Explanation Given data: Refer to Figure 17.86 in the textbook. Formula used: Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (1) Here, T is the period of the function. Write the general expression to calculate the Fourier series of f ( t ) . f ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos n ω 0 t + b n sin n ω 0 t ) (2) Here, a 0 is the dc component of f ( t ) , a n and b n are the Fourier coefficients, n is an integer, and ω 0 is the angular frequency. Calculation: The given waveform is drawn as Figure 1. Refer to Figure 1, the waveform is symmetry about the origin that is the plot is anti-symmetrical about the y -axis. The signal f ( t ) is fold about t = 0 but with the odd symmetry because the plot on negative time axis has opposite sign of the positive time axis signal. Therefore, the waveform is called as an odd function and it is expressed as, − f ( t ) = f ( − t ) Write the expressions for Fourier coefficients for an odd function. a 0 = 0 a n = 0 b n = 4 T ∫ 0 T 2 f ( t ) sin n ω 0 t d t (3) Refer to the Figure 1. The Fourier series odd function of the waveform is defined as, f ( t ) = { 10 , 0 < t < 1 0 , 1 < t < 4 − 10 , 4 < t < 5 The time period of the function in Figure 1 is, T = 5 Substitute 5 for T in equation (1) to find ω 0 . ω 0 = 2 π 5 Substitute 5 for T and 2 π 5 for ω 0 in equation (3) to find b n . b n = 4 5 ∫ 0 5 2 f ( t ) sin n ( 2 π 5 ) t d t = 4 5 ∫ 0 2.5 f ( t ) sin ( 2 n π 5 ) t d t = 4 5 ( ∫ 0 1 f ( t ) sin ( 2 n π 5 ) t d t + ∫ 1 2.5 f ( t ) sin ( 2 n π 5 ) t d t ) = 4 5 ( ∫ 0 1 10 sin ( 2 n π 5 ) t d t + ∫ 1 2.5 ( 0 ) sin ( 2 n π 5 ) t d t ) Simplify the above equation to find b n

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Fourier Transform

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Chapter 17, Problem 72P

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Find the Fourier series representation of the signal shown in Figure 17.86.

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Chapter 17, Problem 71P

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Find the Fourier series representation of the signal shown in Figure 17.86.

Solution Summary: The author explains the Fourier series representation of the signal in Figure 17.86. The waveform is symmetric about the origin that is the plot is anti-symmetrical about y -axis.

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Find the Fourier series representation of the signal shown in Figure 17.86.

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