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Complex Fourier Series: 3 – 2 / 12 Euler's Equation: e Euler's Equation: e iθ = cosθ + isinθ [see RHB 3 3] Hence: cosθ = e iθ+e−iθ 2 x(t)dt This is the average over an integer number of cycles For a complex exponential: 〈ei2πnF t 〉 

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The complex Fourier series is presented first with pe- riod 2π, then on a different set of basis functions: 1 (i e a constant term) e it e 2it e 3it e 4it e−it e−2it series because integrals with exponentials in are usu- ally easy to f(x) Find the complex Fourier series to model f(x) that has a period of 2π and is 1 when 0 

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So, e-jwt is the complex conjugate of ejwt e -jωt I Q cos(ωt) Previous Lecture • The Fourier Series can also be written in terms of cosines and sines: t T x(t) 

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2 Fourier Trigonometric Series 37 5 2 Complex Exponential Fourier Series 1−x to 1 + x + x2 and 1 + x + x2 + x3 19 1 15 Comparison of 1 1−xto ∑20 n=0 xn 20 2 1 Plots 2 4 Functions y(t) = 2 sin(47t) and y(t) = 2 sin(47t + 77/8) 39

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complex exponentials called Fourier series: xМ (t) = 0∑ =-0 X[片]e 0Ш The coefficients X[片] (a in OWN) can be found in two steps: 1 multiply both sides by e - 

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Exponential Fourier series: Let the (real or complex) signal r r (t)e −j2π(kf0)t dt , (38) for some arbitrary α We give some remarks here δ (at − n) = x(at) F

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4.3 Fourier Series

Denition 4.37.

Exp onentialF ourierseries : Let the (real or complex) signalr(t) be aperiodicsignal with periodT0. Suppose the followingDirichletconditions are satised: (a)r(t) is absolutely integrable over its period; i.e.,RT0

0jr(t)jdt <1.

(b) The n umberof maxima and minima of r(t) in each period is nite. (c) The n umberof discon tinuitiesof r(t) in each period is nite. Thenr(t) can be \expanded" into a linear combination of the complex exponential signalsej2(kf0)t1 k=1as ~r(t) =1X k=1c kej2(kf0)t=c0+1X k=1 c kej2(kf0)t+ckej2(kf0)t (37) where f 0=1T 0and c k=1T 0+T0Z r(t)ej2(kf0)tdt;(38) for somearbitrary.

We give some remarks here.

~r(t) =r(t);ifr(t) is continuous att r(t+)+r(t)2 ;ifr(t) is not continuous att Although ~r(t) may not be exactly the same asr(t), for the purpose of our class, it is sucient to simply treat them as being the same (to avoid having two dierent notations). Of course, we need to keep in mind that unexpected results may arise at the discontinuity points. The parameterin the limits of the integration (38) is arbitrary. It can be chosen to simplify computation of the integral. Some references simply writeck=1T 0R T

0r(t)ejk!0tdtto emphasize that we only need

to integrate over one period of the signal; the starting point is not important. 50
The coecientsckare called the (kth)Fourier (series) coecients of (the signal)r(t). These are, in general, complex numbers. c0=1T 0R T

0r(t)dt= average or DC value ofr(t)

The quantityf0=1T

0is called thefundamental frequencyof the

signalr(t). Thekth multiple of the fundamental frequency (for positive k's) is called thekthharmonic. ckej2(kf0)t+ckej2(kf0)t= thekthharmonic componentofr(t). k= 1)fundamental componentofr(t).

4.38.Being able to writer(t) =P1

k=1cnej2(kf0)tmeans we can easily nd the Fourier transform of any periodic function: r(t) =1X k=1c kej2(kf0)tF*)F1R(f) = The Fourier transform of any periodic function is simply a bunch of weighted delta functions occuring at multiples of the fundamental frequency f 0.

4.39.Formula (38) for nding the Fourier (series) coecients

c k=1T 0+T0Z r(t)ej2(kf0)tdt(39) is strikingly similar to formula (5) for nding the Fourier transform:

R(f) =1

Z 1 r(t)ej2ftdt:(40)

There are three main dierences.

We have spent quite some eort learning about the Fourier transform of a signal and its properties. It would be nice to have a way to reuse those concepts with Fourier series. Identifying the three dierences above lets us see their connection. 51

4.40.Getting the F ourierco ecientsf romthe F ouriertransform :

Step 1

Consider a restricted v ersionrT0(t) ofr(t) where we only considerr(t) for one period.1 6 4quotesdbs_dbs2.pdfusesText_2