Digital Signal Processing Reference
In-Depth Information
Figure 15.5
Regularization error occurring as a result of ascribing the signal instantaneous
value
x
k
to the reference function zero-crossing instant
of sampling, all signal sample values should be assigned to the time instants
t
k
which actually are the instants when the reference sine wave
crosses the zero or some other constant level. Thus this regularization approach is
simple and straightforward and leads naturally to the introduction of some noise
ε
,
k
=
0
,
1
,
2
,...,
The signal values at these time instants differ from those corresponding to
the signal reference function crossing instants. As shown in Figure 15.5, the value
of a particular error
(t
k
)
.
ε
k
depends on the angle of the signal/sine-wave crossing and
on the time interval
t
between this crossing instant and the reference function
zero-crossing instant
t
k
. The fact that the time interval
t
is smaller for smaller
values of the sampled signal helps to keep the relative errors close to a constant
level for small to medium signal values.
When the sampling results are regularized as explained, most of the classic DSP
algorithms, including the fast ones like the FFT, could be used for processing
the digital signals obtained in this way. However, the errors introduced at the
sampling result regularization impose some limitations on the applicability of this
approach. To assess these limitations, it is shown in Figure 15.6 how the power
of the regularization noise
(
t
k
) depends on the reference sine-wave frequency.
Actually, the given curve is pessimistic as it has been calculated to conditions that
are near to the worst. Specifically, the regularization errors
ε
k
were calculated for
a full amplitude single-tone signal at the frequency just below the Nyquist limit.
The time intervals
ε
t
and the related regularization errors
ε
k
then tend to their
maximal values.
