MULTIPLEX BUS DATA WAVEFORM SPECTRA. J. Maeks MAY Prepared for DEPUTY FOR DEVELOPMENT PLANS

Transcription

1 ijbki CallSW*. MTR-2963 MULTIPLEX BUS DATA WAVEFORM SPECTRA J. Maeks MAY 1975 Prepared fr DEPUTY FOR DEVELOPMENT PLANS ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE Hanscm Air Frce Base, Bedfrd, Massachusetts Apprved fr public release, distributin unlimited. Prject N Prepared by THE MITRE CORPORATION Bedfrd, Massachusetts Cntract N. F C-0001 ArjUuitfli.

2 When U.S. Gvernment drawings, specificatins, r rhei data are used fr any purpse ther than a definitely related gvernment prcurement peratin, the gvernment thereby incurs n respnsibility nr any bligatin whatsever; and the fact that the gvernment may have frmulated, furnished, r in any way supplied the said drawings, specificatins, r ther data is nt t be regarded by implicatin r the'wise, as in any manner licensing the hlder r any ther persn r crpratin, r cnveying any rights r permissin tc manufacture, use, r sell any patented inventin that may in any way be related theret. D nt return this cpy. Retain r destry. REVIEW AND APPROVAL This technical reprt has been reviewed and is apprved fr publicatin. EMERyF. BOOSE Prject Leader MITRE D-92 FOR THE COMMANDER CITH HANDSAKER Prject Engineer '/Qfrt. & K /CARL A. SEGERS*OM Acting Directr, Technlgy Deputy fr Develpment Plans

3 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) 1. REPORT NUMBER ESD-TR REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) MULTIPLEX BUS DATA WAVEFORM SPECTRA 7. AUTHORf*.) J. Maeks 5. TYPE OF REPORT a PERIOD COVERED 6. PERFORMING ORG. REPORT NUMBER MTR CONTRACT OR GRANT NUMBER(s) F C PERFORMING ORGANIZATION NAME AND ADDRESS The MITRE Crpratin Bx 208 Bedfrd, MA CONTROLLING OFFICE NAME AND ADDRESS Deputy fr Develpment Plans, Electrnic Systems Divisin, AFSC.Hanscm Air Frce Base, Bedfrd, MA PROGRAM ELEMENT. PROJECT, TASK AREA 4 WORK UNIT NUMBERS Prject N REPORT DATE MAY NUMBER OF PAGES MONITORING AGENCY NAME ft ADDRESSfH different frm Cntrlling Olfice) 15. SECURITY CLASS, (f this reprt) 16. DISTRIBUTION STATEMENT (l this Reprt) UNCLASSIFIED 15a. DEC LASSIFI CATION DOWNGRADING SCHEDULE Apprved fr public release; distributin unlimited. 17. DISTRIBUTION STATEMENT (f the abstract entered In Blck 20, It different frm Reprt) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Cntinue n reverse side if necessary and Identity by blck number) DIGITAL DATA BUS SPECTRAL ANALYSIS TIME DIVISION MULTIPLEX 20. ABSTRACT (Cntinue n reverse side It necessary and Identify by blck number) This paper cntains a derivatin f the frmula fr the average energy spectral density f a multiplex bus data wrd. It als cntains graphs f the spectral density and the cumulative distributin f energy versus frequency. DD 1 JAN EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

4 SECURITY CLASSIFICATION OF THIS PAGEfHTien Data Entered; i SECURITY CLASSIFICATION OF THIS PAGEfWhen Data Entered;

5 TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS 2 SECTION I INTRODUCTION 3 SECTION II FUNDAMENTAL RELATIONSHIPS 7 SECTION III CONCLUSIONS 16 APPENDIX DERIVATION OF THE EXPRESSION FOR THE AVERAGE ENERGY SPECTRUM OF THE MULTIPLEX BUS DATA WAVEFORM 18 REFERENCES 31

6 LIST OF ILLUSTRATIONS Figure Number Page 1 Elementary Wavefrms 6 2 Average Energy Spectral Density, r = Micrsecnd 10 3 Average Energy Spectral Density, r = 0.25 Micrsecnd 11 4 Cumulative Distributin f Energy at Lw Frequencies, r = Micrsecnd 12 5 Cumulative Distributin f Energy at Lw Frequencies, r = 0.25 Micrsecnd 13 6 Cumulative Distributin f Energy, r = Micrsecnd 14 7 Cumulative Distributin f Energy, r = 0.25 Micrsecnd 15

