δ(t)e jωt dt = e 0 = 1 e αt u(t)e jωt dt = e (α+jω)t dt = 1 α + jω e (α+jω)t 1 = α + jω, α > 0 1 α 2 + ω 2, X(jω) = tan 1 ( ω α )

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Download "δ(t)e jωt dt = e 0 = 1 e αt u(t)e jωt dt = e (α+jω)t dt = 1 α + jω e (α+jω)t 1 = α + jω, α > 0 1 α 2 + ω 2, X(jω) = tan 1 ( ω α )"

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1 !"#$%#&'()*&+'(,&')-.(/),+ (*?'(+>:+ +'8+'8+'/:(+'-'>8+'*(8δ(t) δ(t)7(.'8'9+9!"#$%#:+ ;<< γx(t) e α t, α > 3456 δ(t)e jωt dt e :()/*+-''8:8 A(/):8(*8+:(+)8/*8/''', < αx(t) δ(t) βx(t) e αt u(t), α > { t < T δx(t) t > T βx(t) e αt u(t), α > X(jω) X(jω) e αt u(t)e jωt dt e (α+jω)t dt α + jω e (α+jω)t α + jω, α > X(jω) α + ω, X(jω) tan ( ω α )

2 X(jω) /α ω B+9'8'-(:+ α α X(jω) π/ π/4 α ω α π/4 π/ C'(&+,+'*(+>'(),D(/),8(*E +,8x(t)-''', X(jω) e α t e jωt dt e αt e jωt dt + e αt e jωt dt α jω + α + jω α α + ω, α > Fx(t) t γx(t) e α t, α >

3 7(.'8)*'9+>:+ X(jω) /α /α α α ω { t < T t > T X(jω) x(t)e jωt dt T /'', + Dsinc(x)''&,)&8+'*(-I8D(/),(*+> e jωt dt T T jω e jωt T ω sin(ωt ωt X(jω) T ωt T ωt sin(x) x X(jω) HT ) ω π/t π/t J((9+'*(+(:)+I('*(-)*K'*π/T 9I8&'-+(9'&'L.''+&)*-:'M')*+'*(sinc(x) δx(t) ('(/)&8G,8 ) ;<F ;<H ).

4 NO'!"#$%#&')+P&'*&' )&')'&'.' '-(88)+''99+'*.(8G(8*&' (/&8(9'++'>8x(t) αx(t) [e αt cos(ω α+jω(.'8'-8 t)]u(t) βx(t) e 3 t αt u(t)' { + cos(πt), t γx(t), t > δx(t) n e t n αx(t) [e αt cos(ω t)]u(t) )P.8)8-888+':+ F{[e αt cos(ω t)]u(t)} F{[e αt u(t)] cos(ω t)} [ α + j(ω ω ) + α + j(ω + ω ) ] α + jω )8!"#$%#8+'*(8x(t)P' α + ω ω + jαω βx(t) e 3 t sint Q*''8(,+E:++8e α t α F α + ω )+9+sin(t) ejt e jt Rj e 3 t sin(t) j ejt e 3 t j e jt e 3 t X(jω) 6 j 9 + (ω ) 6 j 9 + (ω + ) 4ω j ω 4 + ω + 69 { + cosπt t γx(t) t >

5 Q*''8(,+(9>:+ A&8(''8>:+ X(jω) δx(t) n e t n sin(ω) ω ( + cos(πt))e jωt dt ( + ejπt + e jπt )e jωt dt e jωt dt + + sin(ω) π ω sin(ω) π + ω π sin(ω) (π ω )ω e j(π ω)t dt + e j(π+ω)t dt F{e t n } n e jωn F{e t } n e jωn + ω C9/&''(+'(/+>:+ n;-gtd+'*(x(jω)''*'(n )(*& n [ + + ω cos(nω) ] n S?'>'+8+-')*P(&'+'*(x(t) ;αx(jω) ;βx(jω) ( + jω) +jω ;αX(jω) jω x(t) U π jω ejωt dω cos(ωt) π jω dω + sin(ωt) π ω sin(ωt) x(t) dω sgn(t) ω )/9(.),(&'-:'()9dω

