Differentiators - Digital Filter Design Solutions - page 310

100

80

60

90

40

80

20

70 0

-20

60

-40

50

Noiseless simulation
Data w ith added noise
Filtered noisy data

-60

Derivative of noisy data
Derivative of noisef ree data

40 -80

650 652 654 656 658 660 650 652 654 656 658 660

Wavelength (nm)

Wavelength (nm)

Figure 6.1 (a) Simulated part spectrum of an interferometer with and without noise (from [4]) . (b)
Derivative of spectrum with a full-band differentiator showing noise amplification. Dotted line
represents derivative of noisy data while the solid line shows the ideal derivative. (Used by permission
of the Institute of Physics Publishing.)

differentiator is designed to reject out-of-band noise thereby reducing the said
adverse effects. The technique is not new but is perhaps the simplest approach as
it could be designed using the window method. Moreover, given the cut-off
frequency or bandwidth of a differentiating filter, it is possible to predict the noise
amplification factor of the filter [4] . In fact, a foremost consideration in the choice
of a differentiator is its noise amplification.

Terminology

The term differentiating filter will be used to describe a differentiator with low-
pass or band-pass characteristics as well as full-band differentiators. Furthermore,
the term unity-gain filter will be used to mean a filter in which the pass-band gain
in the frequency domain is unity. Thus low-pass, band-pass, and high-pass filters
would fall into this category. The term digital filter or filter will be used in the
generic sense to mean the class of all filters possessing first, second, or m th
derivative functions, unity-gain filters, integrators, and Hilbert transformers.
These distinctions often arise when comparisons need to be made between a
differentiator and its nondifferentiating counterpart. On the other hand, in the
interest of mathematical clarity, the k th value of the m th order differentiating filter
coefficient will therefore be denoted by

( m
k

)

d
In the following sections, the classification, formulas for the filter
coefficients, slope response, and output noise power for low-pass and band-pass
differentiating filters will be given.

.