Numerit[WIN32][1700][1703]% qffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@         Times New RomanArialSymbol Courier New??? G (\P@ C@yx>@>@>@>@>@>@>@ >@ >@ >@>@>@>@>@>@ >@  >==>>??x-1.11.10.50.010.01-.26.510.010.01G (\P@ C@zx>@>@>@>@>@>@>@ >@ >@ >@>@>@>@>@>@ >@  >==>>??x-2210.010.01-.26.510.010.01E c@1@Rf(x)=a02+ancosnx+bnsinnxn=1n=1.!}z}yzyx$z(yyx0|z(zzzz}zzx0|z(zzzz}zzzxyzxyyRRdRR RRRRR E c@X1@-f(x)=sinnxnn=135...!}z}yzyx.$zzz}zzzzxyyyyyyyyyy--d--  ----- E c@X1@:f(x)=(-1)n+1sinnxn.n=123...!}z}yzyx.y}xyy(zxy$zzz}zzzyzxyyyyyyyyy::d::   ::::: G (\P@"~j:@wx>@>@>@>@>@>@>@ >@ >@ >@>@>@>@>@>@ >@  >==>>??-1.11.110.010.01-0.26.520.010.01X ffffff@@nnE UUUUUu:@ @#an=0;bn=1n!z(zxyy}}}z(zx$yz##dSd###### E @@*@,an=0;bn=-1n+1n!z(zxyy}}}z(zx$xy(zxyz,,dSd,,,,,, vvvpvvvpvvvpvvvpvvvpvvvpvvv vvv.Fourier SeriesvvvffB WvvvffB A Fourier series is an expansion of a function in a series of sines and cosines such asvvvffB ! " ^vvvffB One advantage of a Fourier representation over some other representations, such as a Taylor series, is that it may represent discontinuous functions. Two well known examples of such functions are the square wave and the sawtooth wave, both very useful in electronic circuits that handle pulses. The Fourier representation of a square wave is given byvvvffB " # avvvffB namely, the cosine terms have vanished, and only the odd terms in the sine series remain () ).9vvvffB The Fourier representation of a sawtooth wave is given byvvvffB # $ nvvvffB Here again, the cosine terms have vanished, and the sine terms are all there but with alternating sign (* ).vvv The following figures show how a square wave and a sawtooth wave are built as sums of sine waves. The upper traces are the terms added to the sum on each cycle. Note how each added term gets us closer to the desired wave form.vvv vvv3B Term no.: ' NvvvffB &  F ! ! . A square wave !" . A sawtooth wave q8ffffff)@j@fffffvq@?@ffffff9@?@ffffff9@ffffff)@ffffff)@         Times New RomanArialSymbol Courier New =` This sample program demonstrates Fourier series by building:` a square wave and a sawtooth wave as sums of sine waves.;` We see how adding more terms to the sum gets us closer to` the desired wave forms.>` -> Move to the bottom of the Report to see the graph viewers` and then run the program."x = 0 to 2*pi len 200 ` one periody = 0 ` initial squarez = 0 ` initial sawtoothfor n = 1 to 21 w = sin(n*x)/n ` current term if odd(n)` add odd term to bothy += w ` squarez += w ` sawtoothelse$` subtract even term from sawtoothz -= w ` sawtoothnn = n ` display term numberrefresh ` update viewerswait .5 ` pause for a whiled:\num\num1.5\work\ x y z nw nn  7 B f     X8  B C 7 N N6 O   N636(+ 02@200i@1?215@.5?