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5 Methods for Discrete Processing and Analysis of Biosignals
time t / µs
Fig. 5.21: Sine wave of 4 kHz with a DC component of 10 mV sampled at a frequency of 12 kHz.
2.31 mV and a DC component of 10 mV according to
x(t) = 10 mV + 2.31 mV ⋅sin(2π 4 kHz t)
with 12 kHz sampled as well as using a ideal discrete-time high-pass filter with a
purely real transfer function and a cut-off frequency of 2 kHz to suppress the DC
component.
With a sampling frequency of 12 kHz one obtains the following three values per
period:
x(t = 0 μs) = x(0) =
10
x(t = 83.33 μs) = x(1) = 12 x(t = 166.66 μs) = x(2) = 8 .
Since this is a periodic signal, the calculation of the spectrum can be done using the
Fourier-matrix and we obtain as in the example of Equation 5.26:
[[
[
X(0)
X(1)
X(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
X
= [[
[
1
1
1
1
e−j2π/3
e−j4π/3
1
e−j4π/3
e−j8ı/3
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
W
⋅[[
[
x(0) = 10
x(1) = 12
x(2) = 8
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
x
= [[
[
30
−j3.64
j3.64
]]
]
.
The periodic spectrum shows Figure 5.22. It can be seen that the DC component is
evident from a real spectral line at frequency f = 0 Hz. In order to suppress this, the
discrete-time high-pass filter must have a transfer function G that does not pass a DC
component. At the other frequencies a gain of one can be chosen, i.e. with the relation
between output and input spectrum according to Equation 5.53 G(0) = 0, G(1) = 1 and
G(2) = 1, i.e.
[[
[
Y(0)
Y(1)
Y(2)
]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Y
= G ⋅X = [[
[
0
0
0
0
1
0
0
0
1
]]
]
⋅[[
[
30
−j3.64
j3.64
]]
]
= [[
[
0
−j3.64
j3.64
]]
]
.