Changes between Version 10 and Version 11 of SpectraObject

Timestamp:

01/06/15 17:52:00 (11 years ago)

Author:

Jonathan

Comment:

—

SpectraObject

@@ -93 +93 @@
 The spectra object uses a parallelized version of the FFT function.  The 1D FFT calculates
 
-[[latex($F_j=\displaystyle \sum_{j'=1}^{j'=N_x} e^{  \frac{2 \pi i}{N_x} (j-1) (j'-1)} f_{j'}$)]]
-
+[[latex($F_l=\displaystyle \sum_{l'=1}^{N_x} e^{  \frac{2 \pi i}{N_x} (l-1) (l'-1)} f_{l'}$)]]
 
 Here we are using [[latex($j-1$)]] and [[latex($j'-1$)]] because indices in Fortran start at 1.
 
+Well 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$)]] we see that 
+
+[[latex($F(k_x)= \frac{N_x}{L}\displaystyle \int_{x=0}^{L} e^{ik_x x} f(x) dx$)]]
+
+The discrete FFT projects the function onto the basis set [[latex($\{e^{i (l-1)\Delta k}:l=1,N_x\}$)]]
+
+Here are the real parts of the continuous versions of those functions for N_x=10
+[[Image(Screen Shot 2015-01-06 at 3.31.14 PM.png​,width=600)]]
+
+And here is the real part of the discrete form of those same functions.  Note that there are only 6 lines visible!
+
+[[Image(Screen Shot 2015-01-06 at 3.08.22 PM.png​,width=600)]]
+
+
+The 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.  
+[[Image(Screen Shot 2015-01-06 at 3.38.19 PM.png​,width=600)]]
+
+What about the imaginary parts?
+
+[[Image(Screen Shot 2015-01-06 at 4.28.36 PM.png​,width=600)]]
+
+The imaginary parts are different - but only in sign!
+