Question 1.A waveform is periodic with period 27 and over one cycle is defined by4-T

The given problem is to find the values of coefficients of complex Fourier series representation for…

Answered: Question 1. A waveform is periodic with…

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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Question 6. The complex Fourier series representation of a periodic function of period 2 is given by FS(t) = Σ n= ∞ Cne-int where cn = 4+ I 6 n³ (4-5(-1)")j. Find, to three decimal places, the amplitude |c₂| and phase on for n=1,2. Enter the real and imaginary values of c-₁ in the appropriate boxes below. Enter ₁ correct to 3 decimal places: Enter correct to 3 decimal places: Enter |c₂| correct to 3 decimal places: Enter 2 correct to 3 decimal places: Enter the real part of c_₁: Enter the imaginary part of c_₁: C-1
QUESTION 1 80 1.1 f(t) = nt - t, 0
Question 2 please.
c) For a periodic function y(t) with period T, its Fourier series may be represented in complex form as y(t) = Σ n=1∞ where w = 2π/T and all cn are complex constants defined for n integer. derive a formula for Cn. = i. Show that if y(t) is a real function satisfying y where the bar denotes the complex conjugate. ii. Using the fact that 2πT Cneinwt ei(n-m)t dt S 2π { = 0 y, then c_n = if m= n if m‡n Cn for all integers n, = Cn
2. Determine the complex Fourier coefficients co and c„ of the function simulating the periodic heating of a room if it is given that f(t) = e' – 1, 0
4 i want 2nd part only
4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).
10. If it is known that the periodic function f(t) contains even harmonics only, which Fourier coefficients will you calculate?
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
Solve c and d, using the table might help
Question 6. The complex Fourier series representation of a periodic function of period 2π is given by FS(t) = = ∞ n=1x Cne-int I where c = 2 + (5 – 6(−1)”)j . 2 Cn N³ Find, to three decimal places, the amplitude c and phase on for n = 1, 2. Enter the real and imaginary values of c-₁ in the appropriate boxes below. Enter |c₁| correct to 3 decimal places: Enter ₁ correct to 3 decimal places: Enter |c₂| correct to 3 decimal places: Enter 2 correct to 3 decimal places: Enter the real part of c_₁: Enter the imaginary part of c_1:

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Question 1. A waveform is periodic with period 27 and over one cycle is defined by -T