外文翻译-模拟滤波器

Analog Filters Bessel, Butterworth, Chebyshev, elliptic, bilineartransformation1 Functions1.1 bilinear-Bilinear transformation method for analog-to-digital filter conversionSyntaxzd,pd,kd=bilinear(z,p,k,fs) zd,pd,kd=bilinear(z,p,k,fs,fp) numd,dend=bilinear(num,den,fs)numd,dend=bilinear(num,den,fs,fp)Ad,Bd,Cd,Dd=bilinear(A,B,C,D,fs)Ad,Bd,Cd,Dd=bilinear(A,B,C,D,fs,fp)DescriptionThe bilinear transformation is a mathematical mapping of variables. In digital filtering, it is a standard methodof mapping the s or analog plane into the z ordigital plane. It transforms analog filters, designed using classicalfilter design techniques, into their discrete equivalents.The bilinear transformation maps the s-planeinto the z-plane byThis transformation maps the j axis(from = to +) repeatedly around the unit circle (ejw,from = to )bybilinear can accept an optional parameter Fp thatspecifies prewarping. fp, in hertz, indicatesa match frequency, that is, a frequency for whichthe frequency responses before and after mapping match exactly. Inprewarped mode, the bilinear transformation maps the s-planeinto the z-plane withWith the prewarping option, bilinear mapsthe j axis (from = to +) repeatedly around the unitcircle (ejw, from = to ) byIn prewarped mode, bilinear matches the frequency2fp (in radians persecond) in the s-plane to the normalized frequency2fp/fs (in radians per second) in the z-plane.The bilinear function works with three different linear system representations: zero-pole-gain, transfer function, and state-space form.Zero-Pole-Gainzd,pd,kd = bilinear(z,p,k,fs) and zd,pd,kd = bilinear(z,p,k,fs,fp) convert the s-domain transfer function specified by z, p, and k to a discrete equivalent. Inputs z and p are column vectors containing the zeros and poles, k isa scalar gain, and fs is the sampling frequency in hertz. bilinear returns the discrete equivalent in column vectors zd and pd and scalar kd. The optional match frequency, fp is in hertz and is used for prewarping.Transfer Functionnumd,dend = bilinear(num,den,fs) andnumd,dend = bilinear(num,den,fs,fp) convertan s-domain transfer function given by num and den toa discrete equivalent. Row vectors num and den specifythe coefficients of the numerator and denominator, respectively, indescending powers of s. Let B(s) bethe numerator polynomial and A(s) be the denominatorpolynomial. The transfer function is:fs is the sampling frequency in hertz. bilinear returnsthe discrete equivalent in row vectors numd and dend indescending powers of z (ascendingpowers of z1). fp isthe optional match frequency, in hertz, for prewarping.State-SpaceAd,Bd,Cd,Dd = bilinear(A,B,C,D,fs) andAd,Bd,Cd,Dd = bilinear(A,B,C,D,fs,fp) convert the continuous-timestate-space system in matrices A, B, C, Dto the discrete-time system:fs is the sampling frequency in hertz. bilinear returnsthe discrete equivalent in matrices Ad, Bd, Cd, Dd.The optional match frequency, fp is in hertz andis used for prewarping.Diagnosticsbilinear requires that the numerator orderbe no greater than the denominator order. If this is not the case, bilinear displays