Mobile Digital Filter Design Toolbox

Transcription

1 Mobile Digital Filter Deign Toolbox Somak Sukittanon and Joh Pott Univerity of Tenneee at Martin Department of Engineering Martin, TN USA Abtract! Mobile device have become very ueful in many field a portable computation device. They function a data analyi and collection tool. Though they have become common place in many field, there are few application in the area of Digital Signal Proceing (DSP). In thi paper, we have developed a filter deign toolbox for mobile device, particularly ios device. Thi toolbox allow for the deign of filter of uer pecified order and type, export of filter coefficient via , frequency and impule repone plotting, and real time filtered playback. The limited memory and proceing power of the device required pecial mathematic for the filter deign to be derived. I. INTRODUCTION Digital ignal proceing (DSP) i common in today technical world and can be found in many application. Audio data manipulation relie heavily on DSP in the digital age for filtering data, digitization, and analyi. Power ytem alo rely on DSP for application uch a better detecting AC current frequencie. Filter deign, an advanced topic in DSP, i a vital tool that i ued in many field. Currently, filter deign i done motly on dektop computer platform with program uch a Matlab or the open ource alternative Octave. Both offer powerful tool for digital ignal proceing. Smaller ytem for DSP implementation have been developed before for microproceor, and now mobile device. Mobile device are becoming the preferred computational device for many different field, even though they lack the power of a traditional dektop computer. In Phyic [1], data collecting tool uch a compae and high reolution camera are being connected to mobile device to analyze data. In chemitry, the ToiChem application [] i enhancing and reinforcing topic covered in organic chemitry. Mobile application [3] have been developed to help tudent learn introductory programming concept in the engineering field. One of the mot utilized device for thee type of application i the Apple ipad/iphone, which run the ios. The primary difference between the ipad and iphone i the creen ize. Thee device ue the Cocoa Touch programming, which i baed on Objective-C language. Thee device differ from their dektop counterpart in that multiple application cannot be diplayed imultaneouly on the creen, and are limited to one uer interface at a time. The device are alo limited to the memory available to application, and application cannot acce anything that would require adminitrator privilege on a typical dektop. Mobile device offer much more portability than dektop, but uffer from a limited creen ize a a reult. Though mobile device have proliferated thee many field, DSP i till a relatively untouched area on mobile device. One uch application i ijdsp [4], an iphone application that allow the uer to do data filtering and analyi uing graphical block diagram. Matlab [5] alo offer a mobile application, but i chained via internet to the power of a dektop computer. In thi paper we develop a filter deign toolbox for mobile device. The mathematic developed had to be compatible with the limited memory and proceing power of thee device. It provide filter deign, real time playback, and frequency/impule repone plotting. Verion of thi toolbox were created for the iphone and ipad. II. SOFTWARE DESIGN Thi ection decribe the IIR filter deign with Biquad form uing Bilinear tranformation. There are 3 tep, i.e. Lowpa prototype deign, frequency tranformation, and bilinear tranformation. The derivation for mobile implementation i dicued. Thi new method fit for the mall memory contrain on the mobile device and yield the ame reult a other PC oftware, e.g. Matlab Fdatool. A. Analog Prototype Lowpa Filter The goal of thi tep i to deign the analog lowpa filter with cutoff frequency equal to one. The final equation, uing any deign, will be in thi form, where N B i the number of biquad (econd order) tate. N Bi 1 b H () = G k 0 + b k1 + b k (1) a k 0 k=0 + a k1 + a k Three method were deigned in the tool, i.e. Butterworth, Chebyhev I and II, to etimate, B, and A parameter. Given the filter order N, the implified tranfer function H () for Butterworth and Chebyhev I filter can be expreed a following: N 1 G H () = 1 ( + a) r aco N +1 N π + kπ (). k=0 N + p k For the Butterworth filter, ince pole location are located on the unit circle, a,, become one in the equation (). For the Chebyhev I, ince the pole are not placed on the unit circle but intead form an oval hape center at the origin, a,, parameter dep on the ripple magnitude of the paband. For both cae, if the order N i an odd

