Digital Signal Processing Reference
In-Depth Information
floating-point processing. In general, this approach appears to be a reasonable test
of performance for higher-order differentiators. If higher accuracy is required, it
might be necessary to run a first-order differentiator twice over the data.
6.4.2
Quantization of Filter Coefficients and Implementation
The quantization of second-order coefficients is dependent on the peak value
attained by the impulse response function of the differentiator. In this case, the
global maximum occurs at the origin. Thus, for second-order differentiating
filters, the conversion formula from full precision to
B
-bit word size is
3
d
(
2
)
(
2
B
1
1
(6.15)
d
(
2
)
=
ROUND
k
k
,
B
3
c
(
+
Q
/
100
)
512
F
where the symbols have their usual meanings. The filter implementation given in
(6.3) is applied for
m
= 2.
6.4.3
Filter Gain
G
The filter gain
G
from (6.15) is
(
2
B
1
1
3
(6.16)
G
=
3
c
512
(
+
Q
/
100
)
F
Again, dividing the filtered output by
G
produces a properly scaled derivative of
the input signal.
6.5 NOISE CONSIDERATION
6.5.1
The Noise Amplification Factor
The performance of filters in the presence of noise is an important consideration
in digital signal processing. In fact, the choice of cut-off frequencies for the
differentiators presented in this work was determined solely on the nature of their
noise attenuation characteristics. It was shown recently [4] that the output noise
variance (power) σ of an
m
th order differentiator is
2
y
2
y
2
η
(6.17)
σ
σ