Numerator cannot be higher order than denominator.For bilinear to distinguish between the zero-pole-gainand transfer function linear system formats, the first two input parametersmust be vectors with the same orientation in these cases. If thisis not the case, bilinear displaysFirst two arguments must have the same orientation.More AboutAlgorithmsbilinear uses one of two algorithms dependingon the format of the input linear system you supply. One algorithmworks on the zero-pole-gain format and the other on the state-spaceformat. For transfer function representations, bilinear convertsto state-space form, performs the transformation, and converts theresulting state-space system back to transfer function form.Zero-Pole-Gain AlgorithmFor a system in zero-pole-gain form, bilinear performsfour steps:1. If fp is present,it prewarps: fp = 2*pi*fp; fs = fp/tan(fp/fs/2)otherwise, fs = 2*fs.2. It strips any zeros at using z = z(finite(z);3. It transforms the zeros, poles, and gain using pd = (1+p/fs)./(1-p/fs); % Do bilinear transformation zd = (1+z/fs)./(1-z/fs); kd = real(k*prod(fs-z)./prod(fs-p);4. It adds extra zeros at -1 so the resultingsystem has equivalent numerator and denominator order.State-Space AlgorithmFor a system in state-space form, bilinear performstwo steps:1. If fp is present,let If fp is not present, let =fs.2. Compute Ad, Bd, Cd,and Dd in terms of A, B, C,and D using References1 Parks, T.W., and C.S. Burrus. DigitalFilter Design. New York: John Wiley & Sons, 1987.Pgs.209-213.2 Oppenheim, A.V., and R.W. Schafer. Discrete-TimeSignal Processing. Upper Saddle River, NJ: Prentice-Hall,1999, pp. 450-454.1.2 besselap- Bessel analog lowpass filter prototypeSyntaxz,p,k=besselap(n) Descriptionz,p,k = besselap(n) returns the poles and gain of an order n Besselanalog lowpass filter prototype. n must be less than or equal to 25. The function returns the poles in thelength n column vector p and the gain in scalar k. z is an empty matrix because there are no zeros. The transfer function isbesselap normalizes the poles and gain so that at low frequency and high frequency the Bessel prototype is asymptotically equivalent to the Butterworth prototype of the same order 1. The magnitude of the filter is less than at the unity cutoff frequency c = 1. Analog Bessel filters are characterized by a group delay thatis maximally flat at zero frequency and almost constant throughoutthe passband. The group delay at zero frequency isMore AboutAlgorithmsbesselap finds the filter rootsfrom a lookup table constructed using Symbolic Math Toolbox software.References1 Rabiner, L.R., and B. Gold. Theoryand Application of Digital Signal Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1975. Pgs.228-230.1.3 besself -Bessel analog filter designSyntaxb,a=besself(n,Wo)z,p,k = besself(.)A,B,C,D = besself(.)Descriptionbesself designs lowpass, analog Bessel filters,which are characterized by almost constant group delay across theentire passband, thus preserving the wave shape of filtered signalsin the passband. besself does not support the designof digital Bessel filters.b,a = besself(n,Wo) designsan