7 SECTION I INTRODUCTION MITRE Prject 6370, Radi Infrmatin Distributin Systems (RIDS), is investigating the use f multiplex busing techniques in aircraft t facilitate the intercnnectin f avinic subsystems. In August f 1973, the Air Frce issued a standard, MIL-STD-1553 (USAFK, which is a fundamental reference dcument fr this prject. This dcument defines a set f standards fr time divisin multiplex digital data bus systems. It specifies the bus system cnfiguratin, the signal structure, the wrd frmat, data rates and the bus interface circuitry. One aspect f MITRE's wrk n RIDS is t suggest changes t MIL- STD-1553 where this standard in its present frm is unwrkable r unduly restrictive. One area in which it was felt that the standard might present unnecessary difficulties t a system designer is that f the input impedance t the Multiplex Terminal Unit (Paragraph f MIL-STD-1553). Fr the frequency range f dc t 100 khz, this is specified t be at least 6800 hms. Since the impedance is measured at a pair f terminals which are cnnected by a transfrmer winding, it is expected that this specificatin wuld be difficult t meet. It was suspected that because f the way in which the bus wavefrms were specified, very little f the wavefrm energy wuld be fund belw 100 khz. T the extent that this is true, the lw frequency input impedance specificatin culd be relaxed withut serius cnsequence. This, then, is the mtivatin fr the study f multiplex bus wavefrm spectra as described in this paper.

8 Accrding t the standard, there are three types f wrds which are transmitted n the bus: cmmand, data and status. Each wrd cnsists f a sync wavefrm and a sequence f seventeen Manchester II bi-phase level wavefrms representing sixteen infrmatin bits and ne parity bit. Since the wavefrms n the multiplex bus are sample functins frm a randm prcess, their spectral characteristics are best described by a pwer spectrum. The cmputatin f the pwer spectrum fr the cmplete wavefrm prcess n the bus wuld require knwledge f the statistics f ccurrences and timing f all three types f wrds and f the infrmatin bits within each wrd. These statistics are system dependent and cannt be deduced frm the standard. Thus, we cannt cmpute this pwer spectrum, even if we were willing t expend the effrt t d s. Althugh it is nt pssible t cmpute the pwer spectrum f the cmplete wavefrm prcess n the bus, ther types f spectral cmputatins are feasible. The principal criteria fr chice amng such spectra is that the spectrum chsen be reasnably easy t cmpute and representative f bus wavefrm spectra in general. Given these criteria, an apprpriate wavefrm n which t base a spectral cmputatin is the wavefrm crrespnding t a single data wrd. It is relatively simple cmpared t the ttality f wavefrms n the bus, and it is reasnable t assume fr this wavefrm that the underlying infrmatin bits are independent and have equal prbability f being '0' r '1'. These assumptins are nt as reasnable fr the cmmand and status wrds. Since there are 2 pssible data wrds, it will be apprpriate t cmpute the average energy spectral density, where the averaging is dne ver all pssible data wrds. It will als be necessary t make sme assumptins regarding the wave shape f the data wrd wavefrms as these are nt cmpletely specified by the standard.

9 Accrding t MIL-STD-1553, a data wrd has duratin f 20 us. This duratin cmprises 20 bit times f 1 us each. The first three bit times are ccupied by the sync wavefrm, the next sixteen bit times by wavefrms representing the sixteen infrmatin bits and the last bit time by a wavefrm representing the parity bit. The elementary wavefrms representing infrmatin r parity bits are called Manchester II bi-phase level wavefrms. The sync wavefrm is an invalid Manchester wavefrm (i.e., it culd nt be the result f a sequence f Manchester wavefrms representing infrmatin bits). The parity bit is chsen s that the resultant sequence has dd parity. In any infrmatin r parity bit psitin, the wavefrm representing a '1' is the negative f the wavefrm representing a '0'. There is a minimum f 150 ns specified fr the wavefrm rise and fall times, when measured at the ten percent and ninety percent pints f the specified signal vltage limits. We shall make the assumptin that the wavefrms are all trapezidal r triangular in shape and that the rise and fall times f sync, data, and parity wavefrms are all equal. If, in additin, this cmmn rise/fall time is precisely 150 ns as measured at the ten percent and ninety percent pints f the wavefrms, then the assumed waveshape has 187 ns rise/fall times as measured at the 0 and 100 percent pints. Making all these assumptins, the elementary wavefrms ut f which a data wrd is frmed, are shwn in Figure 1. The elementary data r parity wavefrm shwn in the figure represents a '0'.

