2d barcode generator vb.net CXV'. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS in Visual Studio .NET

Drawer Quick Response Code in Visual Studio .NET CXV'. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS

CXV'. 61 FOURIER ANALYSIS OF DISCRETE-TIME 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
QR Code Maker In VS .NET
Using Barcode printer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications.
QR Creator In Visual Basic .NET
Using Barcode drawer for .NET Control to generate, create QR-Code image in .NET applications.
x[n] ePJRn
Code 39 Extended Creation In VS .NET
Using Barcode creator for .NET Control to generate, create Code 39 Full ASCII image in VS .NET applications.
Printing EAN13 In .NET Framework
Using Barcode maker for .NET Control to generate, create EAN / UCC - 13 image in Visual Studio .NET applications.
(6.27)
Bar Code Generator In .NET
Using Barcode creation for .NET framework Control to generate, create barcode image in .NET framework applications.
USPS OneCode Solution Barcode Maker In VS .NET
Using Barcode maker for .NET Control to generate, create 4-State Customer Barcode image in VS .NET applications.
and we say that x[n] and X(R) form a Fourier transform pair denoted by (6.29) Equations (6.27) and (6.28) are the discrete-time counterparts of Eqs. (5.31) and (5.32). C. Fourier Spectra: The Fourier transform X(R) of x[n] is, in general, complex and can be expressed as As in continuous time, the Fourier transform X(R) of a nonperiodic sequence x[n] is the frequency-domain specification of x[n] and is referred to as the spectrum (or Fourier spectrum) of x[n]. The quantity IX(R)I is called the magnitude spectrum of x[n], and #dR ) is called the phase spectrum of x[n]. Furthermore, if x[n] is real, the amplitude spectrum IX(R)I is an even function and the phase spectrum 4((n) is an odd function of R.
Create Barcode In None
Using Barcode creator for Online Control to generate, create barcode image in Online applications.
GS1 - 13 Generator In Java
Using Barcode drawer for Java Control to generate, create EAN13 image in Java applications.
44 ++X(fl)
UPC A Drawer In None
Using Barcode generator for Online Control to generate, create UPC-A Supplement 2 image in Online applications.
UPC - 13 Encoder In Java
Using Barcode encoder for Android Control to generate, create EAN 13 image in Android applications.
D. Convergence of X(R):
Create GS1 - 13 In None
Using Barcode encoder for Software Control to generate, create EAN-13 Supplement 5 image in Software applications.
Printing EAN / UCC - 14 In None
Using Barcode generation for Font Control to generate, create EAN 128 image in Font applications.
Just as in the case of continuous time, the sufficient condition for the convergence of X(R) is that x[n] is absolutely summable, that is,
Scanning ANSI/AIM Code 128 In VB.NET
Using Barcode scanner for .NET Control to read, scan read, scan image in .NET framework applications.
Code 3 Of 9 Encoder In Java
Using Barcode maker for Eclipse BIRT Control to generate, create Code 39 Full ASCII image in BIRT applications.
n = -oo
Ix[n]kw
(6.31)
E. Connection between the Fourier Transform and the z-Transform:
Equation (6.27) defines the Fourier transform of x[n] as X(R) =
x[n] e-jnn
n = -m
The z-transform 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 DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
and X(ejn) mean the same thing whenever we connect the Fourier transform with the z-transform. Because the Fourier transform is the z-transform with z = ein, it should not be assumed automatically that the Fourier transform of a sequence x [ n ] is the z-transform 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 z-transform 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 z-transform of 6 [ n ] is
By definitions (6.27) and (1.45)the Fourier transform of 6 [ n ] is
Thus, the z-transform and the Fourier transform of 6 [ n ] are the same. Note that 6 [ n ] is absolutely summable and that the ROC of the z-transform 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 z-transform 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 z-transform of u[nl is
The Fourier transform of u [ n ] cannot be obtained from its z-transform because the ROC of the
CHAP. 61 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
z-transform 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 continuous-time case. Many of these properties are also similar to those of the z-transform when the ROC of X ( z) includes the unit circle.
Periodicity:
As a consequence of Eq. (6.41), in the discrete-time 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 continuous-time 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:
Copyright © OnBarcode.com . All rights reserved.