R help - Apr 2013 - How to select the scale parameter for Gabor transform (Rwave)?

Dear list,

I am trying to choose the scale parameter for the cgt transform but I don't
know how to do it. In time I would like to be able to separate points 30
samples apart, and in frequency I would like to separate bands 0.04 Hz
apart. I tried the two approaches described below and they gave me
different results.  I would appreciate advise on how to do this.

The Rwave Gabor transform uses a Gaussian window; i.e.,
g(x)=exp(-x^2/(2*sigma)), with radius deltaG=sqrt(sigma/2). The Fourier
transform of this window has a radius deltaGHat=1/sqrt(2*sigma). Solving
for deltaG=10 I get sigma=200, which in turn gives 
deltaGHat=0.05 radians=0.008 Hz. So, sigma=200 should satisfy the 
requirements above.

To make sure sigma=200 works in practice, I computed the Gabor transform of
two sinusoids separated by 0.04 Hz plus two spikes separated by 30 points
using the code below. I obtained an excellent frequency resolution but a
disastrous temporal resolution
http://sccn.ucsd.edu/~rapela/scaleSelectionProblem/gtResultSigma200.jpg. To
obtain decent frequency and temporal resolutions I used sigma=10
http://sccn.ucsd.edu/~rapela/scaleSelectionProblem/gtResultSigma10.jpg.

    fromTime <- 0
    toTime <- 250
    by <- 1
    ts <- seq(from=fromTime, to=toTime)
    f1 <- 0.01
    f2 <- 0.05
    t1 <- 50
    t2 <- 80
    K <- 15
    fMax <- 0.20
    nf <- 100
    df <- fMax/(nf/2)
    scale <- 10

    ys <- sin(2*pi*f1*ts) + sin(2*pi*f2*ts)
    ys[t1] <- K
    ys[t2] <- K

    plot(x=ts, y=ys, xlab="Time", ylab="Signal",
type="b")
    cgtsinwave <- cgt(input=ys, nvoice=nf, freqstep=df, scale=scale,
plot=TRUE)

Many thanks, Joaquin

-- 
Joaquin Rapela, PhD
Swartz Center for Computational Neuroscience
University of California San Diego
9500 Gilman Drive,
San Diego, CA 92093-0559
tel: (858) 822-7536
fax: (858) 822-7556
http://sccn.ucsd.edu/~rapela

Dear list,

It seems that there is a typo in the book "Practical Time-Frequency
Analysis"
that documents Rwave. The window that the cgt function of Rwave seems to be 
using 

g(x)=exp(-x^2/(2*sigma^2))

and not

g(x)=exp(-x^2/(2*sigma))

as stated in Equation (3.23) of the book. Also, it appears that the scale
parameter in the cgt of Rwave equals sigma. In the previous email I derived an
optimal sigma=200, and if in the code below I use sigma=sqrt(200), I obtain a 
Gabor transform with good time-frequency resolution.

I am copying the authors of Rwave in this message. If the previous observation
is correct, it would be a good idea to discuss it in the documentation of Rwave.

Cordially, Joaquin

scale
On Fri, Apr 19, 2013 at 02:36:56PM -0700, Joaquin Rapela
wrote:
> 
> Dear list,
> 
> I am trying to choose the scale parameter for the cgt transform but I
don't
> know how to do it. In time I would like to be able to separate points 30
> samples apart, and in frequency I would like to separate bands 0.04 Hz
> apart. I tried the two approaches described below and they gave me
> different results.  I would appreciate advise on how to do this.
> 
> The Rwave Gabor transform uses a Gaussian window; i.e.,
> g(x)=exp(-x^2/(2*sigma)), with radius deltaG=sqrt(sigma/2). The Fourier
> transform of this window has a radius deltaGHat=1/sqrt(2*sigma). Solving
> for deltaG=10 I get sigma=200, which in turn gives 
> deltaGHat=0.05 radians=0.008 Hz. So, sigma=200 should satisfy the 
> requirements above.
> 
> To make sure sigma=200 works in practice, I computed the Gabor transform of
> two sinusoids separated by 0.04 Hz plus two spikes separated by 30 points
> using the code below. I obtained an excellent frequency resolution but a
> disastrous temporal resolution
> http://sccn.ucsd.edu/~rapela/scaleSelectionProblem/gtResultSigma200.jpg. To
> obtain decent frequency and temporal resolutions I used sigma=10
> http://sccn.ucsd.edu/~rapela/scaleSelectionProblem/gtResultSigma10.jpg.
> 
>     fromTime <- 0
>     toTime <- 250
>     by <- 1
>     ts <- seq(from=fromTime, to=toTime)
>     f1 <- 0.01
>     f2 <- 0.05
>     t1 <- 50
>     t2 <- 80
>     K <- 15
>     fMax <- 0.20
>     nf <- 100
>     df <- fMax/(nf/2)
>     scale <- 10
> 
>     ys <- sin(2*pi*f1*ts) + sin(2*pi*f2*ts)
>     ys[t1] <- K
>     ys[t2] <- K
> 
>     plot(x=ts, y=ys, xlab="Time", ylab="Signal",
type="b")
>     cgtsinwave <- cgt(input=ys, nvoice=nf, freqstep=df, scale=scale,
plot=TRUE)
> 
> Many thanks, Joaquin
> 
> -- 
> Joaquin Rapela, PhD
> Swartz Center for Computational Neuroscience
> University of California San Diego
> 9500 Gilman Drive,
> San Diego, CA 92093-0559
> tel: (858) 822-7536
> fax: (858) 822-7556
> http://sccn.ucsd.edu/~rapela
> 
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
-- 
Joaquin Rapela, PhD
Swartz Center for Computational Neuroscience
University of California San Diego
9500 Gilman Drive,
San Diego, CA 92093-0559
tel: (858) 822-7536
fax: (858) 822-7556
http://sccn.ucsd.edu/~rapela
----------------------------------

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