booklet format - inaf iasf bologna

A.A. 2011/2012
Temporal Data Analysis
Discrete Fourier Transform: Shifting Rules
3.2 Shifted cosine with N = 2 — Page 47
First, let us compute the DFT for the cosine function:
{f k } = {0,1} or
f k = 1 (1 − cosπk) k = 0,1
2
W 2 = e iπ = −1
F 0 = 1 2 (0 + 1) = 1 2
F 1 = 1 2 (0 + 1(−1)) = −1 2
{F j } =
{ 1
2 ,−1 2}
Now we shift the input by n = 1:
{f shifted
k
} = {1,0} or
f k = 1 (1 + cosπk) k = 0,1
2
{F shifted
j
} =
{ 1
2 W 2 −1×0 , 1 } { 1
2 W 2
−1×1 =
2 , 1 2}
3.3 Modulated cosine with N = 2 — Page 48
We want to modulate the input with W −nk
−k
N
, with n = 1. From its definition W2 = (−1) −k ,
therefore
{f shifted
k
} = {0,−1} or
f k = 1 (−1 + cosπk) k = 0,1
2
{
{F shifted
j
} = {F j −1 } = − 1 2 , 1 }
2
M.Orlandini 127