2digitalfilterdesign(a)

2digitalfilterdesign(a)内容摘要：

d filter: Hd(F) is an even function (F is the normalized frequency)  The weighting function is W(F) 2 個限制條件 52 11 1 1[ ( ) ( ) ] ( ) ( 1 ) kk d k kR F H F W F e     0 0 0[ ( ) ( ) ] ( )dR F H F W F e  1 1 1[ ( ( ) ] ( )dR F H F W F e2 2 2[ ( ) ( ) ] ( )dR F H F W F e  3 3 3[ ( ) ( ) ] ( )dR F H F W F e: (Step 1): Choose arbitrary k+2 extreme frequencies in the range of 0  F  , (denoted by F0, F1, F2, ….., Fk+1) Note: (1) Exclude the transition band. (2) The extreme points cannot be all in the stop band. Set E1 (error)   Extreme frequencies: the locations where the error is maximal. (參考 page 56) 53 (Step 2): From page 42, [R(Fm)  Hd(Fm)]W(Fm) = (1)m+1e (where m = 0, 1, 2, ….. , k+1) can be written as where m = 0, 1, 2, ….. , k+1. Expressed by the matrix form: Solve s[0], s[1], s[2], ….. , s[k] from the above matrix (performing the matrix inversion).        10[ ] c os 2 1k mm m d mns n F n W F e H F   0 0 0 01 1 1 12 2 2 21 1 11 c os( 2 ) c os( 4 ) c os( 2 ) 1 / ( )1 c os( 2 ) c os( 4 ) c os( 2 ) 1 / ( )1 c os( 2 ) c os( 4 ) c os( 2 ) 1 / ( )1 c os( 2 ) c os( 4 ) c os( 2 ) ( 1 ) / ( )1 c os( 2 ) c os( 4 ) c os( 2 ) ( 1 )kk k k kkk k kF F k F W FF F k F W FF F k F W FF F k F W FF F k F            012111[][ 0][][ 1 ][][ 2][][]/ ( ) [ ]ddddkk d kHFsHFsHFsHFskW F H Fe                           Square matrix 54 (Step 3): Compute err(F) for 0  F  , exclude the transition band. (Step 4): Find k+2 local maximal (or minimal) points of err(F) local maximal point: if q() q( + ) and q() q(  ), then  is a local maximal of q(x). local minimal point: if q() q( + ) and q() q(  ), then  is a local minimal of q(x). Other rules: Page 58 Denote the local maximal (or minimal) points by 0 1 1, , .. .. .. , ,kkP P P P      0( ) [ ( ) ( ) ] ( ) { [ ] c os 2 }kddne rr F R F H F W F s n n F H F W F   These k+2 extreme points could include the boundary points of the transition band 55 (Step 5): Set E0 = Max(|err(F)|). (Case a) If E1  E0 , or E1  E0 0 (or the first iteration)  set Fn = Pn and E1 = E0, return to Step 2. (Case b) If 0  E1  E0    continue to Step 6. (Step 6): Set h[k] = s[0], h[k+n] = s[n]/2, h[k  n] = s[n]/2 for n = 1, 2, 3, …., k Then h[n] is the impulse response of the designed filter. 01: | ( ) |: ite r a tio n | ( ) |E M a x e r r FE M a x e r r F現在的前一次 的用 MiniMax方法所設計出的 filters， 一定 會滿足以下二個條件 (1) 在 F  [0, ] 的地方有 k+2個以上 的 extreme points (2) 在 extreme points上， 是 定值 56 ( ) | ( ) ( ) |m m d mW F R F H F( ) 1mWF  的情形 T. W. Parks and J. H. McClellan, “Chebychev approximation for nonrecursive digital filterlinear phase”, IEEE Trans. Circuit Theory, vol. 19, no. 2, pp. 189194, March 1972. 證明可參考  2F MiniMax FIR Filter 設計時需注意的地方 57 (1) Extreme points 不要選在 transition band I。