Calculate the Fourier series of the periodic function depicted in the figure below.10.90.80.70.60.50.40.30.20.10-8-7-51a²-21Hint: fx cos(ax) dx = cos(ax) +=x sin(ax),a0X12345671fx sin(ax)dx = sin(ax)81sin(ax) -x cos(ax)a

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Answered: Calculate the Fourier series of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. The figure below is a piecewise continuous function of sine and square wave. Find f(t) representing both the piecewise continuous notation and its Fourier series. 1.5 0.5 -0.5- -1 -1.5 -2 -1.5 -1 -0.5 0.5 1.5 2 Time Answer for the Fourier Series. G- ) + sinnt + En=1( cos 2nnt + E1 ) sin(2n – 1)nt
SHOW COMPLETE SOLUTION. If f(x) = x2 + cos(x) is a periodic function with period 2W, thena. It is an odd function which gives a value of a0 = 0b. Its Fourier series is classified as a Fourier cosine series where a0 = 0c. it is neither odd nor even function, thus no classification can be deduced.d. it is an even function which gives a value of bn = 0
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a f(x) After writing the Fourier representation =90 +1 (an cos + bn sin ) = with ao 2 an = bn = ½ ſ²³3 ƒ (x) sin "™ª dx, 3 use the exponential forms Cos 0 = e²0 te-io sin = 2 2 2i of the cosine and sine functions to put that representation in exponential form: ƒ(x) = Σ‰… An exp (in) where Ao = ²³3 f (x) cos nx dx, n = 0, 1, 2, ... -3 3 n=1,2,... ηπα 3 An an-ibn = 2 eio -i0 A_n = an+ibn 2 (n = 1, 2, ...). b) An = ³₂ f(x) exp (-i) da (n = 0, ±1, ±2, ...) Then use expression of an and bn to obtain a single formula
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Question 7 2a If the exponential Fourier transform of f(x) = e alxl Not yet answered Marked out of What is the Fourier transform of e alx *e bx| Flag question Oa. 4ab (a² +p²)(b² +p²) Ob. (a² + p?)(b² I p?) 2ab (a²+p²)(b² + p²) Od. (a² + p²)(b²+p²) 4.
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Calculate the Fourier series of the periodic function depicted in the figure below. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -8 -7 -6 -1 1 Hint: fx cos(ax) dx = cos(ax) +=x sin(ax), a 0 X 1 2 F 4 3 5 6 7 8 fx sin(ax) dx = sin(ax) -1x cos(ax) a ■