Chapter 4 - The Fourier Series

Transcription

1 M. J. Robrts - 8/8/4 Chaptr 4 - Th Fourir Sris Slctd Solutios (I this solutio maual, th symbol,, is usd for priodic covolutio bcaus th prfrrd symbol which appars i th txt is ot i th fot slctio of th word procssor usd to crat this maual.). Usig MATLAB plot ach sum of complx siusoids ovr th tim priod idicatd. (a) x t sic t ()= =, 5ms < t < 5ms Th MATLAB program ca xactly follow th mathmatical procss i th quatio. For xampl: f = ; T = /f ; fmax = *f ; Tmi = /fmax ; dt = Tmi/ ; t = -.5:dt:.5 ; x = t* ; for = -:, x = x + sic(/)*xp(**pi**t) ; d x = ral(x/) ; plot(t,x,'') ; (Although th vctor, x, should b ral, it may hav som vry small, but o-zro imagiary parts bcaus of roud-off i th calculatios. That is why th ral commad is usd i th xt-to-last li.) x(t) -5 5 t (ms) Solutios 4-

2 M. J. Robrts - 8/8/4 9 + (b) x()= t sic sic 4 =9 t, ms < t < ms. Show by dirct aalytical itgratio that th itgral of th fuctio, is zro ovr th itrval, < t <. ()= g t Asi t Bsi 4t () = g tdt AB si tsi 4tdt Usig si( x) si( y)= cos( x y) cos x + y [ ] AB g() tdt= [ cos( t4t) cos( t+ 4t) ] dt, AB g() tdt= cos( t) cos( t) dt [ ] = AB si t si t () = g tdt AB si si si si = ()=( + ) + ( ). Covrt th fuctio, g t dos ot appar. 4 t 4 t Two vry importat trigoomtric idtitis to rmmbr ar, to a quivalt form i which cos( x)= +, si( x)= x x x x Solutios 4-

3 M. J. Robrts - 8/8/4 4. Usig MATLAB plot ths products ovr th tim rag idicatd ad obsrv i ach cas that th t ara udr th product is zro. (a) x()= t si( t) cos( 4 t), < t < 4 (b) x t si t cos 4 t, < t < ()= (c) x()= t si( t) cos( 4 t), < t < ()= () () (d) x t x t x t whr of ad x () t is a v, 5%-duty-cycl squar wav with a fudamtal priod 4 scods ad a avrag valu of zro x () t is a odd, 5%-duty-cycl squar wav with a fudamtal priod of 4 scods ad a avrag valu of zro x(t) 4 t - ()= () () () x t x t x t ()= () ad x ()= t rct t comb whr x t rct t comb t 5. A si fuctio ca b writt as si( ft)= 4. f t f t t 8 8 Solutios 4-

4 M. J. Robrts - 8/8/4 This is a vry simpl complx CT i which th harmoic fuctio is oly o-zro at two harmoic umbrs, + ad. Vrify that w ca writ th harmoic fuctio dirctly as X[ ]= ( δ[ + ]δ [ ] ). ( ) Writ th quivalt xprssios for si f t ad show that th harmoic fuctio is th complx cougat of th prvious o for si( f t). If th harmoic fuctio is corrct this quality should hold: ft ft ( f ) t si( ft )= = X[ ] = ( δ[ + ]δ[ ] ) = f t = Fiish simplifyig ad s whthr it dos.. For ach sigal, fid a complx CT which is valid for all tim, plot th magitud ad phas of th harmoic fuctio vrsus harmoic umbr,, th covrt th aswrs to th trigoomtric form of th harmoic fuctio. ()= () (a) x t 4rct 4t comb t This form is i Appdix E. But lt s do o dirctly from th dfiitio. 8 ( f ) t t t X[ ]= x() t dt = rct( t) dt = dt T 4 T 4 4 Fiish ad chc your aswr agaist Appdix E. t (b) x()= t 4rct( 4t) comb (c) A priodic sigal which is dscribd ovr o fudamtal priod by ()= x t () < sg t, t, < t < Solutios 4-4

