Explanation Given data: The given diagram is shown in Figure 1. The time period of the given function is T = 6 s . Calculation: The expression for function f ( t ) calculated from the graph is, f ( t ) = { 10 2 < t < 3 0 otherwise The Fourier coefficient of n th term is, a n = 2 T ∫ 0 T f ( t ) cos n ω 0 t d t (1) Here, ω 0 is the angular frequency. The angular frequency is given by, ω 0 = 2 π T Substitute 6 s for T in the above relation. ω 0 = 2 π 6 = π 3 Substitute π 3 for ω 0 , 6 for T and 10 for f ( t ) which lies in the range between t = 2 s to t = 3 s in equation (1). a n = 2 6 ∫ 2 3 10 cos n π 3 t d t = 10 3 [ sin n π 3 t n π 3 ] 2 3 = 10 n π [ sin n 3 π 3 − sin n 2 π 3 ] = 10 n π [ sin n π − sin n 2 π 3 ] Further solve as, a n = − 10 n π [ sin n 2 π 3 ] (2) Substitute 1 for n in equation (2). a 1 = − 10 ( 1 ) π [ sin ( 1 ) 2 π 3 ] = − 10 π ( 0.866 ) = − 2.757 Substitute 2 for n in equation (2). a 2 = − 10 2 π [ sin ( 2 ) 2 π 3 ] = − 10 2 π [ sin 4 π 3 ] = − 5 π ( − 0.866 ) = 1.378 Substitute 3 for n in equation (2)

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Author: Hayt

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Fourier Series

Most of the incidents studied in the field of Engineering and Science as a function of time. For example, current and voltage are in the alternating current circuit. These occasional activities can be analyzed in the activity components (fundamental freq…

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Chapter 17, Problem 7E

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The Fourier coefficient a1, a2, a3, b1, b2 and b3.

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A continuous time signal x is shown below x(t) 2 t 1 2 3 If periodic (t) = (t - 3m) then which of the following statements is true? Select one: O a. O b. O c. O d. periodic is periodic with fundamental period two. periodic has duration two and is periodic with fundamental period three. periodic is not periodic. periodic has duration three.

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