Changes between Version 10 and Version 11 of SpectraObject


Ignore:
Timestamp:
01/06/15 17:52:00 (10 years ago)
Author:
Jonathan
Comment:

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  • SpectraObject

    v10 v11  
    9393The spectra object uses a parallelized version of the FFT function.  The 1D FFT calculates
    9494
    95 [[latex($F_j=\displaystyle \sum_{j'=1}^{j'=N_x} e^{  \frac{2 \pi i}{N_x} (j-1) (j'-1)} f_{j'}$)]]
    96 
     95[[latex($F_l=\displaystyle \sum_{l'=1}^{N_x} e^{  \frac{2 \pi i}{N_x} (l-1) (l'-1)} f_{l'}$)]]
    9796
    9897Here we are using [[latex($j-1$)]] and [[latex($j'-1$)]] because indices in Fortran start at 1.
    9998
     99Well if we make the substitutions [[latex($x=(l'-1) \Delta x$)]], [[latex($N_x=\frac{L_x}{\Delta x}$)]] and [[latex($k_x=(l-1)\frac{2\pi}{L}=(l-1) \Delta k $)]] and take the limit as [[latex($\Delta X \rightarrow 0$)]] we see that
     100
     101[[latex($F(k_x)= \frac{N_x}{L}\displaystyle \int_{x=0}^{L} e^{ik_x x} f(x) dx$)]]
     102
     103The discrete FFT projects the function onto the basis set [[latex($\{e^{i (l-1)\Delta k}:l=1,N_x\}$)]]
     104
     105Here are the real parts of the continuous versions of those functions for N_x=10
     106[[Image(Screen Shot 2015-01-06 at 3.31.14 PM.png​,width=600)]]
     107
     108And here is the real part of the discrete form of those same functions.  Note that there are only 6 lines visible!
     109
     110[[Image(Screen Shot 2015-01-06 at 3.08.22 PM.png​,width=600)]]
     111
     112
     113The real part of the discrete function for l = 2 and l = 10 are coincident! As are 3 and 9, 4 and 8, 5 and 7. 
     114[[Image(Screen Shot 2015-01-06 at 3.38.19 PM.png​,width=600)]]
     115
     116What about the imaginary parts?
     117
     118[[Image(Screen Shot 2015-01-06 at 4.28.36 PM.png​,width=600)]]
     119
     120The imaginary parts are different - but only in sign!
     121