Biomedical Engineering Reference
In-Depth Information
Fig. 1.2
Sketch representation of the FO integral and derivator operators in frequency domain, by
means of the Bode plots (magnitude, phase)
where
F(s)
and
s
is the Laplace complex variable. The Fourier transform
can be obtained by replacing
s
by
jω
in the Laplace transform and the equivalent
frequency-domain expressions are
=
L
{
f(t)
}
cos
π
−
n
cos
nπ
1
(j ω)
n
=
1
ω
n
j
sin
π
2
1
ω
n
j
sin
nπ
2
2
+
=
2
−
(1.10)
ω
n
cos
π
n
ω
n
cos
nπ
j
sin
π
2
j
sin
nπ
2
(jω)
n
=
2
+
=
2
+
(1.11)
Thus, the modulus and the argument of the FO terms are given by
20 log
(jω)
∓
n
=∓
Modulus
(
dB
)
=
20
n
log
|
ω
|
(1.12)
arg
(j ω)
∓
n
=∓
n
π
2
Phase
(
rad
)
=
(1.13)
resulting in:
n
2
, anticlockwise rotation of the mod-
ulus in the complex plain around the origin according to variation of the FO
value
n
;
•
a Nyquist contour of a line with a slope
∓
•
magnitude (dB) vs. log-frequency: straight line with a slope of
∓
20
n
passing
through 0 dB for
ω
=
1;
•
phase (rad) vs. log-frequency: horizontal line, thus independent of frequency, with
value
n
2
.
The respective sketches can be seen in Fig.
1.2
.
∓