6 + P9+'()*&:8 )'-8(.89/&'''>8x(t)'X(jω) {, t > sgn(t), t < ;βx(jω) ( + n 7(9E98e t u(t) /(:8('+ F + tx(t) F j dx(jω) t n x(t) F j n dn X(jω) dω n V@'8X(jω) ( + jω) )(&E'8n dx(jω) ( )j( + jω) dω dx (jω) dω ( )( )j ( + jω) 3 τ)dτ;-''+':g dx n (jω) dω n j n (n )!( + jω) n t n e t (n )! u(t) F ( + n t n e t u(t) F (n )! ( + n +'*(+E9'x(t) tn e t u(t) (n )! WV@&+'*(r(t) x(τ)y(t + ;α@)/(*'!"#$%#r(jω)8r(t)l*+98x(jω), Y (jω) 3456 ;γy&8);l+'r(jω)(.'8':+;-8,>'+'r(jω) +'8 y(t)' t?/9+),(&)'r(t)(x Z &'x(t), ;βp:x(t) y(t) e

7 ;αe'8'>8r(t):+ y(t):+ R(jω) ;β?')p.'88+'(,8x(t), [()*P(&(9'(*I+'+'*(r(t)&8G,8 P:+ +'*(r(t)'e τ e t+τ dτ?'t > J('&'8.('>8+'*(8+(:)+IP:+ Mt < y(t)p-' ;γva&8:++'(&*)>&'+'(,&'x(t), r(t) r(t) t ( + t)e t ( t)e t R(jω) r(t)e jωt dt [ x(τ)y(t + τ)dτ ] e jωt dt x(τ) [ y(t + τ)e jωt dt ] dτ x(τ)y (jω)e jωτ dτ X( jω)y (jω) r(t) e τ e t+τ dτ + e τ e t+τ dτ + t t x(τ)y(t + τ)dτ e τ e t τ dτ + e τ e t+τ dτ + r(t) ( + t )e t r(t)e jωt dt ( t)e t e jωt dt + t e τ e t τ dτ e τ e t τ dτ ( + t)e t e jωt dt + jω + ( + jω) + jω + ( jω) 4 ( + ω ) :X(jω) Y (jω) \ + ω

8 C9/&'(α)(.9 θ);αl('f(xm(t)) X(jω)+F8 R(jω) X( jω)y >8r(t)P'R(jω) 9/&''-8-(/&8P' θ)>8+'*(8+,8 )+/&':(.(β) ]?'F(x(t))!"#$%#)x m ;β?'f (t) + f (t) + f(t) (δ(t + π) + δ(t π))l('f(t) 3456;α?'P&(,++'*(w(t) 7(.'88-9(*'&:8:+ ) x(t) cos(ωt (t) y(t)cos(ω t + θ A>8y(t)P'Y π (t)p'(x(j(ω (jω) )>8x m W(jω) X( jω)y (jω) + ω + ω 4 ( + ω ) m ( e jθ X(j(ω ω )) + e jθ X(j(ω + ω )) ) (t) x(t)cos(ω t)sin(ω t+ x(t)cos(ω t)x ω)) + X(j(ω + ω))) X m (jω) ( e j(θ π ) Y (j(ω ω )) + e j(θ π ) Y (j(ω + ω )) ) X m (jω) j ( e ^ jθ Y (j(ω ω )) + e jθ Y (j(ω + ω )) ) j [ e jθ( X(j(ω ω ω )) + X(j(ω + ω ω )) ) e jθ( X(j(ω ω + ω )) + X(j(ω + ω + ω )) ) ] j [ e jθ X(j(ω ω ω )) + e jθ X(j(ω + ω ω )) 4 ] e jθ X(j(ω ω + ω )) e jθ X(j(ω + ω + ω ))

