Power Spectral Density from jTransforms DoubleFFT_1D

Question

I\'m using Jtransforms java library to perform analysis on a given dataset.

An example of the data is as follows:

980,988,1160,1080,928,1068,1156,11         


        

Answer 1

I think you need to interpret the output data as follows:

10952       Re[0] = sum of all inputs = DC component
 -152       Re[5] - see note about a[1] being special - there is no Im[0]
   80.052   Re[1]
  379.936   Im[1]
 -307.691   Re[2]
   12.734   Im[2]
 -224.052   Re[3]
  427.607   Im[3]
  -48.308   Re[4]
   81.472   Im[4]

The magnitudes are therefore:

spectrum[0] = 10952
spectrum[1] = sqrt(80.052^2 + 379.936^2) = 388.278
spectrum[2] = sqrt(-307.691^2 + 12.734^2) = 307.427
spectrum[3] = sqrt(-224.052^2 + 427.607^2) = 482.749
spectrum[4] = sqrt(-48.308^2 + 81.472^2) = 94.717

[Sorry about there being two separate answers from me now - I think two related questions got merged while I was working on the new answer]

Answer 2

Each entry in the transform represent the (complex) magnitude of the frequency in the sample.

the power density in a given frequency is just the magnitude of the complex amplitude of the transform in that frequency. the magnitude of a complex number is computed from its components and you should not have a problem obtaining this

Each column represents amplitudes for increasing frequencies, starting from 0 (the first entry), then 2 Pi/T (where T is the length of your sample), until the last sample 2*Pi*N /T (where N is the number of samples)

there are other conventions where the transform is returned for the -Pi * N /T frequency up to Pi * N / T, and the 0 frequency component is in the middle of the array

hope this helps

Answer 3

To get the magnitude of bin k you need to calculate sqrt(re * re + im * im), wheer re, im are the real and imaginary components in the FFT output for bin k.

For your particular FFT re[k] = a[2*k] and im[k] = a[2*k+1]. Therefore to calculate the power spectrum:

for k in 0 to N/2 - 1
  spectrum[k] = sqrt(sqr(a[2*k]) + sqr(a[2*k+1]))