濾波器外文翻譯外文文獻(xiàn)英文文獻(xiàn)用改進(jìn)的窗函數(shù)設(shè)計(jì)FIR數(shù)字濾波器.doc

research on fir digital filter design using an improved window functiontan jiajie , luo changyou, huang sanwei ， deng xiaohui ( department of physics and electronic information science, hengyang normal university, hengyang hunan 421008, china)abstract : window function has been used to design a linear phase digital filer for long times, but the use o f optimization techniques for designing digital filter has become widespread in recent year a new met ho d has been proposed to improve fir window function in this paper ，t he window function that combines with co sine sequences in linear is different from previous hann ,hamming and blackman window function the paper also proposes linear programming to optimize characterization of fir digital filter according to its magnitude condition, and gives out t he algorithm to design dig ital filter using the improved window function ,finally , we have designed fir filter using new window for simulation and compared w it h the filter designed by hamming window, blackman window function t he simulation results show that filters designed using this method to meet t he design specifications key words: improved window function; fir digital filter ; window function; linear programming 0 lead speechthe design method of fir digital filters are mainly: window function method, frequency sampling method and the chebyshev etc corrugated approximation method 1-4. window function method is the most commonly used designing fir digital filters, the simplest method of 4-5. the essence of window function method is the truncated ideal impulse response to approximate the method petitions filter index. frequency sampling method is a design optimization method for its shortcoming is when the design that use the variable is limited to a few samples values of transitional, cut-off frequency not easy control 3. chebyshev etc corrugated approximation method is a kind of optimization design, but existing computational complexity, big disadvantage computation 1-2.window function method is simple in design, have closed form of formula, thus very practical. defect is the stopband bandpass, cut-off frequency not easy control 2-3. digital filter, window function of auto-heating window function method of selecting, the key is: design to choose the appropriate window function, choose the right order number of digital filter, improve amplitude frequency characteristics, reduce gibbs phenomenon, solve convergence problem 1-2. 3 choose window function, through to guass guass window function improved, design a low-pass filter has better superiority; 4 the error information, using the known in the iteration process through the window function method continuously revised design result in filter order number, under the condition of invariable frequency response approximation, filter ideal frequency response. 5 use integer sequence, such as where fibonacci sequence, golomb sequence, conway hofstadter recursive sequence, triangular series produce window function to design the filter, its effect is better than that of classical design method. 6 choose dual window window function sequence of structure was system characteristics approximation error is the minimum; 7 will be well saramaki dolph - and chebysheve window with the well designed, its effect fir digital filters than kaiser well; 8 put good effect in hamming reimann well well well and kaiser window. 