Digital Signal Processing Reference
In-Depth Information
However, their frequency response can still be affected by the lack of accuracy of
their coefficients. As an example, we consider the case of the Parks-McClellan
filter designed in Example 7.4. The coefficients for that example have been
truncated to signed 16-bit, 12-bit, and 8-bit numbers as shown in Table 8.1.
Table 8.1
Comparison of Original and Truncated Coefficients
Coefs
Original
16-bit
12-bit
8-bit
h(10)
0.348822
0.348822
0.348822
0.348822
h(9), h(11)
0.004010
0.004013
0.004090
0.002747
h(8), h(12)
-0.268081
-0.268076
-0.268049
-0.269170
h(7), h(13)
-0.008972
-0.008974
-0.009032
-0.008240
h(6), h(14)
0.102739
0.102740
0.102755
0.101625
h(5), h(15)
0.008322
0.008325
0.008350
0.008240
h(4), h(16)
0.012282
0.012285
0.012269
0.010987
h(3), h(17)
-0.004333
-0.004333
-0.004260
-0.005493
h(2), h(18)
-0.033368
-0.033363
-0.033400
-0.032960
h(1), h(19)
0.001195
0.001192
0.001193
0.000000
h(0), h(20)
0.012527
0.012530
0.012610
0.013733
Equation (8.6) shows the procedure for determining the truncated values. The
calculation within the
Round
function actually indicates the procedure of
converting the floating-point coefficient to a signed fraction representation for use
in fixed-point processors. The multiplication outside of the
Round
function
converts the signed fraction back to a truncated decimal:
B
−
1
⎡
⎤
h
(
n
)
⋅
(
2
−
1
h
(
10
)
h
(
n
)
=
Round
⋅
(8.6)
⎢
⎥
trunc
B
−
1
h
(
10
)
(
2
−
1
⎣
⎦
As we would expect, as the number of bits is reduced, the error in the
coefficients increases. The frequency response generated from the 16-bit
coefficients was virtually identical to the original response. However, the
responses due to the 12-bit and 8-bit coefficients were noticeably degraded, as
shown in Figure 8.1. In that figure, the passband response was virtually
unchanged in all cases, but the stopband characteristics did not match those of the
original coefficients (shown as −56.7 dB). The 12-bit coefficients did produce a
response that satisfied the original design specifications of −50 dB. The 8-bit
coefficients, however, caused the frequency response to degenerate completely,
providing as little as 32 dB of attenuation. The reason for this degradation, of
course, is that the truncation has moved the pole and zero locations from their
original positions.
