滤波器外文翻译外文文献英文文献用改进的窗函数设计FIR数字滤波器.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.用改进的窗函数设计fir数字滤波器谭家杰, 罗昌由, 黄三伟, 邓小辉衡阳师范学院 物理与电子信息科学系, 湖南 衡阳 421008摘 要窗函数设计 fir 数字滤波器已多年, 近年来, 用优化技术设计数字滤波器十分流行, 论文提出了一种新方法对窗函数进行改进, 这种窗函数不同于以往的 hanning窗、hamming 窗、blackman 窗, 它先将余弦序列线性组合为窗函数, 然后根据 fir 数字滤波器的特性对数字滤波器的幅度条件进行线性规划。并且给出了改进窗函数的算法。最后, 用该方法设计出了 fir 数字滤波器的仿真实例, 并与用 hamming 窗、blackman 窗方法设计的 fir 滤波器进行对比, 仿真结果表明用该方法设计的滤波器满足设计规格。关键词改进的窗函数 fir 数字滤波器 窗函数 线性规划中图分类号: tn713+.7 文献标志码: a 文章编号: 1673-0313( 2010) 06-0031-040 引 言fir 数字滤波器的设计方法主要有:窗函数法、频率采样法和切比雪夫等波纹逼近法1-4。窗函数法是设计 fir 数字滤波器最常用、 最简单的方法4-5。窗函数法的实质是用截断理想冲激响应的方法来逼近所求的滤波器指标。频率采样法是一种优化设计方法,其缺点是设计时使用的变量仅限于过渡带上的几个采样值,截止频率不容易控制3。切比雪夫等波纹逼近法是一种优化设计, 但是存在计算复杂,计算量较大的缺点1-2。窗函数法设计简单, 有闭合形式的公式, 因而很实用。缺点是通带、 阻带的截止频率不容易控制2-3。数字滤波器的好坏取决于窗函数的选取, 窗函数法设计的关键是: 选择合适的窗函数, 选择合适的阶数,改善数字滤波器的幅频特性, 减少 gibbs现象,解决收敛问题1-2。文献3 选择 guass 窗函数,通过对 guass 窗函数改进, 设计出的低通滤波器具有更好的优越性; 文献4 利用已知的误差信息,在迭代过程中通过窗函数法不断修改设计结果, 在滤波器阶数不变的情况下, 滤波器的频率响应逼近理想频率响应。文献5 用整数序列,如 fibonacci序列、 golomb 序列、 hofstadter conway 序列、 recursive triangular 序列产生窗函数, 来设计滤波器, 其效果优于经典的设计方法。文献6 选择双窗结构得到的窗函数序列使系统特性逼近误差最小; 文献7 将 saramaki window 窗并且与dolph-chebysheve window 结合设计出的fir 数字滤波器, 其效果胜过 kaiser window ; 文献8 提出的reimann window 效果好于 hamming window和kaiser window 窗。文献9 提出一种指数型窗函数,该窗函数具有宽度可调特性, 用该窗设计的数字滤波器有主瓣能量更集中, 旁瓣更少的特点。文献10 采用线性规划法设计线性相位的升余弦数字滤波器能100 %超带宽。文献11则采用线性规划法设计数字滤波器。本文利用已有的窗函数,对其进行加权组合, 参照文献10-11 的线性规划法, 与同长的hamming、 blackman 窗进行比较。这种方法的优点是逻辑性强、 目标明确、 易于实现, 并能探索最佳方案。1 常用的窗函数窗函数选择原则:窗函数能量尽可能集中在主瓣内,过渡带陡峭;减少窗函数频谱的旁瓣高度,增大阻带衰减, 减小通带和阻带的波纹。常用的窗函数有1-4: rectangle窗、 hanning 窗、 hamming 窗、 blackman窗、 kaiser 窗。窗函数法设计 fir 滤波器的思路是1-2:先确定理想滤波器的频率响应, 实际设计滤波器的频率响应来逼近。