DSpace at VNU: Design and Implementation of high - order digital equalizers for audio signal using matlab and dsk tms 320c6711

VNU JOURNAL OF SCIENCE, Mathematics - Physics, T.XXII, N01 2006 D E S I G N A N D IM P L E M E N T A T IO N O F H I G H -O R D E R D IG IT A L E Q U A L I Z E R S F O R A U D IO S IG N A L U S I N G M A T L A B A N D D S K TM S320C6711 S u n g Ho V a n D epartm ent o f Electronics & Telecommunication, College o f Technology V N U A b s t r a c t In this paper, the transfer function of the seventh order digital graphic equalizer is calculated The gain responses of the digital filters, of individual equalizers and of overall graphic equalizer are designed by MATLAB and implemented by DSK TMS320C6711 These gains can be controlled independently by adjusting the parameters c, a and p of each section in the digital graphic equalizer I n tr o d u c tio n In the audio a nd musical in s t r u m e n t s , the equalizers are used to enhance performance of the t r a n s m i t t e r channels or to improve the quality of sound reaching to the listeners A typical equalizer consists of a low frequency shelving filter a nd t h r e e or more peaking filters with a dju sta ble p a r a m e t e r s to provide a d ju s tm e n t of t h e overall equalizer frequency response over a broad range of frequencies in the audio spectrum In a para m etric equalizer, each individual p a r a m e t e r can be varied independently without effecting the p a r a m e t e r s of the other filter blocks in the equalizer In a graphic equaliz er , its consists of a cascade of peak ing filters with fixed center frequencies b ut a d ju sta ble gain levels The major applications of equalizers are to correct and to improve certain types of problems t h a t may have occurred during the processing or the transfe r process a nd to a l t e r or to reduce the noise The a daptiv e equalizers are basically an adaptiv e filter FIR with coefficients t h a t are ad ju sted by the LMS algorithm to compensate c h an n e l distortions caused by intersymbol interference s (ISI) In th is pa per, allpas s filters are employed to design a nd to realize high order equalizers for audio a nd musical signals The purpose of these equalizers is to increase the de sired frequency components and to reduce the u n desired frequency components in the sound ran ge by modifying the gain response S tr u c tu r e o f h ig h o rd er e q u a lizer A high o r d e r equaliz er is created by connecting a cascade of one first-order with one or more second-order equalizers The frequency response of overall eq ualizer can be controlled by adju sting the center frequencies of each section in the cascade Figure ( 1) shows the block schema of a cascade of a seventh equalizer which consist of one first-order and three second-order equalizers In this block Su ng Ho Van 48 schema, A,(z) is t r a n s f e r function of the first-order allpass filter, while A 2(z), A 3(z), A ,(z) a re t r a n s f e r functions of th e second-order allpass filters The first-orde r e q u a liz e r is cre ate d by add ing one low-pass filter a nd one highpass filter with th e m u lt i p li e r coefficients Cj/2 and 1/2, respectively a nd is c h arac terize d by th e following t r a n s f e r function H , ( z ) = —r ^ - = —L[ l - A | ( z ) ] + - [ l + A | ( z ) ] ; X(z) 2 (1) where c, is a positive p a r a m e t e r ; Aj(z) is a first-order allpass t r a n s f e r function given by A,(z) = oti “ z — 1-ocjZ Figure The block schema of a seventh - order graphic equalizer The frequency res p o n s e of th is first section Hj(z) can be varied by varying the values of p a r a m e t e r s Cị a n d Qị P a r a m e t e r c controls the a m o u n t of boost or cut at low frequencies, while t h e c o n s t a n t Ơ Ị controls the boost or cut bandwidth The c o rre sp on d in g in p u t- o u t p u t relation of the first-order equalizer is described by following difference equation y,[n] = - [(C,+l) + ( -C ,)a 1]x[n] + ị [(0,-1) - ( c , + l ) a 1]x[n-l] + a , y i [n-l] • (3) T h a t shows clearly t h a t th e coefficients of difference equatio n can be adjusted by vary in g th e p a r a m e t e r s Ci a n d ƠỊ The t r a n s f e r function of th e ith second-order equalizer is given by H,(z) = — [l - A, (z)] + ỳ [ l + A; (z)] ,1 = , , , 2 where (4) Design a n d I m p l e m e n t a t i o n o f H ig h - o rd e r D i g i t a l A / X _ a i ~ P i ( l +OCj)z + z A,(z) = - L—1 ~ - - P j (1 + a jZ 49 , i = 2, 3, (5) +aịZ The rela tio ns (4) and (5) show t h a t