9 put forward a kind of exponential window function, this window function has the width can be adjusted with the window design characteristics, the digital filters have more centralized, lord disc energy side-lobe less features. 10 using linear programming design linear phase fir filter ascending cosine to 100 % super bandwidth. 11 linear programming method is adopted to design digital filter. this paper uses the existing window function, and carry on the weighted combination, reference 10 or 11, and linear programming with long hamming, blackman window are compared. the advantages of this method is strong logicality, goal clear, easy to achieve, and to explore the best solutions.1 common window functionwindow function select principle: window function as focus on energy, lord disc transitional steep; reduce the window function spectrum side-lobe level, increase stopband attenuation, and reduce the stopband bandpass and ripple effect. common window function have 1-4 : rectangle window, hanning window, hamming window, blackman window, kaiser window. window function method design idea of fir filters is 1-2 : make sure the frequency response of ideal filter.the frequency response of practical design filterto approximate.for againreverse transform get finally use window functionto truncate,mean h( n) =.to truncate ,will produce gibbs phenomenon, all the window function choices to reduce this phenomenon for the purpose. judge ideal window function mainly according to the following three criteria! the lord is high double amplitude and its width should try to narrowthe amplitude side-lobe fast speed, the biggest drop side-lobe relative to the main valve should be as low as possible. # transitional requirements will try to narrow. facts prove the two standard cannot simultaneously satisfy window function should be, so the two compromise 1-3. in order to reduce caused due to add window truncation ripple and transitional grows wider impact in engineering design common hamming window and kaiser window. 2 improved window function 1-2 enumerated window function, hanning window, hamming window, blackman window is cosine sequence and rectangular sequence of linear combination. in order to restrain the amplitude, side-lobe hanning window, hamming window on the basis of the second, add cosine, when the harmonic component design and ideal window function and related to the frequency response of different from blackman window, window function improved form below (1)formula (1) of a, b, c undetermined, their size and given filter technology indexes related. for convenience, this window function length choice for odd. the next several special case discussion this type. case 1, take a = 1, b = c = 0, for rectangular window. condition 2, take a = 0.5, b = - 0.5, c = 0, for hann window. case 3, take a = 0.53, b = - 0.46, c = 0, for hamming window. situation, a = 4, b = 0 0.42 j c = 0.08, 5, blackman window for. by above knowable, the improved window function with these four window function the nature, belong to the general form of the window function.3 improved window function algorithm according to the given filter technology index,determine the backlog filter unit, but by sampling response formula below ask out: (2) calculating the actual filter unit sampling response:h( n) = (3)filter the frequency response is: （4）will formula (1) generation into the formula (4) : (5)reference 1-2 10 or 11, consider fir filters satisfy the first kind of linear phase conditions, for accidentally symmetry, and n an odd number,ordering h ( n) = 。