 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode lib ssrs FOURIER ANALYSIS in Software
FOURIER ANALYSIS Code 128A Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Creating Code 128 Code Set B In None Using Barcode maker for Software Control to generate, create Code 128 Code Set B image in Software applications. [CHAP. 2
Decoding Code 128 Code Set B In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Draw Code128 In Visual C# Using Barcode creator for Visual Studio .NET Control to generate, create Code128 image in .NET framework applications. Derive the upsampling property of the DTFT, which states that if x (ej") is the DTFT of x ( n ) , the DTFT of Code128 Encoder In VS .NET Using Barcode creation for ASP.NET Control to generate, create Code 128 Code Set C image in ASP.NET applications. Code 128 Code Set C Printer In Visual Studio .NET Using Barcode printer for Visual Studio .NET Control to generate, create Code 128B image in .NET framework applications. y(n) = n = O , & L , f 2 L , ... Draw Code128 In VB.NET Using Barcode generation for VS .NET Control to generate, create Code 128 Code Set B image in VS .NET applications. UPC Code Creator In None Using Barcode drawer for Software Control to generate, create UCC  12 image in Software applications. otherwise
ANSI/AIM Code 128 Encoder In None Using Barcode drawer for Software Control to generate, create Code 128 Code Set C image in Software applications. Barcode Creation In None Using Barcode creator for Software Control to generate, create barcode image in Software applications. From the definition of the DTFT, we have
EAN 128 Maker In None Using Barcode encoder for Software Control to generate, create UCC128 image in Software applications. Encoding Barcode In None Using Barcode creator for Software Control to generate, create bar code image in Software applications. Because y(n) is equal to zero except when n is an integer multiple of L, Drawing UPCE In None Using Barcode creator for Software Control to generate, create UPC E image in Software applications. Drawing USS Code 39 In None Using Barcode printer for Online Control to generate, create Code 39 image in Online applications. ,,=m
Encode Bar Code In ObjectiveC Using Barcode generation for iPhone Control to generate, create barcode image in iPhone applications. USS128 Printer In Visual Basic .NET Using Barcode creation for .NET framework Control to generate, create EAN 128 image in Visual Studio .NET applications. Thus, Y(eJW) formed by scaling X(eJW) frequency. is in
Print DataMatrix In ObjectiveC Using Barcode maker for iPhone Control to generate, create ECC200 image in iPhone applications. UPCA Drawer In VS .NET Using Barcode printer for .NET framework Control to generate, create UPCA Supplement 2 image in Visual Studio .NET applications. Find the inverse DTFT of
Bar Code Encoder In Java Using Barcode encoder for Android Control to generate, create barcode image in Android applications. UPC Code Reader In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.  ;ej,h
For this problem, the direct approach of performing the integration
is not easy. However, a simple approach is to recall that the inverse DTFT of
y(n) = ( f ) " u ( n ) and to note that ~ ( e j " is related to X(eju) by scaling in frequency, ) ~ ( e j " = Y (eJIOw ) 1 Therefore, it follows from the upsampling property in Prob. 2.3 1 that otherwise
In other words, the sequence x(n) is formed by inserting nine zeros between each value of y ( n ) . Let x ( n ) be a sequence with a DTFT ~ ( e ; " ) .For each of the following sequences that are formed from x ( n ) , express the DTFT in terms of X ( e J W ) : ( a ) x*(n>
(b) x ( n ) * x * (  n ) (c) x ( 2 n
+ 1) CHAP. 21 (a) The DTFT of x*(n) is
FOURIER ANALYSIS
Bringing the conjugate outside, we have
which leads to the DTFT pair x*(n) Dg x*(ejw) (b) For y(n) = x(n) * x*(n), note that because y(n) is the convolution of two sequences, the DTFT of y(n) is the product of the DTFTs of x(n) and x*(n). As shown in part (a), the DTFT of x*(n) is X*(ejW).Therefore, we have the DTFT pair x(n) * x*(n) ~(ej")~*(e'") 1 ~ ( e j " ) 1 ~ = n=m
(c) For x(2n
+ 1) we have
DTFT(x(2n
+ I)) = r(2n
+ 1)eIn" = n odd n even
n odd
x(n)ejnY
To evaluate this sum, a "trick" is to use the identity 1This allows us to write the DTFT as follows: DTFT(x(2n + 1)) = n odd
x(n)ejn" = n=m
 (lr]x(n)ejnw
Because the first sum is simply X(eJU),and the second is the DTFT of the modulated signal
then
DTFT(x(2n
+ 1)) = f [x(eiw)  x(eicw") )I
Let x ( n ) be the sequence
which has a DTFT
~ ( e j " )= xR(eJ") ~(eJo) , where x R ( e j w )and x l ( e j w ) are the real part and the imaginary part of x ( e j W ) ,respectively. Find the sequence y(n) that has a DTFT given by The key to solving this problem is to recall that if x(n) is real, and if X(eJW) written in terms of its real and is imaginary parts, XR(ejw) the DTFT of the even part of x(n), and Xl(eJW) the DTFT of the oddpart: is is x,(n> = [x(n) + x(n)] D xr(eiw) B
DTFT
x&) = f [x(n)  x(n)] IXl(eJw) FOURIER ANALYSIS
Therefore, the DTFT of j x , ( n ) is X l ( e i " ) , [CHAP. 2
and the DTFT of j x e ( n + 2 ) is
jxe(n
+ 2 ) DTF j .X R ( e j W ) e j z W cT
y ( e j W )= X , ( e J W ) j X R ( e i W ) e J h +
Thus, and it follows that
ixe(n
+ 2 )  jx,(n) & D
y(n) = j x J n
+ 2 )  jx&) Finally, with x e ( n ) and x,(n) as tabulated below.
it follows that y ( n ) , which is formed from these two sequences, is as shown below: Let x(n ) be the sequence
Evaluate the following quantities without explicitly finding X(eJo): (4 x (ejo)lo=O
( b ) 9Mw) (c) x(eJo)dw x(ejo)lo=r IX(ejw)l2d~
(a) Because the DTFT of x ( n ) is
x ( e j W )= )7 x(n)ejnw
n=m
note that if we evaluate X ( e j W )at w = 0, we have
which is simply the sum of the values of x ( n ) . Therefore, for the given sequence it follows that
(b) To evaluate the phase, note that because x ( n ) is real and even, X ( e J W is real and even and, therefore, the phase ) is equal zero or n for all o. CHAP. 2 1
( c ) From the inverse DTFT, FOURIER ANALYSIS
note that when n = 0 : x(0) = Therefore, it follows that
 ,( e l Y ) d w X 2n
x(eM)do = 2nx(0) =6 n
( d ) Evaluating the DTFT of x ( n ) at o = n,we have
which, for the given values of x ( n ) , evaluates to
( e ) From Parseval's theorem, we know that
Therefore, ~ x ( e ~ ~ ) l = 271 'do
n=w
lr(n)12 = 3871 The center of gravity of a sequence x ( n ) is defined by
and is used as a measure of the time delay of a sequence. Find an expression for c in terms of the DTFT of x ( n ) , and find the value of c for the sequence x ( n ) that has a DTFT as shown in the figure below. To find the value of c in terms of X ( e J W ) first note that the denominator is simply the value of X ( e J o ) evaluated at , o=O: For the numerator, recall the DTFT pair
nx(n) " j d do ~ ~ m , W
FOURIER ANALYSIS
[CHAP.2
Therefore, and c may be evaluated in terms of X ( e J m )as follows:

