第五章数字滤波器FIR数字滤波器频率取样设计法
第五章 数字滤波器 FIR数字滤波器 频率取样设计法
设计原理H(k) = Ha(k) = Ha(z)2=e()= Ha(e"H(k)NN-IZV内插公式H(z)WNz-1N-k=0N-1H(z)= H (k)厘k(2)k=01-≥-N1Φ(2)N 1- W-kz-1NZ
( ) ( ) 2 2 1 1 0 1 0 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j k d d z e N j d k N N N k k N N k k N k k N H k H k H z H e H k z H z N W z H z H k z z z N W z p w p w F F = = - - - - = - = - - - = = = - = - ìï = í ï - = - î å å 设计原理 内插公式
设计原理NwsinN元A21jweekN元kwZsin2N)=元1H(k)雪k2元k=0Np-iNwNw中sinNiw-h21APNS11N-17jwH6AenNk=0NwsinN-122ekHN元kwk=0sin2N
( ) ( ) ( ) ( ) 1 0 2 1 1 1 ( ) j N j j k k jN j k j k N z e H e H k e e e N e w w w w w p w F F - = - - - = ìï = í ï - = î - å 设计原理 ( ) ( ) ( ) ( ) ( ) ( ) sin sin sin sin N k j j N k j k Nj N j k Nj N e e N k N e k N N e N H e H k k N e w w p w w w w w w p p w w w w p w F F F F F - - + ç - ÷ - - = ç - ÷ - ç ÷ = ç - ÷ = ç - ÷ ç ÷ Þ = ç ÷ Þ = ç - ÷ = å 2 2 1 2 1 0 1 2 1 2 2 2 1 2 2 2 ( ) sin sin N k j N k N H k e N k N w p w w p - - = ç ÷ - ÷ ç å 1 0 2 2
NwsinTKN222eAHH0NN元kwk=0k=0sin2NNwsin元k22NHs(w,k>eN元kwk=0S112N内插函数H(k)2元/N抽样点上，频率响应严格相等抽样点之间，加权内插函数的延伸叠加变化越平缓，内插越接近理想值，逼近误差较小
( ) j Hd e w ( ) H j e w H(k) 2p/N • 抽样点上，频率响应严格相等 • 抽样点之间，加权内插函数的延伸叠加 • 变化越平缓，内插越接近理想值，逼近误差较小 ( ) ( ) ( ) ( ) ( ) sin sin sin , sin Nj N N k j j N k k k N j j N j N e H e H k k H k e N N k N N s k e H e e N k N w p w p w w p w w p w w w p F ç - ÷ - - - - = = ç - ÷ - - ç ÷ = ç - ÷ = - ÷ ç ç ÷ = Þ = ç - ÷ 1 1 2 1 0 0 1 2 2 2 2 1 2 2 ( ) ( , ) N k H k s k w w - ÷ =å 1 0 内插函数
线性相位约束条件H(w)对第一类线性相位滤波器，h(n)为偶对称，N为奇数H(e ) = H(w)e j0(u)N-1H(w) = a(n)cos [wn]2元w0(w)H(w) = H(2元 -w)2p(w)2元元wH(k) = H(ek = 0,1,2,..",N-12TKNj0 :H(k) =H(N-1)元H(e")10=w)IH,(e)i1H(k)2NN1412H(w)=H(2元 -w)H=Hn-k-N为偶数：H,=-Hn-k
H(w) p 2p w w p 2p (w) -(N-1)p H(w) = H(2p-w) [ ] 1 2 0 2 2 1 2 0 1 2 1 1 2 2 2 ( ) ( ) ( ) ( ) ( )cos ( ) ( ) ( ) , , , , ( ) ( ) ( ) ( ) k j j N n j k N j k k k N k N k k N k H e H e H a n n N H k H e k N H k H e N k N H H H H H H w q w w p w q p w w w w q w w p q q w w p w - = = = - - = ìï = í ï - = -ç ÷ î = = - ìï = í ï - = = -ç ÷ ï ç ÷ î = - Þ = = - å L 对第一类线性相位滤波器，h(n)为偶对称，N为奇数 线性相位约束条件 N为偶数： 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 | H(e j ) | | Hd (e j ) | | H(k) |