Fourier Series Square Wave Example The Fourier series of a ...

1 University of California, San DiegoJ. ConnellyFourier Series Square Wave ExampleThe Fourier Series of a Square wave with period1isf(t)=1+4 1Xn=1noddsin( nt)nIn what follows, we plot1+4 2N 1Xn=1noddsin( nt)nforN=1,2,..,10,25,50,75,100,1000, +4 1Xn=1noddsin( nt)n 2 1 Series of Square wave with 1 terms of sum21+4 3Xn=1noddsin( nt)n 2 1 Series of Square wave with 2 terms of sum31+4 5Xn=1noddsin( nt)n 2 1 Series of Square wave with 3 terms of sum41+4 7Xn=1noddsin( nt)n 2 1 Series of Square wave with 4 terms of sum51+4 9Xn=1noddsin( nt)n 2 1 Series of Square wave with 5 terms of sum61+4 11Xn=1noddsin( nt)n 2 1 Series of Square wave with 6 terms of sum71+4 13Xn=1noddsin( nt)n 2 1 Series of Square wave with 7 terms of sum81+4 15Xn=1noddsin( nt)n 2 1 Series of Square wave with 8 terms of sum91+4 17Xn=1noddsin( nt)n 2 1 Series of Square wave with 9 terms of sum101+4 19Xn=1noddsin( nt)n 2 1 Series of Square wave with 10 terms of sum111+4 49Xn=1noddsin( nt)

2 N 2 1 Series of Square wave with 25 terms of sum121+4 99Xn=1noddsin( nt)n 2 1 Series of Square wave with 50 terms of sum131+4 149Xn=1noddsin( nt)n 2 1 Series of Square wave with 75 terms of sum141+4 199Xn=1noddsin( nt)n 2 1 Series of Square wave with 100 terms of sum151+4 1999Xn=1noddsin( nt)n 2 1 Series of Square wave with 1000 terms of sum161+4 19999Xn=1noddsin( nt)n 2 1 Series of Square wave with 10000 terms of sum17 University of California, San DiegoJ. ConnellyFourier Series Sawtooth Wave ExampleThe Fourier Series of a sawtooth wave with period1isf(t)=12 1 1Xn=1sin(2 nt)nIn what follows, we plot12 1 NXn=1sin(2 nt)nforN=1,2,..,10,25,50,75,100,1000, 1 1Xn=1sin(2 nt)n 2 1 Series of sawtooth with 1 terms of sum212 1 2Xn=1sin(2 nt)n 2 1 Series of sawtooth with 2 terms of sum312 1 3Xn=1sin(2 nt)n 2 1 Series of sawtooth with 3 terms of sum412 1 4Xn=1sin(2 nt)n 2 1 Series of sawtooth with 4 terms of sum512 1 5Xn=1sin(2 nt)n 2 1 Series of sawtooth with 5 terms of sum612 1 6Xn=1sin(2 nt)n 2 1 Series of sawtooth with 6 terms of sum712 1 7Xn=1sin(2 nt)n 2 1 Series of sawtooth with 7 terms of sum812 1 8Xn=1sin(2 nt)n 2 1 Series of sawtooth with 8 terms of sum912 1 9Xn=1sin(2 nt)n 2 1 Series of sawtooth with 9 terms of sum1012 1 10Xn=1sin(2 nt)n 2 1 Series of sawtooth with 10 terms of sum1112 1 25Xn=1sin(2 nt)n 2 1 Series of sawtooth with 25 terms of sum1212 1 50Xn=1sin(2 nt)

3 N 2 1 Series of sawtooth with 50 terms of sum1312 1 75Xn=1sin(2 nt)n 2 1 Series of sawtooth with 75 terms of sum1412 1 100Xn=1sin(2 nt)n 2 1 Series of sawtooth with 100 terms of sum1512 1 1000Xn=1sin(2 nt)n 2 1 Series of sawtooth with 1000 terms of sum1612 1 10000Xn=1sin(2 nt)n 2 1 Series of sawtooth with 10000 terms of sum17 University of California, San DiegoJ. ConnellyFourier Series |sin(t)|ExampleThe Fourier Series of|sin(t)|isf(t)=2 4 1Xn=1cos(2nt)4n2 1In what follows, we plot2 4 NXn=1cos(2nt)4n2 1forN=1,2,..,10,25,50,75,100,1000, 4 1Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 1 terms of sum22 4 2Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 2 terms of sum32 4 3Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 3 terms of sum42 4 4Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 4 terms of sum52 4 5Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 5 terms of sum62 4 6Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 6 terms of sum72 4 7Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 7 terms of sum82 4 8Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 8 terms of sum92 4 9Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 9 terms of sum102 4 10Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 10 terms of sum112 4 25Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 25 terms of sum122 4 50Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)

4 | with 50 terms of sum132 4 75Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 75 terms of sum142 4 100Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 100 terms of sum152 4 1000Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 1000 terms of sum162 4 10000Xn=1cos(2nt)4n2 1 6 4 Series of |sin(t)| with 10000 terms of sum17