5 M. J. Robrts - 8/8/4 7. Usig th CT tabl of trasforms ad th CT proprtis, fid th CT harmoic fuctio of ach of ths priodic sigals usig th tim itrval, T F, idicatd. (a) x()= t si( t), T F = (b) x()= t cos ( ( t. 5) ), T F = 5 I this cas, th rprstatio tim is th sam as th fudamtal priod. From Appdix E, cos( ft ) ( δ[ ]+ δ[ + ] ), TF T = Liarity Proprty: cos( t) δ[ ]+ δ[ + ], TF = T = 5 Tim Shiftig Proprty: cos ( ( t. 5) ) ( δ[ ]+ δ[ + ] ).5, T = F T = 5 ft = 5 = 4 cos ( ( t. 5) ) ( δ[ ]+ δ[ + ] ), T = F T = 5 Usig th quivalc proprty of th impuls, w ca writ g [ ] [ ]= [ ] A g A δ δ[ ], δ[ ]+ δ[ + ]= δ [ ] + δ + [ ] cos ( ( t. 5) ) δ[ + ]δ ( [ ]) From Appdix E, si( t) δ[ + ]δ (c) x t 4cos 5 t, T F = 5 ()= ( [ ]), TF = T =, T = F T = 5 5 Solutios 4-5

6 M. J. Robrts - 8/8/4 f = 5 T = TF =5T 5 Us th tabl try, T F = mt cos( ft) δ[ m]+ δ + m m a itgr ( [ ]) (d) x t d dt t ()= TF = T = 5 Start with, T F = 5 f t δ F = = 5 [ ], T T from Appdix E. Th apply th drivativ proprty of th CT. t () x()= t rct() t comb 4, T F = 4 (f) x()= t rct() t comb() t, T F = Us th fact thatsic= δ [ ]. (g) x()= t tri() t comb() t, T F = 8. If a priodic sigal, x(), t has a fudamtal priod of scods ad its harmoic fuctio is X[ ]= 4sic, with a rprstatio priod of scods, what is th harmoic fuctio of z()= t x( 4t) usig th sam rprstatio priod, scods? Solutios 4-

7 M. J. Robrts - 8/8/4 X, Usig th tim-scalig proprty, Z[ ]= a a itgr a,, othrwis X, sic, Z[ ]= a itgr 4 a itgr = , othrwis, othrwis This ca b writt mor compactly as Z[ ]= 4sic comb [ ] A priodic sigal, x(), t has a fudamtal priod of 4 ms ad its harmoic fuctio is X[ ]= 5 δ[ ]+ δ [ + ], with a rprstatio priod of 4 ms. Fid th itgral of x(). t Start with Liarity cos( ft ) ( δ[ ]+ δ[ + ] ), T = T = F 4ms cos( 5t) 5 δ[ ]+ δ[ + ], T T F = = 4ms Th us th itgratio proprty.. If X[ ] is th harmoic fuctio ovr o fudamtal priod of a uit-amplitud 5%-duty-cycl squar wav with a avrag valu of zro ad a fudamtal priod of µs, fid a xprssio cosistig of oly ral-valud fuctios for th sigal whos harmoic fuctio is X[ ]+ X[ + ]. Fid th tim-domai fuctio which corrspods to th CT harmoic fuctio, X [ ], th apply th frqucy-shiftig (harmoic-umbr-shiftig) proprty. Solutios 4-7

8 M. J. Robrts - 8/8/4. Fid th harmoic fuctio for a si wav of th gral form, Asi( f t). Th, usig Parsval s thorm, fid its sigal powr ad vrify that it is th sam as th sigal powr foud dirctly from th fuctio itslf. ( [ ]) A f t A si( ) δ[ + ]δ, T T F = P x T T T A A = A ( f t) dt= ( ftdt ) = [ ( ft) ] dt T si T si T cos 4 By Parsval s thorm, T T T A P = x P A A A x = ( δ[ + ]δ [ ] ) = ( + )= 4. Chc. =. Show for a cosi ad a si that th CT harmoic fuctios hav th proprty, For a cosi, X[ ]= X * [ ]. X[ ]= δ[ ]+ δ + ( [ ]) X[ ]= ( δ[ ]+ δ[ + ] )= ( δ[ + ]+ δ [ ] )= X[ ]= X * [ ]. Chc.. Fid ad stch th tim fuctios associatd with ths harmoic fuctios assumig T F =. (a) X[ ]= [ ]+ [ ]+ + (b) X[ ]= sic δ δ δ[ ] 4. Fid th v ad odd parts, x () t ad x o (), t of Solutios 4-8

