Mathematical groundwork I: Fourier theory Fundamentals of Radio Interferometry Dr. Gyula I. G. Józsa SKA-SA/Rhodes University NASSP 2016 Basic remarks ● Mathematics in this course is a requirement to understand and conduct interferometric imaging ● Interferometry is a nice field for the mathematically inclined, but required maths is manageable ● Mathematics is presented as tool, proofs partly not complete and used as an exercise to memorize the tool functions ● Principles presented here are fundamental to experimental physics, radio technology, informatics, image processing, theoretical physics etc. ● Unlike the last session, this one will not contain many pictures NASSP 2016 2:56 Important functions ● Some functions that will return over and over again are presented NASSP 2016 3:56 Important functions: Gaussian a: amplitude μ: mean σ: standard deviation Carl- Friedrich Gauß (1777- 1855) NASSP 2016 4:56 Important functions: Gaussian a: amplitude μ: mean σ: standard deviation FWHM (full width at half maximum): NASSP 2016 5:56 Important functions: Gaussian a: amplitude μ: mean σ: standard deviation Area below Gaussian: NASSP 2016 6:56 Important functions: Normalised Gaussian μ: mean σ: standard deviation NASSP 2016 7:56 Important functions: Gaussian 2d NASSP 2016 8:56 Important functions: Gaussian 2d NASSP 2016 9:56 Important functions: Boxcar function NASSP 2016 10:56 Important functions: Rectangle function NASSP 2016 11:56 Important functions: Sinc NASSP 2016 12:56 Important functions: Dirac's Delta ● Not a function but a distribution ● In many ways a function... It follows: Paul Dirac (1902- 1984) NASSP 2016 13:56 Important functions: Dirac's Delta ● More precisely, the Delta function is defined as the “limit” of a suitable series Find set of functions with δα with Then define δ through the integral In this sense: NASSP 2016 14:56 Important functions: Dirac's Delta ● More precisely, the Delta function is defined as the “limit” of a suitable series Find set of functions with δα with Then define δ through the integral In this sense: NASSP 2016 15:56 Important functions: Dirac's Delta ● More precisely, the Delta function is defined as the “limit” of a suitable series ● Two important relations: NASSP 2016 16:56 Important functions: Sha (Shah) or comb function Relations: NASSP 2016 17:56 The Fourier transform ● Definition of the Fourier transform: ● Inverse Fourier transform (via the Fourier inversion theorem): Jean-Baptiste Joseph Fourier (1768 - 1830) NASSP 2016 18:56 The Fourier transform ● Definition of the Fourier transform: ● The Fourier tansform can be seen as decomposition of a function into a wave package Leonard Euler (1707 - 1783) NASSP 2016 19:56 The Fourier transform ● Notation: functions in the “Fourier space” are named by capital letters ● The inverse Fourier transform is the Fourier transform of the reverse function (an inverse Fourier transform is hence a triple forward Fourier transform) NASSP 2016 20:56 The Fourier transform of a Gaussian ● The Fourier transform of a Gaussian with dispersion σx is ... NASSP 2016 21:56 The Fourier transform of a Gaussian ● The Fourier transform of a Gaussian with dispersion σx is a Gaussian with -1 dispersion σs=(2πσx) NASSP 2016 22:56 The Fourier transform of a Delta function ● The Fourier transform of a Delta function is ... NASSP 2016 23:56 The Fourier transform of a Delta function ● The Fourier transform of a Delta function is a sinusoid in the real and the imaginary part, a wave NASSP 2016 24:56 The Fourier transform of a comb function ● The Fourier transform of a sha function with period T is ... NASSP 2016 25:56 The Fourier transform of a comb function ● The Fourier transform of a sha function with period T is a sha function with period T-1 NASSP 2016 26:56 The Fourier transform of a rectangle function ● The Fourier transform of a rectangle function is ... NASSP 2016 27:56 The Fourier transform of a rectangle function ● The Fourier transform of a rectangle function is the sinc function! NASSP 2016 28:56 The Fourier transform of a real-valued function ● The Fourier transform of a real-valued function is a Hermetian function and vice versa Hermetian means: Real-valued means: NASSP 2016 29:56 The n-dimensional Fourier transform ● The n-dimensional Fourier transformation and its inverse is defined as NASSP 2016 30:56 The Convolution ● The convolution o is the mutual broadening of one function with the other ● Mathematical equivalent of an instrumental broadening or “filtering” NASSP 2016 31:56 The Convolution ● The convolution o is the mutual broadening of one function with the other ● Mathematical equivalent of an instrumental broadening or “filtering” NASSP 2016 32:56 The Convolution: rules NASSP 2016 33:56 The Convolution: examples NASSP 2016 34:56 The Convolution: examples NASSP 2016 35:56 The Convolution: examples NASSP 2016 36:56 Cross-correlation NASSP 2016 37:56 Cross-correlation NASSP 2016 38:56 Auto-correlation NASSP 2016 39:56 Fourier transform properties: Linearity and separability NASSP 2016 40:56 Fourier transform properties: Shift theorem Proof: NASSP 2016 41:56 Fourier transform properties: Shift theorem Proof: NASSP 2016 42:56 Fourier transform properties: Convolution theorem NASSP 2016 43:56 Fourier transform properties: Convolution theorem Proof: NASSP 2016 44:56 Fourier transform properties: Convolution theorem Proof: NASSP 2016 45:56 Fourier transform properties: Crosscorrelation theorem Proof: NASSP 2016 46:56 Fourier transform properties: Autocorrelation theorem ● Also: Wiener-Khinchin Theorem Proof: ● Just a special case of the cross-correlation theorem NASSP 2016 47:56 The discrete Fourier transform: definition ● Discrete Fourier transform ● Inverse discrete Fourier transform ● Numerical methods exist to make the expensive FT faster (“Fast FT”) NASSP 2016 48:56 The discrete Fourier transform: Inverse NASSP 2016 49:56 The discrete Fourier transform and the Fourier transform ● Sample a function with the sampling function s in N regularly spaced steps over the interval NASSP 2016 50:56 The discrete Fourier transform and the Fourier transform ● Fourier-transform the sampled function ● Define the set ● With the discrete FT: ● We see that if we define NASSP 2016 51:56 Nyquist's sampling theorem ● Consider a real-valued wave package with a frequency cutoff at Harry Nyquist (1889 – 1976) ● The Fourier transform has the support and it follows NASSP 2016 52:56 Nyquist's sampling theorem ● In an experiment we sample the function with the sampling period Δx NASSP 2016 53:56 Nyquist's sampling theorem ● In an experiment we sample the function with the sampling period Δx ● The Fourier transform repeats itself, it is aliased. ● If we sample a function with the bandwidth , the sampling interval has to fulfil the condition NASSP 2016 54:56 Limited sampling ● The thought experiment is not yet realistic. We can only measure for a limited number of samples ● It follows: ● The Fourier transform is hence always filtered with a sinc function, which gets narrower with increasing number of samples NASSP 2016 55:56 Literature ● Bracewell, R. N. (1999), The Fourier Transform and Its Applications. McGraw-Hill ● Briggham, E. O. (1988) The Fast Fourier Transform and Its Applications. (Pearson Education U.S.) ● Briggs, W. L., Henson, V. E. (1995) The DFT: An Owners' Manual for the Discrete Fourier Transform (Society for Industrial & Applied Mathematics, U.S.) NASSP 2016 56:56.