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Lecture 9- Convolution and sampling (Voltage and Power beams (A triangular…
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
Maths convolution w/ diagrams on pg 12
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
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