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1、1,Chapter 6 Design of FIR Digital Filter,2,IIR vs.FIR,IIR Analog filterDesign simply.Spectrum specification uneasy to control.FIR:Strictly linear spectrum characteristics.h(n)is finite length,no problem of stability and causality.,3,CONTENTS,6.1 Characteristics of linear phase FIR filter,6.2 Design
2、of FIR filter by Windowing,6.3 Frequency sampling method,6.4 Comparison of IIR and FIR filter,4,Introduction,IIR filters phase characteristics is hard to control.FIR filters advantage：Exact liner phase.(Arbitrary amplitude-frequency characteristics)FIR filters impulse response h(n)is a finite length
3、 sequence z-transform convergent on whole z-plane(no stability problem)can delay some time(no causality problem)FIR filters structure is non-recursive generally,there will no problem of instability with finite precision computation.,5,6.1 Characteristic of linear phase FIR filter,（1）Condition for li
4、near phase If impulse response h(n)of FIR filter is real value,and meets condition of even symmetric or odd symmetric,that is:then filters phase characteristic is exact linear.,（2）Linear phase FIR filters amplitude-frequency characteristic,6,7,8,（3）Zero location of linear phase FIR filter,System fun
5、ction:,If z=zi is zero of H(z),its reciprocal zi-1 is also zero of H(z).Because h(n)is real sequence,H(z)s zero are conjugate symmetric.That is:zi*and(zi-1)*are also zero of H(z).Characteristic of zero location:Zeros are reciprocal conjugate symmetric pair.,Exceptional:Zeros are real value,then ther
6、e are only two zeros.Zeros are pure complex and lie on unit circle,then there are only two zeros.Zeros are real value and lie on unit circle,then there are only two zeros.,9,Basic structure,Structure of linear phase FIR filter,（4）Structure of linear phase FIR filter,10,Structure of linear phase FIR
7、filter,Save half multiplier,N is even,N is odd,11,6.2 Windowing,1.Basic idea of windowing method,Infinitely longNot absolutely summable,Window function,Approximate,Unrealizable!,12,单位脉冲响应,矩形窗,h(n)偶对称,理想幅度特性,矩形窗幅度特性,时域相乘,频域卷积,？,1.Basic idea of windowing method,13,时域相乘,频域卷积,理想幅度特性,矩形窗幅度特性,14,15,截止频率,1
8、6,max,17,min,18,矩形窗对理想低通幅度特性的影响,19,20,讨论：增大N对窗函数的影响,N=10,N=20,增加窗函数的长度，能够减少窗函数频谱主瓣宽度；不能改变主瓣和旁瓣的相对特性，该值取决于窗函数的形状。,Gibbs Phenomenon 吉布斯效应,21,2.Effects of windowing on ideal filter specification At disconnected points of cutoff frequency,Hd(w)becomes continuous curve transition region.Its width is equa
9、l to windows main lobe width.Wider windows main lobe wider transition region 4/N Effects of windows side lobe:ripple appear in filters amplitude-frequency characteristics,amplitude of ripple depends on side lobes relative amplitude.Larger area of side lobe larger ripple of passband and stopband.Incr
10、ease windows length can only decrease main lobe width,not relative characteristic between main lobe and side lobe,that value only depends on windows shape.Increase N can only decrease width of transition region,not ripple characteristics.,22,3.Principle of window selection Small side lobe amplitude,
11、especially the first side lobe amplitude.High speed decreasing of side lobe amplitude,in order to increase stopband attenuation.Narrow main lobe width,in order to get narrow transition width.,Trade-off Small side lobe amplitude:Smoothing amplitude-frequency characteristics Transition width increase
12、Narrow main lobe width:Narrower transition width.Ripple of passband and stopband is large.,23,矩形窗,三角窗,哈明窗,Rectangle,Triangle,Hamming,常用窗函数幅频特性,24,汉宁窗,凯塞窗,Hann,Blackman,Kaiser,常用窗函数幅频特性,布莱克曼窗,25,4.Process of FIR filter design using window method(1)Determine filters unit impulse response:,If filters f
13、requency response is Hd(ejw),then:,If edge frequency,attenuation of passband,stopband is known,we can choose ideal filter as approximation function:Hd(ejw)IDFT hd(n),26,(2)Determine window function and its length based on requirement,Suppose width of transition is,it is an approximation of windows m
