Explanation Given data: Refer to Figure 17.83(a) and (b) in the textbook. The complex Fourier series of the function in Figure 17.83(a) in the textbook is, f ( t ) = 1 2 − ∑ n = − ∞ ∞ j e − j ( 2 n + 1 ) t ( 2 n + 1 ) π (1) Formula used Write the general expression to calculate the complex Fourier series. f ( t ) = ∑ n = − ∞ ∞ c n e j n ω 0 t (2) Here, c n is the new Fourier coefficient. n is an integer, and ω 0 is the angular frequency. The equation (2) is also re-written as follows, f ( t ) = c 0 + ∑ n = − ∞ n ≠ 0 ∞ c n e j n ω 0 t (3) Here, c 0 is the dc component of the complex Fourier series. Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (4) Here, T is the period of the function. Calculation: Given waveforms are redrawn as Figure 1 and Figure 2. Refer to the Figure 1. The Fourier series function of the waveform is defined as, f ( t ) = { 1 , 0 < t < π 0 , π < t < 2 π The time period of the function in Figure 1 is, T = 2 π Substitute 2 π for T in equation (4) to find ω 0 . ω 0 = 2 π 2 π = 1 By comparing the equations (1) and (3), value of c 0 of the function f ( t ) can be obtained. c 0 = 1 2 To find c 0 of the function h ( t ) : Refer to equation 17.56 in the textbook, the relationship between the dc components is, c 0 = a 0 (5) Write the expression to calculate the dc component of the function h ( t ) . a 0 = 1 T ∫ 0 T h ( t ) d t (6) Refer to the Figure 2. The Fourier series function of the waveform is defined as, h ( t ) = { 2 , 0 < t < 1 − 2 , 1 < t < 2 The time period of the function in Figure 2 is, T = 2 Substitute 2 for T in equation (4) to find ω 0 . ω 0 = 2 π 2 = π Substitute 2 for T in equation (6) to find a 0

6th Edition

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 59P

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Find the complex Fourier series representation of the function h(t) in Figure 17.83(b).

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