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2d barcode generator vb.net CXV'. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS in Visual Studio .NET
CXV'. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS QR Code 2d Barcode Reader In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Print Denso QR Bar Code In VS .NET Using Barcode generation for Visual Studio .NET Control to generate, create QR image in VS .NET applications. B. Fourier Transform Pair: Decode Quick Response Code In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Maker In VS .NET Using Barcode drawer for .NET framework Control to generate, create bar code image in .NET framework applications. The function X(R) defined by Eq. (6.23) is called the Fourier transform of x[n], and Eq. (6.26) defines the inverse Fourier transform of X(R). Symbolically they are denoted by X ( R ) = F { x [ n ] )= Decode Bar Code In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Printing QR Code JIS X 0510 In Visual C# Using Barcode drawer for VS .NET Control to generate, create QR image in Visual Studio .NET applications. m n= m
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Ix[n]kw
(6.31) E. Connection between the Fourier Transform and the zTransform: Equation (6.27) defines the Fourier transform of x[n] as X(R) = x[n] ejnn
n = m
The ztransform of x[n], as defined in Eq. (4.3), is given by X ( Z )= x[n]z" n m
Comparing Eqs. (6.32) and (6.331, we see that if the ROC of X(z) contains the unit circle, then the Fourier transform X(R) of x[n] equals X(z) evaluated on the unit circle, that is, ~ ( a=) ( z ) l , = , , ~ ~ ~ (6.34) Note that since the summation in Eq. (6.33) is denoted by X(z), then the summation in Eq. (6.32) may be denoted as X(ejn). Thus, in the remainder of this book, both X(R) FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
[CHAP. 6
and X(ejn) mean the same thing whenever we connect the Fourier transform with the ztransform. Because the Fourier transform is the ztransform with z = ein, it should not be assumed automatically that the Fourier transform of a sequence x [ n ] is the ztransform with z replaced by eiR. If x [ n ] is absolutely summable, that is, if x [ n ] satisfies condition (6.311, the Fourier transform of x [ n ] can be obtained from the ztransform of x [ n ] with = e i f l since the ROC of X(z) will contain the unit circle; that is, leinJ= 1. This is not generally true of sequences which are not absolutely summable. The following examples illustrate the above statements. EXAMPLE 6.1. Consider the unit impulse sequence 6 [ n l . From Eq. (4.14) the ztransform of 6 [ n ] is By definitions (6.27) and (1.45)the Fourier transform of 6 [ n ] is
Thus, the ztransform and the Fourier transform of 6 [ n ] are the same. Note that 6 [ n ] is absolutely summable and that the ROC of the ztransform of 6 [ n l contains the unit circle. EXAMPLE 6.2. Consider the causal exponential sequence
x [ n ]=anu[n] a real
From Eq. ( 4 . 9 ) the ztransform of x [ n ] is given by
Thus, X(ei") exists for la1 < 1 because the ROC of X ( z ) then contains the unit circle. That is, Next, by definition (6.27) and Eq. (1.91)the Fourier transform of x [ n ] is
Thus, comparing Eqs. (6.37)and (6.38),we have
X ( R ) =X(z)l,=p
Note that x [ n ] is absolutely summable.
EXAMPLE 6.3. Consider the unit step sequence u[nl. From Eq. (4.16)the ztransform of u[nl is
The Fourier transform of u [ n ] cannot be obtained from its ztransform because the ROC of the
CHAP. 61 FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
ztransform of u[n] does not include the unit circle. Note that the unit step sequence u[n] is not absolutely summable. The Fourier transform of u[n] is given by (Prob. 6.28) PROPERTIES OF THE FOURIER TRANSFORM
Basic properties of the Fourier transform are presented in the following. There are many similarities to and several differences from the continuoustime case. Many of these properties are also similar to those of the ztransform when the ROC of X ( z) includes the unit circle. Periodicity: As a consequence of Eq. (6.41), in the discretetime case we have to consider values of R (radians) only over the range 0 I R < 27r or 7r I R < 7r, while in the continuoustime case we have to consider values of o (radians/second) over the entire range  m < o < m. B. Linearity: C. Time Shifting: D. Frequency Shifting:

