Digital Signal Processing Reference
In-Depth Information
19.3.3 Amplitude Equalization
As shown in the previous section, it is possible to synthesize an analog signal
from scarce data obtained under conditions when the well-known requirements
of the sampling theorem are not met and when the frequencies of the signal to be
synthesized exceed half of the DAC switching rate. However, there are also some
problems and limitations. First of all, the amplitude of the synthesized sinusoidal
signal mainly depends on its frequency. That might cause problems when the
analog signal to be synthesized contains a number of frequencies. As that is
exactly what is typically needed for the generation of multifrequency test signals,
the feasibility of synthesizing such a multifrequency signal will be considered.
Apparently various types of multifrequency test signal are needed. In the con-
text of the considered application, a group of frequencies typically has to be gen-
erated around a certain central frequency. Even in this case, there are a number of
possible variations. Amplitude spectra of some of them are given in Figure 19.7.
Probably the most popular structure of a multifrequency test signal is the one
shown in Figure 19.7(a). All frequencies of it are equidistanced on the frequency
axis and amplitudes of all components are equal. However, as discussed further,
the price of achieving the equality of amplitudes may sometimes be too high.
Quite often it might be better to accept some amplitude variation. Such amplitude
variations, if predetermined, should not represent a problem, especially if all com-
ponent amplitudes are given with sufficiently high precision. This kind of multifre-
quency signal is given in Figure 19.7(b). The third type of multifrequency signals,
illustrated in Figure 19.7(c), differs from the others by the location of the frequen-
cies. In this case the signal components are placed on the frequency axis irregularly
at arbitrary places. How beneficial this possibility could be is another question;
here the fact is simply demonstrated that the suggested synthesis method would
permit that possibility. When the traditional methods for generating such multifre-
quency test signals are used, it certainly is not simple to provide for this functional
feature.
While there are no problems at synthesizing signals with specified frequencies,
except that they cannot be placed too closely together, it is quite another situation
with the synthesis of multifrequency signals with equal or specified amplitudes
of their components. Direct application of the suggested synthesis approach leads
to the synthesis of signals with component amplitudes defined by Equation (19.8)
and illustrated by Figure 19.4. It follows from Equation (19.8) that when the fre-
quency of a component grows its amplitude declines according to
,
where
f
s
is the sampling frequency. In addition, the amplitudes of components
close to frequencies
f
s
,2
f
s
,
|
sinc(
π
f
/
f
s
)
|
f
s
,...
are very small (frequencies
f
s
,
2
f
s
,...
cannot