the ith e q u a liz e r is c re a t e d by combining one b a n d p a s s filter with one bandstop filter T he c e n t e r fr eq uency a n d the 3-dB band w id th of each filter can be varied by varying t h e v a lue s of p a r a m e t e r s k a a nd p, These p a r a m e t e r s of each equalizer can be t u n e d i n d e p e n d e n t l y w i t h o u t effecting the p a r a m e t e r s of the other sections Therefore th e frequency a n d m a g n itu d e response of the overall equalizer can be controlled by a d j u s t i n g t h e s e p a r a m e t e r s The c e n t e r frequency CÙQÌ is controlled by th e p a r a m e t e r Pj, b ecause which is dete rm ined by the following relation cooi = a r c o s ( P ) • The p a r a m e t e r relation (7) Qj d ete rm ines Bw i (6) the 3-dB b a n d w i d t h BW j of each e q ualiz er by arcos (7) 1+ a : The magnitude response of the ith equalizer is controlled by para m eter c = H (ej“0)The t r a n s f e r function of th e overall equalizer a s on fig u re(l) given by Y(z) H( z ) ~ ™ H l ( z ) H ( z ) H ( z ) H (z) (8) N u m e r ic a l r e s u lts G a in r e s p o n s e o f e q u a liz e r a = 0 20 15 m s * - y -20 I -4Ũ □ 0.5 N o r m a liz e d fr e q u e m c y củ/ ti 10 QJ~ IQ I "vT 0 N o r m a liz e d fr e q u e m c y c o /71 Figure Gain response of the bandpass, band stop and secondorder equalizers with the different values of c, a a nd p Sung Ho Van 50 A second-order equaliz er is built by adding one b a n d p a ss filter with one bandstop filter Figure2 shows the gain responses of these filters a nd of the equalizer sim ula ted with different p a r a m e t e r s c, a and p The b a n d p a s s and bandstop filters are designed with the values of a : ƠỊ =0.8; a =0.5; a =0.2 a nd p = 0.315 These filters are employed for im plem enting two second-order equalizers; the first equalizer with th e p a r a m e t e r s : C 10 =1.5; C 20 =2.5; C 30 =5; C 40 =0.5; a = ; p= 0.8 and c , =1; C = ; C =3.5; C =0.7; a, = ; p = 0.315 By connecting in cascade of one first-order equalizer with the second-order equalizers , we can built the higher- order graphic equalizers as plotted on the figurel The figure a nd plot the gain responses of the ba ndpass, bandstop filters, equalizers a nd seventh-order graphic equalizer obtained by synthesizing these filters and individual equalizers from equations (4) a nd (8 ) Figure3 is plot of gain responses with the p a r a m e t e r s of values: a = 0.1584; p = 0.809; p3= 0.309; p.j = - 0.809 and c , = 1.3; C = ; C = 0.95; C = 1.1 and figure4 with a = 0.7267; p = 0.7071; p = 0.1564; p = - 0.7071; and c , = 1.3; C 2=2.75; C3=3.65; C = 3.21 Figure is the impulse response of graphic equalizer which has frequency response given on the figure G ain re sp o n se o f lo w p a ss and b a n d p a ss filGfeam resp on se of h ig h p a ss and ba n d sto p filter 0.5 N o rm a lize d fre q u e n cy ,củ/ h G ain re sp o n se o f th e individual eq u a lize rs "0 0.5 N o rm alized fre q u e n cy co/71 G ain resp on se of th e overal eq ua lize r - 0.5 N o rm a lize d fre q u e n cy , cd/ h 0.5 N o rm a llize d fre q u e n c y ,03/71 Figure G ain response of the bandpass, b a n dsto p filters, individual equalizers a nd seventh-order graphic equalizer with: a = ; p, = 0.809; p3= 0.309; p4= - 0.809; a nd C l = 1.3; C = 1.7; C3 = 1.55; C4=1.31 D esign a n d I m p l e m e n t a t i o n o f H ig h - o rd e r D i g i t a l 51 G a in re s p o n s e o f lo w p a s s and b a n d p a s s filt& o in re s p o n e o f h ig h p a s s an d b a n d s to p filter 05 N o r m a liz e d fre q u e n c y (d/ k G ain re s p o n s e o f individual e q u a liz e rs 0.5 N o r m a liz e d fre q u e n c y 0 N o r m a liz e d fre q u e n c y co/n G a in re s p o n s e o f overal e q u a liz e r , u/ti N o r m a liz e d fre q u e n c y ,co/w Figure Gain response of the b a n dp ass, b andstop filters, individual equalizers and seventh-order graphic equalizer with: a=0.7267; P2=0.7071; (33=0.1564; (34= - 0.7071; and C 1=1.3;C 2=2.75; C =3.65;C4 =3.21 The plots show t h a t the gain response of each equalizer a nd can be regulated independ ently without effecting the p a m e t e r s of th e oth er equalizers and hence the gain response of overall equalizer can be controlled by reg ulating the p a r a m e t e r s of each individual equalizer Therefore, the desired frequency components can be increased or reduced by reg ula tin g the p a r a m e t e r s c, a or p, respectively Impulse response of overall equalizer Sample number.n Sample number.n