First some background for the Biomch-L subscriber. In the GCVSPL

software, VAL is the "smoothing parameter" of the spline, which occurs

as a weighting factor in a cost function which is minimized. This

produces a compromise between how well the spline fits the measurements

and its smoothness (for a more detailed explanation, see my notes in

http://isbweb.org/software/sigproc/bogert/filter.pdf).

Woltring describes the relationship between VAL and frequency domain

filter characteristics in his release notes:

http://isbweb.org/software/sigproc/gcvspl/gcvspl.memo. This states that

a smoothing spline is equivalent to a

"double, phase-symmetric

Butterworth filter, with transfer function H(w) = [1 +

(w/wo)^2M]^-1,

where w is the frequency, wo = (p*T)^(-0.5/M) the filter's

cut-off

frequency, p the smoothing parameter, T the sampling interval,

and 2M

the order of the spline. If T is expressed in seconds, the

frequen-

cies are expressed in radians/second."

The VAL calculations that you cited can be derived from this.

The equation for the transfer function has the property H(w0)= 0.5.

This is because it is a double Butterworth filter (applied twice).

Cut-off frequency is usually defined as the frequency at which

H=1/sqrt(2), and this is why Tony Reina and I have an extra factor in

the equation.

You can easily verify the correctness of your spline smoothing by

processing a sine wave signal at the cut-off frequency. If it comes out

as a sine wave with its amplitude reduced by a factor 1.41 (=sqrt(2)),

you have used the correct VAL. I have done this and it came out as

expected.

--

Ton van den Bogert

> -----Original Message-----

> From: * Biomechanics and Movement Science listserver

> [mailto:BIOMCH-L@NIC.SURFNET.NL] On Behalf Of Rettig, Oliver

> Sent: Friday, September 15, 2006 5:30 AM

> To: BIOMCH-L@NIC.SURFNET.NL

> Subject: [BIOMCH-L] Woltring Filter cutt-off-frequency

> VAL/P-value

>

> Dear all,

>

> we implement an algorithm for gait event detection.

>

> For smoothing and derivating the data we want to use the WOLTRING

> filter. We have found Tony Reina's Matlab MEX interface for the

> GCVSPL and SPLDER packages based on Dwight Meglan's C Code on the web

> address http://www.isbweb.org/software/sigproc.html.

>

> We have the problem to find the correct VAL value for the WOLTRING

> GCVSPL package.

> We found three approaches:

>

> 1. From dan@indy1.motorlab.nsi.eduSun

>

> VAL = sampling_freq/(2*PI*cut_off_freq)**(2*order)

>

> 2. from Ton van den Bogert

>

> w0factor = (sqrt(2.0)-1)**(0.5/m)

>

> val = 1.0/dt*(2*pi*f0/w0factor)**(-2*m)

>

> 3. Tony Reina

>

> VAL = (EMG-sampling_freqency/1000.0) /

> pow(2*PI*EMG_CUTOFF_FREQUENCY/1000.0/pow( (sqrt(2.0)-1),0.5/M),2.0*M)

>

> (including in EMG example)

>

>

> Our questions are:

>

> Do anybody know what is the correct equation for calculating the mode

> value VAL from given cut-off-frequeny?

>

> And why the factor 1000 appears in the equation from Tony Reina?

>

> Many thanks

>

> Kai Kraft

> Oliver Rettig

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