Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
N N
X [ − k]
= X [ + k]
2 2 (6.13)
The proof is as below:
N −1
N
⎛ 2p
( N / 2 − k)
n ⎞
X[
− k]
= ∑ x[
n]exp⎜
− j
⎟
2
n=
0 ⎝ N ⎠
N −1
⎛ 2p
( N / 2 − k − N)
n ⎞
= ∑ x[
n]exp⎜
− j
⎟
n=
0 ⎝ N ⎠
N −1
⎛ 2p
( N / 2 + k)
n ⎞ * N
= ∑ x[
n]exp⎜
j
⎟ = X [ + k]
n=
0 ⎝ N ⎠ 2
N N
Therefore, X [ − k]
= X [ + k]
. This property tells that the modules of the DFT is symmetrical about the vertical
2 2
line
N
n = .
2
8. Complex signal x[n]
If the signal x [n]
is complex, there is no spectral symmetry, and all N coefficients are distinct in general.
6.3 The fast Fourier transform (FFT)
James W. Cooley and JohnW. Tukey in 1965 made a revolutionary invention in calculating the DFT (published in J.W.Cooley
and J.W. Tukey in Math. Comput., vol. 19, April 1965, pp297-301). In the algorithm known as FFT, redundancy in direct
calculating complex DFT due to periodicity in sinusoidal functions has been removed, therefore the computing time has
been remarkably reduced. The principle can be explained in the following.
N −1 kn
For the DFT X[ k] = x[ n]exp
⎛ 2p ⎞
∑ ⎜−
j ⎟
⎝ N ⎠
n=
0
, let the complex function
W N
⎛ 2p
⎞
= exp
⎜ − j
⎟
⎝ N ⎠
then
W kn
N
2π kn
= exp
− j
.
N
If separating x [n]
to an eve and an odd sequences
Download free ebooks at bookboon.com
93