order n lowpass analog Bessel filter, where Wo is thefrequency up to which the filters group delay is approximately constant. Larger values of the filter order (n) producea group delay that better approximates a constant up to frequency Wo. besself returns the filter coefficients inthe length n+1 row vectors b and a,with coefficients in descending powers of s, derivedfrom this transfer function:z,p,k = besself(.) returns the zeros andpoles in length n or 2*n columnvectors z and p and the gainin the scalar k. A,B,C,D = besself(.) returns the filter designin state-space form, where A, B, C,and D areand u is the input, x isthe state vector, and y is the output.ExamplesDesign a fifth-order analog lowpass Bessel filter with an approximateconstant group delay up to 10,000 rad/s and plot the frequency responseof the filter using freqs:b,a = besself(5,10000);freqs(b,a) % Plot frequency responseLimitationsLowpass Bessel filters have a monotonically decreasingmagnitude response, as do lowpass Butterworth filters. Compared tothe Butterworth, Chebyshev, and elliptic filters, the Bessel filterhas the slowest rolloff and requires the highest order to meet anattenuation specification.For high order filters, the state-space form is the most numericallyaccurate, followed by the zero-pole-gain form. The transfer functioncoefficient form is the least accurate; numerical problems can arisefor filter orders as low as 15.More AboutAlgorithmsbesself performs a four-step algorithm:1. It finds lowpass analog prototype poles,zeros, and gain using the besselap function.2. It converts the poles, zeros, and gaininto state-space form.3. It transforms the lowpass prototype intoa lowpass filter that meets the design specifications.4. It converts the state-space filterback to transfer function or zero-pole-gain form, as required.1.4 buttap-Butterworth filter prototypeSyntaxz,p,k=buttap(n)Descriptionz,p,k = buttap(n) returnsthe poles and gain of an order n Butterworthanalog lowpass filter prototype. The function returns the poles inthe length n column vector p andthe gain in scalar k. z is anempty matrix because there are no zeros. The transfer function isButterworth filters are characterized by a magnitude responsethat is maximally flat in the passband and monotonic overall. In thelowpass case, the first 2n-1 derivatives of thesquared magnitude response are zero at = 0. The squared magnitude response function iscorresponding to a transfer function with poles equally spacedaround a circle in the left half plane. The magnitude response atthe cutoff angular frequency 0 is always regardless of the filter order. buttap sets0 to 1 for a normalized result.Algorithmsz = ;p = exp(sqrt(-1)*(pi*(1:2:2*n-1)/(2*n)+pi/2).;k = real(prod(-p);References1 Parks, T.W., and C.S. Burrus. DigitalFilter Design. New York: John Wiley & Sons, 1987.Chapter7.1.5 butter-Butterworth filter designSyntaxz,p,k=butter(n,Wn)z,p,k = butter(n,Wn,ftype)b,a=butter(n,Wn)b,a=butter(n,Wn,ftype) A,B,C,D=butter(n,Wn) A,B,C,D = butter(n,Wn,ftype)z,p,k=butter(n,Wn,s)z,p,k = butter(n,Wn,ftype,s)b,a=butter(n,Wn,s)b,a=butter(n,Wn,ftype,s)A,B,C,D=butter(n,Wn,s) A,B,C,D = butter(n,Wn,ftype,s) Descriptionbutter designslowpass, bandpass, highpass, and bandstop digital and analog Butterworthfilters. Butterworth filters are characterized by a magnitude responsethat is maximally flat in the passband and monotonic overall.Butterworth filters sacrifice rolloff steepness for monotonicityin the pass- and stopbands. Unless the smoothness of the Butterworthfilter is needed, an elliptic or Chebyshev filter can generally providesteeper rolloff characteristics with a lower filter order.Digital Domainz,p,k = butter(n,Wn) designsan order n lowpassdigital Butterworth filter with normalized cutoff frequency Wn.It returns the zeros and poles in length n columnvectors z and p, and the gainin the scalar k.z,p,k = butter(n,Wn,ftype) designsa highpass, lowpass, or bandstop filter, where the string ftype isone of the following:high fora highpass digital filter with normalized cutoff frequency Wnlow fora lowpass digital filter with normalized cutoff frequency Wnstop foran order 2*n bandstop digital filter if Wn isa two-element vector, Wn = w1 w2. The stopband is w1 w2.bandpass for an order 2*n bandpassfilter if Wn is a two-element vector, Wn = w1 w2. The passbandis w1 w2. Specifyinga two-element vector, Wn, without an explicit ftype defaultsto a bandpass filter.Cutoff frequency is that frequency wherethe magnitude response of the filter is . For butter,the normalized cutoff frequency Wn must be a numberbetween 0 and 1, where 1 corresponds to the Nyquist frequency, radians per sample.If Wn is a two-element vector, Wn = w1 w2, butter returnsan order 2*n digital bandpassfilter with passband w1 w2.With different numbers of output arguments, butter directlyobtains other realizations of the filter. To obtain the transfer functionform, use two output arguments as shown below.Note: See Limitations below for informationabout numerical issues that affect forming the transfer function.b,a = butter(n,Wn) designsan order n lowpass digital Butterworth filter withnormalized cutoff frequency Wn. It returns thefilter coefficients in length n+1 row vectors b and a, with coefficients in descendingpowers of z.b,a = butter(n,Wn,ftype) designsa highpass, lowpass, or bandstop filter, where the string ftype is high, low,or stop, as described above.To obtain state-space form, use four output arguments as shownbelow:A,B,C,D = butter(n,Wn) or A,B,C,D = butter(n,Wn,ftype) where A, B, C,and D areand u is the input, x isthe state vector, and y is the output.Analog Domainz,p,k = butter(n,Wn,s) designsan order n lowpass analog Butterworth filter withangular cutoff frequency Wn rad/s. It returns thezeros and poles in length n or 2*n columnvectors z and p and the gainin the scalar k. butters angularcutoff frequency Wn must be greater than 0rad/s.If Wn is a two-element vector with w1 w2, butter(n,Wn,s) returnsan order 2*n bandpass analog filter with passband w1 w2.z,p,k = butter(n,Wn,ftype,s) designsa highpass, lowpass, or bandstop filter using the ftype valuesdescribed above.With different numbers of output arguments, butter directlyobtains other