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Statistics for Astronomy VIII: Fourier methods
B. Nikolic b.nikolic@mrao.cam.ac.uk
Astrophysics Group, Cavendish Laboratory, University of Cambridge http://www.mrao.cam.ac.uk/~bn204/lecture/astrostats.html

17 November 2008


Goals for this Lecture

Fourier analysis theory ­ basics

Example: Simple redshift determination

Noise analysis/filtering + GBT Example


Fourier analysis basics I
For continuous functions: 1 F ( ) = 2 1 f (t ) = 2
- -

dt · f (t ) exp (-i t ) d · F ( ) exp (i t )

(1) (2)

Convolution theorem


g (x ) =
-

dx · f (x )h(x - x )

(3) (4) (5)

G( ) = F ( )H ( )

= (f g )(x )


Fourier analysis basics II
Correlation


g (x ) =
-

dx · f (x )h(x + x )

(6) (7)

G( ) = F ( )H ( ) If h is symmetric, same as convolution

Auto-Correlation & the Power Spectrum


af (x ) =
-

dx · f (x )f (x + x )
2

(8) (9)

Af ( ) = |F ( )|


Discrete transform
In practice often have regularly sampled data, in which case:
N -1

Fk =
n =0

fn exp - Fk exp

2 ikn N 2 ikn N

(10)

fn =

1 N

N -1 k =0

(11)

Algorithms (FFT) for computing the above transforms are extremely efficient: run-time N log N as opposed to N


2

Do not implemented you own ­ use e.g., FFTW (http://www.fftw.org/) If data are not regular can bin, but at loss of accuracy and ability to detect high frequencies Missing data are a big problem






Practicalities


Aliasing




Appearance of signal frequencies higher than 1/(2t ) in the sampled signal Filter data before sampling In physical systems sampling is often proceeded by a non-negligible averaging interval Can be represented by a multiplication by a comb of -functions (sampling) and convolution by the averaging function Due to finite length of data Window / apodisation, e.g.,: h( ) = 1 (1 + cos( /T )) 2 (12)



Finite integration before sampling






Spectral leakage



Fourier analysis applications



`Matched-filter' extraction, convolution Detection of periodic signals, e.g., clustering of sources Cross-correlation, e.g., finding the redshift of galaxies Investigation of noise, system stability


Simple redshift determination
Approach
Cross correlate a template and an observed spectrum of a galaxy. The peak of the cross correlation function reveals the relative shift between the signals.

Redshift vs stretch
Redshift corresponds to a stretch of the spectrum ( scale transformation). Transform into log() space to turn redshift into a translation. o log
bs

= (1 + z )

em em

(13) (14)

obs

= log [1 + z ] + log


Simple redshift determination
Template spectrum
200

Target spectrum
70 60

150

50
F 100

F 40 30 20
0 250 500 750 pixel 1000 1250 1500

50

0

0

250

500

750 pixel

1000

1250

1500

1000

1000 750

500

500
Re[F T F ]

Re[F T F ]

0

250 0

-500

-250 -500

-1000

0

250

500

750

1000

1250

1500

0

250

500

750

1000

1250

1500


Cross correlation result
5.75 · 105

5.7 · 105 Power

5.65 · 105

5.6 · 105

5.55 · 105

0

250

500

750 pixel

1000

1250

1500


Noise
White noise
If noise is completely independent from one time to the next:


af =
-

dx · f (x )f (x + x ) = (x )

(13)

= Af ( ) = |F ( )|2 = constant What if noise is correlated: |F ( )|2 = constant? Often the noise power spectrum is a power law (scale invariance, fractal): |F ( )|2
n

Correlated noise

(14)


Correlated noise types



n = -2, Brownian noise, i.e. a random walk type signal


Characteristic of `drift' in electronics, e.g., cumulative environmental effects Present in wide variety of real life systems, including Atmospheric effects on shor t time scales



n = -1, 1/f noise, intermediate between




n = -11/3 Kolmogorov turbulence



Noise realisations
White noise n = 0
0.1

0.3

1/f noise n = -1

0.2
0.05

0.1
0 f

f 0 -0.1 -0.2

-0.05

-0.1

0

200

400 t

600

800

1000

0

200

400 t

600

800

1000

2

Brownian noise n = -2

20

Ver y steep n = -3

1

10

0

0

f

f -10 -20 -30

-1

-2

0

200

400 t

600

800

1000

0

200

400 t

600

800

1000


Generating Noise

Know what the power spectrum must be: |F ( )|2 n , hence: F ( ) =
n/2

exp [2 (t )]

(15)

where (t ) is uniformly distributed between 0 and 1 and uncorrelated.

Impor tant note
This procedure can not simulate the noise correctly for timescales comparable to the size of the region being generated. For example, the previous examples are periodic.


Example
Dual beam calibration observation of a strong-point source with the GBT:


Example
Dual beam calibration observation of a strong-point source with the GBT:
0

-1

-2

Tb

-3

-4

0

1000

2000

3000 t

4000

5000

6000


Example 1/f spectrum

Shor t section of the data away from the strong source Time stream Power spectral density
-1.75

-1.76

-1.77

-1.78

-1.79

240

260

280

300 t

320

340

360


15 5 5 15 25 0.0 0.2 0.4 Frequency 0.6

Power Spectrum (dB)

Tb

0.8

1.0


Example signal spectrum

Shor t section of the data crossing the source Time stream Power spectral density
0

Power Spectrum (dB)

-1

-2

-3

-4 2900

2950

3000 t

3050

3100

Ў Ў Ў Ў
30 20 10 0 10 20 30 40 0.0 0.2 0.4 Frequency 0.6

Tb

0.8

1.0


Whole example

0

Power Spectrum (dB)

-1

-2

-3

-4

0

1000

2000

3000 t

4000

5000

6000

ў ў ў ў ў
40 20 0 20 40 60 80 100 0.0 0.2 0.4 Frequency 0.6

The whole observation: Time stream

PSD (N = 256)

Tb

0.8

1.0


Whole example

0

Power Spectrum (dB)

-1

-2

-3

-4

0

1000

2000

3000 t

4000

5000

6000

ё ё ё ё ё ё ё
20 10 0 10 20 30 40 50 60 70 0.0 0.2 0.4 Frequency 0.6

The whole observation: Time stream

PSD (N = 16376)

Tb

0.8

1.0


Full power-spectrum

¤ ¤ ¤ ¤ ¤ ¤ ¤
10 0 10 20 30 40 50 60 70 .0 0 0.2

20

Power Spectrum (dB)

0.4

Frequency

0.6

0.8

1.0


Removing low-frequency noise
0 2 -1 1

-2

0

Tb

-3

Tb 2600 2800 3000 t 3200 3400 3600

-1

-4 2400

-2 2400

Raw data

40

20 0

20 40 60 80

100 .0 0

0.2

0.4

Frequency

0.6

0.8

1.0

¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦
2600 2800 3000 t

3200

3400

3600

De-baselined data

Ґ Ґ Ґ Ґ Ґ

0

10 20 30 40 50 60 70 80

Power Spectrum (dB)

Power Spectrum (dB)

90 .0 0

0.2

0.4

Frequency

0.6

0.8

1.0


Bibliography

Recommended for a tutorial on Bayesian methods: D. S. Sivia, "Data Analysis: A Bayesian Tutorial", Oxford University Press