180分析結(jié)果如圖2(c (d 所示。由圖2可明顯看出,全相位頻譜分析得到的相位和信號(hào)的真實(shí)相位誤差極小,而傳統(tǒng)DFT 方法分析得到相位誤差則較大。2全相位相位差法估計(jì)正弦信號(hào)參數(shù)文獻(xiàn)2中給出了關(guān)于相位差法校正的原理,全相位相位差法對(duì)其序列的取法進(jìn)行了改進(jìn)。具體算法步驟如下:a.對(duì)信號(hào)做一定長(zhǎng)度的非整周期采樣,從采樣序列的第i 點(diǎn)開(kāi)始起,取2N -1點(diǎn),得到一個(gè)序列u 1(t ;再?gòu)膇 +N 點(diǎn)開(kāi)始起,也取2N-1點(diǎn),得到另一個(gè)序列u 2(t ;b.對(duì)序列u 1(t 進(jìn)行全

13、相位雙窗(漢寧與漢恩窗的卷積窗N 點(diǎn)頻譜分析,得到頻率表示為f=k f s /N ,頻率分辨率F 0=f s /N (k 為主譜線的序號(hào),頻率校正第1組第2組第3組第4組第5組第6組-0.1736-0.1736-0.1736-0.1736-0.1736-0.1736-0.9903-0.9903-0.9903-0.9903-0.9903-0.1736-0.4384-0.4384-0.4384-0.4384-0.9903-0.99030.71930.71930.7193-0.4384-0.4384-0.43840.88290.88290.71930.71930.71930.7193-0.17360

14、.88290.88290.88290.88290.8829表1取樣信號(hào)Tab.1Sampled signals第1組第2組第3組第4組第5組第6組180.000000180.0000180.0000180.0000132.5107112.9053112.497888.459883.627572.5107-25.769951.729853.4788138.4375161.4769214.23010180.0000180.000000180.000025.7699308.2702306.5212-138.4375-161.4769145.7699-132.5107247.0947247.5022-

15、88.4598-83.6275287.4893表2取樣信號(hào)相位(N=6Tab.2Phases of sampled signals (N=6(°黃曉紅,等:基于全相位頻譜分析的正弦信號(hào)高精度參數(shù)估計(jì)方法第7 期圖2傳統(tǒng)DFT 與全相位頻譜分析多信號(hào)的相位譜Fig.2Phase spectrums analyzed by traditionalDFT and apFFT spectrum analysis1005020406080(c 雙窗頻譜分析相位譜(10°30°50°70°90°20010020406080(d 傳統(tǒng)加窗DFT 相

16、位譜(10°30°50°70°90°/(°/(°f /Hzf /Hz10305070901047.885.7123.6161.520010020406080(b 傳統(tǒng)加窗DFT 相位譜(50°50°50°50°50°/( °10050020406080(a 雙窗頻譜分析相位譜(50°50°50°50°50°/(°f /Hzf /Hz5067.885.7103.5121.4量f=d F 0(d 為泄露誤差系數(shù)

17、;c.信號(hào)校正泄漏誤差后的頻率為f !=f +f =f +dF 0=(k+d F 0同時(shí)得到相位1;d.對(duì)序列u 2(t 進(jìn)行全相位雙窗(漢寧與漢恩窗的卷積窗N 點(diǎn)頻譜分析,得到主譜線處相位2,根據(jù)相位差=2-1=2d ,計(jì)算出泄漏誤差系數(shù)d ;e.校正其頻率、幅值、相位,將d 代入步驟c 后得到校正后的頻率。令主譜線處的粗估幅值為A k ,則幅值校正公式為A=2(1-2A k !2sin ("#2信號(hào)的初始相位公式為0=21-23仿真與結(jié)果單頻余弦信號(hào)為s (t =cos (4.39×2t /f s +40/180對(duì)其進(jìn)行非整周期采樣,采樣頻率f s =1024Hz ,頻

18、譜分析點(diǎn)數(shù)為N=1024點(diǎn),分別用全相位相位差法和文獻(xiàn)6相位差法進(jìn)行參數(shù)估計(jì),結(jié)果如表3所示(表中幅值為歸一化數(shù)值。4結(jié)語(yǔ)在傳統(tǒng)頻譜分析的基礎(chǔ)上,對(duì)輸入數(shù)據(jù)截?cái)喾绞竭M(jìn)行了改進(jìn),得到了全相位頻譜分析,相對(duì)于傳統(tǒng)頻譜分析,具有頻譜泄露小、參數(shù)估計(jì)精度高的優(yōu)點(diǎn)。該算法無(wú)需對(duì)信號(hào)進(jìn)行嚴(yán)格的整周期采樣,可有效抑制DFT 算法中的頻譜泄露和柵欄效應(yīng),且無(wú)需相位校正,算法簡(jiǎn)單,可用于電力自動(dòng)化中的向量測(cè)量以及正弦交流電諧波分析等工程,有良好的實(shí)用價(jià)值。參考文獻(xiàn):1JAIN V K ,COLLINS W L ,DAVID D C.High -accuracy analogmeasurements via in