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11 SECTION II FUNDAMENTAL RELATIONSHIPS If f(t) is a (abslutely integrable) real functin f time with Furier transfrm F(u)), defined as F(u>)«/f(t) e" jwt dt (1) then we may define $ ff (c) " 2^ l F(w) 2 (2) as the energy density spectrum f f(t) Parseval's Therem gives us / f2 W " 'f 27 2 F(u)) dw (3) Thus, the integral ver frequency f the energy density spectrum is the wavefrm energy, as required. Let the (time) autcrrelatin functin, $,,(T), be given by 00 4> ff <T) = / f(t) f(t + x) dt (4) Nte that if g(t) = ±f(t + s), where s is a cnstant, then $ (x) = < > ff (T). Then lf(t) 2 = I 'ff 00 7

12 Thus, CO * ff (>) = 4> ff (T) e di (6) Let (t) be a randm functin f time which is a member f the set A. If 'E' dentes expectatin, we may define S AA (W) = E [*ffh fea (7) as an average energy density spectrum. Fr the situatin f interest t us, f is a particular data wrd and A is the set f all pssible data wrds. We shall defer the derivatin f the average energy spectrum f a multiplex bus data wrd t the Appendix, and here give nly the result, S(w). This is 0 fr t < 0 S(u>) = \ 64. wr 5 7- sm^ = TTCJ r 2. 2 m(3t-r). 2 3wT _, 11 2 e\ iin^ sm^ r +17 sin^ (8) u)(t-r). 2 wt 2" sin T fr 0 < a) < This is a single sided (psitive frequencies nly) spectrum. The frmula is valid fr 0 < r < 0.5T. At the upper limit fr r, the wavefrms representing data bits becme triangular. The average * The develpment, up t this pint, is basically similar t that in Reference 2, pp Thus, we have mitted all prfs. 8

13 energy spectral density with respect t cycle frequency, f, rather than radian frequency, t, may be btained by multiplying the expressin fr S(u) abve by 2TT, and substituting 2iTf fr u. The average energy spectral density is pltted in Figures 2 and 3 fr the tw limiting cases r = ysec. and r = 0.25 ysec. Nte that, in either case, the bulk f the spectral energy is fund between 0 and 2 MHz. When r = ysec, 98.7 percent f the energy is fund between 0 and 2 MHz. Fr r 0.25 ysec, the cmparable figure is 98.0 percent. Appreciable additinal energy is fund between 2 MHz and 3 MHz fr r = ysec. (1.1 percent) and between 2 MHz and 3.26 MHz fr r = 0.25 usec. (1.8 percent). Of particular interest is the prprtin f energy at frequencies less than 100 khz. This cmprises 0.95 percent f the ttal when r = ysec. and 1.16 percent f the ttal when r = 0.25 ysec. Mre cmplete infrmatin abut lw frequency cntent is given by the cumulative distributin f energy fr the range 0 t 100 khz. This is pltted in Figures 4 and 5 fr the cases r = ysec. and r = 0.25 ysec, respectively. Of sme interest als is the cumulative distributin f energy ver the brad range f frequencies which cntain the bulk f the spectral energy. We shall cnsider this range t be 0 t 4 MHz. These brad range cumulative distributins are pltted in Figures 6 and 7 fr the cases r = ysec. and r = 0.25 ysec., respectively.

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20 SECTION III CONCLUSIONS This study was mtivated by a suspicin that perhaps the lw frequency Multiplex Terminal Unit (MTU) input impedance specificatins in MIL-STD-1553 were unnecessarily restrictive. The results f the study supprt this ntin. Since nly abut ne percent f the signal energy lies belw 100 khz, the effect f the MTU input impedance in that frequency range n the bus wavefrm shuld be similarly small. Thus, a lwering f the present 6800 hm minimum impedance specificatin fr the range 0 t 100 khz t a value r values mre easily realizable by the specified MTU input circuit cnfiguratin is indicated. It shuld, hwever, be realized that we have nt prvided a cnclusive theretical prf that the energy belw 100 khz cannt appreciably effect synchrnizatin r detectability. There are tw majr missing ingredients fr such a prf. First, we have nly cnsidered an average energy spectrum. In the analysis f changes in detectability, fluctuatins abut this average may be imprtant, as the prbability f a particular "wrst case" data wavefrm (2 ) is large cmpared t the srt f errr prbabilities we are interested in achieving (apprximately 10 ). Secnd, it might be argued that the frequency cntent belw 100 khz has a disprprtinately large influence upn detectability. In fact, rudimentary analysis indicates that the "wrst case" wavefrm is nt significantly wrse than the average we have cmputed at the lw frequency end f the spectrum. Further, it seems unlikely, in view f the very great distrtin f the wavefrm by the bus, that small amunts f energy in any part f the spectrum culd effect detectability in a significantly disprprtinate way. It wuld, 16