9 J((9(',8('(/&8''('&'/99 π)(9'(*i+ ++j)?')*'+'*+*)* ;<R ;<U )/99 (jω) F(jω) + jf(jω) + F(jω) F{ δ(t + δ(t π)}. _:'8x(t) δ(t + π) δ(t ()9 ω +jω + (jω + j)(jω u(t))(.'8'-8+':g8(x e e à&x(t)(&')88+(e&8g,8 π)p:+:':)-8y(t)--, _:'8'y(t) sin(t)e αj(''8'(*&8x(t)))*''8(,8-88(.++x t '>;-GT)/(*'(*&8x(t)&8*P(-9.' b9 )-,x(t) δ(t + π) δ(t β'>(.'8'-9((*&8x(t) F{x(t)} F(jω) ω + jω +. ω + jω + j jω + j j jω + + j ( ) j j(ω ) + j(ω + ) + '+ t u(t) F + jω jt e t u(t) F + j(ω ) e jt e t u(t) F + j(ω + ) F { ω + jω + } ( e jt e t u(t) e jt e t u(t) ) j sin(t)e t u(t) F(jω) F{x(t)}F{y(t)} f(t) x(t) y(t) f(t) sin(t)e t π u(t + π) + sin(t)e t+π u(t π) { t t < x(t)

10 3456 αd(/),(*8+'*(8x(t)/'', γ'>(.'8'-8+':g8;-gtp&( x(t):(:p+':g-9*&' J('&'8'(*&8(*'&+'*(8)(*''*8':+' ''(/':8(9'(*I+'*P(-9.'c8 G,8(/),(* '&'>+'-+99(8T; T/&8T/-'' '8.'X.' :'8)-8+'-+9;'-G*E)(a 9Tsinc(ωT/)C'(&'+T <K t x(t) y(t)x (t) t t!>'sinc(ω/) sin(ω/) e j ω sin( ω ) ω ω/

11 [(>8y(t)&()8P9*P(&'.'&' t β?'*(+-9((*&+'*(+()9' ))*''8(,8-88(.8:+ e j ω sin( ω ) ω J(/'.8(99)'++'>8-9(8(.+8 P-': -9.'Y sin(ω ) ( ) ω e j ω e j ω 4j ω sin ( ω ) Y (jω) 4 ω sin ( ω ) sinc ( ω γa'8(8(8'++'>8+'*(8+8-''' ):'&8X(jω)P' ) ' y(t) x (t) δ(t + ) + δ(t ) δ(t) Y e + e jω cos ω 4 sin ( ω ) X(jω) (jω) Y (jω) 4 ω sin ( ω ) sinc ( ω ) P&(,+&8:8+':G8-9*&'C+))(:'+'*(x(t) ('()9I+':G-9'-&'.'--,x(t) x (t) x (t) +'8++,8' )?'-8+':G8&8:+ { t < / x(t) X (jω) sinc( ω X(jω) X < (jω)x (jω)8g-++m7?+x (jω) sinc ( ω ) dd-8:'m7?9'x(t) + δ(t ) + sin(πt) { ω < π H(jω) ω > π ;α>x(jω) ;β7(.'8:+;)/(*>y,8)(h(jω)''-/(+,x(t) 9 πt))(

12 ;βq*''8(,8()8+e+'>)*p(+8g-++/(+ π''-:px(t) π):'-')(/( ;γ7(''(/!"#$%#'l('y(t) X(jω) πδ(ω) + e jω + u(ω + u(ω ++'*(u(ω + π) u(ω /9u(ω ;γ7(.'8''(/>+e+':g-y(t)+' J(P:!'88:(+8G-+8('+>8G-+ + +'),8ω c ea'+'98('+m7?9:)(+),)( 3456 α')(+'8++,8 βc-*:()/*8)(8+'8;7(,(l*' αm'++9)'>8)(+),8)(8(+e:+8 γ':g-++,8'/(+'-, >fghfi >8h(t)P' ;<Z <F H(jω) e 3jω [ { ω < π π) u(ω π) ω > π Y (jω) πδ(ω)h(jω) πδ(ω)h(j) πδ(ω) Y (jω) e jω H(jω) e jω [u(ω + π) u(ω π)] Y 3 (jω) [u(ω + π) u(ω π)]h(jω) [u(ω + π) u(ω π)][u(ω + π) u(ω π)] [u(ω + π) u(ω π)] Y (jω) πδ(ω) + (e jω + )[u(ω + π) u(ω π)] sin(π(t )) y(t) + + sin(πt) π(t ) πt h(t) δ(t 3) e 7(t 3) u(t 3) x(t) 7 + 7cos(7t + π) e αt u(t) α + jω 7 + jω ] e 3jω( 6 + jω) 7 + jω

