Lecture 9- Convolution and sampling

Convolution theorum

Fourier approach

Sinusoidal signals input to a linear time invarient system

Unchanged in angular frequency

Different amplitudes

different phases

different weightings and delays

Input signal is superposition of fourier components (sine waves)

Output is a superposition of different fourier components: the transfer function is a recipe for changing input

Linear systems do not generate any extra fourier coomponents i.e harmonics of imputs

Diagrams

Flipped, but no effect if symmetric

Input spectrum extends to all frequencies

system transfer function passes only a limited range of frequencies

output spectrum has only a limited range of frequencies

True sky brightness distribution can be viewed either as

Fine grid of point sources to be convolved with the antenna beam

or a spectrum of angular frequency components with units cycles per radian to be weighted by the angular frequency "transfer function" of the antenna for intensity

Angular frequency transfer function of the antenna for intensity = ACF of the aperture distribution

Angular frequency transfer function

Angular frequency components higher than umax are not passed by the antenna

same as the weighted spatial frequencies which make up the virtual intensity aperture

Diffraction theory means that umax corresponds directly to an angular frequency

maths

Voltage and Power beams

A triangular ACF is appropriate for a uniformly illuminated antenna measuring intensity.

Antenna can be regarded as having a virtual intensity aperture of this shape

This extends between +- umax and its triangular shape implies an increasingly low weighting to the high spatial frequency fourier components making up the virtual aperture and thus to the angular frequency components of the sky

the weighting means that the intensity beam obtained from the FT of the ACF therefore does not have double the formal resolution of the voltage beam obtained from the FT of a uniform distribution with only half the extent

The FWHM of the intensity main beam is however narrower than in the voltage main beam

Maths convolution w/ diagrams on pg 12

Antenna smoothing

Convolution with the antenna beam smooths out fine details on the sky temperatrue distribution on angular scales smaller than lam/d

equivalently the antenna ACF acts as a low pass filter cutting out high angular frequncies, band limiting

Treat sky distribution as a series of impulses (point sources) and convolve with simple symmetric beam

lose detail on smaller scales than beam width

equivalently: the angular spectrum of the map is cut-off beyond

Nyquist-Shannon Sampling theorum

an arbitray function containing no frequencies higher than vmax can be reconstructed exactly (by interpolation) from a series of samples of that function taken at a frequency of at least 2vmax

undersampled is aliasing

Antenna cut-off and sampling in angle

to make 2-d map of sky brightness temp distribution shift antenna beam in angle and collect samples of the antenna temp

Maths