2 number, r become one and when the order N i even, r become zero. Given the paband ripple magnitude in linear cale, the ret of parameter can be etimated by p k α = 1+ = a co N +1 N π + kπ N + b in N +1 N π + kπ N, where R p R p 1 parameter are 0.5 db (or 1.1 in linear cale) and 30 db (or 1000 in linear cale) for Chebyhev I and II, repectively. (a) a = 1 N α N ( 1 α ) b = 1 N α + N ( 1 α ). (3) Finally, the gain of Chebyhev I filter i computed. G = a r N ( R p ) r 1 1 p k k=0 (4) For Chebyhev II, the ripple occur in the topband frequency. Given the topband ripple magnitude in linear cale, the implified H () of the Chebyhev II can be expreed a following: H () = N 1 G + k=1 ( k 1)π co N 1 e r 3N k=n+1 Γ = ( R + R 1) 1 N c k = Γ 1 Γ d k = Γ +1 Γ c k c k + d + 1 k c k + d k in (k 1)π N, where co (k 1)π N (5) where e i equal to with k = 3N +. Finally, the gain of Chebyhev II filter i computed by the ratio of the product of all and the product of all, 1 G = a k b k. Once the mathematic are k k implified to be uitable for a mobile OS, the implementation wa executed in ios (iphone/ipad) ytem. Fig. 1 how the output of the prototype lowpa filter order 10 (5 biquad) from an ipad app. Thee reult are identical to Matlab Fdatool. Each row of the SOS matrix contain B and A parameter for each biquad. The firt 3 number are B parameter and the ret are A parameter. The ripple (b) (c) Fig. 1. Screenhot of analog lowpa prototype output from ios implementation uing (a) Butterworth, (b) Chebyhev I, and (c) Chebyhev II method. All the reult are identical with Matlab. B. Frequency Tranformation The goal of thi tep i to tranform the lowpa filter with cut-off at one to one of four filter type, i.e. lowpa, highpa, bandpa, bandtop, at the deired cut-off frequency. The final form after tranformation to lowpa and highpa filter will be in thi form. H () = f (u) N b H (u) = G k 0 u Bi 1 + b k1 u + b k (6) a k 0 u + a k1 u + Note that the gain and number of biquad doe not change during thi tep. The final output are till in the biquad form. The implementation in thi tep i to etimate and parameter. After derivation, the olution become a imple ubtitution of the previou tep. For the lowpa filter with cut-off frequency Hz, the mapping equation can be expreed a b k0 a k0 k=0 a k = b k0 b k1 = b k1 Ω b k = b k Ω (7). = a k0 a k1 = a k1 Ω a k = a k Ω On the other hand, for highpa frequency with cut-off frequency Hz, the mapping equation can be expreed a b k0 a k0 = b k b k1 = b k1 Ω b k = b k0 Ω (8). = a k a k1 = a k1 Ω a k = a k0 Ω For the bandpa and bandtop, the final form after tranformation i in the fourth order. The decompoition back to biquad form will be applied after bilinear

