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244 chapter 10coefficients a k . If we use the trapezoid rule to approximate the integration in (10.38),we determine the N independent Fourier components asa n ≃ 2hTN∑cos (nωt k ) y(t k )= 2 Nk=1N∑( ) 2πnkcos y k , n=0,...,N. (10.41)Nk=1Because we can determine only N Fourier components from N independent y(t)values, our Fourier series for the y(t) must be in terms of only these components:N∑N∑( ) 2πnty(t) ≃ a n cos(nωt)= a n cos . (10.42)Nhn=0In summary, we sample the function y(t) at N times, t 1 , ..., t N . We see that allthe values of y sampled contribute to each a k . Consequently, if we increase N inorder to determine more coefficients, we must recompute all the a n values. In thewavelet analysis in Chapter 11, “WaveletAnalysis & Data Compression,” the theoryis reformulated so that additional samplings determine higher Fourier componentswithout affecting lower ones.n=010.4.4 AssessmentsSimple analytic input: It is always good to do these simple checks before examiningmore complex problems. If your system has some Fourier analysispackages (such as the graphing package Ace/gr), you may want to compareyour results with those from the packages. Once you understand how thepackages work, it makes sense to use them.1. Sample the even signaly(t) = 3 cos(ωt) + 2 cos(3ωt) + cos(5ωt).Decompose this into its components; then check that they are essentiallyreal and in the ratio 3:2:1 (or 9:4:1 for the power spectrum) and that theyresum to give the input signal.2. Experiment on the separate effects of picking different values of the stepsize h and of enlarging the measurement period T = Nh.3. Sample the odd signaly(t) = sin(ωt) + 2 sin(3ωt) + 3 sin(5ωt).Decompose this into its components; then check that they are essentiallyimaginary and in the ratio 1:2:3 (or 1:4:9 if a power spectrum is plotted)and that they resum to give the input signal.4. Sample the mixed-symmetry signaly(t) = 5 sin(ωt) + 2 cos(3ωt) + sin(5ωt).−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 244