Digital Signal Processing Reference
In-Depth Information
100
80
(
a
)
(
b
)
60
90
40
80
20
70
0
60
-20
50
-40
Noiseless simulation
Data w ith added noise
Filtered noisy data
40
Type IV DIFF
Bandlimited DIFF - noisy data
Bandlimited DIFF - noisless data
-60
noiseless
-80
30
620
625
630
635
640
620
625
630
635
640
Wavelength (nm)
Wavelength (nm)
Figure 6.11
(a) Simulated spectrum with added noise. (b) First derivative of spectrum using a low-
pass differentiating filter with cut-off of 0.15 and full-band Type IV differentiator. (From
[4].
Used by
permission of the Institute of Physics Publishing.)
6.5.4
General Performance in Noise
The performance of band-limited differentiators in white noise has been
investigated previously
[4].
Figure 6.11 (a) shows part of a simulated spectrum of
a white light interferometer to which noise was added. The noisy spectrum was
differentiated first by a Type IV full-band differentiator and then a 55-point band-
limited differentiator (
F
c
= 0.15). The noise-free spectrum was differentiated by
the band-limited differentiator only. Figure 6.11 (b) shows a typical differentiating
filter output in the presence of noise. The noise amplification factor
for the
band-limited differentiator in this case is 0.01 (i.e., 1%) with an estimated increase
in SNR of 40 dB. The extent of noise attenuation is clearly seen and apart from
small discrepancies at the turning points, there is good general agreement with the
noise-free derivative. In the next section, we will look at ways of producing first-
and second-order differentiators from unity gain filters. This could be useful when
time is a constraint.
