Changes between Version 15 and Version 16 of SpectraObject
- Timestamp:
- 01/13/15 10:59:00 (10 years ago)
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SpectraObject
v15 v16 101 101 [[latex($F_{k+N_x}=\displaystyle \sum_{x=1}^{N_x} e^{ \frac{2 \pi i}{N_x} \left(k+N_x\right) x} f_{x} = \displaystyle \sum_{x=1}^{N_x} e^{2 \pi i x} \times e^{ \frac{2 \pi i}{N_x} k x} f_{x}=\displaystyle \sum_{x=1}^{N_x} e^{ \frac{2 \pi i}{N_x} kx} f_{x} =F_k$)]] 102 102 103 We can visually see this by plotting 103 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($N_x=10$)]] 104 104 105 [[Image(Screen Shot 2015-01- 06 at 3.38.19 PM.png​,width=600)]]105 [[Image(Screen Shot 2015-01-13 at 10.49.34 AM.png​​,width=600)]] 106 106 107 107 108 Now the DFT returns the array of transforms with [[latex($k=[0,1,2,...,N_x-1]$)]] 109 110 [[latex($F_k=\displaystyle \sum_{x=1}^{N_x} e^{ \frac{2 \pi i}{N_x} k x} f_{x}$)]] 111 112 but we can interpret [[latex($F_N 108 113 109 114 so while we can calculate the transform for any k, only N of them will be unique.
