FFT Amplitude Scaling

The relative amplitude of a sideband follows the relative amplitude of the peak in the FFT.



In [1]:

    
from numpy.fft import *



In [2]:

    
import numpy



In [3]:

    
t = numpy.arange(0,100)



In [4]:

    
data = 5*numpy.sin(t) + 3*numpy.sin(0.5*t)



In [5]:

    
%pylab inline









    



Populating the interactive namespace from numpy and matplotlib






    



WARNING: pylab import has clobbered these variables: ['fft', 'bench', 'test', 'info']
`%matplotlib` prevents importing * from pylab and numpy



In [6]:

    
plot(data)









    Out[6]:





[<matplotlib.lines.Line2D at 0x7f68c820af50>]



In [7]:

    
fft_out = abs(fft.fft(data))



In [8]:

    
fft_out.max()









    Out[8]:





247.96434997097225



In [9]:

    
plot(fft_out)









    Out[9]:





[<matplotlib.lines.Line2D at 0x7f68c8172590>]



In [20]:

    
from scipy.signal import find_peaks_cwt



In [32]:

    
peak_ind = find_peaks_cwt(fft_out, numpy.arange(1,3))



In [33]:

    
fft_out[peak_ind]









    Out[33]:





array([   1.45920534,    1.46277576,  247.96434997,   21.86061547,
          8.34397026,  147.79532401,  247.96434997,   20.09975115,
          1.46277576])



In [34]:

    
247/147.0









    Out[34]:





1.6802721088435375



In [35]:

    
5/3.0









    Out[35]:





1.6666666666666667

Note, this also holds for interference between waves of different amplitudes. I've verified this with beam simulations.

FFT Frequency Scaling

Off-axis component of the wave is $k_x = k \sin\theta = \frac{2\pi}{780 nm}\sin\theta$



In [39]:

    
K_x = 2*pi/780e-9 * numpy.sin(0.005) # in radians per meter

The step size $\Delta k$ is given by $\frac{2\pi}{(20\mu m)(1340)}$ based on pixel size and pixel count



In [38]:

    
delta_K = 2*pi/(20e-6*1340) # deltaK in radians per meter

So the horizontal axis scale should reflect these values. Note, they will not change with the setup so we can hard-code them into our graphing routines.

For example, if a peak is found with index 166 in the FFT output, that corresponds to:



In [42]:

    
166*delta_K # radians per meter









    Out[42]:





38918.23735044072

Conclusion

Scale the horizontal and vertical axes accordingly: horizontal is in units of $\Delta K$ and vertical should be normalized to the DC component on a log scale



In [ ]: