EECE 301 Signals & Systems Prof. Mark FowlerFourier Series ExamplesEECE 301 Signals & SystemsProf. Mark FowlerDiscussion #5• Fourier Series ExamplesFourier Series ExamplesExample #1te−)(tx121-2 -1……sec/rad2220ππω==⇒=T choose10)1(10)1(211020112121021)(21)(1000⎥⎦⎤⎢⎣⎡+−==⎥⎦⎤⎢⎣⎡×+===+−+−−−−−+−∫∫∫∫∫tjktjktjktjkttjkTtttjkkejkdtedtedteedtetxdtetxTcπππππωπ[])1(211)1(21110)1(ππππjkeeejkjkjk+−=−+−=−+−⎩⎨⎧−=−koddkevenejk,1,1:Noteπkkjjkee )1()(lyequivalentor −==−−ππ)1(2)1(11πjkeckk+−−=−So…Now we can use Matlab to plot |ck| & ∠ck(rad/sec)(rad/sec)(rad)Example #2 )(tx112-2 -1-3 3Tπω==02T10−=tpick[])1()1()1(0)1()(2211211−−−−−−=≠⎥⎦⎤⎢⎣⎡−−−==−−−−∫πππππππjkjkkktjkjkedttecktjktjkk1)()1( :Note22−==−−=±jjekjkπcancel0)1(TableIntegral From2≠−=∫aaxaedxxeaxax0)1(2)1(222≠−+=−+= kkjkjkckkkπππ0020=≠=ckkckπWhen k < 0:ckis j(neg)(-1)kWhen k > 0:ckis j(pos)(-1)k⇒ Since ∠j(neg) = -π/2Since ∠j(pos) = π/2we get this⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎨⎧⎪⎩⎪⎨⎧−⎪⎩⎪⎨⎧==∠evennegativekoddpositivekoddnegativekevenpositivekkck&&,2&&,20,0ππ∫−==1100tdtcAlso…Easy to get the magnitude results: Harder to get the phase results:(rad/sec)(rad/sec)(rad)Matlab Explorationsdisc_05_FS_example.mfunction Disc_05_FS_example(type)%% Computes FS-based approximation to various signal types and plays sound%% Input: type = 'ramp'% 'rand'% 'tone1' ... no FS used% 'tone2'% 'pulse'% 'square'Fs=60000;Ts=1/Fs;t=(0:60000)*Ts;ft=2*pi*440*t;N=length(t);… etc.
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