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4 Numerical Methods in Structural Dynamics …
Fig. 4.7 Steps of FFT
The DFTs of these two short sequences are
Y n =
T
N /2
N /2 − 1
m = 0
y m exp {−i (2π mn)/(N /2)}
Z n =
T
N /2
N /2 − 1
m = 0
z m exp {−i (2π mn)/(N /2)}
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
n = 0, 1, 2, . . . ,
N
2
− 1
(4.54)
Now the original sequence of F m is written and is split up into two separate sums,
similar to those occurring in Eq. (4.54) for Y m and Z m .
F(ω n ) =
T
N
N − 1
m = 0
F(t m ) e
i(−2πnm/N )
=
T
N
N /2 − 1
m = 0
F 2m e
− i [2 π (2m) n/N ]
+
N /2 − 1
m = 0
F 2m + 1 e
−i [2π(2m + 1)n/N ]
=
T
N
N /2 − 1
m = 0
y m e
−i(2πmn)/ (N /2)
+
N /2 − 1
m = 0
z m e
− i [2 π n/N ] e
− i (2 π mn)/ (N /2)