Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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7.2.2 Decimation-in-Frequency
The development of the decimation-in-frequency FFT algorithm is similar to the
decimation-in-time algorithm presented in the previous section. The first step consists
of dividing the data sequence into two halves of N/2 samples. Then X(k) in (7.1.3) can be
expressed as the sum of two components to obtain
Xk
N=2À1
n0
xnW
nk
N
NÀ1
nN=2
x n
W
nk
N
N=2À1
n0
xnW
nk
N
W
N=2k
N
N=2À1
n0
x n
N
2
W
nk
N
:
7:2:7
Using the fact that
W
N=2
N
eÀj
2p
N N=2
e
Àjp
À1,
7:2:8
Equation (7.2.7) can be simplified to
Xk
N=2À1
n0
xn À1
k
x n
N
2
!
W
nk
N
:
7:2:9
The next step is to separate the frequency terms X(k) into even and odd samples of k.
Since W
2kn
N
W
kn
N=2
, Equation (7.2.9) can be written as
X2k
N=2À1
n0
xn x n
N
2
!
W
kn
N=2
7:2:10a
and
X2k 1
N=2À1
n0
xn À x n
N
2
!
W
k
N
W
kn
N=2
7:2:10b
for 0 k N=2 À 1. Let x
1
n xn x n
N
2
À
Á
and x
2
n xn À x n
N
2
À
Á
for
0 n N=2 À 1, the first decomposition of an N-point DFT into two N=2-point
DFTs is illustrated in Figure 7.8.
Again, the process of decomposition is continued until the last stage is made up of
two-point DFTs. The decomposition proceeds from left to right for the decimation-
in-frequency development and the symmetry relationships are reversed from the
decimation-in-time algorithm. Note that the bit reversal occurs at the output instead
of the input and the order of the output samples X(k) will be re-arranged as bit-
reversed samples index given in Table 7.1. The butterfly representation for the
decimation-in-frequency FFT algorithm is illustrated in Figure 7.9.
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319
Summary :
Then X(k) in (7.1.3) can be expressed as the sum of two components to obtain Xk N=2À1 n0 xnW nk N NÀ1 nN=2 x n W nk N N=2À1 n0 xnW nk N W N=2k N N=2À1 n0 x n N 2 W nk N : 7:2:7 Using the fact that W N=2 N eÀj 2p N N=2 e Àjp À1, 7:2:8 Equation (7.2.7) can be simplified to Xk N=2À1 n0 xn À1 k x n N 2 !
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n2à1,algorithm,samples,into,decomposition,decimationinfrequency,two,bit,step,729,dfts,decimationintime,727