3 tranformation in the next tep. For the frequency tranformation of the bandpa filter with cut-off frequency Hz. and bandwidth Hz., can be expreed a following: H () = = f (u) N b H 4 (u) = G k 0 u 4 + b k1 u 3 + b k u Bi 1 + b k 3 u + b k 4 a k 0 u 4 + a k1 u 3 + a k u k=0 + a k 3 u + a k 4 (9). The mapping function for the bandpa filter can be expreed a following. b k0 = b k0 b k1 = b k1 Δ b k b k3 b k 4 = b k0 Ω + b k Δ = b k1 ΔΩ = b k0 Ω 4 a k0 = a k0 a k1 = a k1 Δ a k a k3 a k 4 = a k0 Ω + a k Δ = a k1 ΔΩ = a k0 Ω 4 (10) For the mapping function of the bandtop filter, the equation i imply the interchange between, and, of the bandpa filter. b k0 = b k b k1 = b k1 Δ b k b k3 b k 4 = b k Ω + b k0 Δ = b k1 ΔΩ = b k Ω 4 a k0 = a k a k1 = a k1 Δ a k a k3 a k 4 = a k Ω + a k0 Δ = a k1 ΔΩ = a k Ω 4 (11) The cut-off frequency and bandwidth are the prewrapping frequency. Given the ampling frequency and deired cut-off frequency in digital domain, the prewrapping frequency i etimated by Ω = F tan( π F c F ). When uing them in bandpa and bandtop, The cut-off frequency i the geometric mean of the high and low prewrapping frequencie, and the bandwidth i the difference between two frequencie. C. Bilinear Tranformation The goal of thi final tep i to tranform from analog domain to digital domain (z domain). For the lowpa and highpa biquad, the final form i imilar to the previou tep, except the mapping function i different. Again the gain and number of biquad doe not change during thi tep. H (u) u= f (z 1 ) N Bi 1 ˆbk H (z) = G 0 + ˆb k1 z 1 + ˆb k z (1) â k 0 + â k1 z 1 + â k z k=0 The mapping function for the biquad form uing bilinear tranformation can be expreed a following: ˆb 0 = 4b 0 F + b 1 F + b ˆb 1 = 8b 0 F + b ˆb = 4b 0 F b 1 F + b â 0 = 4a 0 F + a 1 F + a â 1 = 8a 0 F + a â = 4a 0 F a 1 F + a For the bandpa and bandtop, ince they are fourth order, the tranformation can be expreed a following: H 4 (u) u= f (z 1 ) H (z) = G N Bi 1 k=0 (13). ˆbk 0 + ˆb k1 z 1 + ˆb k z + ˆb k 3 z 3 + ˆb k 4 z 4 â k 0 + â k1 z 1 + â k z + â k 3 z 3 + â k 4 z 4 The mapping function for the fourth form uing bilinear tranformation can be expreed a following. ˆb k0 = 16b k0 ˆb k1 = 64b k0 ˆb k = 96b k0 ˆb k3 = 64b k0 ˆb k 4 = 16b k0 F 4 + 8b k1 F 4 16b k1 F 3 + 4b k F 4 8b k F + 6b k 4 F 4 +16b k1 F 4 8b k1 F + b k3 F + b k 4 F 3 + 4b k3 F + 4b k 4 F 3 4b k3 F + 4b k 4 F 3 + 4b k F b k3 F + b k 4 (15). For the A parameter, the mapping function i the ame fahion of the above equation by interchanging b to a term. Once the bilinear tranformation i finihed, the decompoition back to biquad i applied. The tak i done by finding 4 root of B and A parameter and pairing the ame complex conjugate term. Thi will make the number of biquad double. The final normalization i done by making and equal to one. Fig. how the reult of digital IIR filter running on an iphone. Fig.. Screenhot of digital IIR filter from ios implementation: (a) Highpa filter uing Butterworth order 10 with cut-off 4000 Hz, and (b) Bandpa filter uing Chebyhev II order 4 with cut-off 1000 and 4000 Hz. Both filter have the ampling frequency Hz. (a) (b) (14).

