Fast Fourier Transform - Signal Processing of Random Physiological Signals - page 154

13.6.4 The DF T by Directed Method
In comparison, the direct calculation would be as follows:

− 1

N

j 2 N

( W N ) nk

e − x ( k ) = x ( n ) × k = 0 , 1 ,... N − 1 W N =

n = 0

x (0) W 0 x (1) W 0 x (2) W 0 x (3) W 0 x (4) W 0 x (5) W 0 x (6) W 0 x (7) W 0 X (0) = + + + + + + +

= (1 + × 0) + (0 . 707 + × 0) + (0 + × 0) + ( − 0 . 707 + × 0) + ( − 1 + × 0)

j j j j j

+ ( − 0 . 707 + j × 0) + (0 + j × 0) + (0 . 707 + j × 0) = 0 + j × 0

x (0) W 0 x (1) W 1 x (2) W 2 x (3) W 3 x (4) W 4 x (5) W 5 x (6) W 6 X (1) = + + + + + +

x (7) W 7

+ = (1 + × 0) + ( . 5 + × 0) + (0 + × 0) + (0 . 5 + × 0)

j j j j

+ (1 + × 0) + (0 . 5 + × 0) + (0 + × 0) + (0 . 5 + × 0) = 4 + × 0

j j j j j

x (0) W 0 x (1) W 2 x (2) W 4 x (3) W 6 x (4) W 8 x (5) W 10 x (6) W 12 X (2) = + + + + + +

x (7) W 14

+ = (1 + × 0) + (0 + × 0) + (0 + × 0) + (0 + × 0)

j j j j

+ ( − 1 + × 0) + (0 + × 0) + (0 + × 0) + (0 + × 0) = 0 + × 0

j j j j j

x (0) W 0 x (1) W 3 x (2) W 6 x (3) W 9 x (4) W 12 x (5) W 15 x (6) W 18 X (3) = + + + + + +

x (7) W 21

+ = (1 + × 0) + ( − 0 . 5 + × 0) + (0 + × 0) + ( − 0 . 5 + × 0)

j j j j

+ (1 + × 0) + ( − 0 . 5 + × 0) + (0 + × 0) + ( − 0 . 5 + × 0) = 0 + × 0

j j j j j

x (0) W 0 x (1) W 4 x (2) W 8 x (3) W 12 x (4) W 16 x (5) W 20 x (6) W 24 X (4) = + + + + + +

x (7) W 28

+ = (1 + × 0) + ( − 0 . 707 + × 0) + (0 + × 0) + (0 . 707 + × 0)

j j j j

+ ( − 1 + × 0) + (0 . 707 + × 0) + (0 + × 0) + ( − 0 . 707 + × 0) = 0

j j j j

+ j × 0

x (0) W 0 x (1) W 5 x (2) W 10 x (3) W 15 x (4) W 20 x (5) W 25 x (6) W 30

X (5)

= + + + + + +

x (7) W 35

+ = (1 + × 0) + ( − 0 . 5 + × 0) + (0 + × 0) + ( − 0 . 5 + × 0)

j j j j

+ (1 + j × 0) + ( − 0 . 5 + j × 0) + (0 + j × 0) + ( − 0 . 5 + j × 0) = 0 + j × 0