13 β7(.'8(l*'>fghfi+e+')(+'8++x,8;:()/*(),)) Magnitude.95 H(jω) ω [rad/sec] γ?/99'()+e+':g-g*e'8)*p(8-+ Angle 4 3 H(jω) [rad] '-:8,*88'+:')* H(j7) --,':(8?''-:+P(,+'*(')*'++':G- &(*-+9'8)+8-9;P(*)+''-8+'8 3 (,')P+:(+*M7?9+,+' (9'9/99'M7?):G-8P'X )(+'8+))(:'+'@8/*8G-+P)P+X ;lp(*+'*((:''jk+'c'+))(:'(& 6 ' ω [rad/sec] *E+')(++,8'jk+';ω --,:G-8P'7 H(j) 7 6/7 M''-,-((9,-+:+'ω π/)'-++))(:'++,8.6π@:'&8' <H --,/*π/p'π/ + H(j7).66π ''-:8,7 cos(7t + :G-8P,'7 H(j7) cos(7t + π/ + H(j7)) 6.5 cos(7t.6π) +,x(t) 7 + 7cos(7t + π) :G-8+P'y(t) cos(7t.6π)

14 mn)(+),)('8m7?+,8-': α')(+'8++,8)-*(&'+:(+ βj+'o'9o((9pc+'(*&'+ )/*&8+'*(+ω 3456 :(+8)(8('8:8,8p;D+'+,'/:() )((':, h(t) δ(t) + 5e 5t u(t) α@/(e&'8>')(+),)(+e+')(+'8 γj*':g-8'-8' 7(.'8(L*'>fghfi-*E+')(+'8;:() /*(),))&8/'()*&, &83dB+'-dB))log H(jω) )()'HK<dB P'H(jω) x(t) + cos(t) ;<\ F{δ(t)} + 5F{e 5t u(t)} jω 5 + jω 5 + jω 4 Magnitude squared H(jω).5.5 <R ω [rad/sec] Angle H(jω) [rad] ω [rad/sec]

15 γ?'&e'8)*p(8-+&(*+e+&8)('')* 8)(8+'8 ('LL&P)'(/),(*+(.'++:(+ βy&8/'),9'9(('jk+'ω rad/sec?'98)(8:& cos(t)'+&':g-(: '(*&'+:(+' H(j) 4M'L(9'3dB +'9'+'G& H(jω) + :G-'-8'x(t) :G-8P'y(t)?':&8-'+'*(x(t) 'G:(&:()/*8)(8+'+,8-+ω 3456@/(E+)-9:>):+ ')(+),)(++,8 qa'm7?9((*/'()*&-/(),g& :G-8P'y(t) 'P+,:G-y(t) + c H(j) ω 5 + ω ω 5 7.7rad/sec H(j) + j H(j), 4e j j, 4 cos(t.5)., 8 cos(t.5) d y(t) dt + 6 dy(t) 8y(t) x(t). dt Y + 6jωY 8Y (jω) X(jω) (jω)/x(jω)('+ <U Y H(jω) Y (jω) X(jω) (jω) + 6jω + 8 ( + jω)(4 + jω) c ( + jω) + c (4 + jω) ( + jω)h(jω) jω 4

