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Fourier Theory
CASS Radio School
Joshua Marvil | OCE Postdoctoral Fellow 29 September, 2015
ASTRONOMY & SPACE SCIENCE


This talk is based on:
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The Fourier transform and its applications



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by Ronald N. Bracewell

Essential Radio Astronomy Course
by J.J. Condon and S.M. Ransom




Fourier Transforms are deeply ingrained in many aspects of radio interferometry
Antenna's primary beam Array's synthesized beam Frequency downconversion Correlation, coherence & visibility Convolution & deconvolution
! The goal of this talk is to reinforce key th!eoretical concepts in order to help you better understand applications of Fourier theory


Fourier series
A generalized series expansion of a function based on the special properties of a set of basis functions
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Fourier series
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
This works very well for smooth functions Discontinuities cause ringing (Gibbs phenomenon)


Fourier series in practice
Solve for coefficients

and construct the series


Fourier series in practice
Square wave solution


Aside: even and odd functions
Decompose any function into even and odd parts


Aside: Euler 's formula

Recast the Fourier series using complex notation


The Fourier Transform: Definitions
Generalize the complex Fourier series to work with non-
periodic functions Take the limit where where the period infinity The discrete sum becomes a continuous function


The Fourier Transform: Definitions
The `forward' transform:
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The `reverse' transform:
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The Fourier Transform: Symmetry
The Fourier Transform is a reversible, linear transform between domains, e.g. x and s, where the product of x and s is dimensionless
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The Fourier Transform: Symmetry
real imaginary even odd real and even imaginary and even real and odd imaginary and odd Hermitian anti-
Hermetian even odd real and even imaginary and even imaginary and odd real and odd


The Fourier Transform: Properties
Linearity

If then


The Fourier Transform: Properties
Similarity theorem

If then or equivalently


The Fourier Transform: Properties
Addition theorem

If then


The Fourier Transform: Properties
Shift theorem

If then


The Fourier Transform: Properties
Modulation theorem

If then


The Fourier Transform: Simple Functions


The Fourier Transform: Simple Functions


The Fourier Transform: Simple Functions


The Fourier Transform: Simple Functions


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


The Fast Fourier Transform (FFT)
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a family of algorithms to increase the speed of calculating a DFT
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reduces the number of computations from O(N2) to O(N logN)
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recursive divide and conquer by re-
using intermediate calculations


The Fast Fourier Transform (FFT)
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interferometric images are typically made using an FFT of the gridded visibilities
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performance improves when the grid size is highly factorable
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see also: FFTPACK, FFTW


Convolution
The function f is multiplied by the time-
reversed kernel g


Fourier Transforms are deeply ingrained in many aspects of radio interferometry
Antenna's primary beam Array's synthesized beam Frequency downconversion Correlation, coherence & visibility Convolution & deconvolution
! !


Applications: Array synthesized beam

u-
v coverage

PSF


Applications: Primary beam pattern

1-
d aperture


Applications: Primary beam pattern

2-
d aperture


Applications: Primary beam pattern

aperture illumination

antenna power pattern


van Cittert-
Zernike Theorem

V (u, v) =



A(l , m) B(l , m)e

-2i (ul +mv )

dl.dm

B(l,m) := sky brightness in direction l,m A(l,m) := antenna reception pattern


van Cittert-
Zernike Theorem

V (u, v) =



A(l , m) B(l , m)e

-2i (ul +mv )

dl.dm

B(l,m) := sky brightness in direction l,m A(l,m) := antenna reception pattern


Applications: Convolution

Interferometer Input Output

Output = Input Impulse Response Impulse Response = Point Spread Function With knowledge of the PSF we can undo the! convolution and estimate the original input


Thank you
CSIRO Astronomy & Space Science Joshua Marvil | OCE Postdoctoral Fellow t +61 2 9372 4329 e josh.marvil @ csiro.au