Dr. Guy Simoneau

12-05-1997, 10:11 AM

Thank you to all who responded to my question about filtering.

The answers were most helpful.

************************************************** *******

My original posting was.

In the process of analyzing some wrist kinematic data that we collected

with the use of electrogoniometers, an interesting question was raised

by one of our graduate students.

Background: Our interest is to look at position, velocity and acceleration

of the wrist during various tasks. Since the electrogoniometers provide

position data, we differentiate the signal once for the velocity

and twice for the acceleration data. In order to eliminate noise

in the signal we use a fourth order, zero phase shift, Butterworth filter

with a cut-off frequency of 6 Hz. This filter is applied to the raw position

data.

His question was: should we apply the filter to the data again after each step

where the signal has been differentiated. His logic was that the process

of differentiating the signal would create a higher frequency component

in the signal. So, in essence we would filter the raw position data,

differentiate the signal to get velocity, filter the velocity data, then

differentiate the signal to get acceleration.

************************************************** ********

I believe that the following answers best summarize the responses.

All answers are at the end of this message for those of you with a greater

interest.

Summary of answers:

-------------------------------

you should review the recent work of Giakas and Baltzopoulos, Journal of

Biomechanics (1997), 30 (8), 851-856. Also, on a historical note, Vaughan

(1982) might have been one of the first to publish results based on

acceleration data that had been previously filtered twice.

--------------------------------

Filters such as splines, Butterworth digital filters, etc. act as

linear operators. So does a finite-difference differentiation.

This means that you can interchange the order of these operations

without affecting the result.

So, if you apply the same filter again, after differentiating,

you would get the same result as if you had applied the filter

twice to the raw position data, before differentiating.

This would give you effectively a higher-order filter (though no

longer a Butterworth filter!), with a slightly lower cut-off

frequency.

--------------------------------

My immediate reaction against this idea is that the three resulting signals

(position, velocity and acceleration) would NOT resemble the pos., vel. and acc.

of a single physical figure any more. Position is (per def:-) the integral of

the corresponding velocity, which by definition is the integral of the

corresponding

acceleration (we ignore the constants of integration for clarity). If filtering

is applied between each differentiation stage, this is no longer true for the

resulting signals.

**********************************************

All answers are included below

I am sure you will receive other messages to the same extent, but you

should review the recent work of Giakas and Baltzopoulos, Journal of

Biomechanics (1997), 30 (8), 851-856. Also, on a historical note, Vaughan

(1982) might have been one of the first to publish results based on

acceleration data that had been previously filtered twice. Good luck.

Peter Vint

e-mail: pfvint@homans.uncg.edu

-------------------------------------

In effect the process is quite simple. When you filter twice with a

filter with a transfer function H(jw), the resultant transfer function is

H^2 (jw). For a low-pass filter this means the same cut-off frequency, but

a steeper cut-off of the high frequencies.Differentiation is, like

filtering, a linear process. This means that the order in which things are

done is irrelevant. Filter-

differentiate- filter- differentiate will work out just the same as any

other sequence, e.g. D-D-F-F. Just try it!The only thing that may interfere

maybe round-off errors, which

may make one sequence preferable to the other. I hope this advice is of

some help,

At Hof

e-mail: a.l.hof@med.rug.nl

------------------------------------

I'm glad you raised this issue. I discovered myself that it was, indeed,

necessary to filter after each differentiation, otherwise the result is too

noisy. As I understand it what this does in effect is to sharpen the filter

frequency response, so that the cutoff is, in effect, lowered. I guess you

could calculate the precise effect in MATLAB, but I've never done it. What

I'd like to know is whether the commercical motion systems, such as Vicon

or MAC do this?

Dr. Chris Kirtley (Kwok Kei Chi) MD PhD

email rskirt@polyu.edu.hk

--------------------------------

we also calculate the acceleration by deriving the time

series. A derivation is the equivalent of a first-order high pass

filter. To calculate the acceleration, this first-order high pass is

used twice. As derivation amplifies the higher frequency parts, it is

necessary to limit the spectrum. This is done by smoothing the data

using a low pass filter e. g. with a third-degree smoothing spline.

This is a passband filter to extract the wanted signal from noise.

The influence of the filter shape to the wanted signal depends on the

cut-off frequency, the filter shape and the ripple of the low pass

filter.

If the transfer function (filter shape) is known, it is possible to

determine the influence of once or twice filtering to the wanted

signal.