9 M. J. Robrts - 8/8/4 x()= t cos 4t+. Th fid th harmoic fuctios, X [ ] ad X o [ ], corrspodig to thm. Th, usig th tim shiftig proprty fid th harmoic fuctio, X[ ] ad compar it to th sum of th two harmoic fuctios, X [ ] ad X o [ ]. = to fid th v ad odd parts. Us cos x + y cos x cos y si x si y 5. Usig th dirct summatio formula fid ad stch th DT harmoic fuctio of comb [ ] with =. F F X[ ]= comb [ ] = O ca choos ay covit priod for th summatio. I th priod chos blow thr is xactly o o-zro impuls i th comb fuctio at =. ( F ) X[ ]= comb [ ] = =. Usig th DT tabl of trasforms ad th DT proprtis, fid th DT harmoic fuctio of ach of ths priodic sigals usig th rprstatio priod, F, idicatd. (a) x[ ]= cos, F = (b) Th solutio is X[ ]= 5 comb [ + ]comb [ ] W ca dmostrat that this solutio is corrct by rcostitutig th sigal usig x[ ]= X[ ] = 5 comb [ + ]comb [ ] = = Sic th summatio xtds oly ovr o priod, =, choos th simplst priod, <. I that priod th two comb fuctios ar simply two impulss at =±. Solutios 4-9

10 M. J. Robrts - 8/8/4 5 x [ ]= 5 δ[ + ]δ[ ] = x[ ]= = 5 5 x[ ]=5 si si = W could hav chos a diffrt priod, for xampl 4 <. Th 5 x [ ]= 5 δ[ ]δ[ ] = 4 x[ ]= = 5 5 x[ ]= = {{ = = x[ ]= 5 si si = which is th sam aswr as bfor (with somwhat mor ffort). (c) x, x[ ]= a itgr 8 8, othrwis whr x[ ]= si, F = 48 Us th tim-scalig proprty of th DT. (d) x [ ]=, F = Solutios 4-

11 M. J. Robrts - 8/8/4 otic that this fuctio, although writt as x [ ]= could b mor simply writt as x[ ]= bcaus for ay itgr valu of discrt tim,, th rsult is always th sam,. Thrfor = comb [ ] X[] -4 4 Phas of X[] If w ta a mor straightforward approach to fidig th DT harmoic fuctio usig F = m comb m F [ ] w gt [ ] comb Obsrv that this aswr ad th prvious aswr ar idtical. That is, as must b tru. comb[ ]= comb[ ], () x[ ]= cos cos, F = (f) x[ ]=si si =, F = 4 4 Th priod of this sigal is 4. Solutios 4-

12 M. J. Robrts - 8/8/4 This form dos ot appar dirctly i Appdix E. Thrfor w must ithr us what is i Appdix E with som proprtis to gt to this form or simply apply th dfiitio of th DT harmoic fuctio dirctly. From Appdix E, comb [ ] Usig th frqucy-shiftig proprty, m [ ] comb m or Th m m ( m + ) comb m + m comb m [ ] + m = cos ( comb [ m comb m ]+ [ + ]) ad m m m = si ( comb [ m comb m + ] [ ]) X[ ]= comb4[ + ]comb4 ( [ ]) W ca dmostrat that this solutio is corrct by rcostitutig th sigal usig x[ ]= X[ ] = comb [ + ]comb [ ] = = 4 4 Sic th summatio xtds oly ovr o priod, = 4, choos th simplst priod, <. x[ ]= ( comb [ + ]comb [ ] ) 4 4 = Solutios 4-

13 M. J. Robrts - 8/8/4 W must ow dtrmi for which valus of, <, th comb fuctios ar ot zro. Ta th first comb fuctio, comb 4 [ + ]. Its impulss occur whvr + is a itgr multipl of 4. Th oly valu of i th rag, <, for which that is tru is =. Similarly, for comb 4 [ ] th oly valu for which it is o-zro is =. Thrfor w ca writ th summatio as 4 = x[ ]= δ[ ] δ[ + ] = x[ ]= = si si si = 4 4 = But this ca also b writt as x[ ]= = = = x[ ]= si si = 4 If ths two rsults ar to both b corrct si =si for ay itgr valu of. W ca writ 4 si = si = si = si =si Solutios 4-

