Lecture 12 •Complex numbers –an alternate view •The

Lecture 12

• Complex numbers – an alternate view

• The Fourier transform

• Convolution, correlation, filtering.

NASSP Masters 5003F - Computational Astronomy - 2009

Complex numbers

1−=i

iIRz +=


REAL IMAGINARY

1−=i

Complex numbers

1−=i

iIRz +=


REAL IMAGINARY

1−=i

There IS no √-1.

Let’s ‘forget’ about complex numbers for a bit...

...and talk about 2-component vectors instead.

y

vy

=

=

θθ

sin

cosv

y

xv


xx

θ

What can we do if we have two of them?

y

v1

v2

There are lots of operations one could define, but only a few of themturn out to be interesting.

vsum


x

+

+=⊕

21

21

21yy

xxvv

We could define something like addition:

I use a funny symbol to remind us that this is NOTaddition (which is an operationon scalars); it is just analogous to it.:

The following operation has interesting properties:

y

v1

+

−=⊗=

1221

2121

21prodyxyx

yyxxvvv

v2

But it isn’t very like scalarmultiplication except whenall ys are zero.

vprodθ2

θ

θprod=θ1+θ2


x

[ ][ ]

+

+=

21

21

21prodsin

cos

θθθθ

vvv

It’s fairly easy to show that:

θ1

Vectors? These are just complex numbers!

vv

⊗

=

∂∂

1

0

θ

Note that:

This, plus the angle-summing properties of theproduct, leads to the following typographicalshorthand:

( )θiv exp=v


Instead of the mysterious

1−=iwe should just note the simple identity .

0

1

1

0

1

0

−=

⊗

Notation:

iIRz +=

=I

Rz


( )φizz exp=( )RIarctan=φ

( )φφ sincos izz +=

where

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

Some important reals:• Phase

• Power

( )RIarctan=φ =atan2(I,R)

22 IRzzzzP +=== ∗∗

• Amplitude, magnitude or intensity


22 IRzzzzP +=== ∗∗

PzA ==

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


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.

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

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:

{ } ( ) ( ) ( )∫∞

∞−

−== titgdtGg ωω expF

• 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.)


{ } ( ) ( ) ( )∫∞

∞−

== tiGdtgG ωωω exp1-F

Typical transform pairs

point (delta function) � fringes.


By the way, ‘the’ reference for the Fourier transform is Bracewell R, “TheFourier Transform and its Applications”, McGraw-Hill


‘top hat’ � sinc function



wider � narrower



gaussian � gaussian



Hermitian � real


Practical use of the FT:• Periodic signals hidden in noise

• Processing of pure noise:– Correlation

– Convolution

– Filtering

• Interferometry• Interferometry


Periodic signal hidden in noise


The eye can’t see it… …but the transform can.

Transforming pure noise


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

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

22* IRzzP +==

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

Red, brown or 1/f noise


It’s fractal – looks the sameat all length scales.

Nature…?


No, it is simulated – 1/f2 noise.

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 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.


Another example – bandpass filtering:


Convolution

( ) ( ) ( )∫∞

∞−

′−′′=∗= ttgtftdgfth

• It is sort of a smearing/smoothing action.


* =

A very important result:

• This is often a quick way to do a convolution.

{ } { } { }gfgf FFF =∗

• An example of a convolution met already:– Sliding-window linear filters used in source

detection.


Correlation

• It is related to convolution:

( ) ( ) ( )∫∞

∞−

′+′′== ttgtftdgftR gf ,

{ } { } { }gfgf ∗= FFF

• 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

{ } { } { }gfgf = FFF

( ) ( ) ( )∫∞

∞−

′+′′== ttftftdfftR ff ,

How to make the autocorrelation converge for a noise signal?

• First recognize that it is often convenient to normalise by dividing by R(0):

( )( ) ( )

( )[ ]∫∫

′′

′+′′=

2tftd

ttftftdtγ

• 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.


( )( ) ( )

( )[ ]∫∫

−

−

∞→ ′′

′+′′=

2

2

2

2

2lim

T

T

T

T

Ttftd

ttftftdtγ (1)

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.

{ } { } { } { } ( ) 22ωFfffff === ∗

FFFF

• Again, in practice, we normalize the PSD by R(0) and estimate the result over a finite bandwidth.


{ } { } { } { } ( )ωFfffff === FFFF

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Lecture 12 •Complex numbers –an alternate view •The