We had this problem by determing the motion symmetry of horses,

therefore we developed a system matched filter. This was published

1996 (Peham C., Scheidl M., and Theresia Licka: "A method of signal

processing in motion analysis of the trotting horse".)in the J.

Biomechanics, Vol. 29, No. 8, pp. 1111-1114.

* Christian Peham *

* email: Christian.Peham@vu-wien.ac.at *

------------------------------------

My immediate reaction against this idea is that the three resulting signals

(position, velocity and acceleration) would NOT resemble the pos., vel. and acc.

of a single physical figure any more. Position is (per def:-) the integral of

the corresponding velocity, which by definition is the integral of the

corresponding

acceleration (we ignore the constants of integration for clarity). If filtering

is applied between each differentiation stage, this is no longer true for the

resulting signals.

The spectre of the acceleration and velocity SHOULD be richer in high

frequencies

than the corresponding position signal. Maybe the right thing to to would be to

filter the original pos. data "three times" (by applying the filter three

times in

"cascade" or by tripling the filter's order) _before_ the two

differentiations are

carried out.

If noise due to round-off errors during differentiation is a problem,

you could follow the procedure you suggested except that after both

differentiations

are carried out, the velocity should be filtered again once and the pos.

data twice

so that all signals are filtered an equal number of times. In this case

"pure integral"

relationships would not be strictly preserved, but in the presence of round-off

errors it wouldn't anyway.

Oyvind Stavdahl (Siv.ing., Dr.ing. student)

Email: Oyvind.Stavdahl@itk.ntnu.no

------------------------------

I have the "intuitive" impression that applying a filter to the

velocity (after the first differentiation) is equivalent to applying

a stronger filter to the position data. The same with filtering the

acceleration itself: it is equivalent to a stronger filter to the

velocity data.

To demonstrate this you can apply Fourier or Laplace transformation.

A(s)=s·V(s)=s·s·X(s).

So if you apply filter H(s) to V(s), you get:

A'(s)=s·H(s)·V(s)=s·(s·H(s))·X(s), and see that s·H(s) is a much

stronger filter for X(s).

Ruben Lafuente-Jorge

E-Mail: rlafuent@ibv.upv.es

---------------------------

This was the issue in one of our papers recently published. We

examined different filtering+differentiation sequences and one of

them was the one you mentioned (Procedure 3 in the paper).

The problem with this sequence is that it presents big problems at

the endpoints. Moreover we found that the acceleration was quite

oversmoothed (like a flat pattern).

The important finding from that study was that the normal sequence

(Procedure 1 - filtering raw displacement data and then differentiate

for the velocity and acceleration), is not the best we can use.

Instead the data should be filtered using different cut-off

frequencies (Procedure 2) for each derivative domain. However they

have to be filtered in the 0th derivative. The cut-off for

acceleration should be (most of the cases) lower than the one used

for velocity, which in turn has to be lower than the one used for

displacement.

How much lower ? This is a question !!! Based on the findings and

experience I personaly use a cut-off decreased by 10% in each

derivative domain (so if CFd=10 Hz for displacement, then CFv=9Hz and

CFa=8).

Giannis

Giakas G and V Baltzopoulos (1997). Optimal digital filtering

requires a different cut-off frequency strategy for the determination

of the higher derivatives. Journal of Biomechanics 30(8), 851-855

Giannis Giakas PhD

Email: g.giakas@staffs.ac.uk

-------------------------------------

I don't think you should filter the velocity data again before differentiate

velocity data for acceleration if you are interested in all position,

velocity, and acceleration data, because the differentiations will not be the

true differentiations of the position data if you filter the velocity data

again before the second differentiation. You may find the following two

recent references helpful:

Giakas, G. and Baltzopoulos, V. A comparison of automatic filtering

techniques applied to biomechanical walking data. Journal of Biomechanics,

1997, 8: 847-850.

Giakas, G. and Baltzopoulos, V. Optimal digital filtering requires a

different cut-off frequency strategy for the determination of the higher

derivatives. Journal of Biomechanics, 1997, 8: 851-856.

Bing Yu, Ph.D.

--------------------------------

differentiating a signal strongly amplifies high frequency components of

the signal

(because that what's differentiating is all about). Look in textbooks of

signals analysis *) what happens in the frequency-domain and you'll

understand.

Clearly this procedure will worsen the signal to noise ratio.