21 hwever, require a quite substantial analytical effrt t prve that bth these beliefs are crrect. Frtunately, experimental evidence supprting the ntin that the missin f lw frequencies des nt seriusly effect synchrni- zatin r detectability has recently been btained by T. Allen. * Allen s wrk invlved high^pass filtering the data wavefrm befre detectin^ His results shw that fr filter cut-ff frequencies belw 100 khz, synchrnizatin and detectin may be accmplished very nearly as well as when n filter is present. This was fund t be the case despite the fact that the filters used caused significant distrtin in regins f the spectrum abve the cut-ff frequency. * T be published. 17

22 APPENDIX DERIVATION OF THE EXPRESSION FOR THE AVERAGE ENERGY SPECTRUM OF THE MULTIPLEX BUS DATA WAVEFORM Using ur equatins (6), (7) and (4) we may btain SAA(.) = E J fea OO if[ OO if.,. -JUT, ff (x) e dx E * ff (t) e jwt fea -I OO dr (Al) -= J 00 fea L f(t) f(t + T) dt dx We shall define R AA (T) ' E *ff< T > fea L -I OO = E / f(t) f(t + x) dt / fea (A2) c E f(t) f(t + x) dt fea ] Hence, frm (Al) and (A2), S AA ( " ) = 27/" R (T) e_jwtdt AA v (A3) 18

23 Nte that if we define f d (t) by f d (t) A f (t + d) (A4) fr sme cnstant, d, and all fea, then R d *>) AA" ' = R AA (0 (A5) and S<1 AA (a,) = S (a)) (A6) AA Thus, the average autcrrelatin and average energy spectrum are un- affected by a shift in the time rigin. If, fr real functins f time, f(t), g(t), we define a (time) crsscrrelatin, $ f (T) as *fg (T) "/ f Ct) g(t + x) dt (A7) we have * fg (x) - f <-T) (A8) Fr f(t), fea, g(t), geb, we may define R AB (T > " E *fg( T) fea geb = E J f(t) g(t + T) dt fea -0 geb 19

24 E [f(t) g (t + T)] dt fea geb (A9) Assuming the wavefrm time rigin t be fixed, let D be the set cnsisting f the 2 pssible infrmatin wavefrms (i.e., wavefrms ccupying bit times 4 thrugh 19), let P be the set cnsisting f the tw pssible parity wavefrms, let Y be the set cnsisting f the ne sync wavefrm apprpriate t a data wrd, and let W be the set cnsisting f the 2 pssible (entire) data wrd wavefrms. Let a member f the sets D, P, Y and W be dented by d(t), p(t), y(t) and w(t), respectively. Then we have w(t) = y(t) + d(t) + p(t) (A10) Using (4), (A2), (A9), and the fact that the expected value f a sum f randm variables is equal t the sum f the expected values, we have R^CO = ^(T) + R^CT) + Rjpd) + i^ct) + R^CO + R^T) + Rp Y (T) + Rpjjd) + Rp p (T) (All) Let h, (t) be the wavefrm representing a '0' in the k infrmatin bit (k = 1,..., 16). Let h, 7 (t) be the wavefrm representing a '0' in the parity bit. Then, -hj c (t) is the wavefrm representing a '1' in the k fc infrmatin bit and -h. 7 (t) is the wavefrm representing a '1' in the parity bit. We shall let b,, k = 1, 16, be the k tn infrmatin bit, and b. 7 be the parity bit. We shall dente the prbability f ccurrence f an event V by Pr {V}. 20

25 Lemma 1 E [d(t)] =0 all t (A12) ded Prf If t des nt lie within bit times A thrugh 19, then d(t) = 0. If fr sme k, k = 1,..., 16, t lies within bit time k + 3, then E [d(t)] = Pr{b k = 0}-h k (t) + P{b k = l}- "-h k (t)l ded since Pr{b, = 0} = ^ = Pr{b, = 1}, by assumptin. Lemma 2 R^d) = 0, all T (A13) R^d) = 0, all T (A14) Frm (A9), CO R^d) - E yey - ded / y (t) d (t + T) dt CO = / y(t) E [d(t + T)] dt ded = 0 by (A12) 21

