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Fourier Analysis for GPSASEN5190P. AxelradOctober 2003Periodic Functions A periodic function of period Tp = 1/fp Can be expressed by a Fourier Trigonometric Series as: pf tnTf t 01cos2sin2npnpnf taanf tbnf tChange of variable 01Let 2,so for , 2 The function is then expressed as:cossinppnnnxf ttTxff xaanxbnxFourier Coefficients 0121cos 1sin nnaf x dxaf xnx dxbf xnx dxExamples 2111Constant ( )cos 0sin 02111cos cos 0cos 1cos sin 02111cos ()cos 0coscos 0cos sin 02sin AAdxAAnx dxf xnx dxnxnx dxnx dxnxnx dxmxmnmx dxmxnx dxmxnx dxn2111sin 0sin cos 0sin 12xnx dxnxnx dxnx dxa0anbnSquare Wave if -0 if 0Axf xAx 0000000123411351102211cos cos 0 for all 112sin sin 1 cos44 , 0 , , 0 ,.34so sinsin3sin5.nnaAdxAdxaAnx dxAnx dxnAbAnx dxAnx dxnnAAbbbbetcAf xxxxx2A4Fourier Transform of a Pulse 011111110 if 11 if 1( )( )cos( )sin( )( )cos ( )( )cos sin2sin2( )cos sincxf xxf xAxBx dAfdBfdAdSinc and Sinc2-5-4-3-2-1012345-0.500.51-5-4-3-2-101234500.20.40.60.81Fourier series for a signal that is periodic in P 01/210/2/21/2/2/21/2/2cos2sin2cos 2 sin 2 npnpnPPPPnpPPPnpPPf taanf tbnf taf t dtaf tnf t dtbf tnf t dt 2T2/2/2110/2/2/2/2/211/2/2/2/2/20 if NT=P fA if sin 2cos 2 cos 2 2/22 sin2sin/2sin/TPTPTPPPTTPTpnppPPpPTTpntf tNftATAaf t dtAdtPNAnf taf tnf t dtAnf t dtnf PAnf TAnNAanNnNNcnNxAT/2T/2P=NTPeriodic rectangular pulseFourier representation of a periodic rectangular pulse 12sinccos2pnAAnf tnf tNNN-2.5-2-1.5-1-0.500.511.522.5x 106-505101520 x 10-4-2-1.5-1-0.500.511.52x 10400.511.52x 10-3Zoomed in to show linesSeries for 1/T=1.023e6 and fp=1e3
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