There is nothing wrong in filtering the signal with a low pass filter,

to increase the signal -to noise ratio. The only thing you'll have to

know is that this procedure limits the bandwith of events you'll want to

interprete. It is for YOU to decide whether this might hide important

information, or just removes noise. (There is more to say, but this

requires a priori knowledge about the original signal).

So if you filter the signal each time with 6 Hz after differentiating,

no conclusions about components of velocity or acceleration above 6 Hz

can be drawn. This might not be a problem for a student class, but it is

clear that fast transients in position data contain high frequency

components in the accelaration. Again: this is the nature of the

problem.

Conclusion: filtering is OK as long as every statement about

accelerations etc. is done in the context of a specified bandwidth.

(Which should always be the case, from an electrical engineer's point of

view.)

*) I would recommend "designing digital filters" from Charles S.

Williams, [prentice-hall, 1986]

---------------------------

Theoretically, it is not right to filter again after each derivation. Becase

the high frequency components in the velocity and acceleration are natural

and from the actual movement, so that you can not simply throw them away. If

you do, then you will deal with three different movements represented by the

position, the velocity and the acceleration, respectively. We know that the

velocity should be 0 when the position shows an extreme value, and so on.

But by filtering the data repeatedly after each derivation, you would ruin

this relationships among the physical quantities.

So the rationale here would be: (a) choose the physical quantity (either

position, velocity or acceleration) you wish to filter, (b) filter this

particular quantity, and (c) do derivation or integration to obtain the

other quantities.

In this way, you are basically dealing with the same movement represented in

different forms by different mechanical quantities.

Young-Hoo Kwon, Ph.D.

Internet: y-hkwon@sports.re.kr

ykwon@wp.bsu.edu (English only)

--------------------------------------------------------------

I can't answer your question, but you could probably answer it yourself

by doing a numerical simulation (this would also probably make a nice

reference technical note for the biomechanics community).

Here is what you could try (if you felt so inclined):

* Come up with several unperturbered position signals which represent

typical movements (e.g., wrist movements in your case). Since you

know the real underlying position signal, you can easily determine

the actual corresponding velocity and acceleration signals (e.g.,

using analytic techniques or cubic interpolating splines).

* Contaminate the simulated position signals with appropriate noise

(you would have to determine what appropriate noise is). You would

probably have to generate 10 or more perturbed position signals for

each unperturbed position signal.

* Try your new multiple-filtering approach on the perturbed signals, and

see how well you reproduce the true velocity and acceleration data

relative to how well one can do with the traditional approach.

I don't know much about signal processing, so I will be curious to see

what other BIOMCH-L readers have to say in response to your question.

There is probably some good theoretical reason why your student's idea

would or would not be a good approach.

Benjamin J. Fregly, Ph.D.

E-mail: bj@sj.ptc.com

> ------------------------------

Filtering and differentiating are both linear operations (ie, it doesn't

matter which order you apply them in), so there's no benefit to refiltering

prior to the second differentiation (unless you want your velocity and

acceleration signals filtered differently). The net effect is of filtering

with a single filter of higher order and lower cutoff frequency (say ~ 4 Hz

- you'd have to check a Bode plot to be sure). Your student is right that

differentiating amplifies higher frequencies - you might want to check into

some form of cross-validation procedure for estimating accelerations.

Cheers,

Tony

----------------------------

Filters such as splines, Butterworth digital filters, etc. act as

linear operators. So does a finite-difference differentiation.

This means that you can interchange the order of these operations

without affecting the result.

So, if you apply the same filter again, after differentiating,

you would get the same result as if you had applied the filter

twice to the raw position data, before differentiating.

This would give you effectively a higher-order filter (though no

longer a Butterworth filter!), with a slightly lower cut-off

frequency.

In my opinion, this would only lead to confusion, and could also

be sub-optimal. Higher-order filters don't work well on signals

with a broad spectrum, and if you need a lower cut-off frequency

you can simply design the (first) filter that way, rather than

apply it twice.

So I would say, apply a filter only once, and design it so that

it gives you good results at the end of the analysis.

-- Ton van den Bogert

************************************************** ******************

Guy G. Simoneau, Ph.D., P.T., A.T.C.

Assistant Professor, Physical Therapy Department

Marquette University

Walter Schroeder Complex, Room 346

P.O. Box 1881

Milwaukee, WI 53201-1881

e-mail: simoneaug@vms.csd.mu.edu

Phone: 414-288-3380

Fax: 414-288-5987

************************************************** ******************

The answers were most helpful.

