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) |
A summary:线性相位FR滤波器的四种情况h(n) = h(N-n-1)h(n)奇数NH(w)=a(n)cos(wn)中(w)=-w(N-1)/21）时域：a(n)Φ(w)1)2元1-2元7a(0) = h[(N -1) /2)h(n) = ±h(N - n -1)(N-1)/2a(n)= 2h[(N -1)/2-n),n =1 ~(N -I)/2偶数N2频域：H(w)=b(n)cos[w(n-1 / 2))h(n)N-12(N-1)元b(n)H(ew)= H(w)eH(2) = (-1)° 2-(N-1) H(z-1)b(0) = 0TN/2b(n)=2h[N/2-n],n=1~N /2点线h(n) = -h(N-n-1)性相位FIR数字滤波器特Hw）为实丽数1/奇数Nh(n)H(w)=c(n)sin(wn)n)偶对称：L=0中(w)=亦/2-w(N-1)/2hn）奇对称：L=1Φ(w)c(n)3)元/2c(0) = 03零点：2元T(N-1)/2c(n) = 2h[(N - 1)/2-n),n = 1 ~(N-I)/201偶数N成倒易对出现H(w) =d(n)sin[w(n - 1 / 2)]h(n)(N3/2)元d(n)4)元2d(0) = 0IN/2d(n)=2h[N/2-nl,n=1~N/2
A summary: ① 时域： ② 频域： ③ 零点： h(n) = ±h(N -n -1) 1 2 2 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) L N j j L N H e H e H z z H z p w w w ç - ÷ - - - - = = - 为实函数 偶对称：L = 0 奇对称：L = 1 线性相位FIR滤波器的四种情况 h(n) = h(N-n-1) h(n) = -h(N-n-1) 1) 2) 3) 4) ()=-(N-1)/2 ()=/2-(N-1)/2 2 0 -(N-1) 2 0 -(N-3/2) () () /2 ( ) [( )/ ] ( ) [( )/ ], ( )/ a h N a n h N n n N = - = - - = - 0 1 2 2 1 2 1 1 2 ( ) ( ) [ / ], / b b n h N n n N = = - = 0 0 2 2 1 2 ( ) ( ) [( )/ ], ( )/ c c n h N n n N = = - - = - 0 0 2 1 2 1 1 2 ( ) ( ) [ / ], / d d n h N n n N = = - = 0 0 2 2 1 2 h(n) a(n) h(n) b(n) (N-1)/2 N/2 h(n) c(n) (N-1)/2 h(n) d(n) N/2 p 2p w p 2p w p w 2p p w 2p 1 2 0 ( )/ ( ) ( )sin( ) N n H w c n wn - = = å 2 0 1 2 / ( ) ( )sin[ ( / )] N n H w d n w n = = å - 1 2 0 ( )/ ( ) ( )cos( ) N n H w a n wn - = = å 2 0 1 2 / ( ) ( )cos[ ( / )] N n H w b n w n = = å - 奇数N 奇数N 偶数N 偶数N
频率抽样两种方法N=8:偶数jIm[z]jIm[z]N=8:偶数2元NRe[z]Re[z]N元N=9:奇数jIm[z]jIm[z]N=9:奇数Re[z]Re[z]It maybe closertoacertain edge frequencyII型型
2 N -1 1 Re[z] N=8:偶数 jIm[z] 2 N -1 1 Re[z] N=9 jIm[z] :奇数 2 N -1 1 Re[z] N=8:偶数 jIm[z] N 2 N -1 1 Re[z] N=9:奇数 jIm[z] N It may be closer to a certain edge frequency 频率抽样两种方法
1）第一种频率抽样H(k)= H.(k)= Hdk = 0.1.....N - 122元k27N+jIm[z]N-8:偶数NN-IH(k)1-Z系统函数：TH(z)NWk-fwN+Re[Z]sinTR2N122频率响应：HkHeCeNkwk=0sin2N2）第二种频率抽样H(k)k = 0,1,...,N - 1=HHQ2元k+NN jIm[z]N=8属数N-1H-N1+z系统函数：H(z) =DN27+k=0pp.245eZwNRe[z]+1cOSN-1H2N-Ike1频率响应：H2>eN1k=0w元kjsin22N
1）第一种频率抽样 系统函数： 频率响应： 2）第二种频率抽样 系统函数： 频率响应： -1 2 N -1 1 Re[z] jIm[z] N=8:偶数 2 N 1 Re[z] jIm[z] N=8:偶数 N ( ) ( ) ( ) ( ) , ,., j k N j d d d k z e N H k H k H z p H e k N w p w= = = = 2 = 2 = 0 1 - 1 1 1 0 1 1 ( ) ( ) N N k k N z H k H z N W z - - - - = - = å - ( ) ( ) 1 1 2 0 1 2 2 sin sin N N k j j j N k N H e e H k e N k N p w w w w p - ç - ÷ - - = ç ÷ = ç - ÷ å ( ) ( ) ( ) ( ) , ,., j k N N j d d k z e N N H k H z p p H e k N w p p w + = + = = 2 = 2 = 0 1 - 1 ( ) 1 2 1 0 2 1 1 1 ( ) N N j k k N z H k H z N e z p - - ç ÷ = + - + = - å ( ) ( ) 1 2 1 1 2 0 2 1 2 2 cos sin j k N N j N j k N H k e H e e N j k N p w w w w p ç ÷ - + - ç - ÷ - = ç ÷ = - ç + ÷ å
线性相位约束条件第一种抽样方法h(n)中心奇对称h(n)中心偶对称h(n)中心偶对称h(n)中心奇对称N为奇数N为奇数N为偶数N为偶数IH(k)|=|H(N-)IH()|=[H(N-)[H(k) /=|H(N-k) [[H(k) /=|H(N-k)|幅度k = 0~(N /2-1)k=0~(N-1)/2k = 0~(N/2- 1)k = 0 ~ (N -1)/2约東IH(O) /= 0IH(0) /= 0[H(N /2) /= 0p(k) = -(N -k)(k) = -(N - k)p(k) =-p(N-k)p(k) = -P(N - k)相=-k(1-N-1)元位= -k(1 - N-1)=-k(1 - N-1)π=-k(1-N-1)约k = 0~(N /2-1)k= 0~(N-1)/2k= 0~(N /2-1)k= 0 ~ (N -1)/2東(N /2) = 0
线性相位约束条件
对于第一种抽样方式，当h(n)为实数时h(n) = h*(n)h(n)H(k) = DFT[h(n)]根据P91（3-79)[H()]H*(N -k) = DFT[h*(n)于是H(k) = H*(N-k)(p(k)IH(k)/=|H(N-k)|0(k) = arg[H(k)] = -0(N - k)023N中心以k=2
于是对于第一种抽样方式，当h(n)为实数时 - 1 0 1 2 3 4 5 6 05 1 0 1 5 2 0- 1 0 1 2 3 4 5 6 012345 h(n) |H(k)| -1 0 1 2 3 4 5 6 - 3 - 2 - 10123 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 012345 (k) H(k) = DFT[h(n)] H (k) H (N k) * = - | ( )| | ( )| ( ) arg[ ( )] ( ) H k H N k k H k N k N k q q = - = = - - = 2 以 中心 h (n) h (n) * = 根据P91 （3-79） H (N k) DFT[h (n)] * * - =