Frequency Response
A data signal
(positiontime 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 lowpass filter passes relatively low frequency components in the signal but
stops the high frequency components. The socalled 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 lowpass 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 socalled frequency response function, H_{c}. The frequency response
function of the Butterworth lowpass 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 magnitudesquared function (H_{c}^{2}) shown in [1]
and Figure 1 becomes 1 and the frequency component
will be completely passed. When
= , H_{c}^{2} becomes
0 and the frequency component will be completely stopped. Between the pass band
and the stop band, there is the transition band (1 > H_{c}^{2}
> 0) in which the frequency component will be partially passed but partially
stopped at the same time. When
= , H_{c}^{2}
always becomes 0.5 (halfpower) regardless of the order of the filter.
As shown in Figure 1, a
Butterworth lowpass 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 H_{c}, not H_{c}^{2}.)
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 magnitudesquared function is the product of the response function
pairs H_{c}(s) and H_{c}(s):
[3]
where
.
[4]
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ContinuousTime System Function
For the factorization of [3],
the poles (the roots of the denominator) of the magnitudesquared 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 evenorder filter:
[8]
where N = 2, 4, 6...., and k = 0, 1, 2, ...., 2N 
1. Figure 3a & 3b
show the poles of the magnitudesquared function for N = 2 & 4,
respectively. The horizontal axis of the splane 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 H_{c}(s) since [6]
(or [8]) was derived from the magnitudesquared
function. Since the poles always occur in pairs, one may choose the poles in the
left half of the splane based on the following relationship:
.
[9]
Therefore, from [3]
& [8], the frequency
response, H_{c}(s), for a 2ndorder filter is
,
[10]
since
.
[11]
Similarly, if N = 4 (4thorder):
.
[12]
H_{c}(s) shown in [10]
and [12] are called the continuoustime
system function of the filter. A 4thorder lowpass filter is a cascade of two 2ndorder
lowpass 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). Discretetime
signal processing. Englewood Cliffs, NJ: Prentice Hall.
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