4 Fig. 3. Screenhot of mobile digital filter deign toolbox on ios device: (a) main menu on an iphone app where the uer can chooe each method, (b) the ytem where the uer can the deign coefficient in floating and fixed point format, and (c) graphing on an ipad app. III. RESULTS Thi mobile digital filtering toolbox wa implemented on ios device, both iphone and ipad. The ios SDK verion 4.3 wa ued in thi project. The OS i compried of 4 layer including the core OS, Core Service, Media, and Cocoa Touch layer. Mot of the program in thi paper were implemented uing Cocoa Touch and Objective-C code. Cocoa Touch i the et of Objective-C framework that provide the building block for ios application. Contained are all the uer interface widget, event and event loop management, API to repond to touch, geture, movement, acce to the camera, file ytem, and other device feature. Fig. 3 illutrate the creenhot of mobile app running on both device. Due to the creen ize, the additional feature uch a real-time audio playback, frequency repone plotting are not implemented in every device. On the ipad app (DSP Aitant app [6] on the app tore ), the program can plot the impule and frequency repone of the deigning filter. The reult can be hown in both floating and fixed point format (16 bit) and ent out via . The iphone app (DSP digital filter app [7] on the app tore ) function identically to the ipad app, except it doe not plot. The iphone program can alo playback the audio paed thru the filter in real-time. Uer can turn on/off the filter function in real-time. Thi mobile tool can alo etimate FIR filter coefficient, which we did not dicu the deign in thi paper. There are 6 different type, e.g. ame 4 type a IIR and the additional Hibert, Differentiator, of FIR filter and 8 window type, e.g. Kaier, Gauian, Hann, that can be elected. The plotting, function, both fixed and floating format, are imilar to IIR tool. The implementation on Android mobile i ongoing. IV. CONCLUSION A filter deign toolbox wa deigned and implemented for mobile device. Different verion were created for the iphone and ipad device becaue of difference in creen ize. The iphone verion allow a uer to deign a filter and filter real time audio playback. The ipad verion i identical, with addition of plotting function for both impule and frequency repone. There are multiple filter type and window that can be implemented in the ytem. The toolbox upport the deign of both FIR and IIR filter type. Once a filter i deigned, the application allow uer to the filter coefficient in a format that can be implemented in Matlab, along with the order, type of filter and other pertinent information. REFERENCES [1] ipad App demontration with phyic profeor Mike Cavagnero, [] Mai Yin Toi, Dekhane, S., ToiChem: A Mobile Application to Facilitate Student Learning in Organic Chemitry, IEEE ICALT, page , 011 [3] Mahmoud, Q.H.; Popowicz, P., A mobile application development approach to teaching introductory programming, IEEE Frontier in Education, page T4F-1 - T4F-6, 010. [4] Jinru Liu, et al., iphone/ipad baed interactive laboratory for ignal proceing in mobile device, ASEE, 011. [5] [6] [7]

5 6/7/1 1:4 PM /Uer/ukitta/Dektop/ButterAnalog.m 1 of 1 function [Bc,Ac,G] = ButterAnalog(N); %% Uage: [Bc,Ac,G] = ButterAnalog(N); %% Deign the normalized analog butterworth filter prototype with OMEGA = 1 %% input i N (order): calar %% output Biquad coefficient with B, A, vector and G %%%%%%%%%%%%% STEP 1 %%%%%%%%%%%%%%%%%%% %%%%%% Create a output martix %%%%%%%%%%%%%%%%% Ac = zero(ceil(n/),3); %% prepare the matrix Bc = zero(ceil(n/),3); %% prepare the matrix G = 1; %%%%%%%%%%%%% STEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% make the incomplete biquad if N = odd %%%%%% %%%%%%%%%%%%% STEP 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% Compute the complete biquad %%%%%%%%%%%%%%%% for k = 0:floor(N/) 1 Bc(k+1,3) = 1; %% the numerator i alway 1/Ac Ac(k+1,1) = 1; %% ^ Ac(k+1,3) = 1; %% 1 Ac(k+1,) = *co(((n+1)*pi/(*n))+k*pi/n); if (rem(n,) == 1) %% coefficient for, even cae k = floor(n/); Bc(k+1,3) = 1; %% the numerator i alway 1/Ac Ac(k+1,1) = 0; %% ^ Ac(k+1,3) = 1; %% 1 Ac(k+1,) = 1;

