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SUMMARY OF FILTER CHARACTERISTICS
structure of Figure 5-15 realizes the equivalent of a
complex conjugate pair of zeros at s
=
a
±
jw via the
(FORTRAN) equations
V(k)
=
AO*YO
+
Al *YI
+
Al *Y2
AO, Al and A2 must meet the following requirements
(where the value of AO is arbitrary):
Al
=
2'AO'e
oT
cos
wt
A2
=
AO . e-
2oT
of AD
IS
arbitrary
A1
=
2 AD
e-
oT
cos
wt
Figure 5-15. Realization of a Complex
Conjugate Zero Pair
T(s)
=
s2+a 2 s+(a 2 +w 2 )
T(z)
=
AO+AIZ-I+A2Z-2
MAX GAIN
=
IAol+IAII+IA21
5.6.5 Some Practical Considerations
The procedures described above show how second order
filter sections can be realized. In selecting the gain for
the filter, the user should consider the scaling of the
variables within the filter. Improper scaling can result in
a number of problems.
If the variables are very small, it is possible that the
25-bit word width will not provide enough resolution,
and significant truncation noise will be introduced.
Because a second order filter of this type may perform
the equivalent of integrations in which results are
obtained by summing many small values, roundoff
error can occur in unexpected ways.
If the variables are scaled too large, overflow saturation
may result, with behavior very similar to that occurring
in an analog circuit when the signals exceed the dynamic
5-15
range of the amplifiers. However, an additional con-
sideration may be important in 2920 realizations of
second order sections. As coefficient products are
developed by series of additions and subtractions,
intermediate values may be larger than those finally
obtained.
In general, it is necessary to provide sufficient margins
when scaling input variables to ensure that overflow
saturation does not occur for intermediate values.
Sometimes the sequence of calculations can be ordered
to minimize potential overflow saturation.
A third method to prevent intermediate overflow
saturation is to compute some fraction of YO, restoring
it to full value when it is transferred to Y 1, such as
shown in Figure 5-16. This of course adds some noise to
the final output, lowering the accuracy somewhat.
The coding generated by the Signal Processor Applica-
tions Software/Compiler is already ordered and scaled
in this manner to minimize overflow.
Figure 5-16. Method for Preventing Intermediate
Overflow
(If overflow occurs, it will be when YO is increased and
loaded to YI.)
No additional instructions are necessary in general,
because the extra multiplications shown in Figure 5-16
can be performed by modifying the instructions of the
original realization.
When a filter consists of a cascade of second order sec-
tions, code can be saved by performing the gain adjust-
ment calculations at just one point in the cascade.
However, to maintain properly scaled variables, the
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