濾波器外文翻譯 -----用改進(jìn)的窗函數(shù)設(shè)計(jì)fir數(shù)字濾波器

1、<p>　　Research on FIR Digital Filter Design Using an Improved Window Function</p><p>　　TAN Jiajie , LUO Changyou, HUANG Sanwei ， DENG Xiaohui </p><p>　　( Department of Physics and Electroni

2、c Information Science, Hengyang Normal University, Hengyang Hunan 421008, China)</p><p>　　Abstract : Window function has been used to design a linear phase digital filer for long times, but the use o f optim

3、ization 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 sequenc

4、es in linear is different from previous Hann ,Hamming and Blackman window function The paper also proposes linear programming to optimize characterization of FIR </p><p>　　Key words: improved window functio

5、n; FIR digital filter ; window function; linear programming</p><p>　　0 lead speech</p><p>　　The design method of FIR digital filters are mainly: window function method, frequency sampling method

6、 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 tru

7、ncated 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 t</p><p>　　Window function met

8、hod 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

9、 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 proble

10、m [1-2]. [3] choose window function, thro</p><p>　　1 common window function</p><p>　　Window function select principle: window function as focus on energy, Lord disc transitional steep; Reduce th

11、e 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, Blackma

12、n 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 </p><p>　　2 improved window function

13、 [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

14、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

15、 (1)</p><p>　　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 s

16、pecial 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. Situati

17、on, a = 4, b = 0 0.42 j c = 0.08, 5, Blackman window for. By above knowable, the improved window function wi</p><p>　　3 improved window function algorithm</p><p>　　According to the given filter

18、technology index,Determine the backlog filter unit, but by sampling response formula below ask out: (2)

19、 </p><p>　　Calculating the actual filter unit sampling response:</p><p>　　h( n) = (3)</p><p>　　Filter the frequency response is:</p>

20、<p><b>　　（4）</b></p><p>　　Will formula (1) generation into the formula (4) :</p><p><b>　　(5)</b></p><p>　　Reference [1-2] [10 or 11], consider FIR filt

21、ers 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: written</p><p>　　Will continue to have the type

22、 of Jane [1-2]</p><p><b>　　In form </b></p><p><b>　　(6)</b></p><p>　　Then type (6) can be encoded</p><p><b>　　(7)</b></p><

23、;p><b>　　In form</b></p><p><b>　　(8)</b></p><p><b>　　(9)</b></p><p><b>　　(10)</b></p><p>　　If filter conditions for th

24、e amplitude, the pass-band with satisfaction,Stopband inside meet,The pass-band with can list below equation:</p><p><b>　　(11)</b></p><p><b>　　(12)</b></p><p&g

25、t;　　Stopband inside can list equation:</p><p><b>　　(13)</b></p><p><b>　　(14)</b></p><p>　　For bandpass corrugated, For stopband corrugated. For the convenien

26、ce 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 af

27、ter 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] :</p>

28、;<p><b>　?。?5）</b></p><p><b>　?。?6）</b></p><p><b>　　（17）</b></p><p><b>　?。?8）</b></p><p>　　Including d, e, f, g,

29、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 filt

30、er algorithm according to the following steps:</p><p>　　Ⅰ etc. In frequency domain, and the interval take N points for the odd number of frequency domain, the sequence do discrete Fourier inverse transform,

31、digital filters out the ideal sampling response unitsⅡ. By formula (8) (9) (10) ask out respectively、、。Ⅲ. From the formula (11), (12) (13) (14) list optimization constraints. Ⅳ. Solving linear equations, find out a, b, c

32、, so as to determine the optimal solution digital filter h(n)。Ⅴ. Checking technology index whether meet the requirements.</p><p>　　4 application examples</p><p>　　Design one low tong digital fil

33、ters, its technical index is as follows: Bandpass cut-off frequency , Stopband cut-off frequency , passband ripple ,Stopband corrugated .</p><p>　　,Cut-off frequency for,Blackman window FIR digital filters f

34、or 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

35、 shown in figure 2.</p><p>　　Figure 1 Hamming window and Blackman window design range characteristics</p><p>　　Figure 2 improvement window range characteristics graph</p><p>　　In fi

36、gure 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 str

37、ictly 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 req

38、uirements consistent, But the Hamming window and Blackm</p><p>　　5 conclusion</p><p>　　FIR digital filters with window function design method used, this paper adopts improved, and window functio

39、n 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

40、 window function, they have their respective comparative advantages and disadvantages of window functions, thus improving design FIR filter is feasible.</p><p>　　References:</p><p>　　[1] ChengPe

41、iQing. Digital signal processing tutorial 3 edition [j]. Beijing: tsinghua university press, 2001:1395-368.</p><p>　　[2] DingYuMei. Digital signal processing, GaoXiQuan [j] 2 edition. Xian: xian university o