************************************************** *******

My original posting was.

In the process of analyzing some wrist kinematic data that we collected

with the use of electrogoniometers, an interesting question was raised

by one of our graduate students.

Background: Our interest is to look at position, velocity and acceleration

of the wrist during various tasks. Since the electrogoniometers provide

position data, we differentiate the signal once for the velocity

and twice for the acceleration data. In order to eliminate noise

in the signal we use a fourth order, zero phase shift, Butterworth filter

with a cut-off frequency of 6 Hz. This filter is applied to the raw position

data.

His question was: should we apply the filter to the data again after each step

where the signal has been differentiated. His logic was that the process

of differentiating the signal would create a higher frequency component

in the signal. So, in essence we would filter the raw position data,

differentiate the signal to get velocity, filter the velocity data, then

differentiate the signal to get acceleration.

************************************************** ********

I believe that the following answers best summarize the responses.

All answers are at the end of this message for those of you with a greater

interest.

Summary of answers:

-------------------------------

you should review the recent work of Giakas and Baltzopoulos, Journal of

Biomechanics (1997), 30 (8), 851-856. Also, on a historical note, Vaughan

(1982) might have been one of the first to publish results based on

acceleration data that had been previously filtered twice.

--------------------------------

Filters such as splines, Butterworth digital filters, etc. act as

linear operators. So does a finite-difference differentiation.

This means that you can interchange the order of these operations

without affecting the result.

So, if you apply the same filter again, after differentiating,

you would get the same result as if you had applied the filter

twice to the raw position data, before differentiating.

This would give you effectively a higher-order filter (though no

longer a Butterworth filter!), with a slightly lower cut-off

frequency.

--------------------------------

My immediate reaction against this idea is that the three resulting signals

(position, velocity and acceleration) would NOT resemble the pos., vel. and acc.

of a single physical figure any more. Position is (per def:-) the integral of

the corresponding velocity, which by definition is the integral of the

corresponding

acceleration (we ignore the constants of integration for clarity). If filtering

is applied between each differentiation stage, this is no longer true for the

resulting signals.

**********************************************

All answers are included below

I am sure you will receive other messages to the same extent, but you

should review the recent work of Giakas and Baltzopoulos, Journal of

Biomechanics (1997), 30 (8), 851-856. Also, on a historical note, Vaughan

(1982) might have been one of the first to publish results based on

acceleration data that had been previously filtered twice. Good luck.

Peter Vint

e-mail: pfvint@homans.uncg.edu

-------------------------------------

In effect the process is quite simple. When you filter twice with a

filter with a transfer function H(jw), the resultant transfer function is

H^2 (jw). For a low-pass filter this means the same cut-off frequency, but

a steeper cut-off of the high frequencies.Differentiation is, like

filtering, a linear process. This means that the order in which things are

done is irrelevant. Filter-

differentiate- filter- differentiate will work out just the same as any

other sequence, e.g. D-D-F-F. Just try it!The only thing that may interfere

maybe round-off errors, which

may make one sequence preferable to the other. I hope this advice is of

some help,

At Hof

e-mail: a.l.hof@med.rug.nl

------------------------------------

I'm glad you raised this issue. I discovered myself that it was, indeed,

necessary to filter after each differentiation, otherwise the result is too

noisy. As I understand it what this does in effect is to sharpen the filter

frequency response, so that the cutoff is, in effect, lowered. I guess you

could calculate the precise effect in MATLAB, but I've never done it. What

I'd like to know is whether the commercical motion systems, such as Vicon

or MAC do this?

Dr. Chris Kirtley (Kwok Kei Chi) MD PhD

email rskirt@polyu.edu.hk

--------------------------------

we also calculate the acceleration by deriving the time

series. A derivation is the equivalent of a first-order high pass

filter. To calculate the acceleration, this first-order high pass is

used twice. As derivation amplifies the higher frequency parts, it is

necessary to limit the spectrum. This is done by smoothing the data

using a low pass filter e. g. with a third-degree smoothing spline.

This is a passband filter to extract the wanted signal from noise.

The influence of the filter shape to the wanted signal depends on the

cut-off frequency, the filter shape and the ripple of the low pass

filter.

If the transfer function (filter shape) is known, it is possible to

determine the influence of once or twice filtering to the wanted

signal.

