Supplementary Material
to:
An Introduction to Radio Astronomy
4th edition Cambridge University Press 2019
Last updated 05/12/2021
Appendix 1: Fourier Transforms
The wide applicability of convolution
Convolution is a smoothing operation relating an
input to an output through a linear and
translationally-invariant system or process i.e.
· the output
amplitudes Aout are simple linear
multiples of the input amplitudes Ain
o Aout= aAin rather than some non-linear relationship e.g. Aout=[Ain]a
·
the output does not depend on the absolute arrival time (or
equivalently the absolute angle or location) i.e. the same relative input changes produce the
same output variations regardless of where they start à preserves the relative
positions of temporal, spatial, or angular features of the input hence
the relative phases of the
frequency components of that signal
Such processes are very common e.g.
·
radioastronomy: sky brightness (θ,φ) smoothed by
telescope beam
·
image processing: 2-D photograph (x,y)
“blurred” due to motion or out-of-focus
·
electrical
circuits: time varying voltages passing through filters, amplifiers etc
there are
many more.
Convolution can be viewed in the context of a response to impulses with the output being the
convolution with the system impulse response. Analytically one of the functions
(either) must be flipped around before step-wise integration.
More commonly the system response to input Fourier components is
utilized. The system’s complex “Transfer Function” weights the components of
the input spectrum to give the output spectrum; both amplitudes and phases but not the frequencies, of the output
Fourier components are altered.
In the specific example below an input signal is the
combination of two temporal sine waves (frequency components) and the output as a superposition of two
different frequency components:
For a Iinear time invariant system individual sinusoidal signals
are passed through:
•
unchanged in angular frequency ω1
becomes ω1 etc thus no harmonics generated
• with
different amplitudes A becomes A’ etc (often smaller but can be larger in case of amplifier)
•
with different phases φ1
becomes φ2; θ1 becomes θ2 etc
(typically
a delay)
[Spectrum of input signal] x
[Transfer Function] = [ Spectrum of output signal]
Correlation
Correlation is multiplication and integration. Correlation
functions involve multiple shifts; analytically the integral formulation is
identical to that for convolution except neither of the two functions is
flipped around. If the two input functions are identical the convolution and
correlation functions are the same. In general this is
not the situation which applies and the flip involved in convolution changes
the outcome.
Cross-correlation: measures similarities between
independent signals and preserves relative phase information – most useful in interferometry.
Autocorrelation: measures the similarity of a single function at
different delays compared with the undelayed version thus picks out periodicities in the
function but loses all phase information.
Wiener-KhinchinTheorem: FT [ACF] = power spectrum of signal (or
vice versa with inverse FT).
Pictorially multiply a function f(x)
(here a rectangle of length l )
with a copy f(x-s) shifted by an amount s and
integrate (i.e. form an “overlap”
integral) then repeat for all shifts s and
plot the results of each integration as a function of s: this
is the ACF(s).
Appendix 2: Celestial Coordinates and Time
· Celestial
coordinates: The
University of Nebraska Lincoln’s
“Rotating Sky Lab” (requires Adobe Flash Player) http://astro.unl.edu/naap/motion2/motion2.html provides a pictorial explanation of
astronomical coordinates but there are
many tutorials available on the web. We advise the reader to carry out their
own search.
·
Time: UTC is coordinated by the Bureau International de Poids et Mesures (BIPM) - for more information see https://www.bipm.org/en/bipm-services/timescales/time-server.html
Appendix
3: Digitization
There are many pedagogic animations of the effects of sampling and
digitization on the Web. We advise the
reader to carry out their own search.
There are many excellent books on digital signal processing. S.W. Smith’s book The Scientist and Engineer’s Guide to Digital Signal Processing can
be freely downloaded from http://www.dspguide.com/
Appendix 4:
Calibrating Polarimeters
The paper by Robishaw and Heiles (2018) Measurement of Polarisation in
Radioastronomy https://arxiv.org/abs/1806.07391 gives an up-to-date and comprehensive overview of single dish
polarisation measurements.
The paper by Wolleben et al (2006) “An absolutely calibrated
survey of polarized emission from the northern sky at 1.4 GHz”
https://www.aanda.org/articles/aa/pdf/2006/10/aa3851-05.pdf provides a description of a specific set
of measurements using a single dish.
Interferometric polarisation calibration (in the CASA analysis
package) is explained in https://casa.nrao.edu/casadocs/casa-5.1.0/synthesis-calibration/instrumental-polarization-calibration
Interferometric
calibration for the VLA is covered in https://science.nrao.edu/facilities/vla/docs/manuals/obsguide/calibration
All the above are of general relevance.
Appendix 5:
Spherical Harmonics
The HEALPix
Primer: https://healpix.jpl.nasa.gov/pdf/intro.pdf