realizations of the analog filter. To obtain the transferfunction form, use two output arguments as shown below:b,a = butter(n,Wn,s) designsan order n lowpass analog Butterworth filter withangular cutoff frequency Wn rad/s. It returns thefilter coefficients in the length n+1 row vectors b and a,in descending powers of s, derived from this transferfunction:b,a = butter(n,Wn,ftype,s) designsa highpass, lowpass, or bandstop filter using the ftype valuesdescribed above.To obtain state-space form, use four output arguments as shownbelow:A,B,C,D = butter(n,Wn,s) orA,B,C,D = butter(n,Wn,ftype,s) where A, B, C,and D areand u is the input, x isthe state vector, and y is the output.ExamplesHighpass FilterFor data sampled at 1000 Hz, design a 9th-orderhighpass Butterworth filter with cutoff frequency of 300 Hz,which corresponds to a normalized value of 0.6:z,p,k = butter(9,300/500,high);sos,g = zp2sos(z,p,k); % Convert to SOS formHd = dfilt.df2tsos(sos,g); % Create a dfilt objecth = fvtool(Hd); % Plot magnitude responseset(h,Analysis,freq) % Display frequency responseLimitationsIn general, you should use the z,p,k syntaxto design IIR filters. To analyze or implement your filter, you canthen use the z,p,k output with zp2sos and an sos dfilt structure. For higher order filters(possibly starting as low as order 8), numerical problems due to roundofferrors may occur when forming the transfer function using the b,a syntax.The following example illustrates this limitation:n = 6; Wn = 2.5e6 29e6/500e6;ftype = bandpass;% Transfer Function designb,a = butter(n,Wn,ftype);h1=dfilt.df2(b,a); % This is an unstable filter.% Zero-Pole-Gain designz, p, k = butter(n,Wn,ftype);sos,g=zp2sos(z,p,k);h2=dfilt.df2sos(sos,g);% Plot and compare the resultshfvt=fvtool(h1,h2,FrequencyScale,log);legend(hfvt,TF Design,ZPK Design)Algorithmsbutter uses a five-step algorithm:1. It finds the lowpass analog prototypepoles, zeros, and gain using the buttap function.2. It converts the poles, zeros, and gaininto state-space form.3. It transforms the lowpass filter intoa bandpass, highpass, or bandstop filter with desired cutoff frequencies,using a state-space transformation.4. For digital filter design, butter uses bilinear to convert the analog filterinto a digital filter through a bilinear transformation with frequencyprewarping. Careful frequency adjustment guarantees that the analogfilters and the digital filters will have the same frequency responsemagnitude at Wn or w1 and w2.5. It converts the state-space filter backto transfer function or zero-pole-gain form, as required.1.6 cheb1ap-Chebyshev Type I analog lowpass filter prototype1.7 cheb2ap-Chebyshev Type II analog lowpass filter prototype1.8 cheby1-Chebyshev Type I filter design (passband ripple)1.9 cheby2-Chebyshev Type II filter design (stopband ripple)1.10 ellip-Elliptic filter design1.11 ellipap-Elliptic analog lowpass filter prototype1.12 freqs-Frequency response of analog filters1.13 lp2bp-Transform lowpass analog filters to bandpass1.14 lp2bs-Transform lowpass analog filters to bandstop1.15 lp2hp-Transform lowpass analog filters to highpass1.16 lp2lp-Change cutoff frequency for lowpass analog filter模拟滤波器贝塞尔，巴特沃斯，切比雪夫，椭圆形，双线性变换1 功能1.1双线性-用于模拟-数字转换滤波器的双线性变换法用法zd,pd,kd=bilinear(z,p,k,fs) zd,pd,kd=bilinear(z,p,k,fs,fp) numd,dend=bilinear(num,den,fs)numd,dend=bilinear(num,den,fs,fp)Ad,Bd,Cd,Dd=bilinear(A,B,C,D,fs)Ad,Bd,Cd,Dd=bilinear(A,B,C,D,fs,fp)说明双线性变换是变量的数学映射。