180

|

5 Methods for Discrete Processing and Analysis of Biosignals

Then by applying the inverse Fourier-transform to the spectrum of the output signal

Y according to Equation 5.24 in the time domain, it follows:

y = W⁻¹ ⋅Y = ¹

3

[[

[

1

1

1

1

e^j2π/3

e^j4π/3

1

e^j4π/3

e^j8ı/3

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

W⁻¹

⋅^[[

[

0

−j3.64

j3.64

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Y

= ^[[

[

0

2

−2

]]

]

.

From the values 10, 12 and 8 of the sampled sinusoidal input signal with a DC com-

ponent, we have now obtained the values 0, 2 and −2 for the output signal after filter-

ing with an ideal high-pass filter, which corresponds to a sampled sinusoidal signal

without a DC component, cf. Figure 5.23.

Note

The output signal y(n) can also be determined directly by cyclic convolution. How-

ever, for this one needs the pulse response g(n)⁸, which are obtained analogously to

continuous-time signals by Fourier-back-transforming the transmission vector G :=

[G(0), G(1), G(2)]^T:

[[

[

g(0)

g(1)

g(2)

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

g

= ¹

3

[[

[

1

1

1

1

e^j2π/3

e^j4π/3

1

e^j4π/3

e^j8ı/3

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

W⁻¹

⋅^[[

[

G(0) = 0

G(1) = 1

G(2) = 1

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

G

= ¹

3

[[

[

2

−1

−1

]]

]

.

Now according to Equation 5.55 the cyclic convolution can be performed:

[[

[

y(0)

y(1)

y(2)

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

y

= ¹

3

[[

[

2

−1

−1

−1

2

−1

−1

−1

2

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

cycle{g}

󳐂^[[

[

10

12

8

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

x

= ^[[

[

0

2

−2

]]

]

.

As can be seen, the results obtained by Fourier-back transformation and cyclic convo-

lution are identical. In contrast to the continuous-time domain, the cyclic convolution

is less complex in the digital representation and much easier to calculate.

In many measured biosignals, the unwanted interference signal does not only

consist of a sinusoidal 50 Hz mains hum, as shown in Figure 5.19, which can be eas-

ily detected in the frequency range and suppressed with the help of a filter. Noise-like

interference signals (e.g. from electromagnetic radiation sources such as neon lamps,

TV and radio transmitters or mobile phones) also occur, which cannot be described

precisely and of which only the power or statistical quantities such as the correlation

can be measured. These quantities can also be investigated in the frequency domain.

8 Since it is known that signal and transfer function are periodic, in this example the index p at gp

and Gp is omitted for simplification.