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*k[.I in Visual Studio .NET
*k[.I Read QR Code In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Paint QR In VS .NET Using Barcode creator for .NET Control to generate, create QR image in .NET applications. =*k+o~N,,[~l
Scan Quick Response Code In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Painting Barcode In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. integer
Decoding Bar Code In VS .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications. QR Code ISO/IEC18004 Maker In Visual C#.NET Using Barcode generator for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. Thus, the sequences q k [ n ]are distinct only over a range of No successive values of k.
Denso QR Bar Code Encoder In .NET Framework Using Barcode creator for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Painting QRCode In VB.NET Using Barcode generator for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
UPC Code Maker In .NET Framework Using Barcode creation for .NET framework Control to generate, create UPCA Supplement 2 image in Visual Studio .NET applications. EAN / UCC  13 Drawer In .NET Using Barcode creation for Visual Studio .NET Control to generate, create EAN / UCC  13 image in VS .NET applications. B. Discrete Fourier Series Representation: Barcode Creator In .NET Using Barcode encoder for .NET framework Control to generate, create bar code image in .NET framework applications. USPS POSTNET Barcode Drawer In .NET Using Barcode generation for Visual Studio .NET Control to generate, create Postnet 3 of 5 image in Visual Studio .NET applications. The discrete Fourier series representation of a periodic sequence x[n] with fundamental period No is given by Reading USS128 In Visual Basic .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. USS128 Drawer In Java Using Barcode generator for BIRT Control to generate, create EAN 128 image in BIRT reports applications. where c, are the Fourier coefficients and are given by (Prob. 6.2) Code 3 Of 9 Generation In None Using Barcode maker for Online Control to generate, create Code 39 Extended image in Online applications. Barcode Creator In None Using Barcode creator for Online Control to generate, create barcode image in Online applications. Because of Eq. (6.5) [or Eq. (6.6)], Eqs. (6.7) and (6.8) can be rewritten as
Bar Code Creation In Java Using Barcode creation for BIRT Control to generate, create barcode image in BIRT reports applications. Code 128 Generation In VS .NET Using Barcode generation for ASP.NET Control to generate, create Code 128B image in ASP.NET applications. where C , denotes that the summation is on k as k varies over a range of No successive integers. Setting k = 0 in Eq. (6.101, we have Generate Code 128 In Visual Basic .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Code128 image in .NET framework applications. Encoding Code128 In Java Using Barcode maker for BIRT Control to generate, create Code 128B image in Eclipse BIRT applications. which indicates that co equals the average value of x[n] over a period. The Fourier coefficients c, are often referred to as the spectral coefficients of x[n]. C. Convergence of Discrete Fourier Series: Since the discrete Fourier series is a finite series, in contrast to the continuoustime case, there are no convergence issues with discrete Fourier series. D. Properties of Discrete Fourier Series: I. Periodicity of Fourier Coeficients: From Eqs. ( 6 . 5 ) and (6.7) [or (6.911, we see that
C,+N, = Ck
which indicates that the Fourier series coefficients c, are periodic with fundamental period No.
2. Duality: From Eq. (6.12) we see that the Fourier coefficients c, form a periodic sequence with fundamental period No. Thus, writing c, as c[k], Eq. (6.10) can be rewritten as FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
Let n
 m in Eq. (6.13). Then
Letting k
and m = k in the above expression, we get
Comparing Eq. (6.14) with Eq. (6.91, we see that (l/N,,)x[k] of c[n]. If we adopt the notation x [ n ]B c k= c [ k ] are the Fourier coefficients (6.15) to denote the discrete Fourier series pair, then by Eq. (6.14) we have DFS 1 ~ [ nc) ] x[k] No Equation (6.16) is known as the duality property of the discrete Fourier series. 3. Other Properties: When x[n] is real, then from Eq. (6.8) or [Eq. (6.10)] and Eq. (6.12) it follows that * (6.17) C P k =CN,,k= ck where denotes the complex conjugate.
Even and Odd Sequences: When x[n] is real, let x[nl =xe[nl + ~ o [ n l where xe[n] and xo[n] are the even and odd components of x[n], respectively. Let x[n] S c k Then xe[n] Re[ck] (6.18~) (6.186) xo[n] 2%j Im[ck] Thus, we see that if x[n] is real and even, then its Fourier coefficients are real, while if x[n] is real and odd, its Fourier coefficients are imaginary. E. Parseval's Theorem: If x[n] is represented by the discrete Fourier series in Eq. (6.9), then it can be shown that (Prob. 6.10) Equation (6.19) is called Parseval's identity (or Parseual's theorem) for the discrete Fourier series. CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
6 3 THE FOURIER TRANSFORM
A. From Discrete Fourier Series to Fourier Transform: Let x [ n ] be a nonperiodic sequence of finite duration. That is, for some positive integer N , , Such a sequence is shown in Fig. 6l(a). Let x,Jn] be a periodic sequence formed by repeating x [ n ] with fundamental period No as shown in Fig. 6l(b). If we let No  m, we , have No+ lim x N o [ n ]= x [ n ] The discrete Fourier series of x N o [ n ] given by is
where
Fig. 61 ( a ) Nonperiodic finite sequence x [ n ] ; 6 ) periodic sequence formed by periodic extension of ( xhl. FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
[CHAP. 6
Since xN,,[n] x [ n ] for In1 I and also since x [ n ] = 0 outside this interval, Eq. (6.22~) = N, can be rewritten as Let us define X(R) as
Then, from Eq. (6.22b) the Fourier coefficients c , can be expressed as
Substituting Eq. (6.24) into Eq. (6.21), we have
From Eq. (6.231, X(R) is periodic with period 27r and so is eJRn.Thus, the product X(R)e*'" will also be periodic with period 27r. As shown in Fig. 62, each term in the 1" summation in Eq. (6.25) represents the area of a rectangle of height ~ ( k R , ) e ' ~ ~ 1and width R,. As No + m, 0, = 27r/N0 becomes infinitesimal (R, + 0) and Eq. (6.25) passes to an integral. Furthermore, since the summation in Eq. (6.25) is over N,, consecutive intervals of width 0, = 27r/N,,, the total interval of integration will always have a width 27r. Thus, as NO+ a: and in view of Eq. (6.20), Eq. (6.25) becomes 1 x [ n ] =  ~ ( 0ejRnd R ) (6.26) 27r 2 v Since X(R)e 'On is periodic with period 27r, the interval of integration in Eq. (6.26) can be taken as any interval of length 27r. Fig. 62 Graphical interpretation of Eq. (6.25).