Figure Impulse response of the graphic equalizer with frequency response given on the figure 52 Sung Ho Van The coefficients of overall graphic equalizer are printed in the following table coeff.h = { 2.2877; 0.2110; -1.2788; -0.2758; -0.1347; 0.4113; 0.1148; -0.3444; 0.3659; 0.2325; -0.1208; -0.1567; -0.2313; 0.1193; 0.1500;-0.0649; 0.0196; 0.0172; 0.0234; 0.0031; -0.0916; -0.0076; 0.0445; 0.0136; 0.0105; -0.0173; -0.0002; 0.0143; -0.0162; 0.0092; 0.0069; -0.0021; -0.0059; 0.0012; 0.0017; 0.0037; -0.0006; -0.0037; 0.0007; 0.0005; -0.0003; -0.0002; -0.0006; 0.0012; 0.0006; -0.0008; -0.0003; -0.0001; 0.0003; 0.0002; -0.0003; 0001; 0002 ; -0.0000; -0.00QJ; -0.0002; 0.0000; 0.0001; -0.0000; 0000 } Im p le m e n ta tio n o f a h ig h - o rd er e q u a liz e r u s in g DSK TMS320C6711 The above se venth-ord er graphic equalizer can be implemented by employing DSK TMS320C6711 In this in s t r u m e n t , the four sets of coefficients of graphic equalizer designed by MATLAB in th e above table is contained in th e file graphicEQcoeff.h Both th e inp u t samples and the set of coefficients are trans formed into the frequency domain Because the filtering is implemented by fast convolution with overlap-add method The complex FFT and IFFT are carried out on th e floating point DSK TMS320C6711 The progra m graphicEQ.C which im ple m e n ts this seventh-order equalizer is tested using an i n p u t voice file Theforce.wav added a sinusoid of the frequency 950Hz which is g e n e r a te d by bass frequency generator In the o utput of overall equalizer, this sinusoidal signal is a tt e n u a t e d , because the dip of the gain response of equalizer occurs a t this frequency component The slider file graphicEQ.gel allows to control four frequency bands of overall equalizer independently The input, o u u t signa ls a nd th eir spectrum of th e overall equalizer can be obtained with a digital oscilloscope, with a signal analyzer, with th e CCS-window or with an earphone C o n clu sio n By using th e first-order and second-order allpass filters , the lowpass, highpass, b a n d p a s s a nd bandstop filters are built These filters are the basic components to stitute the individual equalizers Therefore the overall graphic equalizer has a very simple stru ctu re T h a t m ea n s t h a t the impl ementing the FIR filter is carried ou t rapidly not only on the software b ut also on the hardware Because, it allows to reduce a great n u m b e r of computations as well as the num ber of delays, a d d e r s a nd the coefficient multipliers The MATLAB and DSK TMS320C6711 p r o g r a m s p erm it to control flexibly th e p a m e te r s of each individual equaliz er a nd hence the gain response of the overall graphic equalizer can be controlled flexibly in a desired range of frequency D esign a n d I m p l e m e n t a t i o n o f H ig h - o rd e r D i g i t a l 53 R efe r e n c e s Hồ Văn Sung, x lý sơ tín hiệu Phương p h p truyền thống kết hợp với p h n m ềm M A T L A B T1&2 Nhà Xuất Bản Giao dục H Nội 2003, 2005 Hồ Văn Sung, Thực h n h x lý sơ'tín hiệu m y tín h PC với M A T L A B Nhà Xuất Bản Khoa học Kỹ T h u ậ t Hà Nội 2005 TM S320C 6000 CPU a n d Instruction Set Reference Guide, SP RU 189F, Texas I n str u m e n ts, Dallas, TX, 2005 4.G Pallot, Processeurs de signaux et logique program m able, cours e t TP CNAM/MEDIAS 2002 S K Oppenheim a nd at.all, Discrete-time S ig n a l Procesing, Prentice 1999 Sanjit K Mitra, Digital S ig n a l Processing: A Computer- B a se d McGraw - Hill Irwin 2001 R Chassaing, D S P A pplications using c a n d the T M S C X D SK , J oh n Wiley & Sons , INC 2002 J a n ie s H.McClellan, c Sidney B u r r u s a nd at., Com puter - B a se d Exercises for S ig n a l Processing Using M A T L A B 5, Prentic e Hall, 1998 TM S320C 6000 Code Composer I n s t r u m e n t s , Dallas, TX, 0 S tu d io Tutorial, Hall Approach, SP R U 301C , Texas ... 0.0105; -0 .0173; -0 .0002; 0.0143; -0 .0162; 0.0092; 0.0069; -0 .0021; -0 .0059; 0.0012; 0.0017; 0.0037; -0 .0006; -0 .0037; 0.0007; 0.0005; -0 .0003; -0 .0002; -0 .0006; 0.0012; 0.0006; -0 .0008; -0 .0003; -0 .0001;... p= 0.8 and c , =1; C = ; C =3.5; C =0.7; a, = ; p = 0.315 By connecting in cascade of one first -order equalizer with the second -order equalizers , we can built the higher- order graphic equalizers. .. 2.2877; 0.2110; -1 .2788; -0 .2758; -0 .1347; 0.4113; 0.1148; -0 .3444; 0.3659; 0.2325; -0 .1208; -0 .1567; -0 .2313; 0.1193; 0.1500 ;-0 .0649; 0.0196; 0.0172; 0.0234; 0.0031; -0 .0916; -0 .0076; 0.0445;