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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