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From: iframe on 14 Jun 2010 07:34
Hi,

I'm writing a program, in awk, for doing periodogram analysis of time
series.
The Fourier coefficients are calculated according to:

a[k] += (2/n) * y[t] * cos(f)
b[k] += (2/n) * y[t] * sin(f)

with f= (2*pi)*k*t/n
n = number of samples (odd number)
k = (n-1)/2

an thus the periodogram is given by:

I[k] = ((a[k]^2) + (b[k]^2))*n/4

My question is, if I calculate the periodogram with k an integer I get the
"regular" periodogram, but if I let k take floating point values, e.g. 1.1,
1.2, 1.3, ..., (n-1)/2, there will be other lines that were not present
when k was allowed to take integral values only, sometimes the periodogram
shows lines with a higher intensity. What is the significance of letting k
take values that are not integer?


From: Rune Allnor on 14 Jun 2010 08:00
On 14 Jun, 13:34, "iframe" <i.frame.0(a)n_o_s_p_a_m.gmail.com> wrote:
> Hi,
>
> I'm writing a program, in awk, for doing periodogram analysis of time
> series.
> The Fourier coefficients are calculated according to:
>
> a[k] += (2/n) * y[t] * cos(f)
> b[k] += (2/n) * y[t] * sin(f)
>
> with f= (2*pi)*k*t/n
>      n = number of samples (odd number)
>      k = (n-1)/2
>
> an thus the periodogram is given by:
>
> I[k] = ((a[k]^2) + (b[k]^2))*n/4
>
> My question is, if I calculate the periodogram with k an integer I get the
> "regular" periodogram, but if I let k take floating point values, e.g. 1.1,
> 1.2, 1.3, ..., (n-1)/2, there will be other lines that were not present
> when k was allowed to take integral values only, sometimes the periodogram
> shows lines with a higher intensity. What is the significance of letting k
> take values that are not integer?

No.

Rune
From: Jason on 14 Jun 2010 08:21
On Jun 14, 7:34 am, "iframe" <i.frame.0(a)n_o_s_p_a_m.gmail.com> wrote:
> Hi,
>
> I'm writing a program, in awk, for doing periodogram analysis of time
> series.
> The Fourier coefficients are calculated according to:
>
> a[k] += (2/n) * y[t] * cos(f)
> b[k] += (2/n) * y[t] * sin(f)
>
> with f= (2*pi)*k*t/n
>      n = number of samples (odd number)
>      k = (n-1)/2
>
> an thus the periodogram is given by:
>
> I[k] = ((a[k]^2) + (b[k]^2))*n/4
>
> My question is, if I calculate the periodogram with k an integer I get the
> "regular" periodogram, but if I let k take floating point values, e.g. 1.1,
> 1.2, 1.3, ..., (n-1)/2, there will be other lines that were not present
> when k was allowed to take integral values only, sometimes the periodogram
> shows lines with a higher intensity. What is the significance of letting k
> take values that are not integer?

All this does is shift the frequencies corresponding to the centers of
each bin that you are calculating. It doesn't accomplish anything that
you couldn't do by just multiplying the signal by a complex sinusoid
to shift its frequency before you perform the FT calculation. It won't
provide any additional information relative to what you're already
calculating.

Jason
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