再对进行反变换得最后用窗函数来截断 ,即h( n) = 。对截断后,会产生吉布斯现象,所有的窗函数选择都以减少这种现象为目的。判断较理想的窗函数主要根据以下三个标准!主瓣的幅度要高,且其宽度应该尽量的窄。旁瓣的幅度下降速度快,最大旁瓣相对于主瓣应该尽量小。#过渡带要求要尽量窄。事实证明前面两条标准不可能同时满足,因此窗函数应该是这两条的折中1-3。为减少由于加窗截断引起的波纹和过渡带变宽影响,在工程设计中常用hamming窗和kaiser窗。2 改进的窗函数文献1-2 列举的窗函数中, hanning 窗、 hamming 窗、 blackman 窗是余弦序列与矩形序列的线性组合。为了抑制旁瓣的幅度, 在 hanning 窗、 hamming 窗的基础上, 再增加余弦的二次谐波分量, 此时设计的窗函数又与理想的频率响应有关, 又不同于blackman 窗,对窗函数改进后的形式如下： (1)公式( 1)中的a, b, c 待定,其大小与给定的滤波器的技术指标相关。为方便起见,该窗函数长度选择为奇数。接下来探讨该式的几个特例。情况1,取a= 1, b= c= 0, 为矩形窗。情况2, 取a= 0.5, b= - 0.5, c= 0,为hann 窗。情况3, 取a=0.53, b= - 0.46, c = 0, 为 hamming 窗。情况 4,a= 0.42, b= - 0. 5, c= 0.08,为blackman 窗。由上述可知,改进后的窗函数具备上述四种窗函数的性质,属于上述窗函数的一般形式。3 改进的窗函数算法根据给定滤波器的技术指标 , 确定待定滤波器的单位取样响应, 可由下列公式求出: （2）计算实际滤波器的单位取样响应: h( n) = （3）滤波器的频率响应为: （4）将公式( 1)代入公式( 4)得: (5)参考文献1-2 10-11 ,考虑fir滤波器满足第一类线性相位条件, 对偶对称, 且 n 为奇数, 令 h ( n) = 。则的幅度写成下式:将上式继续化简得 1-2式中 (6)则式( 6) 可以写成 (7)式中 (8) (9) (10)如果滤波器的幅度条件为:通带内满足, 阻带内满足, 通带内可以列出如下方程: (11) (12)阻带内可列出方程: (13) (14) 为通带波纹, 为阻带波纹。为方便问题的讨论,这里取系数 a, b, c 大于零作为约束条件。该问题可以转化为线性规划问题10-12, 公式(11) (12) (13) (14) 可以作为约束条件。经此转换后, 问题求解变得方便了。公式(11) (13) 和公式(12) (14) 分别添加松弛变量、 惩罚变量变换成标准的线性规划方程组12 : （15） （16） （17） （18）其中d, e, f , g 全大于0。解决这个问题的方法可以用最小二乘法求解或迭代法求解。算出 a, b, c, 便可以确定窗函数。根据上述过程,这种滤波器的设计算法按照下列步骤:.在频率域等间隔取 n 点, 且其为奇数, 对频率域序列做离散傅里叶反变换,求出理想数字滤波器的单位取样响应。.由公式(8) (9) (10)分别求出、。.由公式(11) (12) (13) (14)列出优化的约束条件。.解线性方程组,求出a, b, c 的最优解, 从而确定数字滤波器 h(n)。.验算技术指标是否满足要求。4 应用实例设计一低通数字滤波器, 其技术指标如下:通带截止频率 , 阻带截止频率,通带波纹,阻带波纹。, 截止频率为, blackman 窗的 fir 数字滤波器窗口长度为, hamming 窗的长度为 41。用 hamming 窗和 blackman 窗设计的 fir 滤波器的幅度特性如图1。用改进窗法设计 fir 滤波器如图2。图 1 hamming窗和 blackman窗设计的幅度特性图 2 改进窗的幅度特性图从图1 和图2 的滤波