资源描述
Chapter 10 FIR Digital Filter Design Filter design:constructing the transfer function of a filter that meets prescribed frequency response specifications.The design finally gives H(z)or h(n)(in the case of FIR filters)Choosing FIR or IIR filter?FIR filter:easy to achieve linear phase property；guaranteed stability；high computational cost when meeting sharp filter specifications 10.1 Window Method 10.1.1 Ideal Filters Symbols used:ideal filter:frequency response)(D impulse response)(kd designed filter:frequency response)(H impulse response)(nh Examples of ideal filters:Fig.10.1.1:lowpass(LP)低低通通，highpass(HP)高高通通，bandpass(BP)带带通通，bandstop(BS)带带阻阻 The highest frequency to be processed is 2sff,corresponding to;No transition band(过过渡渡带带),only passband（通通带带）and stopband（阻阻带带）;The phase response 0)(argD Fig.10.1.2:differentiator 微分器微分器(jD)(，njnjnjejeDe11111)()Hilbert transformer 希尔伯特变换器希尔伯特变换器（ideal 90o phase shifter）The impulse response of an ideal filter is generally infinite,double-sided.ideal lowpass filter：kkdekdckjLPcc)sin(21)(k (10.1.2)cLPd)0(10.1.3)ideal highpass filter：ccdedekdkjkjHP212)(kkkkkkkccc)sin()()0()sin()0(1 or 1)()(HPLPDD（when LP and HP have the same c）)()()(kkdkdHPLP kkkkdcHP)sin()()(d(k)s of other ideal filter:（10.1.4）(10.1.6)These)(Dand)(kd imply the property of DTFT:(p.544)real and even)(kd real and even)(D symmetric class of filter real and odd)(kd imaginary and odd)(D antisymmetric class of filter 10.1.2 Rectangular Window Window Method designing FIR filter:Truncating the infinite,double-sided d(k)to a finite length,which is the FIR impulse response approximating the ideal response.Problems concerned：对理想对理想 d(n)截取哪段作为截取哪段作为 FIR 滤波器的滤波器的 h(n)？截取多长，即截取多长，即 FIR 滤波器的阶数取多少？滤波器的阶数取多少？FIR 频率特性频率特性)(H能在多大程度上近似理想频响能在多大程度上近似理想频响)(D？近似程度与什么有关？近似程度与什么有关？Steps:(p.544)1.Pick an odd length N=2M+1,and let M=(N-1)/2.2.Calculate the N coefficients 2)()(deDkdkj,MkM,(10.1.7)3.Make them causal by the delay )10()()(NnMndnh(10.1.10)Equivalent forms of h(n):)()()(nwMndnh If)()(0)()(0nwndothersMnMndnd （windowed,double-sided d(k)）)()(Mndnh Example 10.1.1：N11 Linear phase property of)(H )()(Mndnh kjMMkekdD)()(（10.1.13）kjMMkMjMjekdeDeH)()()(（10.1.16）magnitude response:|)(|)(|DH phase response of)(H：a)in the symmetric case real and even)(kd real and even)(D 0)(0)(0)(argDDD)(2)(1Dsign )()(argMH （10.1.17）piece-wise linear phase response b)antisymmetric case:(10.1.18)Additional material(关关于于)(H幅度形状)Since)(H is complex,we consider its real magnitude)(D)()(DeHMj（10.1.16）（)(D：real and even）)()()(0nwndnd，where notherMMnnw01)(0 ccdWWDD2)()(*)()(00(for lowpass filter)(D:estimating the area of)(0W located in the,cc When)2(Nc,)(Dshows negative overshoot.When c,the magnitude is 1/2 of that of 0 When)2(Nc,)(Dshows positive overshoot.Additional material )(H:lowpass response with cutoff c(a smeared version);ripples(纹波纹波)within the passband and stopband;transition band width N4 Fig.10.1.5 As the truncating width N increases,(p.549)1.For the frequencies within the passband or stopband,)()(DD as N;2.The transition width decreases with N;3.The largest ripples near the discontinuity of)(D get squeezed onto the discontinuity at cand do not get smaller with N.Their size remains 8.9%(Gibbs phenomenon).Gibbs phenomenon：nnjendD)()(（Fourier series expansion of the periodic)(D）MMnnjendD)()(（10.1.13）(truncating the infinite Fourier series expansion to the finite sum will cause overshoot at the discontinuity)Fig.10.1.5 10.1.3 Hamming Window To eliminate 8.9%ripples,we can use a non-rectangular window.Using Hamming window:overshoot drops to 0.2%;transition width becomes widerFig.10.1.7 作业：作业：10-1，10-8，10-18（只做（只做N11）
展开阅读全文