14 M. J. Robrts - 8/8/4 provig that th two xprssios ar idd quivalt for itgr valus of. [ ]= [ ] [ ] (g) x rct comb 5 W + Usig rctw[ ] comb [ ] drcl, W + Ad, from Appdix A, drcl, m comb m m + = + [ ] ad th fact that + W + =, X[ ]= comb [ ] Agrs with comb bcaus x rct comb 5 ad i comb [ ], ca b arbitrarily chos. Th maig of [ ] th rsult is th sam rgardlss of th choic of. [ ]= [ ] [ ]= This rsult was obtaid for a priod of =. But wh th fuctio is a costat, w ca choos ay priod w li ad gt a quivalt rsult (bcaus a costat rpats xactly i ay priod you choos). For xampl, if w lt = 4 th trasform pair, comb [ ], yilds X[ ]= comb [ ] 4. Th, rcostitutig th sigal from its DT, [ ]= [ ] = [ ] x X comb = = Summig ovr th priod, <, yilds 4 4 [ ]= 4[ ] = = = 4 x comb. If w chos th priod, < w would gt 5 ( 4) 4 [ ]= 4[ ] = = = 4 x comb, which is xactly th sam rsult. I gral, for ay choic of priod ad ay rag of covrig o priod, q + = [ ]= [ ] = = x comb Solutios 4-4

15 M. J. Robrts - 8/8/4 whr th itgr, q, lis i th rag, q< +. [ ]= [ ] [ ] (h) x rct comb, F = 7. Fid th DT harmoic fuctio of x[ ]= comb[ m]comb m m = 8. Fid th avrag sigal powr of with = F =. [ ]= [ ] [ ] x rct comb 4 [ ] dirctly i th DT domai ad th fid its harmoic fuctio, X[ ], ad fid th sigal powr i th domai ad show that thy ar th sam. I th DT domai: P x = 9 I th domai: Do th summatio umrically i MATLAB to gt P x = Usig th frqucy shiftig proprty of th DT fid th DT-domai sigal, x[ ], corrspodig to th harmoic fuctio, 7 X[ ]= drcl, 7. This frqucy shiftig causs th sig of th DT-domai fuctio to altrat.. Fid th DT harmoic fuctio for [ ]= [ ] [ ] x rct comb 8 Solutios 4-5

16 M. J. Robrts - 8/8/4 with th rprstatio priod, F = 8. Th, usig MATLAB, plot th DT rprstatio, x F 7 [ ]= X[ ] = 8 ovr th DT rag, 8 < 8. For compariso, plot th fuctio, x F [ ]= X[ ] = 8 ovr th sam rag. Th plots should b idtical.. A priodic sigal, x(t), with a priod of 4 scods is dscribd ovr o priod by x ()= t t, < t< 4. Plot th sigal ad fid its trigoomtric CT dscriptio. Th plot o th sam scal approximatios to th sigal, x (), t giv by + [ ] ( ) x ()= X c []+ X c [ ] cos ( F ) X s si F = t f t f t for =, ad. (I ach cas th tim scal of th plot should covr at last two priods of th origial sigal.) Fid th trigoomtric harmoics sris by dirct itgratio usig Us or X c t + T [ ]= x() t dt T, t + T t Xc [ ]= x() t cos( ( f ) t) dt T t t + T Xs [ ]= x() t si( ( f ) t) dt T t = x cos x dx x si x x si x dx, Solutios 4-

17 M. J. Robrts - 8/8/4 = + x si x dx x cos x x cos x dx from Appdix A, ad ma th chag of variabl to fiish th itgrals. λ = t. A priodic sigal, x(t), with a priod of scods is dscribd ovr o priod by ()= x t si ( t), t <, < t <. Plot th sigal ad fid its complx CT dscriptio. Th plot o th sam scal approximatios to th sigal, x (), t giv by x ( F ) ()= t X[ ] = f t for =, ad. (I ach cas th tim scal of th plot should covr at last two priods of th origial sigal.). Fid ad plot two priods of th complx CT dscriptio of cos( t) (a) Ovr th itrval, < t <, ad (b) Ovr th itrval, < t <. 5. Th priod of cos( t) is, thrfor th complx Fourir sris dscriptio is (a) simply th xpotial dscriptio of th cosi fuctio t t + cos( t)=. That is, X[]= X[ ]= ad X [ ]=,. Solutios 4-7

