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5 Methods for Discrete Processing and Analysis of Biosignals
with equation:
f p : = (fp(0), fp(1), . . . , fp(N −1))
F : = (F(0), F(1), . . . , F(N −1))
W : = {wmn} ,
wmn = e−j2πmn/N
W−1 = { 1
N w−1
nm} ,
w−1
nm = ej2πmn/N
m, n : = 0, 1, . . . , N −1.
(5.25)
The inverse transformation is thus carried out by means of the inverse Fourier-matrix
W−1. With the help of computer-algebra-systems (CAS), which can perform matrix op-
erations directly, such as Octave, Scilab or Matlab, the DFT is particularly easy to cal-
culate.
Example
for N = 3: Forward transformation:
[[
[
F(0)
F(1)
F(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
F
= [[
[
1
1
1
1
e−j2π/3
e−j4π/3
1
e−j4π/3
e−j8ı/3
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
W
⋅[[
[
fp(0)
fp(1)
fp(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
f
,
(5.26)
Reverse transformation:
[[
[
fp(0)
fp(1)
fp(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
f
= 1
3
[[
[
1
1
1
1
ej2π/3
ej4π/3
1
ej4π/3
ej8π/3
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
W−1
⋅[[
[
F(0)
F(1)
F(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
F
.
(5.27)
5.2.3 Discrete Laplace Transform and z-Transform
The z- transform is particularly well suited for describing linear digital systems con-
sisting only of linear components, since here the relationship between input and out-
put signals in the frequency domain can be described by a simple fractional rational
function in the new frequency variable z := ejωTa (cf. subsubsection 5.3.4.1). For a
causal discrete-time signal f(n) (i.e. f(n) = 0 for n < 0), the z-transform is then de-
scribed by the new frequency variable as follows:
FD(f) = F(z = ejωTa) ,
bzw.
F(z) = FD (jω = 1
Ta
ln z) .
(5.28)
It then follows according to Equation 5.9 for the z-transformation:
f(n) =
1
2πj ∮F(z) zn−1dz ,
F(z) =
∞
∑
n=0
f(n) z−n .
(5.29)