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Frequency Response
 A data signal
(position-time curve in motion analysis) normally has a mixture of different
frequency components in it. The frequency contents of the signal and their
powers can be obtained through operations such as the Fast Fourier Transform (FFT).
A low-pass filter passes relatively low frequency components in the signal but
stops the high frequency components. The so-called cutoff frequency divides the
pass band and the stop band. In other words, the frequency components higher
than the cutoff frequency will be stopped by a low-pass filter. This type of
filter is especially useful since the random errors involved in the raw position
data obtained through reconstruction are characterized by relatively high
frequency contents.
The behavior of a filter can be summarized by
the so-called frequency response function, Hc. The frequency response
function of the Butterworth low-pass filter has the following form:
[1]
where
 ,
[2]
= the frequency (rad/s),  =
the cutoff frequency (rad/s), and N = the order of the filter. When
= 0, the magnitude-squared function (Hc2) shown in [1]
and Figure 1 becomes 1 and the frequency component
will be completely passed. When
=  , Hc2 becomes
0 and the frequency component will be completely stopped. Between the pass band
and the stop band, there is the transition band (1 > Hc2
> 0) in which the frequency component will be partially passed but partially
stopped at the same time. When
= , Hc2
always becomes 0.5 (half-power) regardless of the order of the filter.
As shown in Figure 1, a
Butterworth low-pass filter does not completely pass the frequency components
lower than the cutoff frequency, nor completely stops those higher than the
cutoff frequency. Figure 2 shows the effects of the
filter order on the frequency response. As the filter order increases, the
transition from the pass band to the stop band gets steeper. (Note that the
vertical axis in Figure 2 is Hc, not Hc2.)
At = ,
H = 0.707, regardless of the order of the filter.
The frequency response function of the Butterworth filter
involves complex numbers since it is a function of j.
Thus, the magnitude-squared function is the product of the response function
pairs Hc(s) and Hc(-s):
[3]
where
 .
[4]
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Continuous-Time System Function
For the factorization of [3],
the poles (the roots of the denominator) of the magnitude-squared function must
be obtained:
 .
[5]
Therefore, from [5]:
 ,
[6]
where k = 0, 1, 2, ...., 2N - 1, since
 ,
[7]
For N = 2, 4, 6...., "2k + N -
1" is
| N = 2 |
1/4, 3/4, 5/4, 7/4 |
| N = 4 |
3/8, 5/8. 7/8, 9/8, 11/8, 13/8, 15/8,
17/8 (or 1/8) |
| N = 6 |
5/12, 7/12, 9/12, 11/12, 13/12, 15/12,
17/12, 19/12, 21/12, 23/12, 25/12 (or 1/12), 27/12 (or 3/12) |
Generalizing the results presented in the table above for any even-order filter:
[8]
where N = 2, 4, 6...., and k = 0, 1, 2, ...., 2N -
1. Figure 3a & 3b
show the poles of the magnitude-squared function for N = 2 & 4,
respectively. The horizontal axis of the s-plane is the real
axis while the vertical axis is the imaginary axis in Figures
3a & 3b.
Note here that only half of the poles shown in Figure
3 can be used in the factorization of Hc(s) since [6]
(or [8]) was derived from the magnitude-squared
function. Since the poles always occur in pairs, one may choose the poles in the
left half of the s-plane based on the following relationship:
 .
[9]
Therefore, from [3]
& [8], the frequency
response, Hc(s), for a 2nd-order filter is
 ,
[10]
since
 .
[11]
Similarly, if N = 4 (4th-order):
 .
[12]
Hc(s) shown in [10]
and [12] are called the continuous-time
system function of the filter. A 4th-order low-pass filter is a cascade of two 2nd-order
low-pass filters as shown in [12]. Generalizing [12]
for any even order:
 .
[13]
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References & Related Literature
Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time
signal processing. Englewood Cliffs, NJ: Prentice Hall.
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