Fourier Transforms

1. FourierTransforms John Reynolds Joseph Fourier 1768-1830

2. Outline • Basic properties of the Fourier transform • Discrete form and the FFT • Simple example applications • Applications in radio astronomy; • Synthesis imaging and the u-v plane • Frequency conversion • The Sampling Theorem • Advanced signal processing (DSP) • filter-banks, spectroscopy

3. Fourier Integral Transform Fourier integral transform Inverse transform Mutant forms; i j (Engineering) 2πf  ω (Pure maths or theoretical physics) f,t  x,y Individual cosine, sine transforms

4. Basic Properties I a . h(t)  a . H(f) linearity h(t) + g(t)  H(f) + G(f) linearity h(t) is real  H(-f) = H(f)* symmetry h(t) is imag’ry  H(-f) = -H(f)* h(-t) = h(t)  H(-f) = H(f) h(t) real,even  H(f) real,even

5. Basic Properties II Scaling; “broad   narrow” h(at)   H(f/a) / |a| Shifting; “shift   phase roll/gradient” h(t-t0)   H(f) * exp(2πi f t0) Convolution; “convolution   multiplication” h(t) * g(t)   H(f) G(f)

6. Some well-known examples

7. The Dirac delta

8. The humble sinusoid

9. Dirac comb or “shah” Ш dt df=1/dt

10. Basic Properties II Scaling; “broad   narrow” h(at)   H(f/a) / |a| Shifting; “shift   phase roll/gradient” h(t-t0)   H(f) * exp(2πi f t0) Convolution; “convolution   multiplication”   H(f) G(f)

11. Convolution – a simple example * =

12. More convoluted example After J. J. Condon and S. M. Ransom ESSENTIAL RADIO ASTRONOMY http://www.cv.nrao.edu/course/astr534

13. Parseval and correlation theorems Correlation function: Corr (g,h)  G(f)H(-f) = G(f)H(f)* for real g(t),h(t) Corr (g,g)   |G(f) |2 (Wiener-Khinchin) (Parseval)

14. Parseval Energy is conserved! |H(f)|2 := power spectral density PSD

15. Consumer applications Fourier Transform Processing With ImageMagick Introduction One of the hardest concepts to comprehend in image processing is Fourier Transforms. There are two reasons for this. First, it is mathematically advanced and second, resulting images, which do not resemble the original image, are hard to interpret.

16. 2-D and beyond “Top Hat”   Airy disk

17. Practical realisation Periodic   Discrete (“Fourier series”)   df = 1/period period Periodic & Discrete   Periodic & Discrete N.dt.df = 1 period = N.df period = N.dt

18. DFT: Discrete Fourier Transform Periodic, discretely sampled functions with; t = k.dt, f = n.df, (where N.dt.df = 1) Replace indefinite integral with summation over N values; All aforementioned properties of Fourier integrals carry over, e.g.; Discrete form of Parseval * One or other of h(t), H(f) function is generally “band-limited”

19. FFT – the Fast Fourier Transform Simple DFT requires ~N2 multiplications Gets very slow with large N Decompose the NxN matrix into a product of N sparse matrices Have reduced to 2 DFTs of order N/2 Keep going until you get to order 1. Number of mults now ~N.logN

20. Why phase is important Original image 2D (3D) Transform  Spatial Frequency domain Filter: Filter: Amplitude only Phase only

23. error correction by spatial masking

24. Applications in radio astronomy • Aperture synthesis imaging • Frequency conversion • Sampling theorem • Signal processing (spectrometers, PFBs)

25. u-v plane Synthesis interferometer: we cross-correlate each pair of antennas spatial auto-correlation  1-1, 2-2 etc excluded! 3-1 3-2 2-1 1-2 2-3 1-3 1 2 3 East  u  aperture plane u-v plane Distribution function A(x,y) in antennas  Transfer function W(u,v) For n antennas  n(n-1)/2 baselines (points) in u-v plane

26. ASKAP – Australian SKA Pathfinder

27. ASKAP u-v coverage

28. Fourier transform of sky brightness is a function in the u-v plane   λ/a

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

30. Mixing it down – Frequency Conversion Mixer (Multiplier) Signal 1 Signal 1 × Signal 2 Signal 2 cos(ω1t)cos(ω2t)=½[cos((ω1+ω2)t)+ cos((ω1-ω2)t)] cos(ωt) = ½[exp(iwt)/2 + exp(-iwt)] 1*2 Power Difference Sum Frequency  Frequency  Δf Δf

31. Mixing it down II– Frequency Conversion(aka superheterodyne principle) Mixer (Multiplier) Signal 1 Local Oscillator Band pass filter flo Frequency Frequency Δf Δf

32. Image rejection Unwanted image response Δf Frequency * -flo flo 2flo -2flo Frequency CSIRO. Receiver Systems for Radio Astronomy

33. Negative frequencies: learn to love them! -ω ω Analytic signal of real f(t); h(t)  h(t) + i.H(f)(t) H(f) := Hilbert transform cos(ωt)  cos(ωt) + i.sin(ωt)

34. Single Sideband Mixers √2cos[(ωLO- ω1)t] (USB) 0 (LSB) 2√2cos(ω1t) Signal Upper sideband Local Oscillator Signal Lower sideband CSIRO. Receiver Systems for Radio Astronomy

35. Sampling Theorem – History The theorem is commonly called the Nyquist sampling theorem; since it was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others, it is also known as Nyquist Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling theorem, as well as the Cardinal Theorem of Interpolation Theory. It is often referred to simply as the sampling theorem. (From Wikipedia)

36. Sampling Theorem (Shannon) If a function x(t) contains no frequencies higher than Bhertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart. tsamp < 1 / 2B

37. Sampling Theorem “band-limited”

38. ts=1/B (x2 undersampled) ts=1/2B (Nyquist)  “aliased response”, or “aliasing”

39. Sampling Theorem continued Also; Radiotelescopes – Christiansen and Högbom Radio Astronomy – J.D. Kraus Principles of Interferometry and Synthesis in Radio Astronomy - Thompson, Moran, Swenson

40. Aliased sampling 3rd Nyquist zone Baseband Frequency B * -2fs -fs fs= 1/tsamp 2fs 3fs Sampling theorem: fs = 1/tsamp > 2B

41. Recent Trends • Faster, cheaper, samplers • Faster, cheaper processing, data storage • Wider sampled bandwidths • Fewer downconversion stages • “direct conversion” (no downconversion) e.g. DRAO receiver at Parkes)

42. CABB signal path

43. This talk ends here!