6 6/7/1 1:3 PM /Uer/ukitta/Dektop/Cheby1Analog.m 1 of 1 function [Bc,Ac,G] = Cheby1Analog(N,Rp); %% uage: [Bc,Ac,G] = Cheby1Analog(N,Rp); %% deign the normalized analog cheby1 filter prototype %% input i N (order): calar %% Rp: ripple value, default at 0.5 %% output i SOS (B,A) and Gain for biquad: matrix %% make a default for Rp if nargin == 1 Rp = 0.5; Rp = 10^(Rp/10); %% linear %%%%%% Create a output martix %%%%%%%%%%%%%%%%% Ac = zero(ceil(n/),3); %% prepare the matrix Bc = zero(ceil(n/),3); %% prepare the matrix G = 1; %%%%%%%%%%%%% STEP 1 %%%%%%%%%%%%%%%%%%% %%%%%% Compute the parameter %%%%%% %% compute alpha alpha = (1+qrt(Rp))/qrt(Rp 1); %% compute a and b a = (1/)*((alpha^(1/N)) ((1/alpha)^(1/N))); b = (1/)*((alpha^(1/N))+((1/alpha)^(1/N))); %%%%%%%%%%%%% STEP,3 %%%%%%%%%%%%%%%%%%% %%%%%% Make a biquad and find G %%%%%%%%%%%%%%%%%%% %% tart to compute biquad for k = 0:floor(N/) 1 %% elect the left hand pole, N/ pole tart with k=0 pr = a*co((n+1)*pi/(*n) + k*pi/n); pii = b*in((n+1)*pi/(*n) + k*pi/n); Bc(k+1,3) = 1; %% the numerator i alway 1. Ac(k+1,1) = 1; %% pair with conjugate pole, S^ Ac(k+1,) = *pr; %% *re{p}s Ac(k+1,3) = (pr.^)+(pii.^); %% ab(p)^ G = G*Ac(k+1,3); if (rem(n,) == 1) k = floor(n/); Bc(k+1,3) = 1; %% the numerator i alway 1. Ac(k+1,1) = 0; %% pair with conjugate pole, S^ Ac(k+1,) = 1; %% *re{p}s Ac(k+1,3) = a; %% ab(p)^ G = (G*(a)^(rem(N,)))/(Rp)^((1 rem(n,))/);

7 6/7/1 1:3 PM /Uer/ukitta/Dektop/ChebyAnalog.m 1 of 1 function [Bc,Ac,G] = ChebyAnalog(N,R) %% uage: [Bc,Ac,G] = ChebyAnalog(N,R); %% deign the normalized analog cheby filter prototype %% input i N (order): calar %% R: ripple value, default at 30 %% output i SOS (B,A) and Gain for biquad: matrix %% make a default for Rp if nargin == 1 R = 30; W = 1; R = 10^(R/10); gm = (qrt(r)+qrt(r 1)); Gr = gm^(1/n); G = 1; ctr = 1; for k = N+1:floor(3*N/); a = in(((*k) 1)*pi/(*N)); b = co(((*k) 1)*pi/(*N)); a = a*((gr^) 1)/(*Gr); b = b*((gr^)+1)/(*gr); Ac(ctr,:) = [1 *a/(a*a+b*b) 1/(a*a+b*b)]; G = G*Ac(ctr,3); ctr = ctr+1; if (rem(n,) == 1) k = floor((3*n+)/); a = in(((*k) 1)*pi/(*N)); a = a*((gr^) 1)/(*Gr); Ac(ctr,:) = [0 1 1/a]; G = G*Ac(ctr,3); ctr = 1; for k = 1:floor(N/) b = co(((*k) 1)*pi/(*N)); Bc(ctr,:) = [1 0 1/(b*b)]; G = G/Bc(ctr,3); ctr = ctr+1; %% tet code with built in matlab code %% [BB,AA] = cheby(n,30,1, ); %% [SOS,G] = tfo(bb,aa)

8 6/7/1 1:7 PM /Uer/ukitta/De.../LowpaToLowpa.m 1 of 1 function [Bt,At] = LowpaToLowpa(B,A,Omega); %% Uage: [Bt,At] = LowpaToLowpa(B,A,Omega); %% mapping a prototype LP to a pecific LP with correct cutoff frequency. %% input are B, A vector (1 by 3) for each biquad tate, Omega i freq cutoff. %% output are Bt, At vector (1 by 3) for each biquad tate. b0 = B(1); %% ^ b1 = B(); %% b = B(3); %% 1 a0 = A(1); %% ^ a1 = A(); %% a = A(3); %% 1 if (a0 == 0) Bt(1) = 0; Bt() = b1; Bt(3) = b*omega; At(1) = 0; At() = a1; At(3) = a*omega; ele Bt(1) = b0; Bt() = b1*omega; Bt(3) = b*omega*omega; At(1) = a0; At() = a1*omega; At(3) = a*omega*omega; %% incomplete biquad %% complete biquad

