Dealing with Noise

42 Pages·2010·1.91 MB·English

Dealing with Noise
Stéphane GUILLOTEAU
Laboratoire d’Astrophysique de Bordeaux
Observatoire Aquitain des Sciences de l’Univers
I - Theory & Practice of noise
II – Low S/N analysis
Outline
Basic Theory
1.
Point source sensitivity
1.
Noise in images
2.
Extended source sensitivity
3.
Available Tools
4.
Low S/N analysis
2.
Continuum data
1.
Line data
2.
Examples
3.
Advanced tricks: filtering & stacking
4.
System Temperature
The output power of the receiver is linked to the Antenna System

Temperature by:
P = ° k T ¢ º
N ant
On source, the power is P + P with

N a
P = ° k T ¢ º
a a
T is called the antenna temperature of the source.

a
This is not a purely conventional definition.

It can be demonstrated that P is the power the receiver(+antenna) would deliver when
a
observing a blackbody (filling its entire beam pattern) at the physical temperature T .
a
Thus, T is the temperature of the “equivalent’’ blackbody seen by the

ant
antenna (in the Rayleigh Jeans approximation)
System Temperature
T is given by (just summing powers…)

ant
T = T cosmic background
ant bg
+ T ¼ ´ (1-exp(-¿ ) T sky noise
sky f atm atm
+ T ¼ (1-´ -´ ) T ground noise pickup
spill f loss ground
+ T ¼ ´ T losses in receiver cabin
loss loss cabin
+ T receiver noise
rec
This is a broad-band definition. It is a DSB (Double Side Band)

noise temperature
Many astronomical signals are narrow band. g being the image to

signal band gain ratio, the equivalent DSB signal giving the same
antenna temperature as a pure SSB signal is only
P = (1 x P + g x 0) / (1 + g)
DSB SSB
System Temperature
We usually refer the system temperature and antenna

temperature to a perfect antenna (´ = 1) located
f
outside the atmosphere, and single sideband signal:
T = (1+g) exp(¿ )T / ´
sys atm ant f
T * = (1+g) exp(¿ )T / ´
A atm a f
This antenna temperature T * is weather independent,

A
and linked to the source flux S by an antenna
º
dependent quantity only
T * = ´ A S / 2k
A a º
Noise Equation
The noise power is T , the signal is T * , and there are

sys A
2¢º ¢t independent samples to measure a correlation
product in a time ¢t, so the Signal to Noise is
R = ( ¢º ¢t)1/2 T * / T
2
sn A sys
On a single baseline, the noise is thus

this is √2 less than that of a single antenna in total power

but √2 worse than that of an antenna with the same total collecting

area
this sensitivity loss is because we ignore the autocorrelations

Noise Equation
With quantization

With ´ the quantization efficiency

q
Noise is uncorrelated from one baseline to another

There are n(n-1)/2 baselines for n antennas

So the point source sensitivity is

Where is the Jy/K conversion factor of one antenna

Noise on Amplitude and Phase
For 1 baseline, this varies with Signal to Noise ratio

On Amplitude

On Phase

Source detection is much easier on the phase than on the

amplitude, since for S/N = 1, ¾ = 1 radian = 60°.
Á
Noise in Images
The Fourier Transform is a linear combination of the visibilities

with some rotation (phase factor) applied. How do we derive the
noise in the image from that on the visibilities ?
Noise on visibilities

the complex (or spectral) correlator gives the same variance on the

real and imaginary part of the complex visibility <ε 2> = <ε2> = <ε2>
r i
Real and Imaginary are uncorrelated <ε ε> = 0

r i
So rotation (phase factor) has NO effect on noise

Noise in Imaging: first order
In the imaging process, we combine (with some weights) the individual

visibilities V . At the phase center:
i
I = (Σ w V ) / Σw
i i i
for a point source at phase center, V = V +ε , ε being the real part of

i Ri Ri
the noise
I = (Σ w (V+ ε ) ) / Σw
i Ri i
So its expectation is I = V, as < ε > = 0

Ri
As <ε ε > = 0, its variance is

Ri Rj
σ2 = <I2> -<I>2 = = (Σ w 2 < ε 2 > ) / (Σw )2
i Ri i
Now using < ε 2 > = σ 2 and the natural weights w = 1/ σ 2 we have

Ri i i i
1/σ2 = Σ (1/σ 2)
i
Which is true anywhere else in the image by application of a phase shift
