Biomedical Engineering Reference
In-Depth Information
The amplitude terms for each spectral component can be calculated by integrating the
product of the time-domain signal with a sample phasor of the correct frequency
t
2
T
a
n
=
v(
t
)
cos
(
n
ω
t
)
dt
0
t
2
T
b
n
=
v(
t
)
sin
(
n
ω
t
)
dt
(5.60)
0
a
n
+
c
n
=
b
n
tan
−
1
a
n
b
n
θ
n
=
The magnitude,
c
n
,ofthe
n
-th harmonic can be calculated, along with its phase,
θ
n
,as
shown in (5.60).
If there is a component of the signal at or near the selected frequency, this phase-
sensitive integration process will produce a nonzero amplitude. This process examines the
signal only at discrete integer frequencies determined by the value of
n
and at DC where
the term
a
0
/2 is the average value of
v(
t
)
over a complete cycle.
Care must be used when interpreting the results of this spectral analysis, as the process
assumes that individual harmonics are present with constant amplitude and phase for
the duration of the measurement. However, in reality harmonics are seldom constant
in amplitude or phase, so the coefficients, which represent only the averages, may be
significantly in error.
Depending on the nature of the signal and the information that must be reproduced,
the number of frequency components that are required will vary. For example, a square
wave may require up to 1000 harmonics to reproduce the sharp transitions that define
the switching points (Carr, 1997). The harmonic analysis in Figure 5-63 shows that the
amplitudes of the coefficients are reduced by progressively smaller amounts with the result
that their individual contributions become progressively smaller.
However, as discussed earlier is this chapter, an ECG trace can be reproduced using
spectral components up to about 100 Hz.
The effect of truncating the series is demonstrated in Figure 5-64, where the waveform
is reconstructed by summing the appropriately scaled sinusoidal components. For example,
when five components are used, the signal is reconstructed as
5
v(
t
)
=
a
n
cos
(
n
ω
t
)
+
b
n
sin
(
n
ω
t
)
(5.61)
n
=
1
Figure 5-64 shows the effectiveness of the reconstruction for 5, 50, and 500 coefficients.
It can be seen that the transitions are still not perfect even after the largest number.
The spectrum of a repeated time-domain sequence such as an ECG produces a comb
of frequency components at intervals corresponding to the reciprocal of the repeat period.
As shown in Figure 5-65, these fall within the envelope of the spectrum of a single ECG
cycle.
To understand this, consider that the repeated time-domain signal can be generated by
convolving a single ECG cycle with a sequence of impulses at the correct time intervals,
t
. It can be shown that the spectrum of a sequence of impulses is a regularly spaced
