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x * [ n ] X*(R) where in Visual Studio .NET
x * [ n ] X*(R) where Scan Quick Response Code In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Generate QR Code 2d Barcode In .NET Framework Using Barcode maker for VS .NET Control to generate, create QRCode image in VS .NET applications. * denotes the complex conjugate.
QR Code ISO/IEC18004 Reader In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. Paint Bar Code In Visual Studio .NET Using Barcode maker for VS .NET Control to generate, create bar code image in .NET applications. F. Time Reversal: Bar Code Decoder In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. QR Code ISO/IEC18004 Generator In Visual C#.NET Using Barcode printer for .NET framework Control to generate, create QR Code image in .NET framework applications. FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
Denso QR Bar Code Printer In .NET Framework Using Barcode generator for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. QR Code Encoder In VB.NET Using Barcode printer for VS .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. G. Time Scaling: UPCA Generator In .NET Framework Using Barcode encoder for VS .NET Control to generate, create UPC Code image in .NET framework applications. EAN / UCC  13 Encoder In .NET Framework Using Barcode drawer for .NET framework Control to generate, create EAN13 image in VS .NET applications. In Sec. 5.4D the scaling property of a continuoustime Fourier transform is expressed as [Eq. (5.5211 1D Maker In .NET Framework Using Barcode generator for Visual Studio .NET Control to generate, create Linear image in VS .NET applications. USS93 Encoder In .NET Using Barcode maker for Visual Studio .NET Control to generate, create Code 9/3 image in .NET framework applications. However, in the discretetime case, x[an] is not a sequence if a is not an integer. On the other hand, if a is an integer, say a = 2, then x[2n] consists of only the even samples of x[n]. Thus, time scaling in discrete time takes on a form somewhat different from Eq. (6.47). Let m be a positive integer and define the sequence x[n/m] = x [ k ] Then we have if n = km, k = integer ifn#km GS1 128 Generator In None Using Barcode encoder for Software Control to generate, create EAN / UCC  14 image in Software applications. Print UPCA In C#.NET Using Barcode drawer for VS .NET Control to generate, create UCC  12 image in VS .NET applications. Equation (6.49) is the discretetime counterpart of Eq. (6.47). It states again the inverse relationship between time and frequency. That is, as the signal spreads in time (m > I), its Fourier transform is compressed (Prob. 6.22). Note that X(rnR) is periodic with period 27r/m since X ( R ) is periodic with period 27r. Code 39 Extended Scanner In Visual Studio .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Encoding ECC200 In None Using Barcode encoder for Font Control to generate, create Data Matrix ECC200 image in Font applications. H. Duality: ECC200 Reader In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications. Code128 Encoder In Java Using Barcode drawer for Java Control to generate, create Code 128B image in Java applications. In Sec. 5.4F the duality property of a continuoustime Fourier transform is expressed as [Eq. (5.5411 Reading Barcode In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Code39 Drawer In C# Using Barcode printer for Visual Studio .NET Control to generate, create Code 3/9 image in .NET applications. There is no discretetime counterpart of this property. However, there is a duality between the discretetime Fourier transform and the continuoustime Fourier series. Let From Eqs. (6.27) and (6.41) Since fl is a continuous variable, letting R = t and n
k in Eq. (6.51), we have
CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Since X ( t ) is periodic with period To = 27r and the fundamental frequency oo= 27r/T0 = 1 , Eq. (6.53) indicates that the Fourier series coefficients of ~ ( twill be x [  k ] . This duality ) relationship is denoted by ~ ( t )c , = x [  k ] B
where FS denotes the Fourier series and c, are its Fourier coefficients.
(6.54) I. Differentiation in Frequency: J. Differencing: The sequence x [ n ]  x [ n  11 is called the firsf difference sequence. Equation (6.56) is easily obtained from the linearity property (6.42) and the timeshifting property (6.43). K. Accumulation: Note that accumulation is the discretetime counterpart of integration. The impulse term on the righthand side of Eq. (6.57) reflects the dc or average value that can result from the accumulation. L. Convolution: As in the case of the ztransform, this convolution property plays an important role in the study of discretetime LTI systems. M. Multiplication: FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
where @ denotes the periodic convolution defined by [Eq. (2.70)] The multiplication property (6.59) is the dual property of Eq. (6.58). N. Additional Properties: If x[n] is real, let
where x,[n] and xo[n] are the even and odd components of x[n], respectively. Let
, x [ n ] t X ( n ) = A ( R ) +jB(R) IX(R)leJe(n) (6.61) Then
Equation (6.62) is the necessary and sufficient condition for x[n] to be real. From Eqs. (6.62) and (6.61) we have A(  R ) = A ( R ) Ix(fl)I= Ix(R)I B(R) +a) B(R) 9(~) (6.64a) (6.646) From Eqs. ( 6 . 6 3 ~ (6.636), and (6.64~) see that if x[n] is real and even, then X(R) is )~ we real and even, while if x[n] is real and odd, X(R) is imaginary and odd. 0. Parseval's Relations: Equation (6.66 ) is known as Parseual's identity (or Parseual's theorem) for the discretetime Fourier transform. Table 61 contains a summary of the properties of the Fourier transform presented in this section. Some common sequences and their Fourier transforms are given in Table 62. CHAP. 6 FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS 1
Table 61. Properties of the Fourier Transform
Property
Sequence
Fourier transform
Periodicity Linearity Time shifting Frequency shifting Conjugat ion Time reversal Time scaling Frequency differentiation First difference Accumulation Convolution Multiplication Real sequence Even component
Odd component
Parseval's relations
FOURIER ANALYSIS OF DISCRETETIME SlGNALS AND SYSTEMS
[CHAP. 6
Table 62. Common Fourier Transform Pairs
sin Wn ,o<w<sr 7 n 7
THE FREQUENCY RESPONSE OF DISCRETETIME LTI SYSTEMS Frequency Response: In Sec. 2.6 we showed that the output y [ n ] of a discretetime LTI system equals the convolution of the input x [ n ] with the impulse response h [ n ] ; that is, Applying the convolution property (6.581, obtain we
CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
where Y(R), X(R), and H(R) are the Fourier transforms of y [ n ] , x [ n ] , and h [ n ] , respectively. From Eq. ( 6.68) we have

