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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 ).
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+ 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) dω @:'&8(.'8'-:+ 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@)/(*'!"#$%#r(jω)8r(t)l*+98x(jω), Y (jω) 3456 ;γy&8);l+'r(jω)(.'8':+;-8,>'+'r(jω) +'8 y(t)' t?/9+),(&)'r(t)(x Z &'x(t), ;βp:x(t) y(t) e
;αe'8'>8r(t):+ y(t):+ R(jω) ;β?')p.'88+'(,8x(t), [()*P(&(9'(*I+'+'*(r(t)&8G,8 P:+ +'*(r(t)'e τ e t+τ dτ?'t > J('&'8.('>8+'*(8+(:)+IP:+ Mt < y(t)p-' ;γva&8:++'(&*)>&'+'(,&'x(t), r(t) r(t) t ( + t)e t ( t)e t R(jω) r(t)e jωt dt [ x(τ)y(t + τ)dτ ] e jωt dt x(τ) [ y(t + τ)e jωt dt ] dτ x(τ)y (jω)e jωτ dτ X( jω)y (jω) r(t) e τ e t+τ dτ + e τ e t+τ dτ + t t x(τ)y(t + τ)dτ e τ e t τ dτ + e τ e t+τ dτ + r(t) ( + t )e t r(t)e jωt dt ( t)e t e jωt dt + t e τ e t τ dτ e τ e t τ dτ ( + t)e t e jωt dt + jω + ( + jω) + jω + ( jω) 4 ( + ω ) :X(jω) Y (jω) \ + ω
C9/&'(α)(.9 θ);αl('f(xm(t)) X(jω)+F8 R(jω) X( jω)y >8r(t)P'R(jω) 9/&''-8-(/&8P' θ)>8+'*(8+,8 )+/&':(.(β) ]?'F(x(t))!"#$%#)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 + @:'&8(E++'*(y(t) (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 ))
;β@/(e'8>)-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 P*(+ @:'&8P:+ 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)
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(/),(* '>+)*9@'-))*-'+',9/'' '&'>+'-+99(8T; T/&8T/-'' ')*''8(,8-888('(9'++ @G*E'8'G&(*)*P8(.+(9'99' '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/
[(>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))(
3456;α@/(E'8!"#$%#x(t)('+ ;βq*''8(,8()8+e+'>)*p(+8g-++/(+ π''-:px(t) π):'-')(/( ;γ7(''(/!"#$%#'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
β7(.'8(l*'>fghfi+e+')(+'8++x,8;:()/*(),)) Magnitude.95 H(jω).9.85 5 4 3 3 4 5 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 '- 4 5 5 5 5 ω [rad/sec] *E+')(++,8'jk+';ω --,:G-8P'7 H(j) 7 6/7 M''-,-((9,-+:+'ω 7[rad/sec] @/(E'8.('(,8+:(P8),+'M7?9/(X 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) 6 + 6.5 cos(7t.6π)
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)} + 5 + jω 5 + jω 5 + jω 4 Magnitude squared 3.5 3 H(jω).5.5 <R 5 5 5 5 ω [rad/sec] Angle.4.3.. H(jω) [rad]...3.4 5 5 5 5 ω [rad/sec]
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) :+ @/(E'8'(,8+:(P8+'-+*E+8:(+8G-+8)('+ + :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.5 5 + 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
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--, 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)( + 4 + ω H(jω) ω + 9 ω + H(jω) arctan( ω ) + arctan( ω 3 ) + arctan( ω) {-+9'8-8+'8:+:G-8++,8P'Y (jω)
+':G8:+ M'++'L),)(++,8(9'-8 Frequency (rad/s) 4 6 8 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) 6 3 + 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)
':('+(/),(*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
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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.4...4.6.8 8 6 4 4 6 8 H<