Bài toán mô phỏng xử lý tín hiệu số dsp trong hệ định vị vô tuyến radar đề tài nckh qt 06 07

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC T ự NHIÊN BÀI TOÁN MỒ PHỎNG x LÝ TÍN HIỆU s ố DSP TRONG HỆ ĐỊNH VỊ VÔ TUYẾN RADAR (DSP simulations in Radar System) MÃ SỐ : QT-06-07 CHỦ TRÌ ĐỂ T À I: THS Đ ỗ TRUNG KIÊN CÁC CÁN B ộ THAM GIA: THS TRẦN VĨNH THẮNG CN LÊ QUANG THẢO CN NGUYỄN ANH ĐỨC THÂN THANH ANH TUÂN OAI H O C Q U O C G IA HÁ N O ' TRUNG TẦM THÔNG TIN THƯ VIỀN DT7 HÀ N Ộ I - 2006 m Báo cáo tóm tắt (tiếng Việt) a Tên đề tài, mã số BÀI TOÁN MƠ PHỎNG x LÝ TÍN HIỆU s ố DSP TRONG HỆ ĐỊNH VỊ VÔ TUYÊN Mã số : QT-06-07 b Chủ trì đề tài ThS ĐỖ Trung Kiên, Khoa Vật lý, Trường ĐHKHTN c Các cán tham gia ThS Trần Vĩnh Thắng CN Lê Quang Thảo CN Nguyễn Anh Đức Thân Thanh Anh Tuấn, K48 d Mục tiêu nội dung nghiên cứu Thực thuật toán xử lý tín hiệu siêu cao tẩn radar mơi trường mô e Các kết đạt ■ 01 báo cáo Hội nghị Vơ tuyến tồn quốc, REV2006: Estimation of Radar Detection and False Alarm Probability in the Presence of Noise ■ 02 báo Tạp chí Khoa học VNU, 2006: Design waveform generators and filters in radar system R adar digital filters using TMS320C6416T DSK f Tình hình kinh phí đề tài Chi phí hết kinh phí tạm ứng đề tài là: 15.000.000VNĐ - Thuê khoán chuyên môn: 9.000.000VNĐ Seminar 500.000VNĐ Dịch vụ Internet 1.100.000VNĐ Mua sắm vặt tư, linh kiện điện tử 3.212.000VNĐ Chi phí điện, nước, sở vật chất 600.000VNĐ - Quản lý phí 600.000VNĐ KHOA QUẢN LÝ CHỦ TRÌ ĐỂ TÀI fCs^S, TS Nguyễn Thế Bình ThS Đỗ Trung Kiên TRƯỜNG ĐAI HOC KHOA HỌC T ự NHIÊN HIẼU iruóng 1) and non-integration («,=1) p =—— = ! = 10 1; ‘ T;J T h e fu n c tio n " sw e rh n g m " calcu lates the Po for S w e rlin g V targ ets u sing Eq (21) 6 X X 109 T h e F ig (5 ) s h o w s th e (S N R )i = dB N ole th at the o f p c tic a l d a r are q u ite sm all, due to th e ir n a rro w filter b a n d w id th B and large the Tm Tfi, is v ery se n sitiv e to v a ria tio n s in the th resh o ld le v e l V T due to the ex p o n en t re la tio n sh ip sh o w e d in E q (7 ) F o r ex am p le V t :/ u t 12.95 d B 14.72 d B Ipdj = swerling5(np,snr) snr signal lo noise ratio Tfj m in u lc s ,0 0 h o u rs Pulse integration investigation T he re q u ire m e n ts o f re m e m b e rin g th e p h a se o f cach tra n sm itte d p u lse a s w e ll as m a in ta in in g "f c o h crcn c y d u rin g p ro p a g a tio n is very c ostly and c h alle n g in g to a ch iev e In p c tic e, m o st radar sy stem s u tiliz e n o n -c o h c rc n t in te g tio n P eebles re p o rte d an e m p iric a lly SNR - dB F/g n - Pi, versus S \'R , p /u I :' for no integration and non-cuhcrcnt integration in Swcrlmg y model The Fig notes ứiat It requires less SNR with derived 10 pulses integrated non-cohercntly 10 achievc ihe sam e P u as in the case o f a single pulse For exam ple, w ith P p = and p,„ = (SNR)iu = 55 d B c o m p a re s w ith (SN R )i = 15 77 dB e x p re ssio n fo r th e im p r o \e m e n l fa c to r in Eq.(13): IH n , )L , - w lo g in In g ( / p,„ ) , + U-235/>„ ), Vu Anh P hi °\ B ach Gia D uong b> a) F a c u lty o f P hysics, H anoi U n iversity o f Science, V N U b) College o f Technology, H a n o i N a tio n a l U niversity Abstract T h is a p p lic a tio n re p o rt u se s th e tool of G oldw ave so ftw a re to in v e s tig a te th e basic p rin c ip le s of a ty p ic a l r a d a r sy stem T he r a d a r w aveform s such a s L F M an d c o h e re n t p u lse t r a in a re g e n e te d e asily w ith o u t of d esig n in g com plicated c irc u its T h e d e la y tim e a n d D oppler frequency a re e x tra c te d from th e r e tu rn sig n a ls to c a lc u la te th e n g e a n d velocity of ta rg e ts T he filte rs a re also in tro d u c e d t h a t gives s p e c ta c u la r im p ro v em en t in sig n a l to noise tio s In tro d u ctio n C hoosing a p a rtic u la r w aveform type a n d a sig n a l p ro cessin g tec h n iq u e in a d a r system d e p e n d s h e a v ily on th e r a d a r ’s specific m ission a n d role R a d a r sy ste m s can use C o n tin u o u s W av efo rm s (CW) o r pulse w aveform s w ith o r w ith o u t m odulation M odulation te c h n iq u e s c a n be e ith e r a n alo g or digital [1, 2], In th e la b o to ry scale, it is difficult to re se a rc h on th e r a d a r field b ecause of the expensive of h ig h freq u e n c y devices an d com plicated c irc u it if we w a n t to m ake One of the fav o rite a p p lic a tio n s n o w a d ay s is u se every capacity of p erso n al c o m p u te r to design any electronic sy ste m s It is s im ila r th e FPG A, A S I C or D S P technologies w ith all th e needed basic logic fu n c tio n s a n d a n alo g -d ig ital, d ig ita l-a n alo g c o n v e rte rs inside T h a t is th e re a so n for th is re p o rt focus on th e d e sig n a tio n of w aveform g en era to rs, the re tu rn e d s ig n a ls w ith delay, a tte n u a tio n , D oppler effects a n d filte rs for r a d a r system u sing G oldw ave V Õ so ftw a re of G oldw ave Inc [3], T h eo ries o f w aveform s u sin g in radar tec h n iq u e c w r a d a r s c o n tin u o u sly e m it e le ctro m ag n e tic energy, a n d u se s e p a te tr a n s m it and receive a n te n n a s U n m o d u la te d c w r a d a rs can a c c u te ly m e a su re ta rg e t d ia l velocity (D oppler sh ift) a n d a n g u la r position T a rg e t n g e in fo rm a tio n c a n n o t be e x tra c te d w ith o u t u tiliz in g so m e form of m odu latio n P u lse d d a rs (P R) u se a tr a in of pulsed w aveform s (m ain ly w ith m od u latio n ) In th is category, r a d a r s y s te m s can be classified on th e b asis of the P u lse R e p e titio n F re q u e n c y (P R F ) Low P R F d a r s a re p rim a rily u se d for n g in g w here ta r g e t velocity is n o t o f in te re s t H igh P R F d a rs a r e m ain ly u se d to m ea su re ta rg e t velocity, c w a s w ell a s PR can m e a su re both ta rg e t n g e a n d d ia l velocity by u tilizin g d iffere n t m o d u la tio n schem es R a d a r w a v e f o r m s [2] cv v a n d n u lse d w aveform s: c w is given by ( Fi g l a ) : f t ( t ) = A c o s o ì 0t N ex t c o n sid e r th e tim e d o m ain of sig n a l f J t ) given by (Fig lb): 19 (1) f 2( t ) = A R e c t ( - ) = \ A [0 N ow is th e c o h e re n t g a te d 2** otherw ise (2) cw w aveform f 3(t):f 3( t ) = ỵ ^ f 2( t - nT) (3) n=-co C learly f 3(t) is periodic, w h e re T is period (recall t h a t f r - 1IT is th e P R F ) _ N T h e fu nction f 4(t) (lim ite d d u tio n f / t ) ) (Fig.lc): f A( t ) =Ydf ( t - n T ) (4) n=0 Fig A m p litu d e spectrum o f c w a n d pulsed, w aveform s (a) CW ; (b) sin g le p u lse; (c) coherent p u lse tra in o f in fin ite length L in e a r F re q u e n cy M o d u la tio n W aveform s F re q u e n c y o r p h a s e m o d u la ted w aveform s can be u se d to ach iev e m uch w id e r o p e ratin g b a n d w id th s L in e a r F re q u e n c y M od u latio n (LF M ) is com m only used In th is case, th e frequency is sw e p t lin e a rly across th e pu lse w id th , e ith e r u p w a rd (up-chirp) o r dow nw ard (dow n-chirp) T h e F ig.2a, b show s a typical exam ple of a n L F M w aveform T he p u lse w id th IS r, a n d th e b a n d w id th is B F ig.2 T yp ica l L F M w aveform s, (a) up-chirp; (b) d o w n -c h irp ; (c) m a g n itu d e spectrum f A ty p ic a l L F M c an be expressed: s1( t ) = R e c t( - ) e j 2n ( ’ J =e , s (t) (5) T s ( l) = R e c t(—) e mu T is env elo p e fu n ctio n of s,(t) T he sp e c tru m is sh o w n in Fig.2c c w r a d a r s m ay u se LFM w aveform s so t h a t b o th n g e a n d D oppler in fo rm a tio n c an be m ea su re d In one sp e c ia l te c h n iq u e of d a r, th e p u lse c o m pression is accom plished by a d d in g freq u e n c y m o d u la tio n to a long pu lse a t tra n s m iss io n , a n d by u sin g a m atc h ed filter re ce iv e r in o rd e r to co m p re ss th e received sig n al U s in g long p u lse s a n d w ideband L F M m od u latio n w e can ach iev e la rg e com pression ratio s E xp erim en t R esu lts U sin g th e G o l d w a v e v5.1 , w e can use th e com plex D ig ital S ig n a l P rocessing (DSP) in sid e a c o m p u te r to m a k e a n y re q u ire m e n t of d ig ita l sig n a l (Fig.3) S o u n d c ard is u se d for p lay b ack a n d re c o rd in g w ith sa m p le freq u e n c ie s of u p to 44.1 kH z S a m p le frequencies o u tsid e th is n g e can p roduce u n e x p ec te d re su lts So th e r a d a r sig n a ls th is ap p licatio n re p o rt m ad e w ill also h a v e th e lim ite d frequency of 44.1 kH z T h ere fo re it m ay be said th a t th e s e r e s u lts o f th e re p o rt is j u s t for la b -ra d a r m odel 20 Oscilloscope PC Yokoqawa Digital Oscilloscope DL1720E Soundcard L in e o u t Fig.3 Block d ia g m o f experim ent R a d a r S ig n a l G en erators a n d Processing A im p o rta n t sig n a l of r a d a r is L F M w aveform (F ig ) T h e p u lse w id th r a n d the bandwidth B c a n b e c o n tro lled e asily for th e b e st com pression g a in B t W ith p u lse d r a d a r , w e c a n c re a te a sim p le s q u a re p u lse s (Fig.5) T he pulse w id th r a n d P u lse R e p e titio n In te r v a l (P R I) c an be a d ju ste d to ach iev e h ig h sig n a l to noise tio {SN R ) be ca u se o f th e p e a k tr a n s m itte d pow er p , in [i] is p ro p o rtio n a l w ith pu lse w idth As m en tio n ed above, L F M w aveform s h a v e long p u lse a t tra n s m is s io n ( T:ransmiUed long), a n d by u sin g a m a tc h e d filte r re c e iv e r in o rd e r to com press th e received sig n a l in o rd e r to achieve h ig h re so lu tio n (r„ MlW sh o rt) PtG 2}?a R a d a r e q u a tio n : ( S N R ) IIUI = ( ); ( k ) k T B F L R »6^07^17 ze 17:61 (OGMMA♦ cx c R a n g e reso lu tio n : AR = — = —— (7) 2B ,yt ■ -— ■. - ,k I ■—5 Transmitted F ig.4 U p-chirp L F M ob served by Yokogaw a D ig ita l O scilloscope D L E Pultts n im r p - iji ave n i't observed n h s p ru p d h nht Fig.5 SS im im.npl.t* le asquare-w byv Y Yokogaw a D igital O scilloscope D L Ỉ7 E G oldw ave a lso c a n p ro g ram to m ak e c om plicated c o h e re n t p u lse tr a in in w aveform s of sine, 111V) O sq^U uU a re l V) , U o ri esp VOj/VViUll e cially L F M t r a in (F ig.6a, b, c) rt lS Si t - ~ Wra ^ ** tàS/-t W OCH I » I 1M» jf -f -M (a ) (b) (c) F ig.6 C oherent p u ls e tra in , (a) S in e w aveform ; (b) S q u a re w aveform ; (c) Ỉ F \Í w a te fvrm T h e t a r g e t ’s n g e , R , is co m p u ted by m e a su rin g th e tim e delay, A t\ it ta k e s a pu lse to tra v e l th e tw o -w ay p a th b e tw ee n th e r a d a r a n d th e ta rg e t: R = (8) F ig.7a sh o w s t h a t w e n e e d to n o te a b o u t th e le n g th of th e p u lse w id th sh o u ld not be to long, o r o v e rla p e asily to occur for a n e a r ta rg e t cases Fig 7c desc rib e s th e r e tu r n s in the cases h av e m o re th a n one ta rg e ts W e m u st be careful in c alc u la tio n s to avoid false a la rm s of w ro ng ta rg e ts , or w ro n g b e tw ee n ta rg e ts a n d s ta tio n a ry c lu tte rs Transm it Echo Return 'rom t.1 It '• -Vi'—"i-v.v-j ill III iij if III III in ! ! !> ■ n '■ O ve rla p d ue to s h o rt d e la y tim e (a ) (b) (c) Fig R a d a r echoes, (a) O verla p d u e to sh o rt delay tim e m s; lb) J f = 1.8ms, m ore atten u a tio n : R e tu rn s fro m targets a t d ifferent positions fci R a d a rs u se D oppler freq u e n c y f d to e x tra c t ta r g e t d ia l velocity, a s well a s to d istin g u ish b e tw e e n m oving a n d s ta tio n a ry ta rg e ts : f j = - v/ (Fig a , b) (9) 21 ****»♦ |lrQoemq argot cautts frie highg fraqm ncy MM, I'.'- Li i it " * Racadinfltargst causes the l o w frequency (a) (b) (c) F ig.8 (c) D oppler effects, (a) P ositive D oppler frequency f d; Cb) N eg a tive D oppler frequency fd F ilters F ilte rs a re n e c e ssa ry p a r ts in th e r a d a r system u sin g to im prove th e r a d a r detectin g te c h n iq u es, e sp e cially in c re a s in g th e S N R w hen sig n al is p re s e n t alo n g w ith th e noise In our w ork, w e c re a te a sin e w ave sig n a l 1.000 kH z plu s w ith th e w h ite noise, a n d desig n th e b a n d ­ p a ss filte r w ith freq u e n c y n g e from 995 Hz to 1005 H z to rem ove a lm o st t h a t w h ite noise (Fig.9) ty ệ ệ ặ m tệ ệ m * I s i g n a l ♦ W h lta N o is a O u u t of the B P fl Fig Effect o f the filte r in d a r detection techniques C on clu sion s U sin g th e G oldw ave softw are, w e so m e w h a t overcom e difficulties of h a v in g expensive hig h freq u e n c y devices in r a d a r in v estig a tio n s All th e w aveform s m ad e a n d th e e stim a tio n s of n g e s, velo cities a n d th e v a rio u s filte rs give u s th e ex cellent tool to d esign a la b -ra d a r m odel T h e f u r th e r w o rk s w ill use th e s e w aveform s a s th e in p u ts of a D SP b o ard for d a r sig n a l processing R eferen ces: [1], Do T rung Kien, Bach Gia Duong, T ran Van T uan “Im provem ents o f Sig n a l to Noise Ratio, R ange Detection a n d Resolution, o f R a d a r", N ational Conference on Physics 2005 [2] Bassem R M ahafza, "R adar System s A nalysis and D esign Using M atlab", C hapm an & Hal] /CRC, 2000 [3] GoldW ave M anual, C opyright by 2006 GoldWave Inc 22 RADAR DIGITAL FILTERS USING TM S320C6416T DSK Do T ru ng Kien a>, Than T hanh Anh Tuan °\ Vu Anh P h i a), Bach Gia D u o n g 01 a) F a cu lty o f P hysics, H anoi U n iversity o f Science, V N U b) College o f Technology, H anoi N a tio n a l U niversity A bstract T h is r e p o rt d isc u sse s th e im p le m e n ta tio n of F in ite Im p u lse R esponse (FIR) filte rs, o n e o f th e m o st c ritic a l p a rts of a r a d a r sy ste m , u s in g th e TM S320C 6416T D S P S t a r te r K it o f T ex as In s tru m e n ts T he c codes a re w ritte n w ith th e h elp of M a tla b ’s S P tool to c re a te th e filte r’s coefficients a sso c iated w ith th e c irc u la r a d d re s s s tr u c tu r e in DSK By a lte rin g th e se ts of coefficients, we c a n m ak e the filte r re sp o n d in d iffe re n t w ays to th e d iffere n t freq u e n c ie s ju s t in one generic F IR p ro g ram In tro d u ctio n T h e a b ility of a n F IR filte r to o p e te sa tisfa c to rily in a n u n k n o w n e n v iro n m en t and tra c k tim e v a ria tio n s of in p u t sta tis tic s m ake th e F IR filte r a pow erful device for signalprocessing a n d c ontrol a p p lic atio n s Indeed, F IR filte rs h a v e b een successfully ap plied in such div erse field s a s co m m u n ica tio n s, d a r, so n ar, seism ology, a n d biom edical engineering A lthough th e s e a p p lic a tio n s a re in d ee d q u ite d iffere n t in n a tu re , n e v e rth e le ss, th e y h av e one basic com m on fe a tu re : a n in p u t vector a n d a d e sire d re sp o n se a re u se d to com pute an e stim a tio n e rro r, w hich is in tu r n u se d to control th e v a lu e s of a se t of a d ju s ta b le filter coefficients T h eory o f filte r w ith TM S320C6416T D SP S ta rter Kit F o r a la rg e v a rie ty of ap p lic atio n s, d ig ita l filte rs a re u s u a lly b ased on th e following re la tio n s h ip s b e tw e e n th e filte r in p u t sequence x(n) a n d th e filte r o u u t seq u en ce y 'n ) [4] : N M y ( n ) = Y i a k x ( n - k i - Ỵ 'b Jy ( n - j ) (1) *=0 7=1 E q (l) is re fe rre d to a s a lin e a r co n sta n t coefficient difference equation C o n c e p ts o f t h e F I R [1-3] A d isc re te sig n a l x(n) c a n be e x p ressed as: x (n )= Y ^ x (m )b t n - m i (2) m = r, w h e re $ n - m ) is im p u lse se q u e n ce ã n ) d elayed by m T h e sig n a ls a n d sy ste m s t h a t we deal w ith a re lin e a r a n d tim e in v a ria n t, w h ere both su p e rp o sitio n a n d sh ift-in v a ria n c e apply If th e in p u t is a u n it im p u lse ã n ) th e re s u ltin g o u u t re sp o n se is h(n), h (n) is d e sig n a te d as th e im p u lse re sp o n se T h e n x ( m )ẵ n -m ) -> x(m )h (n -m ) by th e sh ift-m v a ria n c e property U sing (2), th e re sp o n se becom es: y (n )= ỵ ^x(m )h (n -m ) L e ttin g k = n-m y ie ld s :y i n ) = ị í h ( k ) x ( n - k ) T h is convo lu tio n e q u a tio n is very useful *=0 for d esign of F IR filte r sin ce w e can a p p ro x im a te it w ith a fin ite n u m b e r of te rm s, or 23 N-1 y (n )= ^ h ( k ) x ( n - k ) (3) *=0 Eq.(l) reduces to Eq (3) with ak = h(k) and bj = F eatures o f FIRs T h e F IR g e n e lly h a s “lin e a r phase" A sig n a l p a ssin g th o u g h th e filte r w ill be d elayed by a fixed tim e period, so th e re la tio n s h ip b e tw ee n h ig h freq u en cy a n d low frequency p a ssin g though the filter stays the same T h e F IR is “in h e r e n tly stable" A nalogue filte rs (a n d In fin ite Im p u lse R esp o n se IIR filte rs) a re v e ry s im ila r to oscillato rs G et o u r design w rong, th e filte r m ay oscillate th is c a n n o t h a p p e n w ith a n d F IR filter Im plem en tin g filte rs on the TMS320C6416T D SP S ta rte r K it T h e T M S 320C 6000 fam ily processors a re good a t filte rin g , h a v in g b een designed for the ty p es of o p e tio n com m on in r a d a r sig n a l processing sy ste m s In th e s e sy ste m s, sa m p le s are ta k e n in a c o n tin u o u s s tre a m - ty p ically from a r a d a r o p e tin g in re a l-tim e It is im p o rta n t th a t th e o u u t is c a lc u la te d a s quickly a s possible T h is case is called “real-tim e processing” H