overshoot in butterworth low-pass filters

Abstract. Demonstrate overshoot for butterworth filters in R.

1. Introduction.

Butterworth filters with order other than 1 have an overshoot phenomenon that can be problematic in some cases. For example, if smoothing is used on an estimate of kinetic energy, overshoots might yield negative values that are nonphysical. This post simply illustrates this with made-up data that the reader can experiment with.

2. Methods.

The code given below creates the graph shown in the next section

n <- 100
x <- 1:n
y <- ifelse(0.3*n < x & x < 0.7*n, 1, 0)
if (!interactive()) png("butter.png", width=7, height=5, unit="in", res=150, pointsize=9)
plot(x, y, type='o', pch=20, ylim=c(-0.1, 1.1))
b1 <- butter(1, 0.1)
y1 <- filtfilt(b1, y)
points(x, y1, type='o', pch=20, col='red')

b2 <- butter(2, 0.1)
y2 <- filtfilt(b2, y)
points(x, y2, type='o', pch=20, col='blue')

b3 <- butter(3, 0.1)
y3 <- filtfilt(b3, y)
points(x, y3, type='o', pch=20, col='forestgreen')
if (!interactive()) dev.off()

3. Results

The graph illustrates; for the colour coding, see the code above.

Demonstration of first, second, and third order butterworth low-pass filters.

4. Conclusions

Caution is recommended in using butterworth filters of order 2 and higher. The advantage of the rapid cutoff in frequency space may be outweighed by the disadvantage of overshooting in time space.


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