数字信号处理-第五章

kn X (k ) x(n)WN n 0
N / 2 1 n 0
N 1
n even
x(n)W

N / 2 1 n 0 k N
kn N
kn x(n)WN n odd
X (k )
x(2n)W
2 kn N
k ( 2 n 1) x ( 2 n 1 ) W N N / 2 1 n 0 kn x ( 2 n 1 ) W N /2
N / 2 1 n 0
x(2n)W
kn N /2
W
(5.1)
k G ( k ) WN H (k )
where
G(k )
N / 21 n 0
x(2n)W
kn N /2
H (k )
N / 21 n 0
k ( 2 n 1) x ( 2 n 1 ) W N
(5.2)
Both G(k) and H(k) are N/2-point DFT. Although the index k ranges over N values, k = 0 , 1, …, N – 1, each of the sums needs only be computed for k between 0 and N/2 – 1 since G(k) and H(k) are periodic in k with period N/2. After the two DFTs corresponding to the two sums are computed, they are then combined to yield the N-point DFT, X(k). We have seen that the direct computation without exploiting symmetry, N2 complex multiplications (CM)
5.2 Decimation-in-time FFT algorithms
Let N be an integer power of 2, i.e., N = 2m. Since N is an even integer, we can compute X(k) by separating x(n) into two N/2-points sequences consisting of the even numbered points in x(n) and the odd-numbered points in x(n). That is or
N-point computation into the two N/2-point computation. If N/2 is even, as it always is when N is a power of 2, then we can continue to decompose each of the N/2-point DFTs into N/4-point DFTs. That is, G(k) and H(k) can be computed as follows:
G(k )
N / 21 n 0
or
g (n)W
kn N /2

N / 4 1 n 0
g (2n)W
kn N /4
kn N /4

N / 4 1 n 0
k ( 2 n 1) g ( 2 n 1 ) W N /2
G (k )
N / 4 1 n 0
x(4n)W
kn kn X (k ) Rex(n)Re WN Imx(n)Im WN n 0 kn N N 1
j Rex(n)ImW Imx(n)Re W

kn N
The above equation shows that the direct computation of X(k) for each k requires 4N real multiplications (RM) and 4N – 2 real additions (RA). The computation of all the coefficients needs 4N2 RM and N(4N – 2) RA. The computation is large, especially when N is large.
Chapter 5. Computation of the discrete Fou
The discrete Fourier transform (DFT) plays an important role in the analysis, the design, and the implementation of digital signal processing algorithms and systems. One of reasons that Fourier analysis is of such wide-ranging importance in digital signal processing is because of the existence of efficient algorithms for computing the discrete Fourier transform. Recall that the DFT of a finite-duration sequence x(n) is N 1 kn X (k ) x(n)WN , k = 0, 1, …, N – 1, (5-1) n 0 j 2 / N where WN e . The IDFT is
W
k N /2
N / 4 1 n 0
kn x ( 4 n 2 ) W N /4
Similarly,
H (k )
N / 4 1 n 0
x(4n 1)W
kn N /4
W
k N /2
N / 4 1 n 0
kn x ( 4 n 3 ) W N /4
We can continue this procedure until left with only twopoint transforms, this requires m (m = log2N) stages of computation. We have seen that in the original decomposition of an N-point transform into two N/2point transforms, the number of CM and CA required was N+N2/2. When the N/2-point transforms are decomposed into N/4-point transforms, the factor (N/2)2 is replaced by N/2+2(N/4)2, so the overall computation requires N+N+4(N/4)2 CM and CA. If N = 2m, this can be done at most m = log2N times, thus, the number of CM and CA is equal to N log2N.
For this reason, it is important to develop efficient algorithms. Many efforts have been paid on this field. The possibility of greatly reduced computation was overlooked by Cooley and Tukey (1965). This publication touched off a flurry of activity in the application of the DFT to signal processing and resulted in the discovery of a number of computational algorithms known as fast Fourier transform (FFT). The fundamental principle of these algorithms is based on the decomposition of a sequence with length N into successively smaller DFTs. The manner in which this principle is implemented leads to a variety of different algorithms, all with comparable improvements in terms
1 x ( n) N
kn X ( k ) W N k 0 N 1
1 N 1 * kn X ( k ) W N N k 0
*
To indicate the importance of efficient computation schemes, it is instructive to consider the direct computation of the DFT equations. Since x(n) may be complex, we can write
computational efficiency. We will present two basic classes of FFT algorithms. The first one, called decimation-in-time (DIT), derives its name from the fact that in the process of arranging the computation into smaller transformations, the sequences x(n) is decomposed into successively smaller subsequences. The second one is termed as decimation-in-frequency (DIF), which consists of decomposing the DFT coefficients X(k) into smaller subsequences.