Digital Signal Processing Reference
In-Depth Information
Figure 19.2
Generalized structure for continuous signal
x
(
t
) synthesis from digital data
and when the classical requirements are met, specifically when the corresponding
discrete data block to be transformed into an analog signal is actually represented
by a sample value sequence taken from a certain basis function under conditions
where at least two sample values are taken in the time interval equal to the period
of the highest frequency present in this function. Then no aliasing takes place
and there are no problems. Therefore this is the preferred way to perform analog
signal synthesis whenever the mentioned conditions can be satisfied. However, the
operational speed of DACs is limited and, consequently, the highest frequency
of the signal to be synthesized in this way cannot exceed a certain boundary
determined by the switching speed of the sufficiently high bit rate DAC available.
The accumulated experience of successful suppression of various negative effects
due to aliasing suggests that it is worth looking for a way to avoid this synthesis
limitation.
The generalized model of the continuous signal
x
(
t
) synthesis from digital
data is given in Figure 19.2. To obtain a continuous or analog signal
x
(
t
), a
related basis function
x
0
(
t
) is formed. Typically that is done by a computer,
forming it as a sum of a finite number of sinusoidal components. As parameters
(frequencies, amplitudes and phase angles) depend in some way on the analog
signal to be synthesized, their values, as well as the sampling frequency
f
s
, should
be calculated on the basis of the algorithm used for synthesis. The approach to such
analog signal synthesis will be discussed further. At this stage it should simply be
noted that these signal component parameters are fixed and, consequently, sample
values
x
0
(
t
n
)
of the basis function can be calculated and stored
in a memory. These digital values are than passed at the sampling rate
f
s
to a
DAC performing their conversion into a kind of analog signal, which changes its
values in an idealized stepwise way, as shown in Figure 19.3. The synthesized
signal is taken off the output of the filter.
It can be seen from the diagrams in Figure 19.3 that the values of the basis
function change continuously while the DAC output signal remains at some level
for relatively long time intervals, which are longer than the period of the basis
function. The DAC output signal certainly does not look similar to the given basis
function. The question arises as to what kind of properties this staircase signal
prossesses and whether it is possible to obtain such a signal with the required
properties.
,
n
=
1
,
2
,...,
N
,
