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Given an FFT program to find the N-point DFT of a sequence, how may this program be used to find the inverse DFT
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As we saw in Prob. 6.9, we may tind . r ( n ) by first using the DFT program to evaluate the sum
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CHAP. 71
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THE FAST FOURIER TRANSFORM
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which is the DFT of X*(k). Then, x(n) may be found from x(n) as follows:
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Alternatively, we may find the DFT of X(k),
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and then extract x(n) as follows:
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Let x ( n ) be a sequence of length N with
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where N is an even integer.
(a) Show that the N-point DFT of x ( n ) has only odd harmonics. that is.
X(k) =0
k even
(b) Show how to find the N-point DFT of x ( n ) by finding the N/2-point DFT of an appropriately modified sequence.
(a) The N-point DFT of x(n) is
Because x(n) = -x(n
+ N/2), if k is even, each term in the sum is zero, and X(k) = 0 fork = 0 , 2 , 4 , . . ..
(b) In the first stage of a decimation-in-frequency FFT algorithm, we separately evaluate the even-index and oddindex san~ples X(k). If X(k) has only odd harmonics, the even samples are zero, and we need only evaluate of the odd samples. From Eq. (7.4) we see that the odd samples are given by
With x ( n ) = -x(n
+ N/2) this becomes
which is the N/2-point DFT of the sequence y(n) = 2WE;x(n). Therefore, to find the N-point DFT of x(n), we multiply the first N/2 points of x(n) by 2W;.
and then compute the N/2-point DFT of y(nh The N/2-point DFT of x(n) is then given by
THE FAST FOURIER TRANSFORM
[CHAP. 7
FFT Algorithms for Composite N
When the number of points in the D F T is a power of 4, w e can use a radix-2 FFT algorithm. However, when N = 4", it is more efficient to use a radix-4 FFT algorithm.
(a) Derive the r a d i x 4 decimation-in-time FFT algorithm when N = 4".
(b) Draw the structure for the butterfly in the radix-4 FFT, and compare the number of complex multiplies and adds with a radix-4 F F T to a radix-2 FFT.
(a) To derive a decimation-in-time radix-4 FFT. let NI = N/4 and N2 = 4. and define the index maps
We then express X ( k ) using the decomposition given in Eq. (7.7) with NI = N/4 and N2 = 4,
The inner summation.
is the N/4-point DFT of the sequence x(4nI
+ n2),and the outer summation is a 4-point DFT,
Since W4 = -j, these 4-point transforms have the form
for kl = 0. 1 , 2 . 3 , and n2 = 0, I , . . . (N/4) 1. If N2 = N/4 is divisible by 4, then the process is repeated. In this way, we generate v = Iog, N stages with N/4 butterflies in each stage.
(b) The 4-point butterflies in the radix-4 FFT perform operations of the form
CHAP. 71 With
THE FAST FOURIER TRANSFORM
Since multiplications by ij only requires interchanging real and imaginary parts and possibly changing a sign bit, then each 4-point butterfly only requires 3 complex multiplications. With v = log, N stages, and N/4 butterflies per stage, the number of complex multiplies for a DFT of length N = 4" is
N 3N 3 . - log, N = - log, N 4 8 For a radix-:! decimation-in-time FFT, on the other hand, the number of multiplications is
N log, N 2 Therefore, the number of multiplications in a radix-4 FFT is
& times the number in a radix-2 FFT.
Suppose that we would like to find the N-point DFT of a sequence where N is a power of 3, N = 3". ( a ) Develop a radix-3 decimation-in-time FFT algorithm, and draw the corresponding flowgraph for N =9. ( h ) How many multiplications are required for a radix-3 FFT (c) Can the computations be performed in place
(a) A radix-3 decimation-in-time FFT may be derived in exactly the same way as a radix-2 FFT. First, x ( n ) is decimated by a factor of 3 to form three sequences of length Nj3:
THE FAST FOURIER TRANSFORM Expressing the N-point DFT in terms of these sequences, we have
[CHAP. 7
Since w$' =
W k 3 ,then
Note that the first term is the N/3-point DFT o f f (n), the second is W i times the N/3-point DFT of g(n), and the third is w;' times the N 13-point DFT of h(n),
We may continue decimating by factors of 3 until we are left with only 3-point DFTs. The flowgraph for a 9-point decimation-in-time FFT is shown in Fig. 7- 1 I . Only one of the 3-point butterflies is shown in the second stage in order to allow for the labeling of the branches. The complete flowgraph is formed by replicating this 3-point butterfly up by one node, and down by one node, and changing the branch multiplies to their appropriate values.
0 X(8)
W" J Fig. 7-11. Flowgraph for a9-point decimation-in-time FFT (only one butterfly in the second stage is shown).
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