16 C+'.8)(+'8P-': ) )(.'8''(/!"#$%#('+')(+),)( V@&-9M7?+,r<)rF&'&'+()/(*E:8 c -/():8G.8S : (4 + jω)h(jω) 4 jω 4 H(jω) ( + jω) (4 + jω) h(t) F {H(jω)} F { + jω } F { 4 + jω } e t u(t) e 4t u(t) (e t e 4t )u(t) dy(t) + y(t) x(t) dt dy(t) S : + y(t) dx(t) ')9+,8)-*L,P8>fghfi?''-:+' + x(t) dt dt 9+()9:G-8(:'- r$s"t$uv9)ll.'8''),++') _&(.'89'*-(+,8+')(+'8++X L),+'*(+':G-++,8;L),)(C-* x(t) y(t) 3456Q(*8+E+8)(8&'-9+*&'MM7?9S('&X <Z '8>:+(jω)Y (jω) + Y (jω) X (jω) H(jω) Y (jω) X (jω) + jω

17 --, H(jω) P:+ ':G-+P(,w(t)&8/', M'++)(+'8++')9+,8;wt!x%yXt!z'*E+ '.M7?9S (jω)y (jω) + Y (jω) (jω)x (jω) + X (jω) Y(jω) X (jω) + jω + jω. x(t) w(t) y(t) )+''*+'*'((): D(*'&:9')(:')9L(+98(/,8+,8?')P.'88)(8&'++*&'L()++')9: (jω)/x(jω)('+ (jω)c+'-+*e'88-9+:8:8) H (jω)w(jω)+w(jω) H (jω)y 9''8.8(8Y H(jω) Y (jω) X(jω) H (jω) + H (jω)h (jω). )(+'8H(jω) C',/''(/):8(*8:(+)/*8(P),)) +'+)*:()/*8H(jω)('+ ll,p8>fghfi(*'+')(+'8;:()/*?'+x H (jω) + H (jω)h (jω) <\ +jω + +jω +jω +jω + (3 + jω)( ω H(jω) ω + 9 ω + H(jω) arctan( ω ) + arctan( ω 3 ) + arctan( ω) {-+9'8-8+'8:+:G-8++,8P'Y (jω)

18 +':G8:+ M'++'L),)(++,8(9'-8 Frequency (rad/s) Frequency (rad/s) C',(*'L),)(++,8 7(.'8'(/>/(-+('+ + jω ( Y (jω) H(jω)X(jω) (3 + jω)( + jω) jω + πδ(ω)) +*'*+*)*(9'(*I+'Y (jω) Y (jω) jω + + jω + 3 jω + 3 πδ(ω). y(t) [ 3 6 e 3t e t ]u(t) y(t) /3 +8+':8&'+&'9&'+(P,)(','LP+(', M'-*+r$s"t$uv':+'+8('+(9 t!wvx'(. LLP,)'simulink?'P:+'-+(,+:'9)( cn')*continuous)(e+ <^ t +'8H(jω) b + b (jω) b n (jω) n α + α (jω) α m (jω) m (9 t!wvt ransf erf Magnitude Phase (degrees)

19 ':('+(/),(*8G-+{':+))+*E+)P.8 (.'8h}zt}w%'('+!"#$%# ''*'9(8,8+LP9+(P,{r$s"t$uv/'E t!wv ):G-8++,8/''',?''),+9 *'&+8)(.'8scopes('+8(/):8(*8A(E'8) jωc'-++,8(9l),+'*(--, ;'(&8(:(+8)*P t!wv)*'&-)) parameters:'p(.' P:+s t!wvstepf unctionstep time ('s%u"simulation Simulation L:+)',:G-8(:'-&8*&''' '*-(-9'&(8'*-()9)&)++':G-&8 <b

20 N{()*&)9)&--~%u%#:''-')I-, C-*':G-+)+).8)+'>8G-+'*-+ a&i-:':g-8('*::'m7?++)(+'8 jωc-*':g-+y(t)&(8''+'+)* -':vin(t) 3456;αJ('++'!"#$%#8G-+(*'+(/)*'X )':G-'v(t) -)':G-',FK 'H(jω) v(t) v in (t) v(t) (t + )u(t + ) tu(t) + (t )u(t ) V olts { vin (t), v in (t) <, v in (t) H(jω) y(t)