14、ain lobe.is reverse ratio to windows length N:NA/,A window form;For example:Rectangular:A=4;Hamming:A=8;,Principle:Select window function with narrower main lobe when stopband attenuation meets the requirement.,27,(3)Compute filters unit impulse response h(n),（4）Verification Compute designed filters
15、 frequency response:,Selected window.,FFT Algorithm,28,5 KAISER（1）Basic property A most useful and optimum window.,Zeroth-order modified Bessel function,Parameter can control shape of the window.,Characteristic:Different width of transition band under same N.,（1）If=5.658，width of transition band 7.8
16、/N,least attenuation of stop band 60dB,（2）If=4.538，width of transition band 5.8/N,least attenuation of stop band 50dB.,29,30,（2）Design approach,Kaiser obtain a pair of formulas that permit designer to predict in advance parameters needed to meet a given frequency selective filter specification.,p,s
17、and As is known,acquire the following parameter:,Width of transition band,Order of filter,31,Example:Design a digital FIR lowpass filter with Kaiser window,specification is as follows:,p=0.2，s=0.3，As=50dB,Determine impulse response and filters frequency response diagram.,MATLAB demo:KAISERFIR_design
18、.m,32,6.3 Frequency sampling method,Suppose system function of designed filter doesnt have closed form equation.,N points regular interval sampling between=0 to2,Condition for linear phase,33,（1）Performance,IIR filter:Require less memory and calculation to achieve a given filter response characteris
19、tic,high cost efficiency.Better selectivity,phase nonlinearity.,FIR filter:Require more memory and calculation to achieve a given filter response characteristic.Delay time is large.Exact linear phase,Under the same requirement,IIRs complexity will increase dramatically,so FIR outweigh IIR in perform
20、ance and cost.,6.4 Comparison of IIR and FIR filter,34,（2）Structure,IIR filter:Recursive structure,pole must be inside unit circle,or system instability will appear.Finite-precision arithmetic can cause significant problems due to the use of feedback.,FIR filter Non-recursive structure,no instabilit
21、y problem in theory and practice.Low computation error.Fast Fourier Transform(FFT),high speed with same order.,35,（3）Design approach,IIR filter:A variety of frequency-selective filters can be designed using closed form design formulas,so it can save computation time.Limited to frequency-selective fi
22、lters,inflexible.,FIR filter:Closed form design equation does not exit for FIR filter.Although window method is straightforward to apply,some iteration is necessary to meet a prescribed specification.Flexible,arbitrary frequency response with exact linear phase.,36,（4）Application,IIR filter:Applicat
23、ion with no requirement for linear phase.,FIR filter:Application with high requirement for linear phase.,In practical application,requirement,tools and cost must be considered.,37,RIVIEW,Comparison of FIR filter and IIR filter Specification,structure,phase characteristics,design approach,application
24、Condition for FIR linear phase(proof)Window method（1）Idea（2）Conclusion Width of transition band is determined by width of windows main lobe;Peak ripples of bandstop and bandpass region is determined by side lobe characteristic.Length of windows:N Influence main lobe,not side lobe.（3）Selective of win
25、dows Width of main lobe and side lobe characteristics(restrict each other)4.Commonly used windows.,38,Additional experiments:FIR digital filter designRequirement:Window:Kaiser windowFilters length can be adjusted:NF=2MFilter type:lowpass,highpass,bandpass,bandstopBasic parameters computation bandpass,edge frequency,ripple and attenuation,transitionWork propose:Experiment report(Program list,outcomes and analysis)Time:final exam,
