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x[nl in .NET framework
x[nl Decoding QR Code 2d Barcode In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. Printing Quick Response Code In .NET Using Barcode printer for .NET Control to generate, create QR Code image in VS .NET applications. = cosn
QR Code ISO/IEC18004 Recognizer In .NET Framework Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Print Barcode In .NET Framework Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in VS .NET applications. ( b ) x[n]=cosn+sinn 3 4
Barcode Recognizer In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. Create QRCode In Visual C# Using Barcode generation for .NET Control to generate, create QR image in .NET framework applications. FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
Creating Quick Response Code In VS .NET Using Barcode generation for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. QR Code Printer In VB.NET Using Barcode drawer for .NET framework Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications. [CHAP. 6
Drawing Barcode In VS .NET Using Barcode drawer for .NET framework Control to generate, create bar code image in VS .NET applications. Generate Linear In VS .NET Using Barcode printer for .NET Control to generate, create Linear Barcode image in Visual Studio .NET applications. The fundamental period of x [ n ] is No = 8, and Ro = 277/N0 = n / 4 . Rather than using Eq. ( 6 . 8 )to evaluate the Fourier coefficients c,, we use Euler's formula and get EAN 128 Encoder In VS .NET Using Barcode generator for .NET Control to generate, create GS1 128 image in Visual Studio .NET applications. USS93 Creator In .NET Framework Using Barcode printer for .NET framework Control to generate, create ANSI/AIM Code 93 image in .NET framework applications. Thus, the Fourier coefficients for x [ n ] are c l = f , c  , = c  , + , = c 7 = $, and all other c, = 0. Hence, the discrete Fourier series of x(n1 is Printing GS1128 In Visual Studio .NET Using Barcode generator for ASP.NET Control to generate, create EAN 128 image in ASP.NET applications. Making Barcode In Java Using Barcode creator for Android Control to generate, create bar code image in Android applications. ( b ) From Prob. 1.16(i) the fundamental period of x [ n ] is No = 24, and R, Again by Euler's formula we have Printing Barcode In None Using Barcode drawer for Microsoft Excel Control to generate, create bar code image in Office Excel applications. Bar Code Creator In None Using Barcode creator for Font Control to generate, create barcode image in Font applications. = 277/N0 = 77/12. Reading GS1  12 In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Universal Product Code Version A Decoder In Visual C# Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. I Thus, c3 = j(4),c4 = $ , c  , = c  ~ =+ Z~= ~ , c P 3 c  ~ = c 2 ] = I ( I ) , and all other c O = + ~ ~ c, = 0. Hence, the discrete Fourier series of x [ n ] is Print Code 128 Code Set B In Java Using Barcode maker for Java Control to generate, create Code128 image in Java applications. 1D Generation In C#.NET Using Barcode creation for .NET framework Control to generate, create Linear 1D Barcode image in Visual Studio .NET applications. From Prob. l.l6( j ) the fundamental period of x [ n ] is No = 8, and Ro = 277/No = n / 4 . Again by Euler's formula we have I Thus, c0 = f , c 1= a, c Fourier series of x[nl is
= c  + n = c7 =
1 a, and all other c, = 0. Hence, the discrete
Let x[n] be a real periodic sequence with fundamental period N, and Fourier coefficients ck = a k + jb,, where a, and b, are both real. ( a ) Show that a  , = a k and b,= bk. (b) Show that c,,,,, is real if No is even. (c) Show that x [ n ] can also be expressed as a discrete trigonometric Fourier series of the form ( N u  1)/2 x[n]=co+2
k= 1 (a,coskfl,n b,sinkfl,n) (6.123) ( N o  2)/2 if No is odd or x[n] = c, + ( 1)'ch/, (a, cos kR,n  bk sin k f l o n ) (6.124) if N,, is even.
CHAP. 61
FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
( a ) If x [ n l is real, then from Eq. ( 6 . 8 ) we have
Thus, =a_,+jb, a,=a, =(a, + j b k ) *= a , b,=bk
and we have and
( 6 ) If No is even, then from Eq. ( 6 . 8 ) 1 No' NO n = o
(  l ) " x [ n ] = real
If No is odd, then (No  1) is even and we can write x [ n ] as
Now, from Eq. (6.17) Thus, = c, (No1)/2 ( a , cos kRon  bk sin k R o n ) If No is even, we can write x [ n ] as
FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
[CHAP. 6
Then
x [ n ] = c0
+ (  I ) ~ C , ~ /+, (Nu2)/2 2Re(ckejkRun) = c, + ( l)nc,v/2 + 2
(Nu2)/2 k =1 ( a , cos kfl,,n  b, sin k R o n ) Let x , [ n ] and x , [ n ] be periodic sequences with fundamental period No and their discrete Fourier series given by Show that the sequence x [ n ] = x , [ n ] x , [ n ] is periodic with the same fundamental period No and can be expressed as where ck is given by
Now note that
Thus, x [ n ] is periodic with fundamental period No.Let
Nu' Then
Ck =
N .=o o
X[n]eikfM
No' No n = o
z x l [ n ] x 2 [ n ]e',"on
since and the term in parentheses is equal to ek,. CHAP. 61
FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Let x,[n] and x,[n] be the two periodic signals in Prob. 6.8. Show that
Equation (6.127) is known as Parseval's relation for periodic sequences.
From Eq. (6.126) we have
N0I
No 1 ck = Setting k
NO n = o
x , [ n ] x 2 [ n ]ejknon =
m =O
ddkm
in the above expression, we get
6.10. (a) Verify Parseval's identity [Eq. (6.19)] for the discrete Fourier series, that is, (6) Using x [ n ] in Prob. 6.3, verify Parseval's identity [Eq. (6.19)]. ( a ) Let
1 No' Then
d   NO n = O
X*[n~e~k~on=  ] No' X[n]e~kRon
(6.128) NO n = o
Equation (6.128) indicates that if the Fourier coefficientsof x [ n ] are c , , then the Fourier coefficientsof x * [ n ] are c ,. Setting x , [ n ]= x [ n ] and x 2 [ n ]= x * [ n ] in Eq. (6.1271, we have d k = c k and ek = c , (or e  , = c : ) and we obtain ( b ) From Fig. 67 and the results from Prob. 6.3, we have
and Parseval's identity is verified.
FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
FOURIER TRANSFORM
6.11. Find the Fourier transform of
x[n] anu[n  1 1 a real
From Eq. (4.12) the ztransform of x[n] is given by X(z) = I az' Izl< lal
Thus, X(eJf') exists for la1 > 1 because the ROC of X( z ) then contains the unit circle. Thus, 6.12. Find the Fourier transform of the rectangular pulse sequence (Fig. 610) x [ n ] =u [ n ] u [ n  N ] Using Eq. (1.90), the ztransform of x[nl is given by N 1 1ZN X(Z) = C zn= n=O 12

