1-4> 0 in .NET

Encoding QR Code ISO/IEC18004 in .NET 1-4> 0

1-4> 0
QR Code JIS X 0510 Recognizer In .NET Framework
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications.
Printing Denso QR Bar Code In .NET
Using Barcode creation for Visual Studio .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
(6.131)
Recognize QR Code ISO/IEC18004 In .NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications.
Barcode Drawer In .NET
Using Barcode printer for .NET Control to generate, create barcode image in VS .NET applications.
Thus, ~ ( e ' " )exists because the ROC of X(z) includes the unit circle. Hence,
Bar Code Reader In Visual Studio .NET
Using Barcode reader for .NET Control to read, scan read, scan image in VS .NET applications.
Printing QR Code In Visual C#
Using Barcode creator for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
1 0 .
Creating QR In Visual Studio .NET
Using Barcode drawer for ASP.NET Control to generate, create QR Code image in ASP.NET applications.
QR Code 2d Barcode Generator In Visual Basic .NET
Using Barcode creation for VS .NET Control to generate, create QR Code image in VS .NET applications.
.. ... ..
Encoding Barcode In VS .NET
Using Barcode printer for VS .NET Control to generate, create barcode image in Visual Studio .NET applications.
Linear Barcode Generation In .NET Framework
Using Barcode creator for VS .NET Control to generate, create Linear image in .NET framework applications.
1 2 3
Encode GS1 DataBar Stacked In VS .NET
Using Barcode creator for .NET Control to generate, create GS1 DataBar image in .NET framework applications.
Encoding Interleaved 2 Of 5 In VS .NET
Using Barcode creation for VS .NET Control to generate, create ANSI/AIM ITF 25 image in VS .NET applications.
N- 1
Data Matrix Generator In Visual C#
Using Barcode generation for Visual Studio .NET Control to generate, create Data Matrix ECC200 image in VS .NET applications.
Barcode Printer In Objective-C
Using Barcode generation for iPhone Control to generate, create bar code image in iPhone applications.
Fig. 6-10
UPC-A Supplement 5 Generation In Java
Using Barcode generation for Java Control to generate, create UCC - 12 image in Java applications.
Drawing GS1 128 In Java
Using Barcode creation for Java Control to generate, create UCC - 12 image in Java applications.
6.13. Verify the time-shifting property (6.431, that is,
Barcode Scanner In .NET
Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications.
Linear Barcode Drawer In Visual Basic .NET
Using Barcode encoder for VS .NET Control to generate, create Linear 1D Barcode image in .NET applications.
B definition (6.27) y
Make EAN13 In None
Using Barcode generation for Font Control to generate, create European Article Number 13 image in Font applications.
Code 128 Recognizer In Visual Basic .NET
Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET framework applications.
F ( x [ n - n,])
n = -m
x [ n - n o ] e-j""
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
By the change of variable m = n - no, we obtain
Hence,
6.14. (a) Find the Fourier transform X ( 0 ) of the rectangular pulse sequence shown in Fig. 6-1l(a).
F g 6-11 i.
( b ) Plot X(R) for N, = 4 and N, = 8.
From Fig. 6-11 we see that
x[n] =x,[n
+N,]
where x , [ n ] is shown in Fig. 6 - l l ( b ) .Setting N = 2 N 1 + 1 in Eq. (6.132),we have
Now, from the time-shifting property (6.43)we obtain
( b ) Setting N, = 4 in Eq. (6.133), we get
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
which is plotted in Fig. 6-12(a). Similarly. for N ,
we get
which is plotted in Fig. 6.12(b).
Fig. 6-12
6.15. ( a ) Find the inverse Fourier transform x[n] of the rectangular pulse spectrum X ( n ) defined by [Fig. 6-13(a)]
( b ) Plot x[n] for W = r / 4 .
I xcn)
.tl n
I 4f)
-4-3-2-10 1 2 3 4
Fig. 6-13
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
( a ) From Eq. (6.28)
1 x[n]=- X(R) 27 , Thus, we obtain sin Wn -x(n) 7n
sin Wn elnnd o = dfi = 27 -w rn
IRl s
W C I ~ I S T
( b ) The sequence x[n] is plotted in Fig. 6-13(b)for W = 7 / 4 .
