0 0. 02 0. 04 0. 06 0. 08 0. 1 0.1 2-1-0.8-0.6-0.4-0.200. 20. 40. 60. 81Composite signals (waveform synthesis)=! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= X0+ Re Xkk=1"#ej 2$kf0t& ' ( ) * + k=5! x(t) = 0.8105cos 2"25t +"( )+ 0.0901cos 2"75t +"( )+ 0.0324cos 2"125t +"( )0 0.02 0. 04 0 .0 6 0 .0 8 0. 1 0. 12-1-0.8-0.6-0.4-0.200. 20. 40. 60. 81! Xk=8"2k2ej"k odd0 k even# $ % & % ! x(t) = 0.8105cos 2"25t +"( )+ 0.0901cos 2"75t +"( )+ 0.0324cos 2"125t +"( )+ ...-300 -200 -100 0 100 200 30000. 050 . 10. 150 . 20. 250 . 30. 350 . 40. 4525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0 .0 2 0. 04 0.06 0 .08 0.1 0.1 2-1-0.8-0.6-0.4-0.200. 20. 40. 60. 81spectrum! x(t ) = 0.8105 cos 2"25t +"( )+ 0.0901cos 2"75t +"( )+ 0.0324 cos 2"125t +"( )+ ...-300 -200 -100 0 10 0 200 30000.0 50. 10.1 50. 20.2 50. 30.3 50. 40.4 525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0.0 2 0. 04 0.0 6 0. 08 0.1 0.1 2-1-0.8-0.6-0.4-0.200.20.40.60.81spectrum! x(t) = 0.8105ej 2"25t+"( )+ e# j 2"25t+"( )2+ 0.0901ej 2"75t+"( )+ e# j 2"75t+"( )2+ 0.0324ej 2"125t+"( )+ e# j 2"125t+"( )2+ ...Complex conjugate form-300 -200 -100 0 10 0 200 30000.0 50. 10.1 50. 20.2 50. 30.3 50. 40.4 525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0.0 2 0. 04 0.0 6 0. 08 0. 1 0.1 2-1-0.8-0.6-0.4-0.200.20.40.60.81spectrum! x(t) = 0.8105ej 2"25t+"( )+ e# j 2"25t+"( )2+ 0.0901ej 2"75t+"( )+ e# j 2"75t+"( )2+ 0.0324ej 2"125t+"( )+ e# j 2"125t+"( )2+ ...Complex conjugate form! x(t) = 0.4053ej 2"25t+"( )+ 0.4053e# j 2"25t+"( )+ 0.0450ej 2"75t+"( )+ 0.0450e# j 2"75t+"( )+ 0.0162ej 2"125t+"( )+ 0.0162e# j 2"125t+"( )+ ...-300 -200 -100 0 10 0 200 30000.0 50. 10.1 50. 20.2 50. 30.3 50. 40.4 525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0.0 2 0. 04 0.0 6 0. 08 0. 1 0.1 2-1-0.8-0.6-0.4-0.200.20.40.60.81spectrum! x(t) = 0.4053ej 2"25t+"( )+ 0.4053e# j 2"25t+"( )+ 0.0450ej 2"75t+"( )+ 0.0450e# j 2"75t+"( )+ 0.0162ej 2"125t+"( )+ 0.0162e# j 2"125t+"( )+ ...! x(t) = 0.4053ej"ej 2"25t( )+ 0.4053e# j"e# j 2"25t( )+ 0.0450ej"ej 2"75t( )+ 0.0450e# j"e# j 2"75t( )+ 0.0162ej"ej 2"125t( )+ 0.0162e# j"e# j 2"125t( )+ ...-300 -200 -100 0 100 200 30000. 050 . 10. 150 . 20. 250 . 30. 350 . 40. 4525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0 .0 2 0. 04 0.06 0 .08 0.1 0.1 2-1-0.8-0.6-0.4-0.200. 20. 40. 60. 81! x(t) = 0.0162e" j#e" j 2#125t( )+ 0.0450e" j#e" j 2#75t( )+ 0.4053e" j#e" j 2#25t( )+ 0.4053ej#ej 2#25t( )+ 0.0450ej#ej 2#75t( )+ 0.0162ej#ej 2#125t( )+ ...-300 -200 -100 0 100 200 30000. 050 . 10. 150 . 20. 250 . 30. 350 . 40. 4525 75 125-125 -75 -25! 0.4053e" j#! 0.4053ej"! 0.0450ej"! 0.0450e" j#! 0.0162ej"! 0.0162e" j#! f! X0 0 .0 2 0. 04 0.06 0 .08 0.1 0.1 2-1-0.8-0.6-0.4-0.200. 20. 40. 60. 81! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= Re Xkn=0"#ej2$kf0t& ' ( ) * + = Zkk=,""#ej2$kf0t“two sided Fourier Series”Fourier SeriesFor a given signal, how do we findfor each k in the Fourier Series ?Fourier Analysis! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= X0+ Re Xkk=1"#ej 2$kf0t& ' ( ) * + ! X0=1T0x(t)dt0T0"where! Xk=2T0x(t)e" j 2#ktT0dt0T0$! Xk= Akej"k! T0= 1/ f0:fundamental frequency! f0Fourier Series: Sawtooth! X0=T02! Xk=T0"kej"2! x(t) = t 0 " t < T0! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= X0+ Re Xkk=1"#ej 2$kf0t& ' ( ) * + ! x(t) =T02+T0"kk=1#$cos 2"kf0t +"2( ) ! x(t) =T02+T0"cos 2"f0t +"2( )+T02"cos 2"2 f0t +"2( )+ KDefined between 0<t<0.04Periodic with period 0.040.04 0.02 0 0.02 0.0400.020.040.040y tt0.040.04 t! x(t) =1 0 " t < T02#1 T02 " t < T0$ % & ! Xk=" j4k#k odd0 k even$ % & ' & ! X0= 0! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= X0+ Re Xkk=1"#ej 2$kf0t& ' ( ) * + ! x(t) =4"cos 2"f0t #"2( )+43"cos 2"3 f0t #"2( )+ K! Xk=4k"e# j"2k odd0 k even$ % & ' & 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0410.500.511.21.2y tz t0.040 tFourier Series:Square Wave ! x(t) =4"sin 2"f0t( )+43"sin 2"3 f0t( )+ K! Xk=" j4k#k odd0 k even$ % & ' & ! X0= 0! x(t) = A0+ Akk=1"#cos 2$kf0t +%k( )= X0+ Re Xkk=1"#ej 2$kf0t& ' ( ) * + Fourier Series:Square Wave Spectrum ! x(t) =4"sin 2"f0t( )+43"sin 2"3 f0t( )+ K! x(t) = X0+ Xkk=1"#ej 2$kf0t+ e% j 2$kf0t2& ' ( ) * + ! Z0= 0! Zk=Xk2=2k"e#"2kk = ±1,±3…0 k = ±2,±4…$ % & ' & -10 -8 -6 -4 -2 0 2 4 6 8 1 000.20.40.60.811.21.4ampphase-10 -8 -6 -4 - 2 0 2 4 6 8 10-2-1.5-1-0.500.511.52! Xk=4k"e# j"2k odd(1,3,5…)0 k even(2,4,6…)$ % & ' & Complex conjugate form! x(t) = Xkk="##$ej 2%kf0t2& ' ( ) * + = Zkk="##$ej 2%kf0t! Zk=" j2k#k = ±1,±3,…0 k = ±2,±4…$ % & ' & “two sided Fourier Series”Properties of Fourier Series! x(t) =1 0 " t < T02#1 T02 " t < T0$ % & ! X0= 0 ! x(t) =4"cos 2"f0t #"2( )+43"cos 2"3 f0t #"2( )+ K! Xk=4k"e# j"2k odd0 k even$ % & ' & ! x(t) =4"sin 2"f0t( )+43"sin 2"3 f0t( )+ KEx. Square wave0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.0410.500.511.21.2y tz2 t0.040.04 ,t todd functionOdd functions consist only of sums of sines ( )Even functions consist only of sums of cosines ( )! "= #$2! "= 0! f ("x) = " f (x)! f ("x) = f (x)Properties of Fourier Series! x(t) =1 0 " t < T02#1 T02 " t < T0$ % & ! X0= 0 ! x(t) =4"cos 2"f0t #"2( )+43"cos 2"3 f0t #"2( )+ K! Xk=4k"e# j"2k odd0 k even$ % & ' & ! x(t) =4"sin 2"f0t( )+43"sin 2"3 f0t( )+ KEx. Square waveodd function0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.0410.500.511.21.2y tz2 t.4!sin..2 !tT0.4.3 !sin...2 ! 5tT00.040.04 ,t tOdd functions consist only of sums of sines ( )Even functions consist only of sums of cosines ( )! "= #$2! "= 0! f ("x) = " f (x)! f ("x) = f (x)Properties of Fourier Series! x(t) =1 0 " t < T02#1 T02 " t < T0$ % & ! X0= 0 ! x(t) =4"cos 2"f0t #"2( )+43"cos 2"3 f0t #"2( )+ K! Xk=4k"e# j"2k odd0 k even$ % & ' & ! x(t) =4"sin 2"f0t( )+43"sin 2"3 f0t( )+ KEx. Square waveodd function0.04 0.03 0.02 0.01 0 0.01 0.02 0.03 0.0410.500.511.21.2y tz2 tsin..2 !tT0cos..2 !tT00.040.04 ,t tOdd functions consist only of sums of sines ( )Even functions consist only of sums of cosines ( )! "= #$2! "= 0! f ("x) = " f (x)! f ("x) = f (x)Properties of Fourier Series! x(t) =4t + T0"T0/2 # t < 0"4t + T00 # t < T02$ % & ! X0= 0Ex.Triangle waveeven functionIn[1]:= 1/T*Integrate[-4*t-T,{t,-T/2,0}]+1/T*Integrate[4*t-T,{t,0,T/2}] Out[1]= 0! X0=1T0"4t "
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