21 M'++>+,8G-+/(E+'( v in (t) v(t) t t V (jω) v(t)e jωt dt ;βd)(+'8++,88/'(.'():'9-x /(,+:G-8'(*&88-+''-:+' (t + )e jωt dt + e jωt dt (t )e jωt dt jω [(t + )e jωt ] + e jωt dt jω jω [e jωt ] + jω [(t )e jωt ] e jωt dt jω jω ejω (jω) [ejω e jω ] + jω [ejω e jω ] + jω e jω + (jω) [e jω e jω ] (jω) [ejω + e jω ] (jω) [ejω + e jω ] ω[cos(ω) cos(ω)] Sl(-),+'*()(+).'.';z%#$!y$w$sz"tx%#}$u(E'9 v(t)':g-p:+,+()*&,8t y(t) ++(&,F< x(t) δ(t nt ) n ;αc-*, 3T t 3T ;βj'l),+'ω 'T 7('T

22 ;γ+8+':88(*8 ;ɛ?'+'*(x(t)'-8'8+,8)(+'8 k''(* ':G-8++,8p!"#$%#c ;δc-*/*+,8 4ω :G-8'(&+,p 3T,'(', ω co π/t ;στ?'ω co 3π/T 3456;αM-* 3T t ;γm'+++8+':88)p),8(*8 L),+'P'!"#$%#)(.'+ ;β@/'t C+'.8+'*(x(t)('(/ (-ck k c k e jkω t ω 4ω { e 4jω ω < ω H(jω) co ω > ω co x(t) 3T T T T T 3T ω π π T 5 T T T x(t)e jkωt dt δ(t)e jk(π/5)t dt δ(t)dt t x(t) e jkπ t F5 k

23 &()8:+?'P&(,++'*(+,&8L,('+ω)&8L),+' ;δm'(,+/*8+'*(8()'++>?/9δ(t) P9δ(t [(/*8+'*(8P:'-(/,'(*8+'*(8 (.'8':)/(8+'*(8)P),(*!"#$%#('+ nt ) e jnωt :+ T ω 4ω n T n δ(t nt ) F n n T e jnωt ω T n n e jnωt e jnωt e jnωt T δ(ω nω ) ('''(,+ 4ω ;ɛa+)*9)p+('-)*r'*-8('+-.''8l:l X(ω)/ω (*8:G-8P' π/5):g-8' π/c+'.8+')(' (88 8+'8+'9(8ω co +o('*o9'+8jk+'8;ω,k ;στc'(&+ω co 3π/T (+o('*'o/('k )FH --,+'8 π/5,, 4π/5 3π/5 π/5 π/5 π/5 π/5 3π/5 4π/5 y(t) H(j) e j()(π/5)t.,,, y(t) H(j) ej()(π/5)t + H(jπ/5) ej()(π/5)t + H( jπ/5) ej( )(π/5)t. +.e 4jπ/5 e j(π/5)t +.e 4jπ/5 e j(π/5)t (. + cos ( π 5 t 4π ) ) 5 (. + cos ( π (t 4)) 5 ω

24 W (L),(-)L),+'+()*&,8)+++X ':8c +++8+':88(*8!"#$%#(9'98)P),8(*8 5('):8;+':8 π/tm' + (L*'>fghfi-*,+()95, 5, )A 3456J(/'.8L),(-8+,8'T)L),+'ω ck Mk :+c T T T/ x(t) T A k x(t) A/ T/ c k e jkω t T/ π jk x(t)e T t dt T/ T/ FR x(t)dt T T/ T/ A T T/ T t + A dt A [ t ] T/ T T/ + A T A [ t ] T/ T/ T t