19、terpolated FFT J .IEEE Trans on InstrumMeas ,1979,28(2:113-122.2曹國(guó)劍,黃純,謝雁鷹,等.基于時(shí)移相位差校正的準(zhǔn)周期算法的研究J .繼電器,2004,32(13:1-5.CAO Guojian ,HUANG Chun ,XIE Yanying ,et al.Research on quasi-synchronization algorithm based on time -shifting and phase difference correcting J .Relay ,2004,32(13:1-5.3RENSERS H ,SC

20、HOUKENS J ,VILAIN G.High-accuracy spectrumanalysis of sampled discrete frequency signals by analytical leakage compensation J .IEEE Transactions on Instrumentation and Measurement ,1984,33(4:287-292.4丁仁杰.電力系統(tǒng)同步相量動(dòng)態(tài)測(cè)量技術(shù)的研究與實(shí)現(xiàn)D .北京:清華大學(xué),1995.5王兆華,侯正信,蘇飛.全相位FFT 頻譜分析J .通信學(xué)報(bào),2003,24(11:16-19.WANG Zhaohua

21、 ,HOU Zhengxin ,SU fei.All phase FFT spectrum analysis J .Jounal of Communications ,2003,24(11:16-19.6丁康,朱小勇,謝明,等.離散頻譜綜合相位差校正法J .振動(dòng)工程學(xué)報(bào),2002,15(1:114-117.DING Kang ,ZHU Xiaoyong ,XIE Ming ,et al.Synthesized correcting method of phase difference on discrete spectrum J .Journal of Vibration Engineerin

22、g ,2002,15(1:114-117.7朱小勇,丁康.離散頻譜校正方法的綜合比較J .信號(hào)處理,2001,17(1:91-97.ZHU Xiaoyong ,DING Kang.The synthetical comparision of cor -recting methods on discrete spectrum J .Signal Processing ,2001,17(1:91-97.8吳國(guó)喬,王兆華,黃曉紅.離散頻譜的全相位校正法J .數(shù)據(jù)采集與處理,2005,20(3:286-290.WU Guoqiao ,WANG Zhaohua ,HUANG Xiaohong.All p

23、hase correction method for discrete spectrum J .Journal of Data Acqui -sition &Processing ,2005,20(3:286-290.9RIFE D C ,VINCENT G A.Use of the discrete Fourier transformin the measurement of frequencies and levels of tones J .Bell Syst Tech J ,1970,49(2:197-228.10PALMER L C.Coarse frequency esti

24、mation using the discreteFourier transform J .IEEE Trans Inform Theory ,1974,20(1:104-109.11齊國(guó)清,賈欣樂(lè).基于DFT 相位的正弦波頻率和初相的高精度估計(jì)算法J .電子學(xué)報(bào),2001,29(9:1164-1167.QI Guoqing ,JIA Xinle.High -accuracy frequency and phase estimation of signal -tone based on phase of DFT J .Acta Electronic Sinica ,2001,29(9:1164

25、-1167.(責(zé)任編輯:康魯豫作者簡(jiǎn)介:黃曉紅(1973-,女,黑龍江海倫人,副教授,博士,研究方向?yàn)閿?shù)字信號(hào)處理及CPLD (E-mail :tshxh ;王兆華(1937-,男,上海人,教授,博士研究生導(dǎo)師,研究方向?yàn)閿?shù)字信號(hào)處理及數(shù)字圖像處理。方法幅值頻率/Hz相位/(°全相位相位差法0.99999994.390000039.9999999文獻(xiàn)6中相位差法0.99881614.386835040.5486832理想值1.00000004.390000040.0000000表3無(wú)噪情況下全相位相位差法與相位差法參數(shù)校正結(jié)果比較Tab.3Comparison of paramete

26、r corrections between apFFT phase difference method and phase differencemethod under no-noise conditions第28卷電力自動(dòng)化 設(shè)備High accuracy parameter estimation of sinusoidal signalbased on all -phase spectrum analysisHUANG Xiaohong 1,WANG Zhaohua 2(1.Hebei Polytechnic University ,Tangshan 063000,China ;2.Tia

27、njin University ,Tianjin 300072,China Abstract :The apFFT (all phase FFT spectrum analysis and its implementation is put forward to im-prove the input data truncation of traditional spectrum analysis.It is theoretically proved that the amplitude spectrum of apFFT spectrum analysis is the square of t

28、hat of traditional spectrum analysis ,which reduces leakage greatly.The apFFT has perfect phase analysis performance with less impact of frequency shift.The phase analyzed by apFFT has extremely small error even if the sampling frequency is not the multiplication of signal frequency.An apFFT phase d

29、ifference algorithm is proposed to estimate the amplitude ,frequency and phase of sinusoidal signal.Simulation results prove that the algorithm is more accurate and practicable than the existing ones.The estimation is nearly error -free under no-noise conditions ,and the phase error is less than 0.001%.Key words :all -phase spectrum analysis ;parameter estimation ;phase difference ;correction電力期貨市場(chǎng)的價(jià)格發(fā)現(xiàn)功能劉思東1,2,楊洪明1,童小嬌1(1.長(zhǎng)沙理工大學(xué)電氣與信息工程學(xué)院,湖南長(zhǎng)沙410076;2.五邑大學(xué)數(shù)理系,廣東江門(mén)529020摘要:價(jià)格發(fā)現(xiàn)功能是期貨市場(chǎng)的基本功能之一,它在期貨市場(chǎng)的發(fā)揮程度直接反