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Fourier Transforms
Astro193 April 6, 2015


Fourier Transforms Applications

· · · ·

Filtering Convolution and Deconvolution Correlation and Autocorrelation Power Spectrum

RR Lyrae lightcurve and Fourier modes


Examples: Boxcar

1

2T

0

-T

0

T

0

t


/T
0

Time-bandwidth product:

Uncertainty principle: The shorter the signal the broader the spectrum.


Boxcar can filter
a low pass filter:

0/

1 -
0



t
0

/

0


Examples: Impulse
(t)

F()
constant

t



( )

F(t)

1/2 t



( )

shift

-

0



0

modulation


f(t) (-0) t -
0

F() (+0) +

0


Basic Fourier Theorems
· Integral

· Linearity


Basic Fourier Theorems
· Shift

Delay => Phase shift Amplitudes stay the same


Basic Fourier Theorems
· Similarity or Scale:

The uncertainty principle: The shorter the pulse the broader spectrum


Basic Fourier Theorems
· Symmetry
f(t) F()

interchange t &


Basic Fourier Theorems
· Derivative


Convolution

f - convolving pattern g - signal

Convolution Theorem