TALLINN UNIVERSITY OF TECHNOLOGY. IRO0140 Advanced Space Time-Frequency Signal Processing. Individual Work

Transcription

1 TALLINN UNIVERSITY OF TECHNOLOGY IRO14 Advancd Spac Tim-Frquncy Signal Procssing Individual Work Toomas Ruubn Tallinn 1

2 Thory about sprad spctrum scanning signals: W will start our practical work with analysing and modlling of sprad-spctrum signals usd in radar and sonar systms Radars or sonars that us scanning signals without intra-puls modulation can discovr objcts locatd at diffrnt distancs. Fig 1. Principl of radar Thir masurmnt prcision dpnds on th duration of th puls and th wav's disprsion spd. On rcption scattrd signals from objcts, th highr th rgistrd signal nrgy, th highr th dtction probability of th objct. Basd on th principl of uncrtainty, in th cas of optimal scanning signals, th prcision of objcts is dtrmind by its ffctiv working rang within th frquncy band. Accordingly, on rcption scattrd signals from an objct w can crat nrgtically quivalnt situations in two diffrnt ways 1. Scanning signal is short, without intra-puls modulation and uss high powr E = P T. (1) S Scanning signal has high duration, is complx with rgard to intra-puls modulation (i.. wid in th frquncy band), but with rducd powr E = P T. () S Hr E S is th nrgy of th scanning signal, P is signal powr and T is th duration. Both signals ar nrgtically quivalnt whn th following conditions ar fulfilld T = >, (3) B, T T1 T1

3 P = >, (4) 1 B, P1 P P E = E, (5) S1 S Fig. Enrgtically quivalnt signals As a rul, a complx scanning signal (signals ar complx with rgard to intra-puls modulation) can b xprssd by KV 1 KH 1 (6) s( l) = A( l τ, k+ ( p KH )) W ( l, p, k). p= k= Hr, valus of th phas manipulatd complx scanning signal k componnt (chip) at discrt tim lar: A( l, k) = a( l k LDi), (7) whr and 1 l ( LDi l) a( l) = othrwis (8) W l p k l = + fs (,, ) cos ω φ p φk (9) l -discrt tim, l =,( L 1), L -duration of th signal in sampls, LDi -duration of th signal lmnt in sampls, KH -amount of (nstd cod) intrnal componnts, KV -amount of (nstd cod) xtrnal componnts,

4 τ -signal dlay, ω -support (cntr) frquncy, fs -sampling frquncy, φ p -phas of p -th xtrnal componnt p {,18 } φ -phas of k -th intrnal componnt φ k k {,18 }. φ, Whn switching to complx amplitud thn quation (6) can b xprssd as KV 1 KH 1 p k (1) p= k= s( l) = A( l τ, k+ ( p KH )) xp( φ φ ). By manipulating th signal phass φpand φ k using a Barkr s cod of 5 lmnts as + 1, + 1, + 1, 1, + 1; 1, 1, 1, 1, 1; , + 1, + 1, 1, + 1; 1, 1, 1, + 1, 1; + 1, + 1, + 1, 1, + 1; (11) w can obtain a vry good signal for distanc masuring in radar or sonar systms. Changs to th phass ar mad according to th valus of th Barkr s cods. Evry xtrnal lmnt of th cod includs an intrnal cod. Th graph for this signal (in continuous tim) is dpictd in Fig. 1. Fig. 5. Scanning signal

5 In thory th Barkr s cods ar short uniqu cods that xhibit vry good corrlation proprtis. Concrning th limitd numbr and th lngth of Barkr s cods, th nstd Barkr s cods shown abov allow us to furthr incras th systm rsolution at th sam powr lvl. Ths short lmntary- and ithr nstd cods ar in principl Dirct Squnc Sprad Spctrum (DSSS) signals. Th trm sprad spctrum simply mans that th nrgy radiatd by th transmittr is sprad out ovr a widr amount of RF spctrum than would othrwis b usd. In cas of DSSS, th transmittr actually sprads th nrgy out ovr a widr portion of th RF spctrum and do not jump from frquncy to frquncy as Frquncy Hoppd Sprad Spctrum (FHSS) systms. As w can s from Figs. to 1, th sprading procss would caus th transmittd spctrum to incras in with by a factor of ( KH KV ) :1. Hr τ is th lngth of th Barkr s cod lmnt (chip). Fig.. Scanning signal without intra-puls modulation f = 5 khz, T = 5 µ s Fig. 3. Frquncy spctrum of th signal without intra-puls modulation f = 5 khz, T = 5 µ s Fig. 4. Scanning signal manipulatd with 5 lmnt Barkr s cod f = 5 khz, T = 5 µ s, τ = 1 µ s

6 Fig. 5. Frquncy spctrum of th signal manipulatd with 5 l. Barkr s cod f = 5 khz, T = 5 µ s, τ = 1 µ s, KH KV = 5 Fig. 6. Frquncy spctrum of th signal without intra-puls modulation f = 5 khz, T = 11 µ s Fig. 7. Scanning signal manipulatd with 11 lmnt Barkr s cod f = 5 khz, T = 11 µ s, τ = 1 µ s Fig. 8. Frquncy spctrum of th signal manipulatd with 11 l. Barkr s cod f = 5 khz, T = 11 µ s, τ = 1 µ s, KH KV = 11

7 Fig. 9. Frquncy spctrum of th signal without intra-puls modulation f = 5 khz, T = 1, 1 ms Fig. 1. Frquncy spctrum of th signal manipulatd with 11 l. nstd cod f = 5 khz, T = 1, 1 ms, τ = 1 µ s, KH KV = 11 Th masurd charactristics of a scanning signal ar dtrmind by th function of ambiguity. 1 j ωt ψ ( τ, ω) = A( t) A ( t τ ) dt = E 1 E s s * * j ωt A ( t) A( t τ ) dt (5.1.1) Hr t is continuous tim and ω is frquncy shift (Dopplr frquncy). Whn th main purpos is to diffrntiat btwn objcts as prcisly as possibl and to stimat distanc, w ar primarily intrstd in th pattrn of th function of uncrtainty ψ (,) τ which is dpictd in Fig

8 Fig. 11. Th pattrn of th function of uncrtainty ψ (,) τ in cas of KH = 5, KV = 5 Signal quivalnt durability T is dtrmind by th main lob width of th function of ambiguity and is many tims lss than th durability of th scanning signal T. Th quivalnt durability T is dtrmind by th ffctiv bandwidth of th scanning signal, if calculatd using: whr 1 T =, (13) F F = f S ( f ) df S ( f ) df (14) and whr S ( f ) is its powr spctrum. Lt us not that in th cas of a scanning signal without intra-puls modulation, 1 T = T, F =, B= T F = 1. (15) T Th adaptiv optimum filtr stors all th nrgy of th rcivd signal at a singl momnt in tim by narrowing th bandwidth to 1/T for a simultanous nois signal. Thus, w can synthsiz diffrnt systms using low puls powr. Diffrncs btwn th units can b dtrmind by thir purpos. Lt us discuss som of thm. W assum that th purpos is to incras th systm's rcognition ability, and at th sam tim, dcrasing th powr lvl from th constant powr rsourcs. Whn B=1, thn in th filtr output, th maximum valu of th signal with a right-angl winding lin is

9 A T1 OF 1( t) = ES1=, (16) whr A is th amplitud. Lt us look now at a signal with th sam durabilityt = T 1, th quivalnt duration of which is B tims smallr than T / = T1 B. Accordingly, th ffctiv bandwidth is B tims largr or F = B * F 1. Thrfor, Having found th ratio: B A T OF ( t) = ES =. (17) E B A T B A T H = = = = B, (18) E A T A T B S 1 S1 1 1 w can s that in addition to an incras in th systm s rsolving ability, w can rduc th powr of th transmittr B tims without causing a dtrioration in th systm's charactristics. If a gratr ability to rcognis an objct is not ndd, thn: T = B T, T = T, F = F (19) and And B A T1 OF ( t) = ES = () H E B A T B A T S 1 = = = = B. (1) ES1 A T1 A T1 Thortically, it is possibl to synthsiz signals using a basis B of any siz; consquntly, scanning can b prformd at any puls powr lvl no mattr how small. In practic, if thr is a shifting objct or shifting sonar or both, w must considr th constraints rsulting from this and usually B< 1. In practic, this mans that th radar or sonar puls powr can b rstrictd by som watts to produc much bttr tactical and tchnical charactristics compard to B= 1 unit.

10 Using digital signal procssing tchnology, w can gnrat scanning signals of any complxity and guarant adaptiv-optimum rcption of ths sprad spctrum signals using acoustic powr that is hundrds and thousands of tims lowr. Complmtary (Golay) cods ar anothr approach to rducing tim sidlobs by using two cods with mutually cancling sidlobs. Work don in th 195s by Wlti and Golay introducd th concpt of cod pairs. Complmntary (or Golay) cods yild autocorrlation function with a singl spik and zro sidlobs. Lt A and B b a complmntary cod st, ach of lngth L-bits, thn ( A A) + ( B B) = Lδ ( t). () Th autocorrlation functions ( A A) and ( B B) ach hav sidlobs with magnituds up to 1% of th ( A A) or ( B B) autocorrlation pak. Howvr, whn th complmntary autocorrlations ar summd, th sidlob lvls ar rducd and th bas of th autocorrlation spik is narrowr rsulting in bttr tim rsolution. For xampl, a pair of 16-lmnt cod complmntary cods givn by Golay ar A= {1,,,,1,1,,1,1,,,,,,1,}. (3) B= {,1,,,,,,1,,1,,,1,1,1,} Th complmntary cod is also a way to stablish tim rsolution much shortr than th duration of th signal. Practically spaking, a sonar (or radar) using complmntary cod modulatd basband signals, i.., transmitting a wavform as th cod, would transmit th two cods simultanously and thn apply ach rfrnc cod to th rturnd signal by splitting th signal and using two corrlators. W can apply th sam tchniqu of phas coding (1) to th complmntary cod squnc and th phas coding dos giv zro tim sidlobs; howvr, th maximum pak valu of th corrlator output givs a smallr maximum output valu. Th tim rsolution ffct is th sam; howvr, th maximum output will b diffrnt. TASK 1 CREATING A MODEL 1. Crat a modl for gnrating th scanning signal with arbritary structur by using formulas 6-1. For that: 1. Log in using domain ELLE. Usrnam : matlab.. Run Matlab R11b by using th shortcut on th dsctop (wait approximatly 1 sconds to s Matlab logo ).

11 3. NB! IMPORTANT!! If you will s window dactivat th licnc or similar, prss CANCEL. 4. Crat foldr for you works as Z:\..\Tim_frquncy_sp\1\your nam. 5. Add th cratd foldr as Matlab working dirctory by using mnu choic St path. 6. Crat your own main script fil (for xampl ivmain.m ) and sav it into th cratd forldr. 7. Crat th following variabls into th ivmain cript fil: a. c - wav spd ( m/s in air, 15 m/s in watr), b. f - carrir frquncy (5 MHz), c. f s - sampling frquncy ( 4* f ), d. λ - carrir wavlngth (calculat it from th carrir frquncy),. p - amount of carrir signal priods in on sprad spctrum signal lmnt (chip), f. τ - duration of th sprad spctrum signal lmnt in tim ( λ * p / c), g. LDi - duration of th sprad spctrum signal lmnt in sampls (you may us th formula cil( τ * f )), s h. KH - amount of (Barkr (nstd) cod) intrnal componnts, i. KV.- amount of (Barkr (nstd) cod) xtrnal componnts, j. A amplitud of th signal. Tabl 1. k. N - amount of snsors ( at th momnt, will b xplaind latr) l. β y.-.signal falling angl (will b xplaind latr), m. d - distanc btwn th snsors (will b discussd latr) n. τ - ovrall signal dlay in tim (in sconds, to dfin, us th formula τ = (( n 1)* d *cos d( β )) / c) (signal dlay in sconds in diffrnt array y channls) o. τ d - ovrall signal dlay in sampls (you may us th formula τ d = round ( f s * τ ),(signal dlay in sampls) p. pa - amount of ovrall signal vctor points/discrts (you may us th formula pa= τ d+ ( LDi * KV * KH) + 1),

12 q. mod -with carrir frquncy, 1- complx form. r. Phas vctorsφ p and φkaccording to th Tabl 1 (+1, -1 π ) 8. Crat a scond script fil signgn.m and add it into your workspac. 9. Crat MATLAB funtion into signgn.m to gnrat sprad spctrum signals with th paramtrs dfind in main modul (abov) taking into account formulas 6-1 with carrir frquncy and without (in complx form). 1. Call th cratd function from th main modul ivmain.m and put th rsult into vctor s. 11. Plot th cratd signal using plot() or stm(functions). 1. Chang diffrnt signal paramtrs and obsrv th rsults. 13. Find th 3D uncrtainty functions and its cuts ψ ( τ,) and ψ (, ) of th following signals: F v a. Monopuls with lngth b. 5 lmnt Barkr cod c. 13 lmnt Barkr cod d. 5 lmnt nstd Barkr cod. 13*5 lmnt nstd Barkr cod f. M-squnc {,,1,1,1,1} Paramtrs. Amplitud A = 1V Carrir frquncy f = 5 MHz Duration of th sprad spctrum signal lmnt in tim τ =.16 µ s 14. Find sidlobs lvls of th uncrtainty functions compard with corrsponding main lob lvls. 15. Find th distanc masurmnt prcision and minimal oprating radius if ths signals usd in data acquisition systm (radar tc.) For array signal procssing modlling w nd dlayd scanning signals. S thory about quidistanc snsor array from lctur 5. It is not nsssary to modl physical array itslf in this courc 16. Crat for cycl (for n=1:n) for gnrating st of dlayd scanning signals. Us th signal gnrator cratd in prvious tasks. Us formulas τ = (( n 1)* d * cos d( β )) / c and/or τ d = round( f s * τ ) to calculat signals dlays in ach array channl n=1:n. Signals mod should b (signals with th carrir frquncy). 17. Dmodulat th dlayd scanning signals by using MATLAB function dmod with mod qam. Put th dmodulation function into th cratd for cycl and aftr th signal gnration function. Dmodulator will giv you IQ componnts of th signal. Crat two dimnsional matrix to stor IQ componnts of th dlayd signals. Sav dlayd signals y

13 into matrix rows (or colums). You may sav IQ componnts as complx numbrs as M(n,o)=I(o)+i*Q(o). Hr o is discrt tim and n is array channl numbr. It is good ida to crat that kind of matrix for ral signals (signals with th carrir frquncy) too. 18. Plot rsults and nsur, that th signals ar proprly dlayd in cas of diffrnt valus of β y. You should plot signals from matrixs rows. 19. Crat nw for cycl (for n=1:n) to find normalizd and cntrd spctrums of th dlayd signals as f=fftshift(fft(m(n,1:pa)))/pa. Crat two dimnsional matrix to stor complx spctrum valus similar to prvious tasks (F(n,o)=f(o)). Also, it is good ida to crat that kind of matrix for normalisd amplitud spctrum. Us function abs to find amplitud spctrum.. Plot amplitud spctrums to nsur corrct rsults. Estmat bandwith of th slctd signals( p. 13 a-f). 1. Crat nw for cycl (for n=1:n) to compnsat dlays of th signals (s thory from lctur 5). You can us th following formula: CF(n,o)=F(n,o)*(xp((*pi*frq(o)*tau1*(- 1j))). Hr: tau1 = τ d / fs, Frquncy vctor frq= ( fs / : ( fs + 1) / pa : fs / ) should b dfind bfor.. Tak invrs Fourir transform from th rows of matrix CF and put th rsults into compnsatd signals matrix as: VS(n, 1:pa)=ifft(CF(n, 1:pa))*pa. To nsur proprs rsults, sparat ral and imaginary part of th compnsatd signals and plot thy sparatly 3. Find th sum of th compnsatd signals, tak th crosscorrlation function sparatly from th ral and imaginary part of th summd signal and aftr that tak th absolut valu from th ral and imaginary part of th crosscorrlation function. 4. Ovsrv th rsults. Why w should procss sparatly imaginary and ral part of th signal? Why it is bttr to us frquncy domain procssing to compnsat tim dlays?