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SUMMARY OF FILTER CHARACTERISTICS
gain at each stage should be adjusted by the appropriate
power of two. The proper scaling factor can be
determined by evaluating the maximum gain from the
input to each point in the cascade, starting with the first
stage. The gain of that stage is adjusted to ensure that
the gain does not exceed unity at any frequency. After
each stage is adjusted, the process is repeated for the
next stage.
5.6.6
Very Low Frequency Filters
As mentioned above, the processes occurring in the
recursive second order section are equivalent to integra-
tion. When very low frequency filters or filters with very
high Q's must be realized, even the 25-bit word width of
the 2920 may not provide adequate protection from
truncation error. In some cases it may be possible to
reduce the clock rate (and therefore sample rate) which
will reduce requirements for coefficient precision.
When other functions prevent reduction of the sample
rate, or when the predicted value of clock rate must be
lower than the minimum permitted for the 2920, alter-
nate programming techniques must be used. (The 2920
word size and the dynamic range of the variables being
processed establish a maximum ratio of sample rate to
frequencies of interest.)
Extended Precision Arithmetic-For very low fre-
quency filters, the effective sampling rate must be
reduced or the effective precision of the processor must
be increased.
One approach,
extended precision
arithmetic, appears possible but cumbersome. When
very low frequencies are being used, the coefficients Bl
and B2 approach very closely the values +2 and
-1 respectively. By realizing the filter as shown in
Figure 5-11, the small terms B
I
-2 and B
2
+1 are isolated
from the large terms and scaled upwards by some power
of two. The equivalent multiplications may then be
done using single precision, which is converted back to
extended precision by a 2-
n
scaling.
Extended precision arithmetic may be executed using
masks derived from the constants, or by conditional
additions. In either case, carries generated by the low
order word are added to the high order word to main-
tain carry propagation. The carries may be simulated in
one of the high order bits of the low order word, tested
via conditional operations or masking, and then
removed by masking or conditional addition of a
negative constant. Table 5-3 shows an extended preci-
sion add routine.
5-16
I_--------~~------------~
t
1
Figure
5-17.
Very Low Frequency Filter
Table
5-3.
Extended Precision Add Routine (48
Bit Precision) Technique Uses
Simulated Carry at 2nd Bit From
Left of Low Order Word.
ADD YL,
XL,
ROO
; add low order word
(25
bits+carry)
LDA TMP, YL,
ROO
; copy word to temporary
location
AND TMP, KP4,
ROO
; mask off simulated carry
bit
SUB
YL,
TMP, ROO
; clear carry from
low
order word
ADD YH,
XH,
ROO
; add high order words
LDA TMP, TMP, R13
; move carry to right
ADD YH,
TMP, RIO
; add carry to high order
word
Submultiple Sampling-When low frequency filters
must be realized, it is in general more convenient to
reduce the sample rate rather than attempt to extend the
precision of the variables. The sample rate may effec-
tively be reduced by using the conditional load opera-
tion triggered by an oscillator run at a submultiple of
the sample rate. The filter calculations go to completion
every nth cycle. Such an oscillator can be realized by the
program shown in Table 5-4.
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