Digital Signal Processing Reference
In-Depth Information
Table 3.1
16-Bit Integer* Representation of LPF99F0.3 (
Q
= 0)
h
1
-
h
5
h
6
-
h
10
h
11
-
h
15
h
16
-
h
20
h
21
-
h
25
h
26
-
h
30
h
31
-
h
35
h
36
-
h
40
h
41
-
h
45
h
46
-
h
50
0 -1 -7 18 76 -149 - 497 807 2446 -4868
1 1 -11 -12 110 94 - 689 - 496 3406 3485
0 4 - 5 - 47 44 348 -264 -1783 1323 16334
-1 4 11 - 50 -102 353 589 -1773 -3055 28043
-2 0 24 0 -210 0 1174 0 - 6447 32767
*
These coefficients could be scaled (i.e., multiplied) with 2
-15
= 3.052×10
-5
to realize a fixed-point
fractional representation.
floating-point filter realization and its 16-bit derivative. In spite of this, it is clear
that phase linearity is maintained on quantization (Figure 3.8(c)).
In most instances, the gain of signals can be readily scaled up or down prior
to filtering and therefore the benefits of using
Q
= 0 is viewed in the context of
whether this scaling would impair the filter output. Moreover, if the application
does not involve large, high slew-rate signals then
Q
should be made zero.
3.2.5
Stop-Band Attenuation after Filter Quantization
The upper bounds of the stop-band attenuation after quantization
A
q
could be
predicted in terms of the bit precision
B
, by the following relation
2
A
q
≤
aB
+
bB
+
c
(3.9)
where
a
,
b
, and
c
are constants. The formula gives the worst-case attenuation
when any low-pass filter is converted from floating-point to
B
-bit precision, and
Q
= 0. In principle, the stop-band attenuation is slightly sensitive to frequency;
however, we have used data corresponding to the poorest (minimum) attenuation
observed over the frequency range (0.1
0.8) to determine the upper bounds.
Figure 3.9(a) shows the minimum attenuation over the mentioned frequency range
in terms of coefficient quantization word size
B
for filters of starting length
L
+1 =
99. Table 3.2 gives the parameters
a
,
b
, and
c
for various starting filter lengths.
≤
F
≤
Table 3.2
Attenuation-Bitsize Relational Coefficients
for Various Filter Starting Lengths
Starting filter
length,
L
+1
a
(dB/bit
2
)
b
(dB/bit)
c
(dB)
99
0.134
-9.33
35.2
155
-
-5.62
14.5
255
-
-5.74
19.7