42、f electronic science and technology press, 2001:195-15.</p><p>　　[3] ChenMingJun, MaoZhangMei. Improve window function in the application of FIR digital filters design [J].journal of relays, 2007,35 (17) : 6

43、5-67</p><p>　　[4] 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

44、, 2008,23 (2) : 228-232.</p><p>　　[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 Te

45、chnology , 2008: 1184-1185.</p><p>　　[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

46、,20 (6) : 273-280-1315.</p><p>　　[7] Saramaki, T. Adjustable window s for the design of FIR filters ，a tutorial: Electro technical Conference,1991 [C][s l] : Proceeding s,6th Mediterranean,1991:28-33.</p&

47、gt;<p>　　[8] Nihal L. Hettiarachchi , Adel A. Sakla. Design of digital FIR filter s via optimized generalized Reimann window function: Circuits and Systems, 1995[C].[s l ]:Proceeding s of the 37th Midwest Sympos

48、ium,1995, 2:1061-1064.</p><p>　　[9] Kemal Avci, Arf Nacaroglu. A New Window Based on Exponential Function[J] .IEEE Trans on signal processing , 2008, 41(12):60-72.</p><p>　　[10] L. F. Lind, B. M

49、.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.</p><p>　　[11] Lawrence R. R

50、abiner . Linear Pro gram Design of Finite Impulse Response Digital Filters [J].IEEE Transactions on audio and electracoustics, 1972, 20 (4):280-288.</p><p>　　[12] the optimization principle and method. Franc

51、i c [M].beijing: Beijing industrial university press, 2004:36-92.</p><p>　　用改進(jìn)的窗函數(shù)設(shè)計(jì)FIR數(shù)字濾波器</p><p>　　譚家杰, 羅昌由, 黃三偉, 鄧小輝</p><p>　　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</p><p>

52、;<b>　　摘 要</b></p><p>　　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

53、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。</p><p><b>　　關(guān)鍵詞</b></p><p>　　改進(jìn)的窗函數(shù) FIR 數(shù)字濾波器 窗函數(shù) 線性規(guī)劃</p><p>　　中圖分類

54、號(hào): TN713+.7 文獻(xiàn)標(biāo)志碼: A 文章編號(hào): 1673-0313( 2010) 06-0031-04</p><p><b>　　0 引 言</b></p><p>　　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

55、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]。</p><p>　　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

56、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,解決收斂問題[1-2]。文獻(xiàn)[3] 選擇 Guass 窗函數(shù),通過對(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,在迭代過程中通過窗函數(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、 Hofst

57、adter 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 Kaiser window ; 文獻(xiàn)[8</p><p>　　1

58、 常用的窗函數(shù)</p><p>　　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

59、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和過渡帶變寬影響,在工程設(shè)計(jì)中常用Hamming窗和Kaiser窗

60、。</p><p>　　2 改進(jìn)的窗函數(shù)</p><p>　　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

61、式如下：</p><p><b>　　(1)</b></p><p>　　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,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= -

62、 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。</p><p>　　3 改進(jìn)的窗函數(shù)算法</p><p>　　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:

63、 （2）計(jì)算實(shí)際濾波器的單位取樣響應(yīng): </p><p>　　h( n) = （3）</p><p>　　濾波器的頻率響應(yīng)為: </p><p><b>　?。?）</b></p&

64、gt;<p>　　將公式( 1)代入公式( 4)得:</p><p><b>　　(5)</b></p><p>　　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) = 。則的幅度寫成下式:</p><p>　　將上式繼續(xù)化簡(jiǎn)得[ 1-2]</p

65、><p><b>　　式中 </b></p><p><b>　　(6)</b></p><p>　　則式( 6) 可以寫成</p><p><b>　　(7)</b></p><p><b>　　式中</b></p>&

66、lt;p><b>　　(8)</b></p><p><b>　　(9)</b></p><p><b>　　(10)</b></p><p>　　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:</p><p><b>　　(11)

67、</b></p><p><b>　　(12)</b></p><p><b>　　阻帶內(nèi)可列出方程:</b></p><p><b>　　(13)</b></p><p><b>　　(14)</b></p><p>

68、　　為通帶波紋, 為阻帶波紋。為方便問題的討論,這里取系數(shù) a, b, c 大于零作為約束條件。該問題可以轉(zhuǎn)化為線性規(guī)劃問題[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] :</p><p><b>　?。?5）</

69、b></p><p><b>　?。?6）</b></p><p><b>　?。?7）</b></p><p><b>　?。?8）</b></p><p>　　其中d, e, f , g 全大于0。解決這個(gè)問題的方法可以用最小二乘法求解或迭代法求解。算出 a, b,