在数字滤波，它是s或模拟平面映射到z或者数字平面的一种标准方法。它把模拟滤波器，采用经典的滤波器设计技术，为他们的离散等价物而设计的。双线性变换由映射s平面到z平面。双线性可以接受一个可选的参数，Fp，指定预畸变。fp，单位为赫兹，表示“匹配”的频率，也就是一个频率的之前和之后的映射匹配频率响应完全相同。在prewarped模式，使用双线性变换映射s平面到Z-平面。随着预畸变选项，双线性映射反复围绕单位圆(ejw,从到)由的虚轴(从 到 +)。在prewarped模式，双线性在s平面的归一化频率2fp/fs（弧度每秒）在z平面上的频率2fp（弧度每秒）相匹配。双线性函数的工作原理有三个不同的线性系统表示：零极点增益，传递函数，状态空间形式。零极点增益ZD，PD，KD=双线性（Z，P，K，FS）和ZD，PD，KD=双线性（Z，P，K，FS，FP）转换为z指定的s域传递函数，p和k以一个离散当量。输入z和p分别含有零点和极点列向量，k是一个标量增益，并且fs是在赫兹的采样频率。双线性返回的列向量ZD和PD和标量KD离散等价的。可选的匹配次数，FP是赫兹和用于预畸变。传递函数numd，DEND=双线性（NUM，书房，FS）和numd，DEND=双线性（NUM，书房，FS，FP）转换由Num和书房的离散相当于给予s域传递函数。行向量Num和书房指定的分子和分母的系数，分别按降序s的能力。设B（s）为分子多项式和A（s）为分母多项式。传递函数是：。FS是赫兹的采样频率。双线性返回离散等效于行向量numd和DEND降序z与权力（升序Z-1的权力）。 fp是可选的匹配频率，以赫兹为单位，对预畸变。状态空间AD，BD，CD，DD=双线性（A，B，C，D，FS）和AD，BD，CD，DD=双线性（A，B，C，D，FS，FP）转换为连续-时间状态空间系统在矩阵A，B，C，D到离散时间系统：FS是赫兹的采样频率。双线性返回离散相当于矩阵AD，BD，CD，DD。可选的匹配次数，FP是赫兹和用于预畸变。诊断双线性要求分子阶数不大于分母阶更大。如果不是这种情况，双线性显示分子不能高阶比的分母。对于双线性到零极点增益和传递函数的线性系统格式区分，前两个输入参数必须是具有相同的取向在这些情况下，向量。如果不是这种情况，双线性显示前两个参数必须具有相同的方向。算法根据您提供的输入线性系统的格式双线性使用两种算法之一。一个算法的工作原理上的状态空间形式的零极点增益形式和其他。对于传递函数表示法，双线性转换为状态空间形式，进行了改造，产生的状态空间系统转换回传输函数形式。零极点增益算法 对于零极点增益形式的系统，双线性执行四个步骤：1. 若fp存在，它具有频率予翘曲的双线性变换(Tustin)法fp = 2*pi*fp;fs = fp/tan(fp/fs/2)否则，fs = 2*fs.2.它去掉任何零点在采用z = z(finite(z);3. 它采用变换的零点，极点和增益PD =（1+ P/ FS）/（1-p/fs）;做双线性变换 ZD=（1+ Z/ FS）/（1-z/fs）; kd = real(k*prod(fs-z)./prod(fs-p);4. 它增加了额外的零在-1使该系统具有等效的分子和分母秩序。状态空间算法 对于状态空间形式的系统，双线性执行两个步骤：1.如果FP存在，让若fp是不存在，取= FS。计算AD，BD，CD和DD在使用A，B，C和D的条款文献1 Parks, Thomas W., and C. Sidney Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, pp. 209213.2 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, pp. 450454.1.2 besselap贝塞尔模拟低通滤波器原型句法z,p,k = besselap(n)描述 Z，P，K= besselap（n）返回两极的N阶贝塞尔模拟低通滤波器原型和增益。 n必须小于或等于25。该函数返回的极点在长度为n的列向量p和标量k中的增益。 z是一个空矩阵，因为没有零。传递函数是besselap标准化的极点和增益，以使在低频率和高频率的贝塞尔原型是渐近相当于巴特沃思原型的顺序相同的1。过滤器的幅度小于在统一的截止频率为c= 1。 模拟贝塞尔滤波器的特征是一组延迟是最平坦的频率为零时，几乎恒定的整个通带。群延迟在零频率更多算法besselap发现过滤器的根从查找表使用符号数学工具箱软件构建。1.3 besself 贝塞尔模拟滤波器设计句法b,a = besself(n,Wo)z,p,k = besself(.)A,B,C,D = besself(.)描述 besself设计低通，模拟贝塞尔滤波器，其特点是在整个通带内几乎恒定的群延迟，从而保持滤波的信号的波形通带。 besself不支持的数字贝塞尔滤波器的设计。 B，A= besself（正，禾）设计了一个n阶低通模拟贝塞尔滤波器，其中禾是主频高达该滤波器的群时延近似为常数。滤波器阶数（n）的值越大，产生群时延能够更加接近于一个常数高达频率窝。 besself返回的长度n +1个行向量b和a的滤波器系数，与降序第权力，从这个传递函数的系数得出：Z，P，K= besself（.）返回零点和极点的长度为n或2* N的列向量z和p和在标量k中的增益。 A，B，C，D= besself（.）返回的滤波器设计的状态空间形式，其中A，B，C和D是公式和u为输入，x是状态矢量，而y是输出。举例一个模拟贝塞尔FilterDesign五阶模拟低通贝塞尔滤波器大约恒定的群延迟至10,000弧度/秒的频率响应。使用FREQS画出滤波器的幅度和相位响应。b,a = besself(5,10000);freqs(b,a)限制低通贝塞尔滤波器有一个单调递减的幅度响应，因为做低通Butterworth滤波器。相较于巴特沃斯，切比雪夫和椭圆滤波器，贝塞尔滤波器具有最慢的衰减和需要最高为满足衰减规范。 对于高阶滤波器的状态空间形式是最准确的数字，后跟零极点增益形式。传递函数系数的形式是最不准确的，可能会出现过滤器订单低至15数值问题。更多Algorithmsbesself进行四步算法： 1. It发现低通模拟原型的极点，零点和增益使用besselap功能。 2. 2的极点，零点和增益转换成状态空间形式。 3. It变换低通原型到符合设计规范的低通滤波器。 4. 4根据需要转换的状态空间滤波反传递函数或零极点增益形式。1.4 buttap 巴特沃斯滤波器原型句法z,p,k = buttap(n)Z，