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2d barcode generator vb.net z 2  3.2 + 1 Ans. ( a ) x[O]=2,x [ w ] = 0 ( b ) x[Ol = 0, x[wl= 1 in Visual Studio .NET
2 z 2  3.2 + 1 Ans. ( a ) x[O]=2,x [ w ] = 0 ( b ) x[Ol = 0, x[wl= 1 Decoding QR Code ISO/IEC18004 In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Paint QRCode In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create QR Code image in VS .NET applications. (b) X(z)= Quick Response Code Reader In .NET Framework Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Bar Code Generator In VS .NET Using Barcode maker for Visual Studio .NET Control to generate, create barcode image in .NET framework applications. , 121 > 1
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Generating Code39 In VS .NET Using Barcode maker for VS .NET Control to generate, create USS Code 39 image in .NET applications. Encode International Standard Book Number In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create ISBN image in Visual Studio .NET applications. In previous chapters we introduced the Laplace transform and the ztransform to convert timedomain signals into the complex sdomain and zdomain representations that are, for many purposes, more convenient to analyze and process. In addition, greater insights into the nature and properties of many signals and systems are provided by these transformations. In this chapter and the following one, we shall introduce other transformations known as Fourier series and Fourier transform which convert timedomain signals into frequencydomain (or spectral) representations. In addition to providing spectral representations of signals, Fourier analysis is also essential for describing certain types of systems and their properties in the frequency domain. In this chapter we shall introduce Fourier analysis in the context of continuoustime signals and systems. EAN 128 Creator In ObjectiveC Using Barcode generation for iPhone Control to generate, create UCC  12 image in iPhone applications. Code 128 Code Set B Encoder In C#.NET Using Barcode creator for VS .NET Control to generate, create ANSI/AIM Code 128 image in VS .NET applications. FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS Periodic Signals: UPCA Supplement 2 Maker In Visual Basic .NET Using Barcode encoder for VS .NET Control to generate, create UCC  12 image in .NET framework applications. Scan Data Matrix In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. In Chap. 1 we defined a continuoustime signal x ( t ) to be periodic if there is a positive nonzero value of T for which x(t + T )= x ( t ) all t (5.1) The fundamental period To of x ( t ) is the smallest positive value of T for which Eq. (5.1) is satisfied, and l / T o =fo is referred to as the fundamental frequency. Two basic examples of periodic signals are the real sinusoidal signal Print Universal Product Code Version A In Java Using Barcode encoder for BIRT Control to generate, create UPC Symbol image in BIRT reports applications. GS1 RSS Generator In Java Using Barcode encoder for Java Control to generate, create GS1 RSS image in Java applications. and the complex exponential signal X ( t )= e~%' where oo= 2n/To = 2nfo is called the fundamental angular frequency. B. Complex Exponential Fourier Series Representation: The complex exponential Fourier series representation of a periodic signal x ( t ) with fundamental period To is given by (5.3) Code 128 Code Set B Encoder In Java Using Barcode encoder for Java Control to generate, create Code 128 Code Set C image in Java applications. EAN 128 Printer In None Using Barcode generator for Excel Control to generate, create UCC128 image in Excel applications. FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
where c , are known as the complex Fourier coefficients and are given by
where 1 denotes the integral over any one period and 0 to To or  T o / 2 to T 0 / 2 is " , commonly used for the integration. Setting k = 0 in Eq. ( 5 3 , we have which indicates that co equals the average value of x ( t ) over a period. When x ( t ) is real, then from Eq. ( 5 . 5 ) it follows that C  , =C, (5.7) where the asterisk indicates the complex conjugate.
C. Trigonometric Fourier Series: The trigonometric Fourier series representation of a periodic signal x ( t ) with fundamental period T,, is given by x(t)=  (a,cos ko,t
+ b , sin k w o t ) w0 =  (5.8) where a , and b, are the Fourier coefficients given by
x ( t ) c o s kw,tdt
(5.9a) The coefficients a , and b, and the complex Fourier coefficients c , are related by (Prob. 5.3) From Eq. ( 5 . 1 0 ) we obtain
+ ( a ,  jb,) = 2 Re[c,] c  , = : ( a k + jb,) b, =0 = (5.11) When x ( t ) is real, then a , and b, are real and by Eq. (5.10) we have
a, Even and Odd Signals:  2 Im[c,] (5.12) If a periodic signal x ( t ) is even, then b, cosine terms: x(t) = a0
and its Fourier series ( 5 . 8 ) contains only
a , cos kwot
o0=  CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
If x ( t ) is odd, then a, and its Fourier series contains only sine terms: m 2T T x(t) = b, sin kwot w0= k= 1 To
D. Harmonic Form Fourier Series: Another form of the Fourier series representation of a real periodic signal x ( t ) with fundamental period To is Equation (5.15) can be derived from Eq. (5.8) and is known as the harmonic form Fourier series of x(t). The term Co is known as the d c component, and the term C, cos(kwot  0,) is referred to as the kth harmonic component of x(t). The first harmonic component ~ C, C O S ( ~ , 8,) is commonly called the fundamental component because it has the same fundamental period as x(t). The coefficients C, and the angles 8, are called the harmonic amplitudes and phase angles, respectively, and they are related to the Fourier coefficients a, and b, by For a real periodic signal ~ ( r ) the Fourier series in terms of complex exponentials as , given in Eq. (5.4) is mathematically equivalent to either of the two forms in Eqs. (5.8) and (5.15). Although the latter two are common forms for Fourier series, the complex form in Eq. (5.4) is more general and usually more convenient, and we will use that form almost exclusively. E. Convergence of Fourier Series: It is known that a periodic signal x(t) has a Fourier series representation if it satisfies the following Dirichlet conditions: x ( t ) is absolutely integrable over any period, that is, 2. x(t) has a finite number of maxima and minima within any finite interval of t . 3. x(t) has a finite number of discontinuities within any finite interval of t, and each of these discontinuities is finite. Note that the Dirichlet conditions are sufficient but not necessary conditions for the Fourier series representation (Prob. 5.8). F. Amplitude and Phase Spectra of a Periodic Signal:

