数字信号处理邵曦lecture20

1、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

2、； 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

3、: 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)() hilbe

4、rt 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 l

5、p 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

6、 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

7、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)()(0nwndoth

8、ersmnmndnd （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

9、)(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 fil

10、ter) )(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 stop

11、band; 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 no

12、t 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 w