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Mathematical groundwork I: Fourier theory

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1 Mathematical groundwork I: Fourier theory
Fundamentals of Radio Interferometry Dr. Gyula I. G. Józsa SKA-SA/Rhodes University NASSP 2016

2 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

3 Important functions Some functions that will return over and over again are presented NASSP 2016

4 Carl- Friedrich Gauß (1777- 1855)
Important functions: Gaussian a: amplitude μ: mean σ: standard deviation Carl- Friedrich Gauß ( ) NASSP 2016

5 Important functions: Gaussian
a: amplitude μ: mean σ: standard deviation FWHM (full width at half maximum): NASSP 2016

6 Important functions: Gaussian
a: amplitude μ: mean σ: standard deviation Area below Gaussian: NASSP 2016

7 Important functions: Normalised Gaussian
μ: mean σ: standard deviation NASSP 2016

8 Important functions: Gaussian 2d
NASSP 2016

9 Important functions: Gaussian 2d
NASSP 2016

10 Important functions: Boxcar function
NASSP 2016

11 Important functions: Rectangle function
NASSP 2016

12 Important functions: Sinc
NASSP 2016

13 Important functions: Dirac's Delta
Not a function but a distribution In many ways a function... It follows: Paul Dirac ( ) NASSP 2016

14 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 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

16 Important functions: Dirac's Delta
More precisely, the Delta function is defined as the “limit” of a suitable series Two important relations: NASSP 2016

17 Important functions: Sha (Shah) or comb function
Relations: NASSP 2016

18 Jean-Baptiste Joseph Fourier
The Fourier transform Definition of the Fourier transform: Inverse Fourier transform (via the Fourier inversion theorem): Jean-Baptiste Joseph Fourier ( ) NASSP 2016

19 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 ( ) NASSP 2016

20 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

21 The Fourier transform of a Gaussian
The Fourier transform of a Gaussian with dispersion σx is ... NASSP 2016

22 The Fourier transform of a Gaussian
The Fourier transform of a Gaussian with dispersion σx is a Gaussian with dispersion σs=(2πσx)-1 NASSP 2016

23 The Fourier transform of a Delta function
The Fourier transform of a Delta function is ... NASSP 2016

24 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

25 The Fourier transform of a comb function
The Fourier transform of a sha function with period T is ... NASSP 2016

26 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

27 The Fourier transform of a rectangle function
The Fourier transform of a rectangle function is ... NASSP 2016

28 The Fourier transform of a rectangle function
The Fourier transform of a rectangle function is the sinc function! NASSP 2016

29 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

30 The n-dimensional Fourier transform
The n-dimensional Fourier transformation and its inverse is defined as NASSP 2016

31 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 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

33 The Convolution: rules
NASSP 2016

34 The Convolution: examples
NASSP 2016

35 The Convolution: examples
NASSP 2016

36 The Convolution: examples
NASSP 2016

37 Cross-correlation NASSP 2016

38 Cross-correlation NASSP 2016

39 Auto-correlation NASSP 2016

40 Fourier transform properties: Linearity and separability
NASSP 2016

41 Fourier transform properties: Shift theorem
Proof: NASSP 2016

42 Fourier transform properties: Shift theorem
Proof: NASSP 2016

43 Fourier transform properties: Convolution theorem
NASSP 2016

44 Fourier transform properties: Convolution theorem
Proof: NASSP 2016

45 Fourier transform properties: Convolution theorem
Proof: NASSP 2016

46 Fourier transform properties: Crosscorrelation theorem
Proof: NASSP 2016

47 Fourier transform properties: Autocorrelation theorem
Also: Wiener-Khinchin Theorem Just a special case of the cross-correlation theorem Proof: NASSP 2016

48 The discrete Fourier transform: definition
Inverse discrete Fourier transform Numerical methods exist to make the expensive FT faster (“Fast FT”) NASSP 2016

49 The discrete Fourier transform: Inverse
NASSP 2016

50 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

51 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

52 Nyquist's sampling theorem
Consider a real-valued wave package with a frequency cutoff at The Fourier transform has the support and it follows Harry Nyquist (1889 – 1976) NASSP 2016

53 Nyquist's sampling theorem
In an experiment we sample the function with the sampling period Δx NASSP 2016

54 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

55 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

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


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