5.3 Methods for Analysis and Processing of Discrete Biosignals
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177
Fig. 5.20: Discrete-time linear system with representation in the image area by its transfer function.
main". E.g. a sum of sine waves is used to represent a periodic signal with the help
of the Fourier series. If these are applied to the input of a linear time-invariant sys-
tem, a sine wave with the same frequency but with different amplitude and phase is
also obtained at the output from each individual sine wave of the input signal. These
differences are described according to magnitude and phase by the complex transfer
function G(m) = |G(m)|ejφ(m) (cf. Figure 5.20). This transfer function is also discrete
for a periodic input signal in the image domain. The relation between input and out-
put signal can then be described for a period with Np frequency points by the matrix-
equation
[[[[[
[
Y(1)
Y(2)
...
Y(Np)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Y
=
[[[[[
[
G(0)
0
⋅⋅⋅
0
0
G(2)
⋅⋅⋅
0
...
...
...
...
0
0
⋅⋅⋅
G(Np)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
G
⋅
[[[[[
[
X(1)
X(2)
...
X(Np)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
X
(5.53)
X and Y are vectors and contain the input and output signals, G is a Np × Np diagonal
matrix whose diagonal elements embody the values of the transfer function. With the
help of the matrix form for the Fourier-transformation in Equation 5.24, the back trans-
formation into the time domain can now be carried out simply by applying the matrix
calculation, i.e., the transfer function can be transformed back into the time domain.
i.e. because of x = W−1 ⋅X and Y = W ⋅y it follows from
Y = W ⋅y = G ⋅X = G ⋅W ⋅x
(5.54)
and by left multiplication with the inverse Fourier-matrix W−1:
y = W−1 ⋅G ⋅W
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Zykl{gp}
⋅x .
(5.55)
Thus, by matrix calculation, a relation between input and output signal in the time
domain is obtained which is identical to Equation 5.52.⁷
Explanatory Example
For a simple illustration of the representation in the frequency domain with filtering
of spectral components there, a sinusoidal oscillation with 4 kHz, an amplitude of
7 It can be mathematically proved that by multiplying a diagonal matrix from the left by the inverse
Fourier-matrix W−1 from the right by the Fourier-matrix W, an cyclic matrix is formed.