A half wavelength dipole is kept in the x-y plane and oriented along $${45^ \circ }$$ from the x-axis. Determine the direction of null in the radiation pattern for $$0 \le \phi \le \pi $$. Here the angle $$\theta \left( {0 \le \theta \le \pi } \right)$$ is measured from the z-axis, and the angle $$\phi \left( {0 \le \phi \le 2\pi } \right)$$ is measured from the x-axis in the x-y plane.

The far-zone power density radiated by a helical antenna is approximated as: $$$\overrightarrow W {\,_{rad}} = \overrightarrow W \,average\, \approx \,\widehat a{}_rC{}_0\,{1 \over {{r^2}}}{\cos ^4}\theta $$$

The radiated power density is symmetrical with respect to $$\phi $$ and exists only in the upper hemisphere: $$0 \le \theta \le {\pi \over 2};\,\,\,\,0 \le \theta \le 2\pi ;$$

$${C_0}$$ is a constant. The power radiated by the antenna (in watts) and the maximum directivity of the antenna, respectively, are

Two lossless X-band horn antennas are separated by a distance of $$200\lambda $$. The amplitude reflection coefficients at the terminals of the transmitting and receiving antennas are $$0.15$$ and $$0.18$$, respectively. The maximum directivities of the transmitting and receiving antennas (over the isotropic antenna) are $$18$$ $$dB$$ and $$22$$ $$dB$$, respectively. Assuming that the input power in the lossless transmission line connected to the antenna is $$2$$ $$W$$, and that the antennas are perfectly aligned and polarization matched, the power ( in mW) delivered to the load at the receiver is ________ .

An antenna pointing in a certain direction has a noise temperature of $$50K$$. The ambient temperature is $$290K$$. The antenna is connected to a pre-amplifier that has a noise figure of 2 dB and an available gain of 40 dB over an effective bandwidth of $$12$$ $$MHz$$. The effective input noise temperature $${T_e}$$ for the amplifier and the noise power $${P_{ao}}$$ at the output of the preamplifier, respectively, are