Digital Signal Processing Reference
In-Depth Information
Figure 19.8
An optional structure for synthesis of a passband signal
frequency
f
s
to sample the basis function so that the frequency band (
f
min
,
f
max
)
can be completely inserted within the interval (
if
s
,
(
i
+
0
.
5)
f
s
), or within the
interval ((
i
1)
f
s
)), where
i
is a positive integer. To obtain such a
signal, the basis function is formed as explained above so that the components
of it (
f
1
+
0
.
5)
f
s
,
(
i
+
,
f
2
,...,
f
K
) cover the frequency band (
f
min
,
f
max
) with the frequency
step
f
. This basis function is sampled and a DAC is used to convert the obtained
digital signal into an analog signal. Then the DAC output signal is passed through
an analog bandpass filter having the same passband (
f
min
,
δ
f
max
). The structure of
such a synthesizer is illustrated by Figure 19.8.
In the case of an ideal filter, the amplitudes of the signal
z
(
t
) components are
determined by Equation (19.8). Therefore if the amplitudes of the basis function
components are taken as
A
0
A
K
=
|
,
(19.25)
sinc(
π
f
k
f
s
)
|
where
k
K
, the components of the synthesized signal
z
(
t
) will have
equal amplitudes
A
0
. The dynamic range of the signal
z
(
t
) is naturally narrower
than that of the DAC output signal (
t
). It is essential to know their ratio. Let
D
x
and
D
z
be the dynamic ranges of signals
x
(
t
) and
z
(
t
). Then their ratio
=
1
,
2
,...,
D
z
.
Figure 19.9 illustrates this approach to signal synthesis where the amplitudes of
the basis function have been calculated on the basis of Equation (19.25). Therefore
they differ for all components, which leads to equalization of the synthesized
signal
z
(
t
) component amplitudes, as shown in Figure 19.9(a). The dynamic
ranges of signals
x
(
t
) and
z
(
t
) are obviously different in comparison with the
respective dynamic ranges when the output signal component amplitudes are not
equalized. In this case
ρ
xz
=
D
x
/
ρ
=
4
.
4.
xz
This effect of
ρ
xz
enlargement rapidly increases if the frequency range
(
f
min
,
f
max
)
approaches
any
of
the
frequency
points
if
s
.
For
instance,
if
(
f
min
,
f
max
) is defined as (1
.
01
f
s
,
1
.
41
f
s
) and again nine components are placed
within it with a frequency step of 0
xz
for the cases
without and with output signal component amplitude equalization are 6.0 and
27.0 respectively. This is due to the fact that the frequency
f
min
has approached
frequency
f
s
.
.
05
f
s
then the named ratios
ρ
This can also be seen from Equation (19.25) as near to
if
s
the