We had this problem by determing the motion symmetry of horses,

therefore we developed a system matched filter. This was published

1996 (Peham C., Scheidl M., and Theresia Licka: "A method of signal

processing in motion analysis of the trotting horse".)in the J.

Biomechanics, Vol. 29, No. 8, pp. 1111-1114.

* Christian Peham *

* email: Christian.Peham@vu-wien.ac.at *

------------------------------------

My immediate reaction against this idea is that the three resulting signals

(position, velocity and acceleration) would NOT resemble the pos., vel. and acc.

of a single physical figure any more. Position is (per def:-) the integral of

the corresponding velocity, which by definition is the integral of the

corresponding

acceleration (we ignore the constants of integration for clarity). If filtering

is applied between each differentiation stage, this is no longer true for the

resulting signals.

The spectre of the acceleration and velocity SHOULD be richer in high

frequencies

than the corresponding position signal. Maybe the right thing to to would be to

filter the original pos. data "three times" (by applying the filter three

times in

"cascade" or by tripling the filter's order) _before_ the two

differentiations are

carried out.

If noise due to round-off errors during differentiation is a problem,

you could follow the procedure you suggested except that after both

differentiations

are carried out, the velocity should be filtered again once and the pos.

data twice

so that all signals are filtered an equal number of times. In this case

"pure integral"

relationships would not be strictly preserved, but in the presence of round-off

errors it wouldn't anyway.

Oyvind Stavdahl (Siv.ing., Dr.ing. student)

Email: Oyvind.Stavdahl@itk.ntnu.no

------------------------------

I have the "intuitive" impression that applying a filter to the

velocity (after the first differentiation) is equivalent to applying

a stronger filter to the position data. The same with filtering the

acceleration itself: it is equivalent to a stronger filter to the

velocity data.

To demonstrate this you can apply Fourier or Laplace transformation.

A(s)=s·V(s)=s·s·X(s).

So if you apply filter H(s) to V(s), you get:

A'(s)=s·H(s)·V(s)=s·(s·H(s))·X(s), and see that s·H(s) is a much

stronger filter for X(s).

Ruben Lafuente-Jorge

E-Mail: rlafuent@ibv.upv.es

---------------------------

This was the issue in one of our papers recently published. We

examined different filtering+differentiation sequences and one of

them was the one you mentioned (Procedure 3 in the paper).

The problem with this sequence is that it presents big problems at

the endpoints. Moreover we found that the acceleration was quite

oversmoothed (like a flat pattern).

The important finding from that study was that the normal sequence

(Procedure 1 - filtering raw displacement data and then differentiate

for the velocity and acceleration), is not the best we can use.

Instead the data should be filtered using different cut-off

frequencies (Procedure 2) for each derivative domain. However they

have to be filtered in the 0th derivative. The cut-off for

acceleration should be (most of the cases) lower than the one used

for velocity, which in turn has to be lower than the one used for

displacement.

How much lower ? This is a question !!! Based on the findings and

experience I personaly use a cut-off decreased by 10% in each

derivative domain (so if CFd=10 Hz for displacement, then CFv=9Hz and

CFa=8).

Giannis

Giakas G and V Baltzopoulos (1997). Optimal digital filtering

requires a different cut-off frequency strategy for the determination

of the higher derivatives. Journal of Biomechanics 30(8), 851-855

Giannis Giakas PhD

Email: g.giakas@staffs.ac.uk

-------------------------------------

I don't think you should filter the velocity data again before differentiate

velocity data for acceleration if you are interested in all position,

velocity, and acceleration data, because the differentiations will not be the

true differentiations of the position data if you filter the velocity data

again before the second differentiation. You may find the following two

recent references helpful:

Giakas, G. and Baltzopoulos, V. A comparison of automatic filtering

techniques applied to biomechanical walking data. Journal of Biomechanics,

1997, 8: 847-850.

Giakas, G. and Baltzopoulos, V. Optimal digital filtering requires a

different cut-off frequency strategy for the determination of the higher

derivatives. Journal of Biomechanics, 1997, 8: 851-856.

Bing Yu, Ph.D.

--------------------------------

differentiating a signal strongly amplifies high frequency components of

the signal

(because that what's differentiating is all about). Look in textbooks of

signals analysis *) what happens in the frequency-domain and you'll

understand.

Clearly this procedure will worsen the signal to noise ratio.