then the magnitude of the next type: writtenwill continue to have the type of jane 1-2in form (6)then type (6) can be encoded (7)in form (8) (9) (10)if filter conditions for the amplitude, the pass-band with satisfaction,stopband inside meet,the pass-band with can list below equation: (11) (12)stopband inside can list equation: (13) (14) for bandpass corrugated, for stopband corrugated. for the convenience of discussion, here take coefficient a, b, c greater than zero as constraints. this problem can be converted into linear programming problem 10-12, formula (11), (12) (13) (14) can be as constraints. through this after converting, problem solving, become convenient. formula (11) (13) and formula (12) (14) add relaxation variables respectively, punish variable transformation into standard linear programming equations 12 : （15） （16） （17） （18）including d, e, f, g, the greater than 0. the way of solving this problem can be solved by least square or iteration method. calculate a, b, c, can determine the window function. according to the above process, this kind of the design of filter algorithm according to the following steps: etc. in frequency domain, and the interval take n points for the odd number of frequency domain, the sequence do discrete fourier inverse transform, digital filters out the ideal sampling response units. by formula (8) (9) (10) ask out respectively、。. from the formula (11), (12) (13) (14) list optimization constraints. . solving linear equations, find out a, b, c, so as to determine the optimal solution digital filter h(n)。. checking technology index whether meet the requirements.4 application examplesdesign one low tong digital filters, its technical index is as follows: bandpass cut-off frequency , stopband cut-off frequency , passband ripple ,stopband corrugated .,cut-off frequency for,blackman window fir digital filters for window length,the length of hamming window for 41. hamming window and blackman window with the fir filters designed the range characteristics as shown in figure 1. with the modified window method design fir filters as shown in figure 2. figure 1 hamming window and blackman window design range characteristicsfigure 2 improvement window range characteristics graphin figure 1 and figure 2 from the perspective of the filter, improve the frequency characteristics of the function of the window is flat bandpass, stopband have ripple; improve the window design cut-off frequency bandpass strictly comply with the design requirements, and traditional window method cutoff frequency is higher. different lies in the transitional zone, improving window functions for transitional zone0.2- 0.4,strict and design requirements consistent, but the hamming window and blackman window for transitional zone0.25-0.4improved traditional window function algorithm of slightly lost window function. in the stopband attenuation window function method, improvement hamming window method, and better blackman window method in weak.5 conclusionfir digital filters with window function design method used, this paper adopts improved, and window function design method of linear programming design the algorithm fir digital filters, design thinking clear effect is good. with the modified window design fir filters strictly meet the design technology index, and traditional window function, they have their