Biomedical Engineering Reference
In-Depth Information
conditions are
1)
the function should have a finite number of maxima or minima,
2)
the function should have a finite number of discontinuities, and
−
t
t
T
3)
the function should have a finite value or
f
(
t
)
dt
is less than infinity.
Conditions for the Fourier Series include the following:
1)
periodic functions (practical is nonperiodic);
2)
can only be applied to stable systems (systems where natural response decays in
time, for example, convergent).
Example Problem
: Consider the sinusoidal waveform of Fig. 13.5.
According to Nyquist sampling theorem, the waveform in Fig. 13.5 should be
sampled at more than two samples per second to successfully reconstruct the signal.
For the sake of convenience, consider eight samples per second of the above wave-
form. Therefore:
N
2
n
2
3
,
N
is the total number of samples, and
n
=
8
=
=
=
3 stages;
T
=
125 milliseconds, sampling interval; Record length
=
1 second (time of recording
data
=
N
×
T) Frequency resolution
=
1Hz(1
/
(
N
×
T
))
The discrete signal is given by (13.19).
cos
2
cos
2
fnT
=
π
fn
N
X
(
nT
)
=
π
(13.19)
1
0.5
0
−
0.5
−
1
FIGURE 13.5
:
Trace of the waveform is
x
(
t
)
=
cos
(
ω
t
)
, where
ω
=
2
π
f
and
f
=
1Hz
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