Page 875 - The Mechatronics Handbook

0066-frame-C29 Page 5 Wednesday, January 9, 2002 7:23 PM

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Time (ms)

FIGURE 29.2 A 1-Hz signal.

Y(t)

t

FIGURE 29.3 A signal sampled and reconstructed using a zero order hold (ZOH).

called an anti-aliasing filter. This is the second and most practical method that is used to satisfy the Nyquist
Sampling Theorem. Thus, a combination of a well-designed anti-aliasing filter as well as a sampling frequency
that is well above the cut-off frequency of the filter will ensure that the Nyquist Sampling Theorem is satisfied.
There are two important points that should be noted when using an anti-aliasing filter. First, it is important
that the anti-aliasing filter be used before the signal is sampled as sampling is what causes aliasing. Basically,
this requires that the anti-aliasing filter is implemented using an analog filter prior to the signal being digitized.
Once a signal has been aliased during sampling, it cannot be corrected using digital filtering. The second
point is that, in practice, the cutoff frequency of the anti-aliasing filter should be a factor of 5–10 below the
value of the Nyquist frequency. It should be noted that an anti-aliasing filter adds phase lag to the measure-
ment, which might deteriorate stability and performance in a feedback loop unless the bandwidth of the
anti-aliasing filter is much higher than that of the closed loop system. Commercially available devices that
perform sampling are analog-to-digital converters (ADCs), and the anti-aliasing filter is used before this
device.
The converse of sampling is reconstruction where a discrete-time signal is converted into a continuous-
time signal. The Nyquist sampling rate ensures that if a continuous-time signal is sampled at a rate that is
at least twice the highest frequency component in the signal, then the continuous-time signal can be recon-
structed exactly from the samples. However, this theorem assumes that an ideal reconstruction process is
available, which is not practical. The most common practical means to reconstruct a signal is a zero-order
hold (ZOH). The ZOH assumes that the value of the signal is constant between samples. This approxi-
mation is quite reasonable if the sampled signal does not change substantially between individual samples.
Figure 29.3 is an example of a signal and its ZOH representation. The gray, smooth line represents the
original analog signal. The black points along the signal indicate sample values of the signal. Each black

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