Changes between Version 16 and Version 17 of SpectraObject

Timestamp:

01/13/15 12:43:53 (11 years ago)

Author:

Jonathan

Comment:

—

SpectraObject

@@ -91 +91 @@
 
 == What does the Spectra calculate? ==
-The Spectra performs a DFT on whatever fields  are assigned to it.  The discrete FFT is defined as
+ First we must understand what the FFT calculates
 
-[[latex($F_k=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} k x} f_{x}$)]]
+=== What does the FFT calculate ===
+The FFT performs a Discrete Fourier Transform (DFT) which is defined as
+
+[[latex($F_k=\displaystyle \sum_{x=1}^{N} e^{  \frac{2 \pi i}{N} k x} f_{x}$)]]
 
 where [[latex($k$)]] is any integer.  
 
-Now [[latex($F_k$)]] is periodic in [[latex($k$)]] since
+We can see how this corresponds to a real fourier transform by making the substitutions [[latex($x'=x \Delta x$)]], [[latex($k'=k \Delta k $)]], [[latex($\Delta k = \frac{2 \pi}{L}$)]]  and [[latex($N=\frac{L}{\Delta x}$)]] and take the limit as [[latex($\Delta x \rightarrow 0$)]]  
+
+[[latex($F(k')= \frac{N}{L}\displaystyle \int_{x'=0}^{L} e^{ik' x'} f(x') dx$)]]
+
+So that the DFT corresponds to a fourier transform where the physical wavelength is normalized by [[latex($\Delta k=\frac{2 \pi}{L}$)]] and the magnitude is normalized by [[latex($N/L$)]]
+
+
+Now the FFT returns the array of transforms with [[latex($k=[0,1,2,...,N_x-1]$)]]
+
+However, [[latex($F_k$)]] is periodic in [[latex($k$)]] since
 
 [[latex($F_{k+N_x}=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} \left(k+N_x\right) x} f_{x} = \displaystyle \sum_{x=1}^{N_x} e^{2 \pi i x} \times e^{  \frac{2 \pi i}{N_x} k x} f_{x}=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} kx} f_{x}  =F_k$)]]
 
-We can visually see this by plotting the discrete sampling of the real part of the continuous functions for [[latex($k=1$)]] and [[latex($k=11$)]] for a grid where [[latex($N_x=10$)]] 
+We can visually see this by plotting the discrete sampling of the real part of the continuous functions for [[latex($k=-1$)]] and [[latex($k=9$)]] for a grid where [[latex($N_x=10$)]] 
 
-[[Image(Screen Shot 2015-01-13 at 10.49.34 AM.png​​,width=600)]]
+[[Image(Screen Shot 2015-01-13 at 10.57.02 AM.png​​,width=600)]]
 
+So we can interpret [[latex($F_9$)]] as representing a signal with wavenumber $9$ or $-1$, or $-11$.  Since we don't expect there to be signals in our data at frequencies above the Nyquist frequency - it is logical to assume that [[latex($F_9$)]] corresponds to a signal with [[latex($k=-1$)]]. 
 
-Now the DFT returns the array of transforms with [[latex($k=[0,1,2,...,N_x-1]$)]]
+Mathematically this means that if [[latex($k > N/2$)]] then  [[latex($k \rightarrow k-N$)]] or equivalently  [[latex($k \rightarrow \left [ \left ( k+N/2 \right ) \bmod N \right ]- N/2 $)]]
 
-[[latex($F_k=\displaystyle \sum_{x=1}^{N_x} e^{  \frac{2 \pi i}{N_x} k x} f_{x}$)]]
+This is why we call the resulting FFT array to be in wrap around order.   That is, the array returned by the FFT corresponds to normalized wavenumbers of $$[0,1,2,...,N/2,-N/2+1,-N/2+2,...,-2,-1]$$
 
-but we can interpret [[latex($F_N
+Or if we index the FFT array starting at 1, and then plot the index of each component on the corresponding grid in k space we get
 
-so while we can calculate the transform for any k, only N of them will be unique.
-
-[[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\}$)]]
-
-Here are the real parts of the continuous versions of those functions for N_x=10
-
-[[Image(Screen Shot 2015-01-06 at 3.31.14 PM.png​,width=600)]]
-
-And here is the real part of the discrete form of those same functions.  Note that there are only 6 lines visible!
-
-[[Image(Screen Shot 2015-01-06 at 3.08.22 PM.png​,width=600)]]
-
-
-The real part of the discrete function for l = 2 and l = 10 are coincident! As are 3 and 9, 4 and 8, 5 and 7.  
-
-
-What about the imaginary parts?
-
-[[Image(Screen Shot 2015-01-06 at 4.28.36 PM.png​,width=600)]]
-
-The imaginary parts are different - but only in sign!
-
-We can see how this happens if we compare [[latex($F_l$)]] and [[latex($F_{N_x+2-l}$)]]
-
-[[latex($F_{N_x+2-l}=\displaystyle \sum_{l'=1}^{N_x} e^{  \frac{2 \pi i}{N_x} (N_x+2-l-1) (l'-1)} f_{l'}=\displaystyle \sum_{l'=1}^{N_x} e^{  \frac{2 \pi i}{N_x} ((-(l-1)) (l'-1)} f_{l'}=F_l^*$)]]
-
-but [[latex($F_{N_x+2-l}=\displaystyle \sum_{l'=1}^{N_x} e^{  \frac{2 \pi i}{N_x} ((-(l-1)) (l'-1)} f_{l'}$)]] 
-
-corresponds to [[latex($F(-k_x)= \frac{N_x}{L}\displaystyle \int_{x=0}^{L} e^{i-k_x x} f(x) dx$)]]