Low-Pass Filter
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Frequency Response
Continuous-Time System Function
References & Related Literature

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:



,    [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):



.    [4]


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:


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]


.    [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]


References & Related Literature

Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.



Young-Hoo Kwon, 1998-