9 6/7/1 1:8 PM /Uer/ukitta/De.../LowpaToHighpa.m 1 of 1 function [Bt,At] = LowpaToHighpa(B,A,Omega); %% Uage: [Bt,At] = LowpaToHighpa(B,A,Omega); %% mapping a prototype LP to a pecific HP with correct cutoff frequency. %% input are B, A vector (1 by 3) for each biquad tate, Omega i freq cutoff. %% output are Bt, At vector (1 by 3) for each biquad tate. b0 = B(1); %% ^ b1 = B(); %% b = B(3); %% 1 a0 = A(1); %% ^ a1 = A(); %% a = A(3); %% 1 if (a0 == 0) Bt(1) = 0; Bt() = b; Bt(3) = b1*omega; At(1) = 0; At() = a; At(3) = a1*omega; ele Bt(1) = b; Bt() = b1*omega; Bt(3) = b0*omega*omega; At(1) = a; At() = a1*omega; At(3) = a0*omega*omega; %% incomplete biquad %% complete biquad

10 6/7/1 1:9 PM /Uer/ukitta/De.../LowpaToBandpa.m 1 of 1 function [Bt,At] = LowpaToBandpa(B,A,Omega,BW); %% Uage: [Bt,At] = LowpaToBandpa(B,A,Omega,BW); %% mapping a prototype LP to a pecific BP with correct cutoff frequency. %% input are B, A vector (1 by 3) for each biquad tate, Omega i freq cutoff and BW i bandwidth. %% output are Bt, At vector (1 by 3) for each biquad tate. b0 = B(1); %% ^ b1 = B(); %% b = B(3); %% 1 a0 = A(1); %% ^ a1 = A(); %% a = A(3); %% 1 if (a0 == 0) %% incomplete biquad Bt(1) = 0; Bt() = 0; Bt(3) = b1; Bt(4) = b*bw; Bt(5) = b1*omega*omega; At(1) = 0; At() = 0; At(3) = a1; At(4) = a*bw; At(5) = a1*omega*omega; ele %% complete biquad Bt(1) = b0; Bt() = b1*bw; Bt(3) = (*b0*omega*omega)+(b*bw*bw); Bt(4) = b1*bw*omega*omega; Bt(5) = b0*(omega^4); At(1) = a0; At() = a1*bw; At(3) = (*a0*omega*omega)+(a*bw*bw); At(4) = a1*bw*omega*omega; At(5) = a0*(omega^4);

11 6/7/1 1:9 PM /Uer/ukitta/De.../LowpaToBandtop.m 1 of 1 function [Bt,At] = LowpaToBandtop(B,A,Omega,BW); %% Uage: [Bt,At] = LowpaToBandtop(B,A,Omega,BW); %% mapping a prototype LP to a pecific BP with correct cutoff frequency. %% input are B, A vector (1 by 3) for each biquad tate, Omega i freq cutoff and BW i bandwidth. %% output are Bt, At vector (1 by 3) for each biquad tate. b0 = B(1); %% ^ b1 = B(); %% b = B(3); %% 1 a0 = A(1); %% ^ a1 = A(); %% a = A(3); %% 1 if (a0 == 0) %% incomplete biquad Bt(1) = 0; Bt() = 0; Bt(3) = b; Bt(4) = b1*bw; Bt(5) = b*(omega^); At(1) = 0; At() = 0; At(3) = a; At(4) = a1*bw; At(5) = a*(omega^); ele %% complete biquad Bt(1) = b; Bt() = b1*bw; Bt(3) = (*b*omega*omega)+(b0*bw*bw); Bt(4) = b1*bw*omega*omega; Bt(5) = b*(omega^4); At(1) = a; At() = a1*bw; At(3) = (*a*omega*omega)+(a0*bw*bw); At(4) = a1*bw*omega*omega; At(5) = a*(omega^4);

