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and so Decoding QR Code JIS X 0510 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Code Creator In None Using Barcode generation for Software Control to generate, create QR image in Software applications. which is the same as in Example 17.1.
QR Code ISO/IEC18004 Recognizer In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. QR Code Maker In C#.NET Using Barcode generator for .NET Control to generate, create QR Code image in Visual Studio .NET applications. WAVEFORM SYMMETRY
Encode QR Code JIS X 0510 In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Painting QR Code In VS .NET Using Barcode encoder for .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications. 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 Print QR Code 2d Barcode In VB.NET Using Barcode drawer for VS .NET Control to generate, create Denso QR Bar Code image in .NET applications. Printing USS Code 128 In None Using Barcode printer for Software Control to generate, create Code 128 Code Set C image in Software applications. FOURIER METHOD OF WAVEFORM ANALYSIS
Code 39 Extended Creation In None Using Barcode printer for Software Control to generate, create ANSI/AIM Code 39 image in Software applications. Creating Barcode In None Using Barcode encoder for Software Control to generate, create bar code image in Software applications. [CHAP. 17
Barcode Creation In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. Drawing EAN13 Supplement 5 In None Using Barcode creator for Software Control to generate, create EAN / UCC  13 image in Software applications. 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! MSI Plessey Creator In None Using Barcode creation for Software Control to generate, create MSI Plessey image in Software applications. Draw Code 128 Code Set B In None Using Barcode generator for Word Control to generate, create Code 128C image in Word applications. 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. 173, the waveforms shown represent even functions of x. They are symmetrical with respect to the vertical axis, as indicated by the construction in Fig. 173(a). Scanning Code 39 Extended In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Creating UCC  12 In Java Using Barcode printer for BIRT Control to generate, create UPCA image in BIRT applications. Fig. 173 1D Creator In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create Linear 1D Barcode image in ASP.NET applications. EAN / UCC  13 Encoder In .NET Framework Using Barcode creator for ASP.NET Control to generate, create EAN / UCC  13 image in ASP.NET applications. 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! 2D Barcode Generation In .NET Using Barcode printer for ASP.NET Control to generate, create Matrix Barcode image in ASP.NET applications. Universal Product Code Version A Reader In Visual C#.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. 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. 174 represent odd functions of x. They are symmetrical with respect to the origin, as indicated by the construction in Fig. 174(a). Fig. 174 Fig. 175 CHAP. 17] FOURIER METHOD OF WAVEFORM ANALYSIS
A periodic function f x is said to have halfwave symmetry if f x f x T=2 where T is the period. Two waveforms with halfwave symmetry are shown in Fig. 175. 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 halfwave 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 halfwave symmetry. Halfwave symmetry, too, may be present only after subtraction of the average value. Fig. 176 Fig. 177 Certain waveforms can be odd or even, depending upon the location of the vertical axis. The square wave of Fig. 176(a) meets the condition of an even function: f x f x . A shift of the vertical axis to the position shown in Fig. 176(b) produces an odd function f x f x . With the vertical axis placed at any points other than those shown in Fig. 176, 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. 177(a) does not meet the requirements of an odd function until the average value is removed as shown in Fig. 177(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.

