Digital Signal Processing Reference
In-Depth Information
Chapter 6
Differentiators
6.1 INTRODUCTION
The ideal differentiator is characterized by a linear transfer function in the
frequency domain, and like all FIR filters, can be designed with either even or odd
filter order. Its magnitude |
D
(
F
)
D
(
F
)
| is perhaps best represented by
|
D
(
F
)
|
=
πF
|
F
|
1
(6.1)
where
F
is the normalized frequency. It is clear from (6.1) that unlike
non
-
differentiating digital filters, the magnitude of the differentiator increases linearly
in frequency, suggesting some form of amplification across its passband. When
the input signal to the ideal differentiator is noiseless, it becomes apparent that the
digital differentiator performs perfect differentiation. However, if white noise is
added to the signal, the high-frequency components of the noise are nonlinearly
amplified across the Nyquist range leading to gross distortion of the differentiated
output. As such, full band differentiators are inherently noisy. To illustrate this
point, Figure 6.1 shows the derivative of a signal without and with a small amount
of added noise. Although the signal-to-noise ratio (SNR) of the input signal is
relatively high, the SNR of the output is markedly poor.
Some effort has been expended in developing digital differentiators that avoid
or reduce the deleterious effects of the non-linear amplification process. For
example, based on white noise assumptions, Vanio and colleagues
[1]
optimized
the differentiator by minimizing the output noise power, while Tseng and Lee
[2]
optimized theirs with respect to the signal-to-noise ratio of the process. Carlsson
when colored noise is present.
The technique we will use to minimize differentiator noise amplification
assumes white noise contamination of the signal and combines low-pass filter
characteristics with that of the differentiator. The low-pass filter inherent in the
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