There is nothing wrong in filtering the signal with a low pass filter,

to increase the signal -to noise ratio. The only thing you'll have to

know is that this procedure limits the bandwith of events you'll want to

interprete. It is for YOU to decide whether this might hide important

information, or just removes noise. (There is more to say, but this

requires a priori knowledge about the original signal).

So if you filter the signal each time with 6 Hz after differentiating,

no conclusions about components of velocity or acceleration above 6 Hz

can be drawn. This might not be a problem for a student class, but it is

clear that fast transients in position data contain high frequency

components in the accelaration. Again: this is the nature of the

problem.

Conclusion: filtering is OK as long as every statement about

accelerations etc. is done in the context of a specified bandwidth.

(Which should always be the case, from an electrical engineer's point of

view.)

*) I would recommend "designing digital filters" from Charles S.

Williams, [prentice-hall, 1986]

---------------------------

Theoretically, it is not right to filter again after each derivation. Becase

the high frequency components in the velocity and acceleration are natural

and from the actual movement, so that you can not simply throw them away. If

you do, then you will deal with three different movements represented by the

position, the velocity and the acceleration, respectively. We know that the

velocity should be 0 when the position shows an extreme value, and so on.

But by filtering the data repeatedly after each derivation, you would ruin

this relationships among the physical quantities.

So the rationale here would be: (a) choose the physical quantity (either

position, velocity or acceleration) you wish to filter, (b) filter this

particular quantity, and (c) do derivation or integration to obtain the

other quantities.

In this way, you are basically dealing with the same movement represented in

different forms by different mechanical quantities.

Young-Hoo Kwon, Ph.D.

Internet: y-hkwon@sports.re.kr

ykwon@wp.bsu.edu (English only)

--------------------------------------------------------------

I can't answer your question, but you could probably answer it yourself

by doing a numerical simulation (this would also probably make a nice

reference technical note for the biomechanics community).

Here is what you could try (if you felt so inclined):

* Come up with several unperturbered position signals which represent

typical movements (e.g., wrist movements in your case). Since you

know the real underlying position signal, you can easily determine

the actual corresponding velocity and acceleration signals (e.g.,

using analytic techniques or cubic interpolating splines).

* Contaminate the simulated position signals with appropriate noise

(you would have to determine what appropriate noise is). You would

probably have to generate 10 or more perturbed position signals for

each unperturbed position signal.

* Try your new multiple-filtering approach on the perturbed signals, and

see how well you reproduce the true velocity and acceleration data

relative to how well one can do with the traditional approach.

I don't know much about signal processing, so I will be curious to see

what other BIOMCH-L readers have to say in response to your question.

There is probably some good theoretical reason why your student's idea

would or would not be a good approach.

Benjamin J. Fregly, Ph.D.

E-mail: bj@sj.ptc.com

> ------------------------------

Filtering and differentiating are both linear operations (ie, it doesn't

matter which order you apply them in), so there's no benefit to refiltering

prior to the second differentiation (unless you want your velocity and

acceleration signals filtered differently). The net effect is of filtering

with a single filter of higher order and lower cutoff frequency (say ~ 4 Hz

- you'd have to check a Bode plot to be sure). Your student is right that

differentiating amplifies higher frequencies - you might want to check into

some form of cross-validation procedure for estimating accelerations.

Cheers,

Tony

----------------------------

Filters such as splines, Butterworth digital filters, etc. act as

linear operators. So does a finite-difference differentiation.

This means that you can interchange the order of these operations

without affecting the result.

So, if you apply the same filter again, after differentiating,

you would get the same result as if you had applied the filter

twice to the raw position data, before differentiating.

This would give you effectively a higher-order filter (though no

longer a Butterworth filter!), with a slightly lower cut-off

frequency.

In my opinion, this would only lead to confusion, and could also

be sub-optimal. Higher-order filters don't work well on signals

with a broad spectrum, and if you need a lower cut-off frequency

you can simply design the (first) filter that way, rather than

apply it twice.

So I would say, apply a filter only once, and design it so that

it gives you good results at the end of the analysis.

-- Ton van den Bogert

************************************************** ******************

Guy G. Simoneau, Ph.D., P.T., A.T.C.

Assistant Professor, Physical Therapy Department

Marquette University

Walter Schroeder Complex, Room 346

P.O. Box 1881

Milwaukee, WI 53201-1881

e-mail: simoneaug@vms.csd.mu.edu

Phone: 414-288-3380

Fax: 414-288-5987

************************************************** ******************