ow ever, a n o th e r sy ste m m ay g a th e r a larg e se t of d a ta before s ta rin g to process it T h is is know n a s “block p ro cessin g " H ow ever, th e F IR is v e ry efficient on th e T M S320C 6xxx so block p ro c essin g is n o t re q u ire d [4\ Table M em ory o rg a n iza tio n for co effic ie n ts an d sam p le 12] i Coefficients h(0) h(l) h(2) Sample x(n) x(n-l) x(n-2) N-l h(N-l) x(n-OM)) Table M em ory o rg a n iza tio n to illu strate u pdate o f sam p les 12] h(0) h(l) h(2) Sample Time n x(n) x(n-l) x( n-2) Time n+1 x(n+l) x(n) x(n-l) Time n+2 x(n+2) x(n+l) x(n) h(N-2) h(N-l) x(n-(N-2)) x(n-Oi-l)) x(n-(N-3)) X(n-(N-2)) x(n-(N-4)) x(n-(N*-3)) Coefficients T he im p o rta n t p o in t in re a l-tim e ap p lic atio n s is w h e n a new sa m p le a rriv e s, it is added to th e sa m p le se t, a n d th e o ld est sa m p le is disposed of T h is can be p erfo rm ed u sin g the c ircu la r a d d re ss in g h a rd w a r e of th e C6000 fam ily E x p erim en t re su lts F r e q u e n c y r e s p o n s e s o f lo w -p a s s , h ig h - p a s s , b a n d - p a s s , b a n d - s to p f i l t e r s T he c so u rc e p ro g m FIR.C im p le m e n ts F IR filte rs by th e Eq.(3) It is a g eneric FIR p ro g ram , sin ce coefficient files specify th e filte r’s c h a c te ris tic s T h is is th e sp e c ta cu la r fe a tu re of F IR filte r By a lte rin g th e se ts of coefficients, we can m a k e th e filte r resp o n d in d iffe re n t w ay s to th e d iffe re n t freq u en cies W ith th e h e lp of SPTool/FD A Tool of M a tla b , we can c a rry o u t th is m issio n easily F o r a n F IR filte r to h a v e lin e a r p h a se , th e coefficients m u st be sy m m etric F o r ex am p le, w e can see in sid e th e b p 00.c o f c re a te d by SPtool S e t o f bp2100.cof sym m etric co effic ie n ts #define N89 short h[N] = (-81 -17, 53, 0, -3, -73, 27 156 -81, -233, 162 287 -257 -303, 345 275, -399 -207, 393, 117 -304 -36 124 142 -48, -468, 208 811, -496, -1120 904, 1341 -1403 -1428, 1944 1354 -2462 Table 1115 2892 734 -3177, 256, 3277 -256 -3177 734 2892 -1115 -2462 1354 1944.-1428.-1403 1341, 904 - 120 -496 811, 208, -468 -48 142, 0, 124 -36, -304 117 393, -207, -399, 275 345, -303 -257 287,' 162 -233 -81,156.27 -73 -3 53.-17,-811; B u ilt a n d ru n th is project in D S K C C S tu d io e n v iro n m e n t F or d e ta il illu stra tio n of the 24 F IR o f b a ss-p a ss 0 H z, w e c a n u s e s q u a re w ave a t th e in p u t T he s q u a re w ave is m ad e up of a fu n d a m e n ta l freq u e n c y a n d all odd h a rm o n ies, th e o re tic a lly to in fin ity T h ere fo re th e F IR filte r j u s t p a sse d th e n a rro w freq u en cy b a n d eq u al th e b a n d -p a ss of filte r I t m ea n s t h a t th e o u u t is a lm o s t th e sin e w av e (F ig l ) T h e a m p litu d e s of th e sin e w ave in Fig lb , c a re s m a lle r th a n th e case 0Ỉ F ig l a because of th e fu n d a m e n ta l fre q u e n c ie s in th e c ases (6) a n d (c) (1900H z a n d 2300H z) a re not 'Ỉ ' • A A A A A A A A /V ’ TJ •A /V W V W W \ A (b) fc) F ig I- In p u t a n d O u u t o f the F IR filte r, observed by Yokogaw a O scilloscope D L Ỉ7 E (a)2100H z (b) 1900Hz (c) 2300H z A v e ry im p o rta n t m eth o d to in v e s tig a te th e freq uency re sp o n se s of th e filte r is u sin g th e w h ite noise a s th e in p u t F or th is purpose, p ro g ram n o is e _ g e n c is w ritte n g e n e te a (1) • 48kH< lunlled by Ihe sim plm g frequency of 96000M1