18 M. J. Robrts - 8/8/4 Th plot is simply th plot of two priods of th cosi bcaus th complx Fourir sris dscriptio is xact vrywhr. (b) T F = ad ff = t + TF ff t X[ ]= x() t dt T F t si si X[ ]= cos cos Usig MATLAB, plot th followig sigals ovr th tim rag, < t <. (a) x ()= t (b) x t x t cos t ()= ()+ ()= ()+ ()= ()+ (c) x t x t cos 4 t (d) x t x9 t cos 4t For ach part, (a) through (d), umrically valuat th ara of th sigal ovr th tim rag, < t <. Th th mmbr of this squc of fuctios ca b rprstd by th gral form ()= ()+ x t x t cos t. Basd o what you obsrvd i parts (a) through (d) what is th limit as approachs ifiity? Fid th trigoomtric Fourir sris xprssio for th uit comb fuctio ad compar it to this rsult. Solutios 4-8

19 M. J. Robrts - 8/8/4 5 Succssiv Cumulativ Sums of Highr Harmoic Cosis so(t), = so(t), = so(t), = so(t), = Tim, t (s) 5. Usig th CT tabl of trasforms ad th CT proprtis, fid th CT harmoic fuctio of ach of ths priodic sigals usig th tim itrval, T F, idicatd. (a) x()= t rctt comb t () 4 Rmmbr:, T F = ()= () () If g t g t δ t Th g( t t )= g ( tt ) δ()= t g () t δ tt Ad g( t t ) g ( tt ) δ( tt )= g( tt ) t (b) x()= t 5[ tri( t) tri( t+ ) ] comb 4 4, T F = 4 t t w Usig tri comb sic w T T T t tri() t comb sic w T, with w = ad T = 4, ( f) t Th usig th tim-shiftig proprty, x( t t ) X t tri( t ) comb sic [ ], Solutios 4-9

20 M. J. Robrts - 8/8/4 ad t tri( t + ) comb sic Th, usig liarity, or t 5 tri( t ) tri( t + ) comb sic 4 4 t 5 tri( t ) tri( t + ) 5 comb 4 4 si sic 4 ()= + (c) x t si t 4cos 8 t, T F = ()= + (d) x t cos 4t 8cos t si t, T F = t () x()= t comb( λ)combλ d λ, T F = ()= (f) x t 4cos t si t, T F = 5 t t t t (g) x()= t rct comb rct comb , T F = 4 Lt Th, usig t t x()= t rct comb 8 t t ad x()= t rct comb 5 8 t t w w rct comb sic w T T T T, X [ ]= ()= () () x t x t x t 8 98 sic, T F = Solutios 4-

21 M. J. Robrts - 8/8/4 8sic, X[ ]= a itgr, othrwis, T F = 4 X [ ]= 5 5 sic, T 8 F = 8 X 5 5sic, a itgr [ ]= 4, othrwis, T F = 4 sic sic, X[ ]= , othrwis a itgr, T F = 4 All th valus of this fuctio ar zro xcpt wh =. Th X[ ]= 948δ [ ], T F = 4. (This rsult idicats that th priodic covolutio of th two sigals is a costat, 948. That ca b cofirmd by graphically priodically covolvig th two sigals. That is by fidig th ara udr th product ovr o cycl ad obsrvig that, as o sigal is shiftd, th ara dos ot chag. Sigal o is a priodic squc of pulss of hight 8, width 8 ad priod. Sigal two is a priodic squc of pulss of hight 5, width 5 ad priod 8. Th ovrall ovrlap width is a costat. That product is 8 5 = 948.) t t t t (h) x()= t rct comb rct comb 8 5, T F = sic sic, X[ ]=9 4, othrwis a itgr, T F =. A sigal, x(), t is dscribd ovr o priod by Solutios 4-

