Structural Health Monitoring Using Smart Sensors - ideals ...

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(a)
Figure 5.25. Signals (a) before and (b) after resampling.
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the filter is expressed in radian normalized to the sampling frequency of an upsampled
signal. This filter is applied to the upsampled signal, which is three times longer than the
original one, and then downsampled by a factor of 2. When a signal sampled at 101 Hz is
resampled at 150 Hz, on the other hand, the rational factor, L
M, is 150/101. Upsampling
by a factor of 101 greatly increases the data size. A lowpass filter with a cutoff frequency
of requires a large number of filter coefficients. If such a filter is applied to the
upsampled signal, the upsampled signal is 101 times longer than the original one. Direct
implementation of such resampling on resource-limited smart sensors is intractable.
Polyphase implementation
An FIR filter can be computationally much more inexpensive as a filter for
resampling if the polyphase implementation is employed. The polyphase implementation
leverages knowledge that upsampling involves inserting many zeroes and that an FIR
filter does not need to calculate the output at all of the upsampled data points. This
implementation of resampling is explained in this section mainly in the time domain
because the extension of the method to achieve synchronized sensing is pursued using a
time domain analysis. Oppenheim et al. (1999) provided a detailed description of the
polyphase implementation in the Z-domain.
Upsampling and lowpass filtering with an FIR filter can be written in the following
manner:
yn
vl
=
N-1
k=0
xm
,
=
,
hk vn-k
l = Lm, m =0, , ,
otherwise
(5.16)
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