fourier transforms cass 2015 · the fourier transform and its applications by ronald n....
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Fourier TheoryCASS Radio School
ASTRONOMY & SPACE SCIENCE
Joshua Marvil | OCE Postdoctoral Fellow 29 September, 2015
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This talk is based on:!!The Fourier transform and its applications by Ronald N. Bracewell !!!Essential Radio Astronomy Course by J.J. Condon and S.M. Ransom
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Fourier Transforms are deeply ingrained in many aspects of radio interferometryAntenna’s primary beam Array’s synthesized beam Frequency downconversion Correlation, coherence & visibility Convolution & deconvolution !!
The goal of this talk is to reinforce key theoretical concepts in order to help you better understand applications of Fourier theory
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A generalized series expansion of a function based on the special properties of a set of basis functions
!!
Fourier series
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For a periodic function f(x) = f(x+T), choose sines and cosines with frequencies that are integer multiples of 1/T
Additional terms refine the approximation
Fourier series
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This works very well for smooth functions
Discontinuities cause ringing (Gibbs phenomenon)
Fourier series
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Fourier series in practice
Solve for coefficients
and construct the series
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Fourier seriesFourier series in practice
Square wave solution
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Decompose any function into even and odd parts
Aside: even and odd functions
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Aside: Euler’s formula
Recast the Fourier series using complex notation
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Generalize the complex Fourier series to work with non-‐periodic functions
The Fourier Transform: Definitions
Take the limit where where the period infinity
The discrete sum becomes a continuous function
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The ‘reverse’ transform: !
The ‘forward’ transform: !
The Fourier Transform: Definitions
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The Fourier Transform is a reversible, linear transform between domains, e.g. x and s, where the product of x and s is dimensionless !!!
The Fourier Transform: Symmetry
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real Hermitian
imaginary anti-‐Hermetian
even even
odd odd
real and even real and even
imaginary and even imaginary and even
real and odd imaginary and odd
imaginary and odd real and odd
The Fourier Transform: Symmetry
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Linearity
then
If
The Fourier Transform: Properties
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Similarity theorem
then
If
or equivalently
The Fourier Transform: Properties
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Addition theorem
If
then
The Fourier Transform: Properties
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Shift theorem
If
then
The Fourier Transform: Properties
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Modulation theorem
If
then
The Fourier Transform: Properties
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The Fourier Transform: Simple Functions
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The Fourier Transform: Simple Functions
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The Fourier Transform: Simple Functions
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The Fourier Transform: Simple Functions
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Discrete Fourier Transform
So far we have been dealing with a continuous function f(x) For a set of N uniformly sampled data we can evaluate the DFT:
Produces N independent bins
Fourier transform properties and symmetries apply
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The Fast Fourier Transform (FFT)
-‐ a family of algorithms to increase the speed of calculating a DFT !-‐ reduces the number of computations from O(N2) to O(N logN) !-‐ recursive divide and conquer by re-‐using intermediate calculations
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The Fast Fourier Transform (FFT)
-‐ interferometric images are typically made using an FFT of the gridded visibilities !-‐ performance improves when the grid size is highly factorable !-‐ see also: FFTPACK, FFTW
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Convolution
The function f is multiplied by the time-‐reversed kernel g
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Fourier Transforms are deeply ingrained in many aspects of radio interferometryAntenna’s primary beam Array’s synthesized beam Frequency downconversion Correlation, coherence & visibility Convolution & deconvolution !!
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Applications: Array synthesized beam
u-‐v coverage PSF
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Applications: Primary beam pattern
1-‐d aperture
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Applications: Primary beam pattern
2-‐d aperture
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aperture illumination
Applications: Primary beam pattern
antenna power pattern
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van Cittert-‐Zernike Theorem
dmdlemlBmlAvuV mvuli .),(),(),( )(2∫∫ +−= π
B(l,m) := sky brightness in direction l,m A(l,m) := antenna reception pattern
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van Cittert-‐Zernike Theorem
dmdlemlBmlAvuV mvuli .),(),(),( )(2∫∫ +−= π
B(l,m) := sky brightness in direction l,m A(l,m) := antenna reception pattern
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Applications: Convolution
Output = Input ✶ Impulse Response
Input Output
Interferometer
Impulse Response = Point Spread Function
With knowledge of the PSF we can undo the!convolution and estimate the original input
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CSIRO Astronomy & Space Science Joshua Marvil | OCE Postdoctoral Fellow t +61 2 9372 4329 e josh.marvil @ csiro.au
Thank you