22 M. J. Robrts - 8/8/4 ()= x t T A, < t < T A, < t<. Fid its complx CT ad th, usig th itgratio proprty fid th CT of its itgral ad plot th rsultig CT rprstatio of th itgral. Fid its harmoic fuctio usig th itgral dfiitio. You should gt X[ ]= cos( ) A Th apply th itgratio proprty. Vrify that th harmoic is corrct at =. (You will d to apply L Hôpital s rul.) 7. I som typs of commuicatio systms biary data ar trasmittd usig a tchiqu calld biary phas-shift yig (BPSK) i which a is rprstd by a burst of a CT si wav ad a is rprstd by a burst which is th xact gativ of th burst that rprsts a. Lt th si frqucy b MHz ad lt th burst width b priods of th si wav. Fid ad plot th CT harmoic fuctio for a priodic biary sigal cosistig of altratig s ad s usig its fudamtal priod as th rprstatio priod. Th sigal ca b rprstd by [ ] x()= t si( t) rct( t) 5 comb( 5 t) Usig mulitplicatio-covolutio duality, 5 5 X[ ]= ( comb[ + ]comb[ ]) sic [ ] δ Rmmbr that a priodic covolutio is th sam as th apriodic covolutio of o priod of o of th priodic sigals with th tir othr sigal. Choos o priod of th diffrc of two combs. Solutios 4-

23 M. J. Robrts - 8/8/4 8. Usig th DT tabl of trasforms ad th DT proprtis, fid th harmoic fuctio of ach of ths priodic sigals usig th rprstatio priod, F, idicatd. [ ]= 4[ ] (a) x comb, F = 48 From th tabl, [ ] comb, F = Thrfor, usig th chag of priod proprty, comb, a itgr, othrwis, F = 48 Also, from th tabl, F = m comb m F [ ] 48 = comb 48 [ ] Th fuctio, comb, is a squc of impulss. Th impulss occur whrvr = m, m ay itgr. Rarragig, th impulss occur whr = 48 m. This dscribs a comb of th form, comb 48 [ ]. Thrfor th two aswrs agr. From th tabl, comb Thrfor, usig th tim scalig proprty, [ ], F = 4 Solutios 4-

24 M. J. Robrts - 8/8/4 comb, a itgr [ ] 4, othrwis, F = 48 Th, sic th two DT s ar both do with rfrc to th sam rprstatio priod, usig th multiplicatio-covolutio duality proprty, comb 48 [ ], a itgr, a itgr X[ ]= 4, othrwis, othrwis Th o-zro impulss i th first harmoic fuctio occur at valus of for which + is a itgr multipl of 48. Thrfor all ths s must b odd. Th valus of for which th scod harmoic fuctio is o-zro ar all v. Thrfor X[ ]=, for all. [ ]= [ ] This rsult implis that th origial DT fuctio, x comb4, is zro. Show that to b tru by actually doig th covolutio dirctly. (b) x[ ]= ( rct 5[ ] comb4[ ] ) si, F = 4 (c) x[ ]= x[ ]x[ ] whr x [ ]= tri comb 8 [ ], F = 9. Fid th sigal powr of 4 x[ ]= 5si 8cos 5. Us Parsval s thorm to fid th sigal powr from th harmoic fuctio. Fid. th DT harmoic fuctio, X[ ], of x[ ]= ( rct[ ]rct[ 4 ]) comb[ ]. Th plot th partial sum, 5 x X, for =,, ad th th total sum, x X. [ ]= [ ] = x[ ]= rct [ ] comb [ ]rct 4 comb [ ] [ ] [ ]= [ ] = Solutios 4-4

25 M. J. Robrts - 8/8/4 Solutios 4-5 Usig rct comb si W W [ ] [ ] + si X si si si si si si [ ]= = 4 4 X si si [ ]= 5 = 5 si si x si si si si [ ]= = = = x[] - = - x[] - = - x[] - = - x[] - Total Sum from to 5. Fid ad stch th magitud ad phas of th DT harmoic fuctio of