70、c, 便可以確定窗函數(shù)。根據(jù)上述過程,這種濾波器的設(shè)計(jì)算法按照下列步驟:</p><p>　?、?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)。</p><p>　?、?由公式(8) (9) (10)分別求出、、。</p><p>　　Ⅲ.由公式(11) (12) (13) (14)列出優(yōu)化的約束條件。&

71、lt;/p><p>　　Ⅳ.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)。</p><p>　?、?驗(yàn)算技術(shù)指標(biāo)是否滿足要求。</p><p><b>　　4 應(yīng)用實(shí)例</b></p><p>　　設(shè)計(jì)一低通數(shù)字濾波器, 其技術(shù)指標(biāo)如下:</p><p>　　通帶截止

72、頻率 , 阻帶截止頻率,通帶波紋,阻帶波紋。</p><p>　　, 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。</p><p>　　圖 1 Hamming窗和 Blackman窗設(shè)計(jì)的幅度特性</p

73、><p>　　圖 2 改進(jìn)窗的幅度特性圖</p><p>　　從圖1 和圖2 的濾波器的頻率特性來(lái)看, 改進(jìn)窗的函數(shù)通帶都是平坦, 阻帶都有波紋; 改進(jìn)窗設(shè)計(jì)的通帶截止頻率嚴(yán)格符合設(shè)計(jì)要求, 而傳統(tǒng)窗法截止頻率較高。不同處在于過渡帶, 改進(jìn)窗函數(shù)的過渡帶為 0.2- 0.4, 嚴(yán)格與設(shè)計(jì)要求一致, 而Hamming 窗和 Blackman 窗的過渡帶為 0.25-0.4, 改進(jìn)的窗函數(shù)算法略輸

74、于傳統(tǒng)的窗函數(shù)。在阻帶衰減方面,改進(jìn)窗函數(shù)法優(yōu)于 Hamming 窗法,而弱于Blackman 窗法。</p><p><b>　　5 結(jié)論</b></p><p>　　FIR 數(shù)字濾波器常用加窗函數(shù)設(shè)計(jì)方法, 文章采用改進(jìn)的窗函數(shù)設(shè)計(jì)方法, 并用線性規(guī)劃法設(shè)計(jì)的FIR 數(shù)字濾波器, 其算法思路明確, 設(shè)計(jì)的效果較好。用改進(jìn)窗設(shè)計(jì)的FIR 濾波器嚴(yán)格滿足設(shè)計(jì)技術(shù)指

75、標(biāo),與傳統(tǒng)的窗函數(shù)比較, 它們具有各自的優(yōu)勢(shì)、劣勢(shì),因而改進(jìn)窗函數(shù)設(shè)計(jì)FIR 濾波器可行。</p><p><b>　　6 參考文獻(xiàn):</b></p><p>　　[1] 程佩青.數(shù)字信號(hào)處理教程[M].3 版.北京:清華大學(xué)出版社,2001:333 -368.</p><p>　　[2] 丁玉美,高西全.數(shù)字信號(hào)處理[M].2版.西安:西

76、安電子科技大學(xué)出版社,2001: 195-15.</p><p>　　[3] 陳明軍,毛樟梅.改進(jìn)窗函數(shù)在FIR數(shù)字濾波器設(shè)計(jì)中的應(yīng)用[J].繼電器,2007,35(17):65-67</p><p>　　[4] 田廣新,高立志,孫春來(lái),等.基于窗函數(shù)法的非選頻型FIR 數(shù)字濾波器的優(yōu)化設(shè)計(jì)[J].數(shù)據(jù)采集與處理,2008,23(2):228-232.</p><p&g

77、t;　　[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.</p><

78、p>　　[6] 黃曉紅,蘇飛,王兆華.基于單窗全相位數(shù)字濾波器和LMS 準(zhǔn)則的窗函數(shù)設(shè)計(jì)[J].傳感技術(shù)學(xué)報(bào),2007,20(6):1312-1315.</p><p>　　[7] Saramaki, T. Adjustable window s for the design of FIR filters ，a tutorial: Electro technical Conference,1991 [C][

79、s l] : Proceeding s,6th Mediterranean,1991:28-33.</p><p>　　[8] Nihal L. Hettiarachchi , Adel A. Sakla. Design of digital FIR filter s via optimized generalized Reimann window function: Circuits and Systems

80、, 1995[C].[s l ]:Proceeding s of the 37th Midwest Symposium,1995, 2:1061-1064.</p><p>　　[9] Kemal Avci, Arf Nacaroglu. A New Window Based on Exponential Function[J] .IEEE Trans on signal processing , 2008, 4

81、1(12):60-72.</p><p>　　[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

82、,32(5):436-437.</p><p>　　[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.</p><