respective comparative advantages and disadvantages of window functions, thus improving design fir filter is feasible.references:1 chengpeiqing. digital signal processing tutorial 3 edition j. beijing: tsinghua university press, 2001:1395-368.2 dingyumei. digital signal processing, gaoxiquan j 2 edition. xian: xian university of electronic science and technology press, 2001:195-15.3 chenmingjun, maozhangmei. improve window function in the application of fir digital filters design j.journal of relays, 2007,35 (17) : 65-674 tianguangxin, gaolizhi, sunchunlai, etc. based on the window function method of frequency selective optimization design of fir digital filters j.journal of data acquisition and processing, 2008,23 (2) : 228-232.5 m.r.arulalan, h. s. jamadani, ashok rao. novel window functions for dig ital filter s: ieee trans 2008 c . s . l : fifth international conference o n information technology , 2008: 1184-1185.6 huangxiaogong, sufei, based on single window. wangzhaohua all phase digital filter and lms standards window function design of j. journal of sensor technology, 2007,20 (6) : 273-280-1315.7 saramaki, t. adjustable window s for the design of fir filters ，a tutorial: electro technical conference,1991 cs l : proceeding s,6th mediterranean,1991:28-33.8 nihal l. hettiarachchi , adel a. sakla. design of digital fir filter s via optimized generalized reimann window function: circuits and systems, 1995c.s l :proceeding s of the 37th midwest symposium,1995, 2:1061-1064.9 kemal avci, arf nacaroglu. a new window based on exponential functionj .ieee trans on signal processing , 2008, 41(12):60-72.10 l. f. lind, b. m.alashimi, w. p. somerset. linear programming design of fir raised co sine filter with 100 % excess bandwidth j . electronics letter s 29th february ,1995 ,32(5):436-437.11 lawrence r. rabiner . linear pro gram design of finite impulse response digital filters j.ieee transactions on audio and electracoustics, 1972, 20 (4):280-288.12 the optimization principle and method. franci c m.beijing: beijing industrial university press, 2004:36-92.用改進(jìn)的窗函數(shù)設(shè)計(jì)fir數(shù)字濾波器譚家杰, 羅昌由, 黃三偉, 鄧小輝衡陽(yáng)師范學(xué)院 物理與電子信息科學(xué)系, 湖南 衡陽(yáng) 421008摘 要窗函數(shù)設(shè)計(jì) fir 數(shù)字濾波器已多年, 近年來(lái), 用優(yōu)化技術(shù)設(shè)計(jì)數(shù)字濾波器十分流行, 論文提出了一種新方法對(duì)窗函數(shù)進(jìn)行改進(jìn), 這種窗函數(shù)不同于以往的 hanning窗、hamming 窗、blackman 窗, 它先將余弦序列線性組合為窗函數(shù), 然后根據(jù) fir 數(shù)字濾波器的特性對(duì)數(shù)字濾波器的幅度條件進(jìn)行線性規(guī)劃。并且給出了改進(jìn)窗函數(shù)的算法。最后, 用該方法設(shè)計(jì)出了 fir 數(shù)字濾波器的仿真實(shí)例, 并與用 hamming 窗、blackman 窗方法設(shè)計(jì)的 fir 濾波器進(jìn)行對(duì)比, 仿真結(jié)果表明用該方法設(shè)計(jì)的濾波器滿足設(shè)計(jì)規(guī)格。關(guān)鍵詞改進(jìn)的窗函數(shù) fir 數(shù)字濾波器 窗函數(shù) 線性規(guī)劃中圖分類號(hào): tn713+.7 文獻(xiàn)標(biāo)志碼: a 文章編號(hào): 1673-0313( 2010) 06-0031-040 引 言fir 數(shù)字濾波器的設(shè)計(jì)方法主要有:窗函數(shù)法、頻率采樣法和切比雪夫等波紋逼近法1-4。窗函數(shù)法是設(shè)計(jì) fir 數(shù)字濾波器最常用、 最簡(jiǎn)單的方法4-5。窗函數(shù)法的實(shí)質(zhì)是用截?cái)嗬硐霙_激響應(yīng)的方法來(lái)逼近所求的濾波器指標(biāo)。頻率采樣法是一種優(yōu)化設(shè)計(jì)方法,其缺點(diǎn)是設(shè)計(jì)時(shí)使用的變量?jī)H限于過(guò)渡帶上的幾個(gè)采樣值,截止頻率不容易控制3。切比雪夫等波紋逼近法是一種優(yōu)化設(shè)計(jì), 但是存在計(jì)算復(fù)雜,計(jì)算量較大的缺點(diǎn)1-2。窗函數(shù)法設(shè)計(jì)簡(jiǎn)單, 有閉合形式的公式, 因而很實(shí)用。缺點(diǎn)是通帶、 阻帶的截止頻率不容易控制2-3。數(shù)字濾波器的好壞取決于窗函數(shù)的選取, 窗函數(shù)法設(shè)計(jì)的關(guān)鍵是: 選擇合適的窗函數(shù), 選擇合適的階數(shù),改善數(shù)字濾波器的幅頻特性, 減少 gibbs現(xiàn)象,解決收斂問(wèn)題1-2。