25 J((9+':8'.8/')(P)E+'(, '.k sx $t%-:'-*8a'(-t),p8n&'+'.' )+P(*++'*-'(),8+'*(/9)+,'(, +P:+ M'(,+(*!"#$%#L,P8>fghfī+(9(*)*& š œš ƒ ˆ ˆƒ Š Œ Ž Ž (/x(t) ž ˆŸ š œš Œ Œ š Œš ŒšŒ ž ˆŸ ˆƒ ˆ š ƒ ˆ ˆ žƒ ˆƒ ª Š«Šƒ ˆ ˆ žˆ Š Ÿ œˆƒ ˆƒ Œ Œ Œ ž žœ ž FU c k T T/ T/ T/ x(t)e jk π T t dt (A T T/ T t + A ) e jk π T t dt A [ T (t T ) + e jk π T jkπ A T ( Te jkπ ) T jkπ j A( )k kπ k c k e jk π T t ] T/ T t A T/ T/ A T π jk T e T t dt jkπ T/ }{{} k,k j A( )k e kπ π jk T t

26 ž ˆ Œ Ž Œˆƒ Ÿ ± ˆ ˆƒ Š Š Š Œ Ÿˆ«Œ Ž² ž ²Ž² ²Ž² ˆ ˆ ²Ž²«² Ž²³ Ÿ ²Ž Œ ŒŽ ² ˆƒ Š ŠƒŠ ² ƒ Š Œ ² ˆ ˆ žˆ Š² C',L:+'(:8+'*(8N 5, 5, 5. Fourier series using 5 harmonic components Fourier series using 5 harmonic components Y&8/'),')*+8((+8+':8(9 )9((E++'*(+P:+FZ Fourier series using 5 harmonic components

27 ] (8(:8 C-*>fghfi,+()9+'(.'85+':8 ;αd+'*(+p:+'(+/'', 3456!"#$%#&'()*&(-).'+'(,&'(-T π π]at D+'*(f')*+',8[ π, { t ( π, ) ;αf(t) t [, π] { t ( π, ) ;βf(t) t t [, π] { t t ( π, ) ;γf(t) f() t t (, π] { ;δf(t) t π t ( π, ) t π t [, π] π f(t) π t α π π α n π b n π π f(t)dt π π π π π π cos(nt)dt π F\ f(t)sin(nt)dt π π sin(nt)dt π nπ sin(nt) π nπ cos(nt) π f(t)cos(nt)dt dt ( )n, n,, 3,... nπ

28 @:'&8+'*(f(t)('((* )5':8+'*((E&8G,8T f(t) + π π])(*'', ;βd+'*(f')*+',8[ ( sin(t) + 3 sin(3t) sin((n )t) +...) n Fourier series using 5 harmonic components π π f(t) π π t α π f(t)dt π tdt π π π π 4 α n π f(t)cos(nt)dt π π π F^ t cos(nt)dt π ( t sin(nt) π π ) sin(nt) π n dt n cos(nt) π π n ( )n n n,, 3,... π

29 '.α V?(+'*(f(t)(E&8G,8 b n π t sin(nt)dt π ( t cos(nt) π ) π π n + cos(nt) dt n [ ( ) n π + sin(nt) π] π n n ( )n, n,, 3,... n f(t) π 4 π cos((n )t) ( ) n sin(nt) (n ) + )5+':8:+'()*&(:n n n Fourier series using 5 harmonic components (, π]) *()/'',f(t) π?/9'*('b n ;γd+'*(+p:+'(+')*+',8[ π, n:+ ) π π π t α π π π f(t)dt π π Fbtdt π

30 [(+'*(('(/&8*P((&'(&'')+P( α n π f(t)cos(nt)dt π π )5+':8:+', π t cos(nt)dt π ( t sin(nt) π π ) sin(nt) π n dt n cos(nt) π π n ( )n n n,, 3,... π f(t) π 4 cos((n )t) π (n ) n /')+P, t Fourier series using 5 harmonic components ;δn+'*(')*+',8[ π, π])(,α n D+'*( π π HK f(t ) π π

31 V?( f(t) n+e'&8g,8 A+':8b b n π π π π π π ( x ) sin(nt)dt π ( x ) sin(nt)dt pi π π π tsin(nt)dt cos(nt) n nπ [( )n ] + ( )n nπ nπ )5+':8('+n π sin(nt) n.6 Fourier series using 5 harmonic components H<

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