数字信号处理第11章

N 1 0
X [k ] x[n]W , 0 k N 1
kn N
where
WN e
j 2 / N
• Direct computation p of all N samples p of {X[k]} requires N2 complex multiplications and N(N N(N-1) 1) complex additions
§11.3.1 11 3 1 Goertzel’s algorithm
§11.3.1 Goertzel’s algorithm
• I In th the realization li ti of f HN-k[z] , th the multiplier lti li in i path is the feedback p 2cos(2π(N-k)/N)=2cos(2πk/N) which is same as that in the realization of Hk[z] vN-k[n] = vk[n] i i.e., e the intermediate variables computed to determine X[k] can again be used to determine X[N-k] • Only O l difference diff between b t the th two t structures t t is i p which is now WN k the feed-forward multiplier k that is the complex conjugate of WN
§11.3.1 Goertzel’s algorithm
• Recall, direct computation of all N samples of {X[k]} requires N2complex multiplications and N(N-1) 1) complex additions • Equivalently, q y, direct computation p of all N samples of {X[k]} requires 4N2 real multiplications and N(4N-2) 2) real additions • Thus, , Goertzel’s algorithm g requires q 2N more real additions than the direct DFT computation t ti
11.3 Discrete Fourier Transform Computation p
• J.W.Cooley and J.W.Tukey are given credit for bringing the FFT to the world in their paper : :“An An algorithm for the machine calculation of complex Fourier Series”, a e a cs Computation, Co pu a o , Vol.19, Vo . 9, 1965. 965. In retrospect, e ospec , Mathematics others had discovered the technique many years before. For instance, the great German mathematician K. F. Gauss(1777-1855) had used the method more than a century earlier. This early work was largely forgotten because it lacked the tool to make it practical: the digital computer. Cooley and Tukey are honored because they discovered the FFT at the right time, the beginning of the computer revolution.
vk [n 2]
X [k ] yk [ N ] vk [ N ] W v [ N 1]
k N k
• Computation p of each sample p of vk[n] involves only 2 real multiplications and 4 real additions nk • Complex p multiplication p by y WN needs to be performed only once at n = N • Thus, Th computation i of f one sample l of f X[k] q (2 ( N+4) ) real multiplications p and requires (4N+4) real additions • Computation of all N DFT samples requires 2N (N+2) 2) real multiplications and 4N (N+1) 1) real additions
Chapter 11 Discrete Fourier Transform Computation
11.3Discrete Fourier Transform Computation p
• The N-point DFT X[k] of a length-N sequence x[n], 0nN-1, is defined by
yk [n] xe [n] W
k N
yk [n 1]
with yk[-1]=0 and xe[N]=0
§11.3.1 Goertzel’s algorithm
• Since a complex multiplication can be implemented with 4 real multiplications and 2 real additions additions, computation of each new value of yk[n] requires 4 real multiplications and 4 real additions • Thus computation of X[k]=yk[n]involves 4N real multiplications and 4N real additions Computation of all N DFT samples requires 4N2 real l multiplications lti li ti and d 4N2 real l additions dditi
§11.3 11 3 Discrete Fourier p Transform Computation
• The definition of DFT:
X (k ) x[n]W
n 0
N 1
nk k N
WN e
2 j N
W
( n mN )( k lN ) N
W
nk N
m, l 0,1, Z { yk (n)} H k ( z) X e ( z) k 1 1 WN z
§11.3.1 Goertzel’s algorithm
• Structural interpretation of the algorithm
• Thus a recursive DFT computation p scheme is
( n l ) k N n
§11.3.1 Goertzel’s algorithm
( n l ) k y [ n ] x [ l ] W • z-transform of k e N n
yields
l 0
kz-1 1) where Hk[z]=Z{hk[n]}=1/(1]}=1/(1 WN-k And Xe[z] ]=Z{ {xe[n]} Thus, yk[n] is the output of an initially relaxed l d LTI digital di i l filter fil Hk[z]with ] i h an input i xe[n] a and, d, w when e n =N, N, yk[N] ]=X[k]
kN N
X ( k ) x[l ]W
l 0
N 1
lk N
W
kN N
x[l ]W
l 0
N 1
lk N
x[l ]W
l 0
N 1
(l N ) k N
§11.3.1 Goertzel’s algorithm
• Define yk [n] xe [l ]W l 0 • Note: yk[n]is the direct convolution of the causal l sequence x[n] 0 n N 1 xe [n] n 0, n N 0 with a causal sequence kn WN n0 hk [n] n0 0 Observe X[k]=yk[n]|n=N
§11.3.1 Goertzel’s algorithm
• Algorithm can be made computationally more efficient by observing that Hk(z)can be rewritten as
k 1 1 WN z 1 H k ( z) k 1 k 1 k 1 1 WN z (1 WN z )(1 WN z ) k 1 1 WN z (1 2 cos( (2k / N ) z 1 z 2 )