Differentiators - Digital Filter Design Solutions - page 323

2

where σ is the input noise power (assumed white) and is the noise

amplification factor, which is given by

2 m 2 m + 1

π F

c

η = (6.18)

2 m + 1

The noise amplification factor is the amount by which the input noise power is
amplified on the output of the filter. Thus for first-order differentiators ( m = 1),
the noise amplification factor becomes

3

F
c = 0.672. This is the highest normalized cut-off frequency a first-order
differentiator could have without exhibiting noise amplification. As such, most of
the differentiating filters have been designed with a cut-off less than 0.67.
Differentiation outside this bandwidth is accomplished using band-pass
differentiators, which is also aimed at managing the output amplification process.
Figure 6.10 shows experimental verification of (6.17) for a 55-point
differentiating filter using various cut-off frequencies F c . The upper bound
predicted by (6.17) is also plotted on the same figure for comparison.
Choice of a particular differentiator is governed mainly by the bandwidth of
the signal, the extent of noise amplification acceptable, and the matching band-
pass error of the filter. For example, the differentiator DIFF99F0.35 has a working
bandwidth of 0.25 when the pass-band error is of the order of 10 -5 , or 0.28 for

= 3.29 . It is not difficult to see that

= 1 when F

4

L+1 = 55
Upper Bound

3.5

3

2.5

2

1.5

1

0.5

0

0 0.25 0.5 0.75 1

Cut-off Frequency, F c 3

Figure 6.10 Output noise variance from a 55-tap differentiating filter compared to the upper bound
predicted by (6.17) (From [4]. Used by permission of the Institute of Physics Publishing).