文獻(xiàn)3 選擇 guass 窗函數(shù),通過(guò)對(duì) guass 窗函數(shù)改進(jìn), 設(shè)計(jì)出的低通濾波器具有更好的優(yōu)越性; 文獻(xiàn)4 利用已知的誤差信息,在迭代過(guò)程中通過(guò)窗函數(shù)法不斷修改設(shè)計(jì)結(jié)果, 在濾波器階數(shù)不變的情況下, 濾波器的頻率響應(yīng)逼近理想頻率響應(yīng)。文獻(xiàn)5 用整數(shù)序列,如 fibonacci序列、 golomb 序列、 hofstadter conway 序列、 recursive triangular 序列產(chǎn)生窗函數(shù), 來(lái)設(shè)計(jì)濾波器, 其效果優(yōu)于經(jīng)典的設(shè)計(jì)方法。文獻(xiàn)6 選擇雙窗結(jié)構(gòu)得到的窗函數(shù)序列使系統(tǒng)特性逼近誤差最小; 文獻(xiàn)7 將 saramaki window 窗并且與dolph-chebysheve window 結(jié)合設(shè)計(jì)出的fir 數(shù)字濾波器, 其效果勝過(guò) kaiser window ; 文獻(xiàn)8 提出的reimann window 效果好于 hamming window和kaiser window 窗。文獻(xiàn)9 提出一種指數(shù)型窗函數(shù),該窗函數(shù)具有寬度可調(diào)特性, 用該窗設(shè)計(jì)的數(shù)字濾波器有主瓣能量更集中, 旁瓣更少的特點(diǎn)。文獻(xiàn)10 采用線性規(guī)劃法設(shè)計(jì)線性相位的升余弦數(shù)字濾波器能100 %超帶寬。文獻(xiàn)11則采用線性規(guī)劃法設(shè)計(jì)數(shù)字濾波器。本文利用已有的窗函數(shù),對(duì)其進(jìn)行加權(quán)組合, 參照文獻(xiàn)10-11 的線性規(guī)劃法, 與同長(zhǎng)的hamming、 blackman 窗進(jìn)行比較。這種方法的優(yōu)點(diǎn)是邏輯性強(qiáng)、 目標(biāo)明確、 易于實(shí)現(xiàn), 并能探索最佳方案。1 常用的窗函數(shù)窗函數(shù)選擇原則:窗函數(shù)能量盡可能集中在主瓣內(nèi),過(guò)渡帶陡峭;減少窗函數(shù)頻譜的旁瓣高度,增大阻帶衰減, 減小通帶和阻帶的波紋。常用的窗函數(shù)有1-4: rectangle窗、 hanning 窗、 hamming 窗、 blackman窗、 kaiser 窗。窗函數(shù)法設(shè)計(jì) fir 濾波器的思路是1-2:先確定理想濾波器的頻率響應(yīng), 實(shí)際設(shè)計(jì)濾波器的頻率響應(yīng)來(lái)逼近。再對(duì)進(jìn)行反變換得最后用窗函數(shù)來(lái)截?cái)?,即h( n) = 。對(duì)截?cái)嗪?會(huì)產(chǎn)生吉布斯現(xiàn)象,所有的窗函數(shù)選擇都以減少這種現(xiàn)象為目的。判斷較理想的窗函數(shù)主要根據(jù)以下三個(gè)標(biāo)準(zhǔn)!主瓣的幅度要高,且其寬度應(yīng)該盡量的窄。旁瓣的幅度下降速度快,最大旁瓣相對(duì)于主瓣應(yīng)該盡量小。#過(guò)渡帶要求要盡量窄。事實(shí)證明前面兩條標(biāo)準(zhǔn)不可能同時(shí)滿足,因此窗函數(shù)應(yīng)該是這兩條的折中1-3。為減少由于加窗截?cái)嘁鸬牟y和過(guò)渡帶變寬影響,在工程設(shè)計(jì)中常用hamming窗和kaiser窗。2 改進(jìn)的窗函數(shù)文獻(xiàn)1-2 列舉的窗函數(shù)中, hanning 窗、 hamming 窗、 blackman 窗是余弦序列與矩形序列的線性組合。為了抑制旁瓣的幅度, 在 hanning 窗、 hamming 窗的基礎(chǔ)上, 再增加余弦的二次諧波分量, 此時(shí)設(shè)計(jì)的窗函數(shù)又與理想的頻率響應(yīng)有關(guān), 又不同于blackman 窗,對(duì)窗函數(shù)改進(jìn)后的形式如下： (1)公式( 1)中的a, b, c 待定,其大小與給定的濾波器的技術(shù)指標(biāo)相關(guān)。為方便起見(jiàn),該窗函數(shù)長(zhǎng)度選擇為奇數(shù)。接下來(lái)探討該式的幾個(gè)特例。情況1,取a= 1, b= c= 0, 為矩形窗。情況2, 取a= 0.5, b= - 0.5, c= 0,為hann 窗。情況3, 取a=0.53, b= - 0.46, c = 0, 為 hamming 窗。情況 4,a= 0.42, b= - 0. 5, c= 0.08,為blackman 窗。由上述可知,改進(jìn)后的窗函數(shù)具備上述四種窗函數(shù)的性質(zhì),屬于上述窗函數(shù)的一般形式。3 改進(jìn)的窗函數(shù)算法根據(jù)給定濾波器的技術(shù)指標(biāo) , 確定待定濾波器的單位取樣響應(yīng), 可由下列公式求出: （2）計(jì)算實(shí)際濾波器的單位取樣響應(yīng): h( n) = （3）濾波器的頻率響應(yīng)為: （4）將公式( 1)代入公式( 4)得: (5)參考文獻(xiàn)1-2 10-11 ,考慮fir濾波器滿足第一類線性相位條件, 對(duì)偶對(duì)稱, 且 n 為奇數(shù), 令 h ( n) = 。則的幅度寫(xiě)成下式:將上式繼續(xù)化簡(jiǎn)得 1-2式中 (6)則式( 6) 可以寫(xiě)成 (7)式中 (8) (9) (10)如果濾波器的幅度條件為:通帶內(nèi)滿足, 阻帶內(nèi)滿足, 通帶內(nèi)可以列出如下方程: (11) (12)阻帶內(nèi)可列出方程: (13) (14) 為通帶波紋, 為阻帶波紋。為方便問(wèn)題的討論,這里取系數(shù) a, b, c 大于零作為約束條件。該問(wèn)題可以轉(zhuǎn)化為線性規(guī)劃問(wèn)題10-12, 公式(11) (12) (13) (14) 可以作為約束條件。經(jīng)此轉(zhuǎn)換后, 問(wèn)題求解變得方便了。公式(11) (13) 和公式(12) (14) 分別添加松弛變量、 懲罰變量變換成標(biāo)準(zhǔn)的線性規(guī)劃方程組12 : （15） （16） （17） （18）其中d, e, f , g 全大于0。解決這個(gè)問(wèn)題的方法可以用最小二乘法求解或迭代法求解。算出 a, b, c, 便可以確定窗函數(shù)。根據(jù)上述過(guò)程,這種濾波器的設(shè)計(jì)算法按照下列步驟:.在頻率域等間隔取 n 點(diǎn), 且其為奇數(shù), 對(duì)頻率域序列做離散傅里葉反變換,求出理想數(shù)字濾波器的單位取樣響應(yīng)。.由公式(8) (9) (10)分別求出、。.由公式(11) (12) (13) (14)列出優(yōu)化的約束條件。.解線性方程組,求出a, b, c 的最優(yōu)解, 從而確定數(shù)字濾波器 h(n)。.驗(yàn)算技術(shù)指標(biāo)是否滿足要求。4 應(yīng)用實(shí)例設(shè)計(jì)一低通數(shù)字濾波器, 其技術(shù)指標(biāo)如下:通帶截止頻率 , 阻帶截止頻率,通帶波紋,阻帶波紋。, 截止頻率為, blackman 窗的 fir 數(shù)字濾波器窗口長(zhǎng)度為, hamming 窗的長(zhǎng)度為 41。用 hamming 窗和 blackman 窗設(shè)計(jì)的 fir 濾波器的幅度特性如圖1。用改進(jìn)窗法設(shè)計(jì) fir 濾波器如圖2。圖 1 hamming窗和 blackman窗設(shè)計(jì)的幅度特性圖 2 改進(jìn)窗的幅度特性圖從圖1 和圖2 的濾波