1. NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 12 Complex numbers – an alternate view The Fourier transform Convolution, correlation, filtering.

2. NASSP Masters 5003F - Computational Astronomy - 2009 Complex numbers REAL IMAGINARY

3. NASSP Masters 5003F - Computational Astronomy - 2009 Complex numbers REAL IMAGINARY NONSENSE! There IS no √-1.

4. NASSP Masters 5003F - Computational Astronomy - 2009 Let’s ‘forget’ about complex numbers for a bit......and talk about 2-component vectors instead. x y v x y θ

5. NASSP Masters 5003F - Computational Astronomy - 2009 What can we do if we have two of them? x y v1v1 v2v2 We could define something like addition: There are lots of operations one could define, but only a few of them turn out to be interesting. v sum I use a funny symbol to remind us that this is NOTaddition (which is an operation on scalars); it is just analogous to it.:

6. NASSP Masters 5003F - Computational Astronomy - 2009 The following operation has interesting properties: x y v1v1 v2v2 But it isn’t very like scalar multiplication except when all ys are zero. It’s fairly easy to show that: v prod θ2θ2 θ1θ1 θ prod =θ 1 + θ 2

7. NASSP Masters 5003F - Computational Astronomy - 2009 Vectors? These are just complex numbers! Note that: This, plus the angle-summing properties of the product, leads to the following typographical shorthand: Instead of the mysterious we should just note the simple identity

8. Notation: NASSP Masters 5003F - Computational Astronomy - 2009 where These are all just different ways of saying the same thing.

9. Some important reals: Phase Power Amplitude, magnitude or intensity NASSP Masters 5003F - Computational Astronomy - 2009 =atan2(I,R)

10. NASSP Masters 5003F - Computational Astronomy - 2009 The lessons to learn: Complex numbers are just 2-vectors. The ‘imaginary’ part is just as real as the ‘real’ part. Don’t be fooled by the fact that the same symbols ‘+’ and ‘x’ are used both for scalar addition/multiplication and for what turn out to be vector operations. This is a historical typographical laziness. –Be aware however that the notation I have used here, although (IMO) more sensible, is not standard. –So better go with the flow until you get to be a big shot, and stick with the silly x+iy notation.

11. The Fourier transform Analyses a signal into sine and cosines: The result is called the spectrum of the signal. NASSP Masters 5003F - Computational Astronomy - 2009

12. The Fourier transform G in general is complex-valued. ω is an angular frequency (units: radians per unit t). the transform is almost self-inverse: But remember, these integrals are not guaranteed to converge. (This is not a problem when we ‘compute’ the FT, as will be seen.) NASSP Masters 5003F - Computational Astronomy - 2009

13. Typical transform pairs NASSP Masters 5003F - Computational Astronomy - 2009 point (delta function)  fringes. By the way, ‘the’ reference for the Fourier transform is Bracewell R, “The Fourier Transform and its Applications”, McGraw-Hill

14. Typical transform pairs NASSP Masters 5003F - Computational Astronomy - 2009 ‘top hat’  sinc function

15. Typical transform pairs NASSP Masters 5003F - Computational Astronomy - 2009 wider  narrower

16. Typical transform pairs NASSP Masters 5003F - Computational Astronomy - 2009 gaussian  gaussian

17. Typical transform pairs NASSP Masters 5003F - Computational Astronomy - 2009 Hermitian  real

18. Practical use of the FT: Periodic signals hidden in noise Processing of pure noise: –Correlation –Convolution –Filtering Interferometry NASSP Masters 5003F - Computational Astronomy - 2009

19. Periodic signal hidden in noise NASSP Masters 5003F - Computational Astronomy - 2009 The eye can’t see it……but the transform can.

20. Transforming pure noise NASSP Masters 5003F - Computational Astronomy - 2009 Uncorrelated noise The transform looks very similar. This sort of noise is called ‘white’. Why?

21. Power spectrum Remember the power P of a complex number z was defined as If we apply this to every complex value of a Fourier spectrum, we get the power spectrum or power spectral density. This is both real-valued and positive. Just as white light contains the same amount of all frequencies, so does white noise. (For real data, you have to approximate the PS by averaging.) NASSP Masters 5003F - Computational Astronomy - 2009

22. Red, brown or 1/f noise NASSP Masters 5003F - Computational Astronomy - 2009 It’s fractal – looks the same at all length scales.

23. Nature…? NASSP Masters 5003F - Computational Astronomy - 2009 No, it is simulated – 1/f 2 noise.

24. Fourier filtering of noise Multiply a white spectrum by some band pass: Back-transform: The noise is no longer uncorrelated. Now it is correlated noise: ie if the value in one sample is high, this increases the probability that the next sample will also be high. I simulated the brown noise in the previous slides via Fourier filtering. NASSP Masters 5003F - Computational Astronomy - 2009

25. Another example – bandpass filtering: NASSP Masters 5003F - Computational Astronomy - 2009

26. Convolution NASSP Masters 5003F - Computational Astronomy - 2009 * = It is sort of a smearing/smoothing action.

27. A very important result: This is often a quick way to do a convolution. An example of a convolution met already: –Sliding-window linear filters used in source detection. NASSP Masters 5003F - Computational Astronomy - 2009

28. Correlation It is related to convolution: Auto-correlation is the correlation of a function by itself. NOTE! For f=noise, this integral will not converge.. NASSP Masters 5003F - Computational Astronomy - 2009

29. How to make the autocorrelation converge for a noise signal? First recognize that it is often convenient to normalise by dividing by R(0): It can be proved that γ (0)=1 and γ (>0)<1. For ‘sensible’ fs, the following is true: A practical calculation estimates equation (1) via some non-infinite value of T. NASSP Masters 5003F - Computational Astronomy - 2009 (1)

30. Autocorrelation and power spectrum From slides 9 and 28, it is easy to show that the Fourier transform of the autocorrelation of a function is the same as its power spectral density. Again, in practice, we normalize the PSD by R(0) and estimate the result over a finite bandwidth. NASSP Masters 5003F - Computational Astronomy - 2009