\begin{center}

Figure: Design of a parabolic Antennae

\end{center}
So the Average collection area ,

\(\langle Ae\rangle\  \equiv \ \int Aed\Omega\ /d\Omega = (\ 1/4\pi)4\pi Aed\Omega\)

\begin{quote}
3.2 Horn Antenna
\end{quote}

The detection of hydrogen lines through a horn-based antenna is another
method for studying the properties of hydrogen gas in the universe. The
horn-based antenna is similar to the dish-based antenna, but it has a
different design and is optimized for certain types of observations. A
horn-based antenna consists of a horn-shaped structure that collects the
radiation emitted by the hydrogen gas and focuses it onto a receiver.
The receiver contains a detector that measures the voltage of the
collected radiation. This voltage signal is then processed by a
spectrometer to create a spectrum, which is a plot of the intensity of
the collected radiation as a function of frequency. One advantage of
using a horn-based antenna is that it has a wider field of view than a
dish-based antenna. This allows astronomers to study large regions of
the sky at once, making it a valuable tool for mapping the distribution
of hydrogen gas in galaxies and the interstellar medium. Another
advantage of using a horn-based antenna is that it can observe at
multiple frequencies simultaneously. This allows astronomers to study
the velocity structure of the gas, which can reveal information about
the motion and turbulence of the gas{[}{\color{blue}19}{]}. The detection of hydrogen
lines through a horn-based antenna is another valuable tool for studying
the properties of hydrogen gas in the universe. The horn-based antenna
has advantages over the dish-based antenna in terms of its wider field
of view and ability to observe at multiple frequencies simultaneously
{[}{\color{blue}14}{]}. With the development of new horn-based antennas and advances
in technology, astronomers will continue to use this technique to gain
new insights into the distribution, motion, and physical properties of
hydrogen gas in the universe.

\begin{center}

\begin{quote}
\includegraphics[width=4.79167in,height=2.19043in]{image2.png}

\end{quote}

Figure: The diagram represent design of the horn shaped antennae.

\end{center}

The gain of the conical horn antenna can be calculated by the following
formula-

$$ Gain(dB) =  (\pi d/\lambda)^{2}*e)A$$

eA = Efficiency of aperture

d = Diameter of aperture

$\lambda$ = Wavelength of radio waves.

\begin{quote}
3.3 Interferometers
\end{quote}

Interferometers are used to combine the signals from multiple radio
telescopes to create a single image with higher resolution and
sensitivity than any single telescope could achieve. This technique is
called aperture synthesis, and it allows radio astronomers to study
objects in the universe with great detail. The basic principle of radio
interferometry is that two or more radio telescopes are pointed at the
same object, and the signals received by each telescope are combined to
create an interference pattern. The interference pattern contains
information about the object\textquotesingle s radio emission, and by
analyzing the pattern, astronomers can reconstruct an image of the
object. The resolution of a radio interferometer is determined by the
distance between the telescopes, and is given by the formula:

$$\Theta = \lambda / D$$

where $\theta$ is the angular resolution, $\lambda$ is the wavelength of the radio
waves being observed, and D is the distance between the telescopes. For
example, if we have two radio telescopes spaced 1000 km apart observing
at a wavelength of 21 cm, the angular resolution of the interferometer
would be:

$$\theta = (21 cm) / (1000 km) = 2.1 arcminutes$$

This means that the interferometer can resolve details in the radio
emission that are separated by at least 2.1 arcminutes in the sky. In
addition to improving resolution, radio interferometry can also increase
sensitivity by combining the signals from multiple telescopes. The
sensitivity of an interferometer is proportional to the square root of
the number of telescopes, so an interferometer with 10 telescopes will
be 3 times more sensitive than one with only 1 telescope {[}{\color{blue}20}{]}.

The formula for Fourier transformation of an interferogram is:

\(I(u,v)\  = \ \int\int\ F(x,y)\ exp\lbrack - 2\pi i(ux + vy)\rbrack\ dx\ dy\)

where I(u,v) is the Fourier transform of the interferogram, F(x,y) is
the two-dimensional function representing the object\textquotesingle s
image, u and v are the spatial frequencies in the x and y directions,
and i is the imaginary unit. {[}{\color{blue}3,13}{]}

\begin{center}