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A.A. 2011/2012
Temporal Data Analysis
mation term e iωt we get the discrete times t k = k ∆t, k = 0,1,2,..., N − 1 with T = N∆t:
(
exp(iωt) → exp i 2πt ) (
k
= exp i
T
2πk ∆t
N∆t
We will use the abbreviation for the “kernel” W N as
( ) 2πi
W N = exp
N
Occasionally we will also use the discrete frequencies ω j
)
= exp
( 2πi k
N
)
≡ W k N
(3.4)
ω j = 2πj
N∆t
(3.6)
related to the discrete Fourier coefficients F j (see below). The kernel W N has the following
properties:
(3.5)
W n N
N = e
2πi n = 1 for all integer n
W N is periodic in j and k with period N
(3.7)
We can define the discrete δ-function as follow:
where δ k,k ′
N−1 ∑
j =0
is the Kronecker symbol with the following property:
W (k−k′ )j
N
= N δ k,k ′ (3.8)
{ 1 for k = k
′
δ k,k ′ =
0 else
This symbol (with prefactor N) accomplishes the same tasks the δ-function had when doing the
continuous Fourier transformation.
3.1.3 Definition of the Discrete Fourier Transformation
Now we want to determine the spectral content {F j } of the series {f k } using discrete Fourier
transformation. For this purpose, we have to make the transition in the definition of the Fourier
series:
c j = 1 T
∫ +T /2
−T /2
f (t)e −2πi j /T dt −→ 1 N
N−1 ∑
k=0
(3.9)
f k e −2πi j k/N (3.10)
with f (t) periodic of period T .
In the exponent we find k∆t/N∆t, meaning that ∆t can be eliminated. The prefactor contains
the sampling raster ∆t, so the prefactor becomes ∆t/T = ∆t/(N∆t) = 1/N. During the transition
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