【正文】 形狀。III型（M為偶數(shù)，h[k]奇對稱）線性相位FIR濾波器在M+1個取樣點值為 (316) 上式表明III型濾波器線性相位FIR濾波器在的值可由在的值確定。在解決問題過程中使用了數(shù)學優(yōu)化中的Remez交換算法。 程序中fir1函數(shù)的用法：b=fir1(n,Wn,’ftype’,window)①n為濾波器的階數(shù)②Wn為濾波器的截止頻率，它是一個0到1的數(shù)。圖423 信號濾波前的時域圖和頻域圖圖424 信號濾波后的時域圖和頻域圖從圖423和圖424的圖像中可以看到：輸入信號是由四個不同頻率的正弦信號疊加而成，信號頻域圖中位于濾波器通帶內(nèi)的頻率分量保留了下來，位于濾波器阻帶內(nèi)的頻率分量被濾除，濾波器的效果符合設計要求。圖411 信號濾波前的時域圖和頻域圖圖412 信號濾波后的時域圖和頻域圖從圖411和圖412的圖像中可以看到：輸入信號是由三個不同頻率的正弦信號疊加而成，信號頻域圖中位于濾波器通帶內(nèi)的頻率分量保留了下來，位于濾波器阻帶內(nèi)的頻率分量被濾除，濾波器的效果符合設計要求。（6）如果，執(zhí)行（7）。表示設計出的線性相位FIR濾波器的幅度函數(shù)。現(xiàn)要求設計一個M階的FIR濾波器h[k]，使得在M+1個取樣點上，F(xiàn)IR濾波器的頻率響應與所需的頻率響應相等，即 (312)由設計的要求給定，h[k]需要通過設計來確定。（1） 矩形窗 (35) 矩形窗的主瓣寬度為。（3）建模過程中忽略了部分次要因素，使得模型仿真結果偏離實際系統(tǒng)。而IIR濾波器在運算中會產(chǎn)生寄生振蕩。實現(xiàn)數(shù)字濾波器的方法一般有兩種：一種是利用計算機的程序編譯，從而仿真實現(xiàn)；另一種是利用硬件來實現(xiàn)。它可以對輸入的離散信號進行一系列運算處理，從輸入的信號中獲得所需要的信息。四種線性相位FIR濾波器的性質(zhì)如表11所示表11 四種線性相位FIR濾波器的性質(zhì)類型IIIIIIIV階數(shù)M偶數(shù)奇數(shù)偶數(shù)奇數(shù)h[k]的對稱性偶對稱偶對稱奇對稱奇對稱關于的對稱性偶對稱偶對稱奇對稱奇對稱關于的對稱性偶對稱奇對稱奇對稱偶對稱的周期00A(0)任意任意00任意00任意可適用的濾波器類型LP,HP,BP,SPLP,BP微分器，變換器，Hilbert微分器，變換器，Hilbert，HP IIR與FIR數(shù)字濾波器的比較(1)在技術指標相同的條件下，IIR濾波器的輸出對輸入有反饋，所以可以用比FIR少的階數(shù)來滿足要求，存儲單元少，運算次數(shù)也少，經(jīng)濟實惠。（3）在現(xiàn)代通信系統(tǒng)協(xié)議的性能研究中，直接試驗幾乎是不可能的，在這種情況下只能通過仿真數(shù)據(jù)來檢驗所選用的對象，以驗證有關的假設。隨著濾波器階數(shù)的增加，幅度函數(shù)在通帶和阻帶振蕩的波紋數(shù)量也隨之增加，波紋的寬度隨之減小，然而通帶和阻帶最大波紋的幅度與濾波器的階數(shù)M無關。隨著參數(shù)的增加，Kaiser窗在兩端的衰減是逐漸加大的。但由于Gibbs現(xiàn)象的存在，使得設計出濾波器在阻帶的衰減一般不能滿足要求。為了減少計算誤差，抽樣間隔應足夠小。 (3) 利用窗函數(shù)法設計多通帶濾波器設計要求：①使用Kaiser窗，采樣頻率200Hz②、③阻帶衰減大于等于30dB，程序參見附錄二中的1（3）利用窗函數(shù)法設計多通帶濾波器圖47 窗函數(shù)法設計多通帶濾波器的增益響應從參考程序及圖47可以得到所設計出濾波器的參數(shù)如下：①濾波器的采樣頻率為200Hz，濾波器的階數(shù)為46②、 、③阻帶衰減為38dB，對比設計要求與所設計出濾波器的參數(shù)可知，其各項參數(shù)均滿足設計指標，所設計出的濾波器即為設計所要求的濾波器。對比figure（1）和figure（4）濾波前后的波形和頻譜，可以看到波形得到了重現(xiàn)②濾波器的采樣頻率為22050Hz，濾波器的階數(shù)為24③，④阻帶衰減為40dB，參考文獻[1]，2006[2]徐明遠，[3]鄒鯤，袁俊泉， [4]張明照，劉政波，[5]劉波，[6]William , Shanmugan,Theodore ,Kurt ，楊光松，許芳，[7][8]Oppenheim A V,Schafer R Signal Processing. Prentice Hall,[9][10]蘇金明，張蓮花，附錄附錄一 外文原文及翻譯外文原文FIR Filter Design TechniquesAbstractThis report deals with some of the techniques used to design FIR filters. In the beginning, the windowing method and the frequency sampling methods are discussed in detail with their merits and demerits. Different optimization techniques involved in FIR filter design are also covered, including Rabiner’s method for FIR filter design. These optimization techniques reduce the error caused by frequency sampling technique at the nonsampled frequency points. A brief discussion of some techniques used by filter design packages like Matlab are also included. Introduction FIR filters are filters having a transfer function of a polynomial in z and is an allzero filter in the sense that the zeroes in the zplane determine the frequency response magnitude z transform of a Npoint FIR filter is given by (1)FIR filters are particularly useful for applications where exact linear phase response is required. The FIR filter is generally implemented in a nonrecursive way which guarantees a stable filter. FIR filter design essentially consists of two parts (i) approximation problem (ii) realization problem The approximation stage takes the specification and gives a transfer function through four steps. They are as follows:(i) A desired or ideal response is chosen, usually in the frequency domain. (ii) An allowed class of filters is chosen ( length N for a FIR filters). (iii) A measure of the quality of approximation is chosen. (iv) A method or algorithm is selected to find the best filter transfer function. The realization part deals with choosing the structure to implement the transfer function which may be in the form of circuit diagram or in the form of a program.There are essentially three wellknown methods for FIR filter design namely: (1) The window method (2) The frequency sampling technique (3) Optimal filter design methods The Window Method In this method, [Park87], [Rab75], [Proakis00] from the desired frequency response specification Hd(w), corresponding unit sample response hd(n) is determined using the following relation (2) (3)In general, unit sample response hd(n) obtained from the above relation is infinite in duration, so it must be truncated at some point say n= M1 to yield an FIR filter of length M (. 0 to M1). This truncation of hd(n) to length M1 is same as multiplying hd(n) by the rectangular window defined as w(n) = 1 0≦n≦M1 (4)0 otherwiseThus the unit sample response of the FIR filter bees h(n) = hd(n) w(n) (5) = hd(n) 0≦n≦M1= 0 otherwise Now, the multiplication of the window function w(n) with hd(n) is equivalent to convolution of Hd(w) with W(w), where W(w) is the frequency domain representation of the window function (6)Thus the convolution of Hd(w) with W(w) yields the frequency response of the truncated FIR filter (7)The frequency response can also be obtained using the following relation (8)But direct truncation of hd(n) to M terms to obtain h(n) leads to the Gibbs phenomenon effect which manifests itself as a fixed percentage overshoot and ripple before and after an approximated discontinuity in the frequency response due to the nonuniform convergence of the fourier series at a the frequency response obtained by using (8) contains ripples in the frequency domain. In order to reduce the ripples, instead of multiplying hd(n) with a rectangular window w(n), hd(n) is multiplied with a window function that contains a taper and decays toward zero gradually, instead of abruptly as it occurs in a rectangular window. As multiplication of sequences hd(n) and w(n) in ti