Structural Health Monitoring Using Smart Sensors - ideals ...

Before this decimation is applied, all of the frequency components above the new Nyquist
frequency need to be eliminated. A discrete-time, lowpass filter with a cutoff frequency
M and gain 1 is applied.
The lowpass filter applied in the downsampling can be combined with the one in the
upsampling process. The cutoff frequency of the filter is set to be the smaller value of
L and M . The gain is L . This filtering process is reviewed more in more detail in
the subsequent paragraphs.
Numerical filters of various types can be represented as in the following equation,
yn
=
N b
N a
bk xn-k –
k=0
k=1
ak yn-k
(5.15)
where xn is the input to the filter, yn is the output from the filter, ak and bk are
filter coefficients for outputs and inputs, respectively. N a
and N b
represent the numbers
of filter coefficients, ak and bk . These filters can be classified as either Finite
Impulse Response (FIR) filters or Infinite Impulse Response (IIR) filters. An FIR filter has
nonzero coefficients corresponding only to the inputs. In other words, N a
is zero for an
FIR filter. An IIR filter has one or more filter coefficients corresponding to the outputs.
FIR and IIR filters have their own advantages over the other. FIR filters are always
stable no matter what coefficients are chosen. Another advantage of an FIR filter is its
linear phase characteristic. The delay introduced by an FIR filter is constant in frequency.
On the other hand, IIR filters with a given performance can be designed using fewer
coefficients, and, thus, are usually less computationally expensive.
One of the possible error sources of this resampling process is imperfect filtering. A
perfect filter, which has a unity gain in the passband and a zero in the stopband, needs an
infinite number of filter coefficients. With a finite number of filter coefficients, passband
and stopband ripples cannot be zero. A filter design with 0.1 to 2 percent ripple is
frequently used. A filter needs to be designed considering these filter characteristics.
Figure 5.25 shows signals before and after filtering. A signal analytically defined as a
combination of sinusoidal waves is sampled at three slightly different sampling
frequencies. Two of the signals are then resampled at the sampling frequency of the other
signal. As can be seen from Figure 5.25, after resampling, the three signals are almost
identical. These signals are, however, not exactly the same due to the imperfect filtering.
Though this signal distortion during filtering is preferably suppressed, this resampling
process is not the only cause of such distortion. AA filtering and digital filtering also use
imperfect filters. The filter in the resampling process needs to be designed so that the filter
does not severely degrade signals as compared with other filters.
The resampling process is considered to be extremely challenging if the upsampling
factor, L , is large. This issue is explained herein with examples. When a signal sampled at
100 Hz is resampled at 150 Hz, the rational factor, L
M, is 3/2. The original signal is
upsampled by a factor of 3. A lowpass filter with a cutoff frequency of can be easily
designed with a reasonable number of filter coefficients. Note that the cutoff frequency of
84