Math 19. Lecture 32Testing for PeriodicityT. JudsonFall 20051 Testing for a Term with Period pCertain phenomena may be the sum of periodic functions. For example,m(t) = md(t) + mm(t) + my(t)might be a function depending on a daily cycle (md), and monthly cycle (mm),and a yearly cycle (my). Yet, m(t) itself may or may not look periodic.• Suppose that we know m(t) on the interval [−τ0, −τ0+ τ]. Here τ0represents the amount o f time before the present at which our databegins and τ is the time span for which we have collected data.• We wish to know if our data m(t) has a periodic component of periodp. If p is much less than τ, we may be able to detect such a perio d.• We must compute the following three integrals:a ≡1τZ−τ0+τ−τ0m(t) cos(2πt/p) dt (1)b ≡1τZ−τ0+τ−τ0m(t) sin(2πt/p) dt (2)σ ≡1τZ−τ0+τ−τ0m(t)2dt. (3)1• With these numbers computed, we then computef =a2+ b2σ, (4)which has a value between 0 and 1. A significant component of theoriginal function m(t) with period p is signified by a large value for f.2 The Power Spectrum FunctionIn general, we may not have an a priori guess of what the periods of thecomponents of m might be (if there are periods). In practice, a standardapproach is to compute many values of f in (4) and see which values of pgive us large f .3 Fourier CoefficientsIt is customary to change va r ia bles from the period p to the frequency ν =1/p. Then (1) and (2) becomea ≡1τZ−τ0+τ−τ0m(t) cos(2πνt) dt (5)b ≡1τZ−τ0+τ−τ0m(t) sin(2πνt) dt. (6)In this case, f becomes a function of ν:f(ν) =a(ν)2+ b(ν)2σ.This function is called the power spectral d ensity function.4 An ExampleSuppose thatm(t) = α cos(2πt/q) + β sin(2πt/q).is periodic with period q. We compute the Fourier coefficients and the powerspectral density function.25 Trigonometric IntegralsFourier coefficients can often b e computed in exact form. We need the fol-lowing indefinite integ r als. If A 6= B, then•Zcos(At) cos(Bt) dt =sin((A − B)t)2(A − B)+sin((A + B)t)2(A + B)•Zsin(At) cos(Bt) dt =cos((A − B)t)2(A − B)−cos((A + B)t)2(A + B)•Zsin(At) sin(Bt) dt =sin((A − B)t)2(A − B)−sin((A + B)t)2(A + B)If A = B, then•Zcos2(At) dt =t2+14Asin(2At)•Zsin(At) cos(At) dt = −14Acos(2At)•Zsin2(At) dt =t2−14Asin(2At)6 ExampleSuppose that m(t) = e−|t|is defined on (−∞, ∞).•a(ν) =1τZ∞−∞e−|t|cos(2πνt) dt=2τZ∞0e−tcos(2πνt) dt=2τ2πνe−tsin(2πνt) − e−tcos(2πνt)1 + 4π2ν2∞0=2τ(1 + 4π2ν2)3•b(ν) =1τZ∞−∞e−|t|sin(2πνt) dt = 0.•σ =1τZ∞−∞e−2|t|dt =2τZ∞0e−2tdt =2τ−e−2t2∞0=1τ.•f(ν) =a(ν)2+ b(ν)2σ=4τ(1 + 4π2ν2)2Readings and References• C. Taubes. Modeling Differential Equations in Biol ogy. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter
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