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