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Ton van den Bogert wrote:
>It is not too hard to determine the transfer function in the
frequency
>domain of such a filter.
After sampling, the discrete-time moving average system (Example 2.2,
p. 17 in Oppenhem, Schafer "Discrete-Time signal processing", Prentice
Hall, 1989):
y[n]=(1/(M1+M2+1))(x[n+M1]+x[n+M1-1]+...+x[n]+x[n-1]+...+x[n-M2]),
where M1 and M2 are numbers of points around x[n]. M1 is the number of
samples received after x[n] - noncausal processing, can be implemented
only off-line, M2 is number of samples received before x[n]. If you
apply moving average such that the 10th point y[10] is the weighted sum
of x[10], x[9],...,x[1] then M1=0, M2=9. Anyway, it's the total number
of averaged points that is important for determining the magnitude of
the frequency response.
Magnitude of frequency response of this discrete-time system (Example
2.15, p. 44, ibid):
H(Wdiscr)=abs( 1/(M1+M2+1)*sin(W discr*(M1+M2+1)/2)/sin(W discr/2)
),
or
H(W discr)=abs ( 1/10*sin(5*Wdiscr)/sin(0.5*Wdiscr) ).
>From a graph of H(Wdiscr) (quick look using Matlab for Wdiscr=0 to pi,
step 0.01):
H(wdiscr)=0.707 for wdiscr=0.28.
A corresponding continuous-time system has equivalent (effective)
frequency response for Wcont=Wdiscr/Tsamp, and
Fcutoff=Wdiscr/2piTsamp.
If sampling at 100Hz:
Tsamp=0.01s, W discr=0.28 => Wcont=28 rad/s, Fcutoff=4.46Hz,
i.e., smoothing with 100ms window in this case is equivalent to
low-pass filtering with 4.46Hz cutoff freq.
Examples for different #points averaged, and for a couple of sample
freqs are given below. After Wdiscr is found and Fcutoff/Window is
calculated for one Fsamp the rest is found by multiplying/dividing by
the ratio F'samp/Fsamp.
#points Wdiscr Fcutoff Hz Fcutoff Hz Fcutoff Hz
averaged window ms window ms window ms
(Fsamp=100Hz) (Fsamp=500Hz) (Fsamp=1000Hz)
3 0.976 15.5 HZ, 30 ms 77.5 Hz, 6 ms 155 Hz, 3 ms
5 0.565 9 Hz, 50 ms 45 Hz, 10 ms 90 Hz, 5 ms
10 0.28 4.5 Hz, 100ms 22.5 Hz, 20ms 45 Hz, 10 ms
20 0.139 2.2 Hz , 200ms 11 Hz, 40ms 22 Hz, 20 ms
Mark Shapiro
***********************
Mark B. Shapiro, Ph.D
School of Kinesiology (M/C 194)
University of Illinois at Chicago
901 West Roosevelt Road
Chicago, Illinois 60608
tel: (312) 942-5414
e-mail: mshapi2@uic.edu
fax (312) 355-2305
***********************
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>It is not too hard to determine the transfer function in the
frequency
>domain of such a filter.
After sampling, the discrete-time moving average system (Example 2.2,
p. 17 in Oppenhem, Schafer "Discrete-Time signal processing", Prentice
Hall, 1989):
y[n]=(1/(M1+M2+1))(x[n+M1]+x[n+M1-1]+...+x[n]+x[n-1]+...+x[n-M2]),
where M1 and M2 are numbers of points around x[n]. M1 is the number of
samples received after x[n] - noncausal processing, can be implemented
only off-line, M2 is number of samples received before x[n]. If you
apply moving average such that the 10th point y[10] is the weighted sum
of x[10], x[9],...,x[1] then M1=0, M2=9. Anyway, it's the total number
of averaged points that is important for determining the magnitude of
the frequency response.
Magnitude of frequency response of this discrete-time system (Example
2.15, p. 44, ibid):
H(Wdiscr)=abs( 1/(M1+M2+1)*sin(W discr*(M1+M2+1)/2)/sin(W discr/2)
),
or
H(W discr)=abs ( 1/10*sin(5*Wdiscr)/sin(0.5*Wdiscr) ).
>From a graph of H(Wdiscr) (quick look using Matlab for Wdiscr=0 to pi,
step 0.01):
H(wdiscr)=0.707 for wdiscr=0.28.
A corresponding continuous-time system has equivalent (effective)
frequency response for Wcont=Wdiscr/Tsamp, and
Fcutoff=Wdiscr/2piTsamp.
If sampling at 100Hz:
Tsamp=0.01s, W discr=0.28 => Wcont=28 rad/s, Fcutoff=4.46Hz,
i.e., smoothing with 100ms window in this case is equivalent to
low-pass filtering with 4.46Hz cutoff freq.
Examples for different #points averaged, and for a couple of sample
freqs are given below. After Wdiscr is found and Fcutoff/Window is
calculated for one Fsamp the rest is found by multiplying/dividing by
the ratio F'samp/Fsamp.
#points Wdiscr Fcutoff Hz Fcutoff Hz Fcutoff Hz
averaged window ms window ms window ms
(Fsamp=100Hz) (Fsamp=500Hz) (Fsamp=1000Hz)
3 0.976 15.5 HZ, 30 ms 77.5 Hz, 6 ms 155 Hz, 3 ms
5 0.565 9 Hz, 50 ms 45 Hz, 10 ms 90 Hz, 5 ms
10 0.28 4.5 Hz, 100ms 22.5 Hz, 20ms 45 Hz, 10 ms
20 0.139 2.2 Hz , 200ms 11 Hz, 40ms 22 Hz, 20 ms
Mark Shapiro
***********************
Mark B. Shapiro, Ph.D
School of Kinesiology (M/C 194)
University of Illinois at Chicago
901 West Roosevelt Road
Chicago, Illinois 60608
tel: (312) 942-5414
e-mail: mshapi2@uic.edu
fax (312) 355-2305
***********************
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To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l
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