Comparing Eq. (6.210) with Eq. (6.2091, we conclude that in .NET framework

Create QR Code ISO/IEC18004 in .NET framework Comparing Eq. (6.210) with Eq. (6.2091, we conclude that

Comparing Eq. (6.210) with Eq. (6.2091, we conclude that
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FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
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[CHAP. 6
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Let W n + , , , + , denote the entry in the ( n + 1)st row and (k + 1)st column of the W4 matrix. Then, from Eq. (6.207)
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and we have
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6.55. ( a ) Find the DFT X [ k ] of x [ n ] = ( 0 , 1 , 2 , 3 ) .
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( b ) Find the IDFT x [ n ] from X [ k ] obtained in part (a).
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Using Eqs. (6.206) and (6.212), the DFT XIk]of x[n] is given by
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Using Eqs. (6.209) and (6.212), the IDFT x[n] of X [ k ] is given by
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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 N-point 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 even-numbered and odd-numbered samples of x [ n ] , respectively.
( a ) Show that
f [ n ] = 41 .
outside 0 s n
-- 1
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
( b ) Show that the N-point 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 DISCRETE-TIME 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. 6-37. ( 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 decimation-in-time 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. 6-37 Flow graph for an 8-point decimation-in-time F l T algorithm.
CHAP. 61
FOURIER ANALYSIS OF DISCRETE-TIME 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 decimation-in-time FFT algorithm. From Figs. 6-3Na) 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 ) ]{
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