 2-H Relations among Filter Length N, Transition Band, and Accuracy

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1  2-H Relations among Filter Length N, Transition Band, and Accuracy
 Suppose that we want:  passband ripple  1,  stopband ripple  2,  width of transition band  F (expressed by normalized frequency) F = (f1  f2)/fs = (f1  f2)T (fs: sampling frequency, T: sampling interval) Then, the estimated length N of the digital filter is: . When there are two transition bands, F = min(F1, F2) 犧牲 transition band 的 frequency response, 換取較高的 passband and stopband accuracies

2 [Ref] F. Mintzer and L. Bede, “Practical design rules for optimum FIR bandpass digital filter”, IEEE Trans. ASSP, vol. 27, no. 2, pp , Apr 問題：假設 1 = 2 =  ，N 為固定， 當 F 變為 A 倍時，  變為多少？

3 N = 17, k+2=10 transition band: 0.24 ~ 0.26 N = 17,
boundary of transition band N = 17, transition band: 0.2 ~ 0.3

4  2-I Relations between Weighting Functions and Accuracy
If we treat the passband more important than the stop band W(f) = 1 in the passband, 0 < W(f) < 1 in the stopband If we treat the stop band more important than the pass band 0 < W(f) < 1 in the passband, W(f) = 1 in the stopband

5 Larger error near the transition band
Hd(F) H(F)

6  2-J FIR Filter in MSE Sense with Weighting Functions
可對照 pages 47~49 F = f / fs n = 0 ~ k when n   問題：

7 τ = 0 ~ k, n = 0 ~ k 可以表示成 (k+1)  (k+1) matrix operation B S = C S = B−1C

8 思考： Is it possible to apply the transition band to the FIR filter in the MSE sense?

9  2-K Four Types of FIR Filter
h[n] = 0 for n < 0 and n  N 點數為 N  Type 1 之前的方法只討論到Type 1 h[n1] = h[n2 − n] and N is odd. (even symmetric) k = (N1)/2

10  Type 1: h[n] = h[N−1− n] (even symmetric) and N is odd.  Type 2: h[n] = h[N−1− n] (even symmetric) and N is even.  Type 3: h[n] = −h[N−1− n] (odd symmetric) and N is odd.  Type 4: h[n] = −h[N−1− n] (odd symmetric) and N is even. k = (N−1)/2

11  Type 2: When h[n] = h[N−1− n] and N is even: (even symmetric)
令 r[n] = h[n + k], where k = (N1)/2 (注意此時 k 不為整數) (比較 page 41) h[0] h[1] h[k-3/2] h[k-1/2] h[k+1/2] h[k+3/2] h[N-2] h[N-1] r[-k] r[-k+1] r[-3/2] r[-1/2] r[1/2] r[3/2] r[k-1] r[k] s[1]/ s[2]/ s[k-1/2]/2 s[k+1/2]/2

12 當 n = 1, 2, …, k+1/2 設計出 s[n] 之後 r[n] = s[n+1/2]/2, h[n] = r[n − k],

13 Design Method for Type 2 由於 n 和 n+1 兩項相加可得 所以可以「判斷」 R(F) 能被改寫成 求 s1[n] 和 s[n] 之間的關係

14 (令 k1 + 1/2 = k) 比較係數可得 for n = 2, 3, …., k−1/2

15 只需將 pages 52-55 的方法當中，Hd(F) 換成
W(F) 換成 k 換成 k − 1/2 = N/2 − 1 注意s1[n] 和 s[n] 之間的關係即可

16 Design Method for Type 3 由於 n−1 和 n+1 兩項相減可得 所以「判斷」可將 R(F) 改寫為

17 令 k1 = k − 1, 比較係數可得 for n = 2, 3, …., k−2

18 將 page 的方法當中，Hd(F) 換成 W(F) 換成 k 換成 k − 1 注意s1[n] 和 s[n] 之間的關係即可 思考：Design Method for Type 4

19 附錄三：Matlab 寫程式需注意的地方 一、各種程式語言寫程式共通的原則 二、Matlab 寫程式特有的技巧
(1) 能夠不在迴圈內做的運算，則移到迴圈外 (2) 寫一部分即測試，不要全部寫完再測試 (縮小範圍比較容易 debug) (3) 先測試簡單的例子，成功後再測試複雜的例子 二、Matlab 寫程式特有的技巧 (1) 迴圈能避免就儘量避免 (2) 儘可能使用 Matrix 及 Vector operation Example: 由 1 加 到100，用 Matlab 一行就可以了 sum([1:100]) 完全不需迴圈

20 三、一些重要的 Matlab 指令 (1) function: 放在第一行，可以將整個程式函式化 (2) tic, toc: 計算時間
(3) find: 找尋一個 vector 當中不等於 0 的entry 的位置 範例：find([ ]) = [1, 4] find(abs([-5:5])<=2) = [4, 5, 6, 7, 8] (因為 abs([-5:5])<=2 = [ ]) (4)  : Hermitian (transpose + conjugation)， .  : transpose (5) imread: 讀圖， image, imshow, imagesc: 將圖顯示出來， (註： 較老的 Matlab 版本 imread 要和 double 並用 A=double(imread(‘Lena.bmp’)); (6) imwrite: 製做圖檔

21 (7) xlsread: 由 Excel 檔讀取資料
(8) xlswrite: 將資料寫成 Excel 檔 (9) aviread: 讀取 video 檔 (10) dlmread: 讀取 *.txt 或其他類型檔案的資料 (11) dlmwrite: 將資料寫成 *.txt 或其他類型檔案