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Comparing Eq. (6.210) with Eq. (6.2091, we conclude that in .NET framework
Comparing Eq. (6.210) with Eq. (6.2091, we conclude that Recognize QR Code In Visual Studio .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. Encoding QRCode In .NET Using Barcode maker for VS .NET Control to generate, create Denso QR Bar Code image in .NET applications. FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
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6.56. Let x [ n ] be a sequence of finite length N such that
x[n]=0 n<O,n>N
Let the Npoint DFT X [ k ] of x [ n ] be given by [Eq. (6.9211 N 1 Suppose N is even and let
The sequences f [ n ] and g [ n ] represent the evennumbered and oddnumbered samples of x [ n ] , respectively. ( a ) Show that
f [ n ] = 41 . outside 0 s n
 1 CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
( b ) Show that the Npoint DFT X [ k ] of x [ n ] can be expressed as
(N/2)1 where
F[k]= f[n]W$2
N k =0,1,...9  1  (6.218~) Draw a flow graph to illustrate the evaluation of X [ k ] from Eqs. ( 6 . 2 1 7 ~and ) (6.2176)with N = 8. ( d ) Assume that x [ n ] is complex and w,"~ have been precomputed. Determine the numbers of complex multiplications required to evaluate X [ k ] from Eq. (6.214) and from Eqs. (6.217a) and (6.217b) and compare the results for N = 2 = 1024. ' ' From Eq. (6.213) f [ n ]= x [ 2 n ]= 0 , n < 0
f: [] = x [ N ] =O
Thus Similarly g[n]=x[2n+l]=O,n<O Thus, ( b ) We rewrite Eq. (6.214) as
g  = x [ N + 1]=O
X [ k ]= x x [ n ]Win+ C x [ n ]W,kn
n even
n odd
With this substitution Eq. (6.219) can be expressed as
FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
[CHAP. 6
(N/2) I
where
F[kl=
f[nlW,k;2 N k = O , l , ...,2 Note that F [ k ] and G [ k ] are the (N/2)point DFTs of f i n ] and g i n ] , respectively. Now
Hence, Eq. (6.221) can be expressed as
The flow graph illustrating the steps involved in determining X [ k ] by Eqs. (6.217~) and (6.2176) is shown in Fig. 637. ( d ) T o evaluate a value of X [ k ] from Eq. (6.214) requires N complex multiplications. Thus, . the total number of complex multiplications based on Eq. (6.214) is N ~The number of complex multiplications in evaluating F [ k ] or G [ k ] is (N/2)2. In addition there are N k ] . multiplications involved in the evaluation of ~ , k ~ [ Thus, the total number of complex multiplications based on Eqs. (6.217~)and (6.217b) is 2 ( ~ / 2 + ~ = ~ ' / 2+ N. For ) N N = 2"'= 1024 the total number of complex multiplications based on Eq. (6.214) is 22"  l o h and is 106/2 + 1024 .= 106/2 based on Eqs. (6.217~)and (6.217b). So we see that the number of multiplications is reduced approximately by a factor of 2 based on Eqs. ( 6 . 2 1 7 ~ and (6.2176). ) The method of evaluating X [ k I based on Eqs. (6.217~) (6.217b) is known as the and decimationintime fast Fourier transform (FFT) algorithm. Note that since N/2 is even, using the same procedure, F [ k l and G [ k ] can be found by first determining the (N/4)point DFTs of appropriately chosen sequences and combining them. Fig. 637 Flow graph for an 8point decimationintime F l T algorithm.
CHAP. 61
FOURIER ANALYSIS OF DISCRETETIME SIGNALS A N D SYSTEMS
6.57. Consider a sequence
x[n]={l,l,1,1,l,l,l,1) Determine the DFT X [ k ] of x [ n ] using the decimationintime FFT algorithm. From Figs. 63Na) and (61, the phase factors W and W,k are easily found as follows: : wb)= 1 w; = w~~  1 = w~~j = W:= 1 1 w,'=  a '7T
1  a,  a  1 Next, from Eqs. ( 6 . 2 1 5 ~and (6.2156) ) f 1.1 = x [ 2 n ] = ( x [ O ] x [ 2 ] ,x [ 4 ] ,x [ 6 ] ) = ( 1 ,  1 ,  1 , l ) , g [ n ] = x [ 2 n +1 ] = { x [ l ] , x [ 3 ] , x [ S ] , x [ 7= } I ,  1 , 1 ,  1 ) ]{

