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ComputingFourierSeriesandPower

SpectrumwithMATLAB

ByBrianD.Storey

1.Introduction

frequencyranges.

2.TheMathPart

timedomaintothefrequencydomain. f(t)=a0+1X n=1(ansin(2¼nt)+bncos(2¼nt))(2.1)

EXPaper

2FourierSeries

end(i.ef(t=0)=f(t=1)). oftheequationtheequalitywillhold. 0Ã f(t)=a0+1X n=0(ansin(2¼nt)+bncos(2¼nt))! sin(2¼mt)dt:(2.7)

2.7willreducetoZ1

f(t)sin(2¼nt)dt=an

2:(2.8)

toobtaintheanalogousresult: f(t)cos(2¼nt)dt=bn

2:(2.9)

interval0FourierSeries3

3.Someexamples

result. sin(2¼t)sin(2¼nt)dt=an:(3.1) 1=2 sin(2¼nt)dt=an:(3.2)

¡2cos(2¼nt)

2¼n¯

¯1=2

=an(3.3) andevaluatingatt=0andt=1=2yields

1¡cos(¼nt)

¼n=an:(3.4)

Thereforewhennisodd,

n=2

¼n(3.5)

andwhenniseven n=0(3.6)

2sin(2¼nt)

2¼n¯

¯1=2

=bn(3.7) andevaluatingatt=0andt=1=2yields sin(¼nt)

¼n=bn;(3.8)

which,foralln,issimply n=0(3.9)

4FourierSeries

00.10.20.30.40.50.60.70.80.91-0.2

0.2 0.4 0.6 0.8 1.2 time f(t) n=20 n=2,000 untilweincludemanypointsintotheseries. thetimeinterval 0=Z 1=2 dt=1=2:(3.10) toconstructthesignalaccurately.

N=2000;

x=[0:100]/100; f=ones(1,101)*1/2; fori=1:2:N a=2/pi/i; f=f+a*sin(2*pi*i*x); end plot(x,f)

4.FourierTransformofDiscreteData

FourierSeries5

00.10.20.30.40.50.60.70.80.91-1

-0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 time f(t)t1t2t3t4t5t6t7t 1A2A3 5A6A up. seeFigure2 j=f(tj)+f(tj+1)

2(tj+1¡tj)(4.1)

¢t.Thereforethetotalareais

A=¢tµf(t1)+f(t2)

(4.2) whichgeneralizesto f(t)dt=¢t0 @f(t1)=2+f(tn)=2+N¡1X j=2f(tj)1

A(4.3)

j=2sin(2¼ntj)f(tj)](4.4)

6FourierSeries

theintegralRx(N) datapoints.quotesdbs_dbs2.pdfusesText_4