12 6/7/1 1:31 PM /Uer/ukitta/.../BilinearZTranform.m 1 of 1 function [bb,aa] = BilinearZTranform(B,A,F); %% uage: function [bb,aa] = BilinearZTranform(B,A,F); %% Bilinear tranform for order %% input i B,A : vector (1 by 3) %% F: a ampling rate %% output i bb and aa in z domain b0 = B(1); %% ^ b1 = B(); %% b = B(3); %% 1 a0 = A(1); %% ^ a1 = A(); %% a = A(3); %% 1 alp = *F; alp_q = alp^; if (a0 == 0) %% firt order bb(1) = (b1*alp)+b; bb() = ( b1*alp)+b; bb(3) = 0; aa(1) = (a1*alp)+a; aa() = ( a1*alp)+a; aa(3) = 0; ele %% econd order bb(1) = (4*b0*F*F)+(*b1*F)+b; bb() = ( 8*b0*F*F)+(*b); bb(3) = (4*b0*F*F) (*b1*f)+b; aa(1) = (4*a0*F*F)+(*a1*F)+a; aa() = ( 8*a0*F*F)+(*a); aa(3) = (4*a0*F*F) (*a1*f)+a; bb = bb/aa(1); %% normalize aa0 to be 1 aa = aa/aa(1);

13 6/7/1 1:31 PM /Uer/ukitta/.../BilinearZTranform4.m 1 of 1 function [bb,aa] = BilinearZTranform4(B,A,F); %% uage: function [bb,aa] = BilinearZTranform4(B,A,F); %% Bilinear tranform for order %% input i B,A : vector (1 by 5) %% F: a ampling rate %% output i bb and aa in z domain b0 = B(1); %% ^4 b1 = B(); %% ^3 b = B(3); %% ^ b3 = B(4); %% b4 = B(5); %% 1 a0 = A(1); %% ^4 a1 = A(); %% ^3 a = A(3); %% ^ a3 = A(4); %% a4 = A(5); %% 1 alp = *F; alp_ = alp^; alp_3 = alp^3; alp_4 = alp^4; if ((a0 == 0) && (a1 == 0)) %% econd order bb(1) = (b*alp_)+(b3*alp)+b4; bb() = ( *b*alp_)+(*b4); bb(3) = (b*alp_) (b3*alp)+b4; bb(4) = 0; bb(5) = 0; aa(1) = (a*alp_)+(a3*alp)+a4; aa() = ( *a*alp_)+(*a4); aa(3) = (a*alp_) (a3*alp)+a4; aa(4) = 0; aa(5) = 0; ele %% 4th order bb(1) = (16*b0*(F^4)) +(8*b1*(F^3)) +(4*b*(F^)) +(*b3*f) + b4; bb() = ( 64*b0*(F^4))+( 16*b1*(F^3))+0 +(4*b3*F) + (4*b4); bb(3) = (96*b0*(F^4)) +0 +( 8*b*(F^))+0 + (6*b4); bb(4) = ( 64*b0*(F^4))+(16*b1*(F^3))+0 (4*b3*F) + (4*b4); bb(5) = (16*b0*(F^4)) (8*b1*(F^3)) +(4*b*(F^)) (*b3*f) + b4; aa(1) = (a0*alp_4) +(a1*alp_3) +(a*alp_) +(a3*alp) + a4; aa() = ( 4*a0*alp_4)+( *a1*alp_3)+0 +(*a3*alp) + (4*a4); aa(3) = (6*a0*alp_4) +0 +( *a*alp_)+0 + (6*a4); aa(4) = ( 4*a0*alp_4)+(*a1*alp_3) +0 +( *a3*alp)+ (4*a4); aa(5) = (a0*alp_4) +( a1*alp_3) +(a*alp_) +( a3*alp) + a4; bb = bb/aa(1); %% normalize aa0 to be 1 aa = aa/aa(1);