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which is the same as in Example 17.1.
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The series obtained in Example 17.1 contained only sine terms in addition to a constant term. Other waveforms will have only cosine terms; and sometimes only odd harmonics are present in the series, whether the series contains sine, cosine, or both types of terms. This is the result of certain types of
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FOURIER METHOD OF WAVEFORM ANALYSIS
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symmetry exhibited by the waveform. Knowledge of such symmetry results in reduced calculations in determining the Fourier series. For this reason the following de nitions are important. 1. A function f x is said to be even if f x f x . The function f x 2 x2 x4 is an example of even functions since the functional values for x and x are equal. The cosine is an even function, since it can be expressed as the power series cos x 1 x2 x4 x6 x8 2! 4! 6! 8!
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The sum or product of two or more even functions is an even function, and with the addition of a constant the even nature of the function is still preserved. In Fig. 17-3, the waveforms shown represent even functions of x. They are symmetrical with respect to the vertical axis, as indicated by the construction in Fig. 17-3(a).
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Fig. 17-3
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2. A function f x is said to be odd if f x f x . The function f x x x3 x5 is an example of odd functions since the values of the function for x and x are of opposite sign. The sine is an odd function, since it can be expressed as the power series sin x x x3 x5 x7 x9 3! 5! 7! 9!
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The sum of two or more odd functions is an odd function, but the addition of a constant removes the odd nature of the function. The product of two odd functions is an even function. The waveforms shown in Fig. 17-4 represent odd functions of x. They are symmetrical with respect to the origin, as indicated by the construction in Fig. 17-4(a).
Fig. 17-4
Fig. 17-5
CHAP. 17]
FOURIER METHOD OF WAVEFORM ANALYSIS
A periodic function f x is said to have half-wave symmetry if f x f x T=2 where T is the period. Two waveforms with half-wave symmetry are shown in Fig. 17-5.
When the type of symmetry of a waveform is established, the following conclusions are reached. If the waveform is even, all terms of its Fourier series are cosine terms, including a constant if the waveform has a nonzero average value. Hence, there is no need of evaluating the integral for the coe cients bn , since no sine terms can be present. If the waveform is odd, the series contains only sine terms. The wave may be odd only after its average value is subtracted, in which case its Fourier representation will simply contain that constant and a series of sine terms. If the waveform has half-wave symmetry, only odd harmonics are present in the series. This series will contain both sine and cosine terms unless the function is also odd or even. In any case, an and bn are equal to zero for n 2; 4; 6; . . . for any waveform with half-wave symmetry. Half-wave symmetry, too, may be present only after subtraction of the average value.
Fig. 17-6
Fig. 17-7
Certain waveforms can be odd or even, depending upon the location of the vertical axis. The square wave of Fig. 17-6(a) meets the condition of an even function: f x f x . A shift of the vertical axis to the position shown in Fig. 17-6(b) produces an odd function f x f x . With the vertical axis placed at any points other than those shown in Fig. 17-6, the square wave is neither even nor odd, and its series contains both sine and cosine terms. Thus, in the analysis of periodic functions, the vertical axis should be conveniently chosen to result in either an even or odd function, if the type of waveform makes this possible. The shifting of the horizontal axis may simplify the series representation of the function. As an example, the waveform of Fig. 17-7(a) does not meet the requirements of an odd function until the average value is removed as shown in Fig. 17-7(b). Thus, its series will contain a constant term and only sine terms. The preceding symmetry considerations can be used to check the coe cients of the exponential Fourier series. An even waveform contains only cosine terms in its trigonometric series, and therefore the exponential Fourier coe cients must be pure real numbers. Similarly, an odd function whose trigonometric series consists of sine terms has pure imaginary coe cients in its exponential series.
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