26 Then we may btain (A14) frm (A8), (A9), and (A13). Lemma 3 Prf Pr{b 17 = 0} = Pr{b 17 = 1} = h (A15) 17 " J iiuu 17 Nte that fr M a psitive integer, 0 <-» k «-*>"u. k=0 y=l (A16) Since b._ is chsen t give dd parity, 16 /16\ 16 /16\ {b 17= 1} =L I )^) 16 =E ) (-D k (^) 16 (M7) Pr k=0 "* k=0 ' k k even k even 16 / 16 \ 16 / 16 \ Pr{b 1? = 0} = ( ) &) 16 = - ^ ) ("D k C5) 16 (A18) k= Xk ' k=0 Vk 7 k dd k dd Subtracting (A18) frm (A17) we have by (A16) that Pr{b 1? = 1} - Pr{b 1? = 0} = 0 (A19) Since 0 and 1 are the nly pssible values fr b. 7, Pr{b 1? = 1} + Pr(b 1? = 0} = 1 (A20 22

27 and cmbining (A19) with (A20) gives us (Al5) Lemma 4 RypOO = 0 all x (A21) Rp Y (T) = 0 all T (A22) Prf The result is a cnsequence f (A15). The prf fllws that f Lemmas 1 and 2. Lemma 5 R Dp (x) =0 all T (A23) Rp D (T) =0 all T (A24) Prf Pick arbitrary values f t and T. If t lies utside bit times 4 thrugh 19, then d(t) = 0 irrespective f the values f the infr- matin and parity bits, {b, },,. Thus E [d(t) p(t + T)] = 0 (A25) ded pep fr t utside bit times 4 thrugh 19 and all T. Suppse t lies in the (k+3) bit time fr sme 1 ^ k 16. Then 23

28 b k b 17 E [d(t) p(t + T)] = E (- 1) K h k (t)-(-l) h 17 (t + T) ded pep b k' b 17 (A26) = E V b 17 [<-.>** + *» h k (t) h 1? (t + T) Let ^k (A27) b 17 is chsen t prduce dd parity in the 17 bit infrmatin and parity bit sequence. Thus, if b, = 0, b., = 0 when {b } has dd parity. If b, = 1, b, 7 = 1 when {b? } has dd parity. Pr{b. + b._ = 0 r b. + b 17-2} k 17 k 17 = Pr{b k = 0} Pr{{b } dd} + Pr{b k = 1} Pr{{b } dd} = Pr{{b } dd} h (A28) where the last step may be btained by an argument similar t that used t prve Lemma 3. (Set M = 15 in (A16)). Since b, + b. -. = 1 is the nly pssibility nt cvered in the left hand side f (A28), Pr{b k + b ly = 1} = h (A29) 24

29 Thus, frm (A26), (A28), and (A29) E [d(t) p(t + T)] = 0 ded pep fr t within bit times 4 thrugh 19 and all T (A30) Taken tgether, (A25) and (A30) cnstitute a statement that E [d(t) p(t + T)] = 0 ded all t and T pep (A31) (A23) and (A24) fllw frm (A9) and (A8). we btain Cmbining (All), (Al3), (Al4), (A21), (A22), (A23), and (A24), R^Ct) = R^Cx) + R DD (x) + ^(T) (A3 2) Nw 16 Kt) = j^ (-i) Uk h k (t> k=l (A33) and E [d(t) d(t + x)] = E b b Z) (-D k -("D l \(t) h (t + x) L J?,=1 k=l ZE «=1 k=l r b k (-1) k -(-l) h l h k (t) h (t + T) (A34) 25

30 if k 4 a, f(-l) k -(-l) * I - E (-1) k l'e (-1) l \ = 0 (A3 5) since the b, 's are independent and have prbability \ f being either 0 r 1. If k =, Since 2 b, is either 0 r 2 and k [<- 2b, E (-1) = 1 (A36) (-1) = (-1) 2 = 1 (A3 7) Thus, [d(t) d(t + T)] = JT] h k (t) h k (t + T ) (A38) k=l Nw each functin h, (t) is merely a time shifted versin f h,(t). The autcrrelatin is unaffected by such time shifts, by the remark fllwing (4). Thus, if we define 00 W T > = I h x (t) h x (t + T) dt (A3 9) we may cmbine (A2), (A38), and (A39) t get R DD (T) = 16 W T) (AAO) 26

