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2d barcode generator vb.net Verify the properties (6.62),(6.63~1, (6.63b);that is, if x [ n ] is real and and in VS .NET
6.27. Verify the properties (6.62),(6.63~1, (6.63b);that is, if x [ n ] is real and and Recognize QR Code In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. Drawing QRCode In .NET Framework Using Barcode creation for .NET framework Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications. x [ n ] = x , [ n ] + x o [ n ] ++X(fl) =A(R) QR Code JIS X 0510 Scanner In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Creation In VS .NET Using Barcode drawer for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. +jB(R) Barcode Decoder In .NET Framework Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. Denso QR Bar Code Drawer In C#.NET Using Barcode generator for VS .NET Control to generate, create QR image in .NET framework applications. (6.140) QR Code Generation In .NET Framework Using Barcode generator for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Encode QRCode In VB.NET Using Barcode creator for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications. where x , [ n ] and x o [ n ] are the even and odd components of x [ n ] , respectively, then X(R) =X*(R) Printing GS1 DataBar Stacked In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create GS1 DataBar14 image in .NET applications. Generate Bar Code In VS .NET Using Barcode creation for VS .NET Control to generate, create bar code image in .NET applications. x,[n] x,[n] UPC  13 Generator In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create EAN13 Supplement 5 image in Visual Studio .NET applications. Code 2 Of 5 Drawer In Visual Studio .NET Using Barcode creator for .NET framework Control to generate, create C 2 of 5 image in .NET applications. Re{X(R)} = A(i2) Paint Bar Code In Java Using Barcode printer for BIRT Control to generate, create barcode image in BIRT reports applications. Making Linear 1D Barcode In VS .NET Using Barcode creation for ASP.NET Control to generate, create Linear Barcode image in ASP.NET applications. j Im{X(i2)} = j B ( f l ) Code 39 Extended Decoder In .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications. Generate Code 3/9 In C#.NET Using Barcode printer for Visual Studio .NET Control to generate, create Code39 image in Visual Studio .NET applications. If x [ n ] is real, then x * [ n ] = x [ n ] , and by Eq. (6.45) we have
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x ( n )= x * (  n ) x(  0 ) = x * ( n ) (6.141) Next, using Eq. (6.46) and Eqs. (1.2) and (1.3), we have
 n ] =x,[n] x,[n] c ,  X(  0 ) = X * ( R ) = A ( R )  jB(R) = Re(X(R)} Adding (subtracting) Eq. (6.141) to (from) Eq. (6.1401, we obtain
x,[n] x,[n] A(R) jB( 0 )= j Im{ X ( R ) ) 6.28. Show that
44 ++X(R>
Now, note that
s [ n ] = u [ n ]  u [ n  11 Taking the Fourier transform of both sides of the above expression and by Eqs. (6.36) and (6.431, we have CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Noting that (1  ejn) = 0 for R = 0, X ( R ) must be of the form
where A is a constant. To determine A we proceed as follows. From Eq. (1.5) the even component of u[nl is given by u,[n] $ + f6[n] f  $[n] Then the odd component of u [ n ] is given by
u o [ n ]= u [ n ]  u , [ n ] = u [ n ]  y { u o [ n l ]= A
 ejn
aS(R)  2 From Eq. (6.63b) the Fourier transform of an odd real sequence must be purely imaginary. Thus, we must have A = a , and 6.29. Verify the accumulation property (6.571, that is, From Eq. (2.132) Thus, by the convolution theorem (6.58) and Eq. (6.142) we get
6.30. Using the accumulation property (6.57) and Eq. (1.501, find the Fourier transform of
4nI.
From Eq. (1.50) Now, from Eq. (6.36) we have
s[n] H 1
Setting x [ k l = 6 [ k ] in Eq. (6.571, we have
x [ n ] = 6 [ n ]H X ( R ) = 1 X(0) FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
FREQUENCY RESPONSE
6.31. A causal discretetime LTI system is described by
y [ n ]  + y [ n  1 + $ y [ n  21 = x [ n ] 1 ( a ) Determine the frequency response H ( n ) of the system.
(6.143) where x[n]and y[n] are the input and output of the system, respectively (Prob. 4.32). ( b ) Find the impulse response h[n]of the system.
( a ) Taking the Fourier transform of Eq. (6.1431, we obtain Y ( R ) i e  ' " ~ ( f l + ;ej2'y ) or (1 i e  i f l + Le  j 2 n ) Y ( R ) X ( R ) = ( a )= X ( R ) Thus, ( 6 ) Using partialfraction expansions, we have H(R)= (1  1  Iein  1  Lein
Taking the inverse Fourier transform of H ( f l ) , we obtain
h [ n ] = [ 2 ( i l n ( f ) " ] u [ n ] which is the same result obtained in Prob. 4.32(6). 6.32. Consider a discretetime LTI system described by
y [ n ]  ;y[n  11 = x [ n ] + ix[n  11 ( a ) Determine the frequency response H ( n ) of the system.
( b ) Find the impulse response h[n] of the system. ( c ) Determine its response y[n] to the input
~ [ n=]cosn 2
( a ) Taking the Fourier transform of Eq. (6.1441, we obtain Y ( R )  i e  j n Y ( R )= X ( R ) + ; e  j n x ( R ) CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Thus, Taking the inverse Fourier transform of H(R), we obtain
(c) From Eq.(6.137) Then
Taking the inverse Fourier transform of Y(R) and using Eq. (6.1351, we get
6.33. Consider a discretetime LTI system with impulse response
Find the output y[n] if the input x [ n ] is a periodic sequence with fundamental period No = 5 as shown in Fig. 617. From Eq. (6.134) we have
and 7r/4, only Since R, = 27r/NO = 2 ~ / 5 the filter passes only frequencies in the range lRl I the dc term is passed through. From Fig. 617 and Eq. (6.11) FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSI'EMS [CHAP. 6
210 1 2 3 4 5 Fig. 617 Thus, the output y[nl is given by ~ [ n= l
all n
634. Consider the discretetime LTI system shown in Fig. 618. (a) Find the frequency response H ( n ) of the system.
( b ) Find the impulse response h [ n ]of the system. ( c ) Sketch the magnitude response IH(n)I and the phase response N R ) . ( d ) Find the 3dB bandwidth of the system. From Fig. 618 we have y [ n ] = x [ n ] + x [ n  11 Taking the Fourier transform of Eq. (6.145) and by Eq. (6.77), we have (6.145)