26 M. J. Robrts - 8/8/4 x[ ]= 4 cos + si 7 which is valid for all discrt-tim. Th last commo priod of ths two sigals is =. Us th tabls ad th chag-of-priod proprty of th DT, to fid th DT harmoic fuctio. X[ ]= ( comb[ ]+ comb[ + ] )+ comb[ + 7]comb 7 X[] ( [ ]) - Phas of X[] - -. Th su shiig o th arth is a systm i which th radiat powr from th su is th xcitatio ad th atmosphric tmpratur (amog may othr thigs) is a rspos. A simplifid modl of th radiat powr fallig o a typical mid-latitud locatio i orth Amrica is that it is priodic with a fudamtal priod of o yar ad that vry day th radiat powr of th sulight riss liarly from th tim th su riss util th su is at its zith th falls liarly util th su sts. Th arth absorbs ad stors th radiat rgy ad r-radiats som of th rgy ito spac vry ight. To p th modl of th xcitatio as simpl as possibl assum that th rgy loss vry ight ca b modld as a cotiuatio of th daily liar radiat powr pattr xcpt gativ at ight. Thr is also a variatio with th sasos causd by th tilt of th arth s axis of rotatio. This causs this liar risad-fall pattr to ris ad fall siusoidally o a much logr tim scal as illustratd blow. Solutios 4-

27 M. J. Robrts - 8/8/4 Radiat Powr Ju.75 Day Dcmbr 5 (a) Writ a mathmatical dscriptio of th radiat powr from th su. t p()= t tri( t) comb() t si (b) Assum that th arth is a first-ordr systm with a tim costat of. yars. What day of th yar should b th hottst accordig to this simplifid modl? Th diffrtial quatio for th rat of hat flow ito th arth is () d dt K t t KT t ( T() )= p() whr K is a proportioality costat rlatig th tmpratur of th arth to its stord rgy (t i days). Sic th radiat powr striig th arth is priodic th tmpratur is also priodic ad both ca b xprssd i a CT with a rprstatio priod of 5 days. Fid th harmoic fuctio for th radiat powr, P[]. Th tmpratur must also b priodic with th sam fudamtal priod as th radiat powr. Solv for th harmoic fuctio of th tmpratur, T[], by substitutig P[] ito th diffrtial quatio. Th simplify it by ralizig that th highr ordr harmoic cott i T[] is gligibl compard to th fudamtal. Th fid th tim-domai tmpratur xprssio, T(t), ad fid th tim at which it rachs a maximum. [ ] sic δ, a itgr 5 + δ[ + ]δ 8 Τ[, othrwis ]= K ( [ ]) Solutios 4-7

28 M. J. Robrts - 8/8/4 Τ t () 8K t t. 44 cos +. 44si (. ). 44K Th maximum tmpratur occurs at about t = So th hottst day should occur about 4 days aftr th summr solstic or about August.. Th spd ad timig of computr computatios ar cotrolld by a cloc. Th cloc is a priodic squc of rctagular pulss, typically with 5% duty cycl. O problm i th dsig of computr circuit boards is that th cloc sigal ca itrfr with othr sigals o th board by big coupld ito adact circuits through stray capacitac. Lt th computr cloc b modld by a squar-wav voltag sourc altratig btw.4 ad. V at a frqucy of GHz ad lt th couplig ito a adact circuit b modld by a sris combiatio of a. pf capacitor ad a 5 Ω rsistac. Fid ad plot ovr two fudamtal priods th voltag across th 5 Ω rsistac. Dscrib th xcitatio as a rctagl covolvd with a comb, plus a costat. Fid its harmoic fuctio. Put that ito th diffrtial quatio for th rspos voltag, [ ( i ) o() ]= o() RC v t v t v t 9 4. sic ad solv for th harmoic fuctio of th rspos, V o[ ]= + RC 9 4. Th plot th rspos voltag by computig its CT. It should loo li th graph blow. Solutios 4-8

29 M. J. Robrts - 8/8/4 v o (t) t - 4. Fid ad plot vrsus F th magitud of th rspos, y[ ], to th priodic xcitatio, i th systm blow. x[ ]= cos( F), x[] y[].9 D or [ ]= [ ]+ [ ] y x 9. y y [ ]9. y[ ]= x[ ]. Th fial graph should loo li th graph blow. Amplitud F This is a lowpass DT filtr. Solutios 4-9