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Lecture 12

Lecture 12. Complex numbers – an alternate view The Fourier transform Convolution, correlation, filtering. Complex numbers. IMAGINARY. REAL. Complex numbers. NONSENSE!. There IS no √-1. IMAGINARY. REAL. Let’s ‘forget’ about complex numbers for a bit.

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Lecture 12

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  1. Lecture 12 • Complex numbers – an alternate view • The Fourier transform • Convolution, correlation, filtering.

  2. Complex numbers IMAGINARY REAL

  3. Complex numbers NONSENSE! There IS no √-1. IMAGINARY REAL

  4. Let’s ‘forget’ about complex numbers for a bit... ...and talk about 2-component vectors instead. y y v θ x x

  5. What can we do if we have two of them? There are lots of operations one could define, but only a few of them turn out to be interesting. vsum y v2 v1 x We could define something like addition: 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. The following operation has interesting properties: y v2 vprod θ2 But it isn’t very like scalar multiplication except when all ys are zero. v1 θprod=θ1+θ2 θ1 x It’s fairly easy to show that:

  7. 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: where These are all just different ways of saying the same thing.

  9. Some important reals: • Phase • Power • Amplitude, magnitude or intensity =atan2(I,R)

  10. 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.

  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.)

  13. Typical transform pairs 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 ‘top hat’  sinc function

  15. Typical transform pairs wider  narrower

  16. Typical transform pairs gaussian  gaussian

  17. Typical transform pairs Hermitian  real

  18. Practical use of the FT: • Periodic signals hidden in noise • Processing of pure noise: • Correlation • Convolution • Filtering • Interferometry

  19. Periodic signal hidden in noise The eye can’t see it… …but the transform can.

  20. Transforming pure noise The transform looks very similar. This sort of noise is called ‘white’. Why? Uncorrelated noise

  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.)

  22. Red, brown or 1/f noise It’s fractal – looks the same at all length scales.

  23. Nature…? No, it is simulated – 1/f2 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.

  25. Another example – bandpass filtering:

  26. Convolution • 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.

  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..

  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. (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.

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