Structural Health Monitoring Using Smart Sensors - ideals ...

dB
A s
Figure 4.4. AA filter design parameters.
fn
fc
fsb
f N
passband transition
Frequency
stopband
component whose frequency is higher than the Nyquist frequency, f N = f s , is folded
back to the frequency range from 0 to f N
, i.e., the signal is aliased. As a result of this
sampling phenomenon, signals in the frequency range above f N
are superimposed onto the
original signal components in the baseband frequency range after the sampling. Once the
signal is contaminated with aliasing components, it cannot be corrected. Therefore, the
high-frequency components above the Nyquist frequency need to be eliminated prior to
the sampling process.
Ideally, a Linear Time Invariant (LTI) analog circuit, whose gain is unity over the
passband range with linear phase response and then attenuates quickly to zero is desirable.
In practice, LTI circuits close to the ideal circuit are employed as AA filters (see Figure
4.4). To completely eliminate aliasing in a digital signal, the following relation needs to
hold
f n
f c
f sb
f N
(4.1)
where f n
is the highest frequency of signal components to be analyzed; f c
is filter’s
passband cutoff frequency; and f sb
is the stopband cutoff frequency. The stopband
attenuation, A s
, is determined so that any signal in the stopband is smaller than the
resolution of the ADC connected to the filter. Other requirements for AA filters include
small-gain variance in the passband; and linear phase over the passband, which keeps the
signals undistorted in the time domain.
There are many variations in LTI circuits for AA filters. Though an arbitrary LTI
system, which satisfies the above mentioned requirement, works as an AA filter, several
filter types have been proposed and used for their specific characteristics. The Butterworth
filter, which is also called the “maximally flat magnitude” filter, has a frequency response
that is as flat as mathematically possible in the passband. The Bessel filter has the
maximally linear phase response. The elliptic filter has an equiripple magnitude response
in both the passband and stopband, minimizing the maximum error in both bands. The
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