6.16. Verify the frequency-shifting property ( 6 . 4 4 , that is,
eJnonx[n] X(n- 0,) tt
By Eq. (6.27)
~ ( ~ j f l t ~=~
,= ,=
] ~ )f l ~ " X [ n ] e
e-~fl"
x [ n ]e - ~ ( f l - f l t ~ ) n X(fl - a , ) =
Hence, e ~ ~ o " x - n ]( R [X
- a,)
6.17. Find the inverse Fourier transform x [ n ] of
x(n)= 2 ~ q - no) n
From Eqs. (6.28)and (1.22) we have
WI, lfiol 5
Thus, we have ejnon
2 r S ( R - R,)
6.18. Find the Fourier transform of
x[n]=1 Setting R ,
all n
in Eq. (6.1351, we get x [ n ] = 1 o2 n S ( R )
I1 I n
Equation (6.136) is depicted in Fig. 6-14.
Fig. 6-14 A constant sequence and its Fourier transform.
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
6.19. Find the Fourier transform of the sinusoidal sequence
x [ n ] = cos R o n
lflolS
From Euler's formula we have
cos Ron = ; ( e i n ~ "+ e-'
X(R)
I n, J
Thus, using Eq. (6.135) and the linearity property (6.42), we get
[ S (- 0 , ) ~
+ 6(R + a , ) ]
IRI, Ia0I5 Ial, IRol r T (6.137)
which is illustrated in Fig. 6-15. Thus, cos Ron
a [ 6 ( R- R o ) + 6 ( R + a , ) ]
Fig. 6-15 A cosine sequence and its Fourier transform.
6.20. Verify the conjugation property (6.45), that is,
x * [ n ] -X*(-R)
From Eq. (6.27)
.F(x*[n]) =
x * [ n ] e-inn =
n= -m
Hence,
x * [ n ] -X*( -0)
6.21. Verify the time-scaling property (6.491, that is,
From Eq. (6.48)
x[n/m] = x [ k ]
if n = km, k
ifnzkm
integer
Then, by Eq. (6.27)
.F(-qrn)b1) =
e-jnn ,= Cm ~(,)bl -
CHAP. 61
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
Changing the variable n = km on the right-hand side of the above expression, we obtain
Hence,
6.22. Consider the sequence x [ n ] defined by
~ [ n=]
In12 otherwise
(a) Sketch x [ n ] and its Fourier transform X ( R ) . ( b ) Sketch the time-scaled sequence x ( , j n ] and its Fourier transform Xo,(R). ( c ) Sketch the time-scaled sequence ~ ( ~ jand ]its Fourier transform Xo,(R). n
Setting N ,
in Eq. (6.1331, we have
The sequence x [ n ] and its Fourier transform X ( 0 ) are sketched in Fig. 6-16(n).
Fig. 6-16
FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
From Eqs. (6.49) and (6.138) we have
The time-scaled sequence xo,[n] and its Fourier transform Xo,(R) are sketched in Fig. 6-16(b). In a similar manner we get
The time-scaled sequence x(,,[n] and its Fourier transform X,,,(R) are sketched in Fig. 6-16(~).
6.23. Verify the differentiation in frequency property (6.55), that is,
From definition (6.27)
Differentiating both sides of the above expression with respect to R and interchanging the order of differentiation and summation, we obtain
Multiplying both sides by j, we see that
Hence,
6.24. Verify the convolution theorem (6.581, that is,
x , [ n l * x 2 b I -X,(~)x,(n)
By definitions (2.35) and (6.27),we have
F { x , [ n ]* x , [ n ] ) =
n- -m
z ( z x,[k]x,[n
] ) e-j""
Changing the order of summation, we get
CHAP. 61
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
By the time-shifting property Eq. (6.43)
Thus, we have
. F { x l [ n ]* x 2 [ n1) =
x , [ k ]e - j n k X 2 ( f l )
6.25. Using the convolution theorem (6.58),find the inverse Fourier transform x [ n ] of
From Eq. (6.37)we have
anu[nI cr
- ae-jn
la1 < 1
Now Thus, by the convolution theorem Eq. (6.58) we get
Hence,
6.26. Verify the multiplication property (6.59),that is,
Let x [ n ] = x l [ n ] x 2 [ n ]Then by definition (6.27) .
By Eq. (6.28)
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
Then Interchanging the order of summation and integration, we get
x,[n]
e'j("-@"
Hence,
Copyright © OnBarcode.com . All rights reserved.