31 Nw, E [p(t) p(t + T)] = E [(-1) 17.(-1) 17 J h 17 (t) h 1? (t + T) = h 1? (t) h 1? (t + T) (AA1) by (A37). Since h 17 is a time shifted versin f h., Rpp(-0 = * hh ( T ) (A42) Since the sync wavefrm, y(t), is deterministic, we may state the equivalence f the ntatins fr its average autcrrelatin and deterministic wavefrm autcrrelatin, i.e., R YY (T) = (T) (M3) *yy Thus frm (A32), (A40), (A42), and (A43) we get By Furier transfrming (A44), we get W T > = *yy (T) + 1? (T) (A44) *hh S._.(u>) = $ (u) + 17» (w) (A45) WW yy hh What we have dne thus far is t reduce the expressin fr the average autcrrelatin f a cmplicated randm functin t a weighted sum f simpler (deterministic) wavefrm autcrrelatins. This has been accmplished using the nature f the wavefrm representatin (i.e., the wavefrm representing a '1' in a particular infrmatin r parity bit is the negative f the wavefrm representing a zer in the same bit), the prbability distributin n the infrmatin bits (i.e., independent bits, equiprbably '0' r ' 1'), the dd parity require- 27

32 merit, and the fact that the sync wavefrm is perfectly deterministic (i.e., independent f the infrmatin bits in a data wrd). The nature f the wavefrms representing the sync and a '0' infrmatin r parity bit has nt entered int the analysis. We shall nw take advantage f the nature f these wavefrms t achieve a further simplificatin. y(t) and h.(t) bth have the prperty f dd symmetry abut their mid-pints. Wavefrms with this prperty have autcrrelatin functins and energy spectra which are derivable frm thse f that prtin f the riginal wavefrm which lies n ne side f the mid-pint. Lemma 6 Let v(t) = u(t) - u(t + Q) (A46) where u(t) is a wavefrm which is zer utside the interval (0,Q). Let U(CJ) be the Furier transfrm f u(t). Then *vv (T) = 2 *uu (T) - *uu (T - Q) (T + Q) (M7) - *uu and 99 $ (u) = - u(u) sin ^ (AA8) VV IT Prf VV CO (T) = / [u(t) - u(t + Q)] [u(t + T) - u(t + T + Q)] dt 28

33 - 2 + (T) - <j> (T - Q) - (T UU UU uu + Q) (A49) Taking the Furier transfrm f (A49) we get $ (t)» 2 * (u) (1 - cs OJQ) w uu (A50) Using (2) and a trignmetric identity we get (A48) frm (A50). Bth h.(t) and y(t) may be decmpsed as per equatin (A46). Fr hjct), Q = h bit time = T. Fr y(t), Q = 3T. (See Figure 1.) u(t) represents the psitive part f each wavefrm. Let b(t) be the psitive (right) half f h.(t) and c(t) be the psitive (right) half f y(t). Let B(c) and C() be the Furier Transfrm f b(t) and c(t), respectively. Then, frm (A45) and (A48), WW 7T C(u)) sin ~ B((D) sm OJT (A51) We have immediately frm (A51) that s ww (0) = (A52) T get the explicit expressin fr S (w), we need the Furier trans- frm f a trapezidal waveshape. The easiest way t derive this is t differentiate the waveshape twice t get fur impulses, transfrm the 2 impulses and multiply the transfrm by -1/ui. The result, fr a trapezid u(t) with base Q and rise and fall times bth r is: IT/ -> 4. (Q-r). ur U(w) = 7 sin a) *s sin r) r 2 2 (A53) Setting Q = 3T in (A53) gives the expressin fr C(w). Setting Q = T 29

34 gives the expressin fr B(ui). Cmbining (A53) and (A51) in this way we get c s < \ 32 2 ur T 4 2 u(3t-r). 2 3cT ww (w) " W^ sin 1 L sin 2 sin 1" + 17 sin 2 as= i sin 2 4] (A54) The abve expressin is a duble sided spectrum (psitive and negative frequencies). The single sided spectrum (8) is btained frm (A54) by dubling the right hand side f (A54) when > 0, and setting it t zer when a) < 0. Use is made here f the fact that lim S T7TT (w) = 0. UH-0 30

35 REFERENCES 1. "Aircraft Internal Time Divisin Multiplex Data Bus," MIL-STD (USAF), 30 August "Statistical Thery f Cmmunicatin," Y. W. Lee, New Yrk: Jhn Wiley & Sns, Inc.,