ECE-GY 9423: Design and Analysis of Communication Circuits and Components (Fall 2024)
Homework # 1 (Due on Thursday, September 17, 2024 before 11:59 pm EST.)
HW: All your calculations should have parametric expressions (wherever possible) and numerical answers (wherever required).
Problem 1. Wireless transmission and Multi-path Scattering
a. Consider a wireless transmitter and a receiver with directional antennas with individual beam patterns being E being separated by a distance of R=1 km.
Calculate the path loss in dB between the transmitter and receiver. Assume frequency of operation is 28 GHz. (6pt)
b. Calculate the input noise of the receiver for a bandwidth of 1 GHz to sustain a 1 Gb/s link. (Optional and extra 6 points. Noise will be covered in future lectures but if you are already familiar … )
c. Assuming transmitter and receiver directivities (GT,GR ) of 3 dB each, calculate the minimum Tx power to ensure (P > -81dBm) at the Rx. Also argue why the field at the Rx is given by ERx = k-1 exp 0−j R1, where k is a constant.
d. Of course, this free-space propagation (point-to-point link) conditions are not satisfied in a cellular settings due to multi-path reflection. Let’s try to analyze what happens in such a scenario. (6pt)
Assume that there are scatterers that are randomly distributed which cause the waves to reflect and they interfere at the Rx, as shown in the figure above. These scatterers could be contributed by buildings in a city, trees, human beings and other reflecting objects. Assume that
1) There is no line-of-sight path (similar to part a.)
2) There are 10 such scatterers which reflect the signals, each path has one scatterer and the waves are attenuated by α=1/10 after reflection.
If the path propagated by each wave is Ri , argue why the field at Rx is given by
e. Assume that the scatters are placed in such a way that all the waves constructively interfere at the Rx. Calculate the total path loss in power (dB) in such a case and compare this to part a, if Ri ~1 km. Calculate the minimum Tx power in this case. (6 pts)
f. Of course, this will almost never happen. Assume that the scatterers are randomly
distributed, such that the phase shifts suffered due to each scatterer
uniformly distributed between [-π, π ]. Since Ri ~1 km the amplitude portion of each
scattering wave does not change dramatically with Ri varying anywhere between 1km±100 meters, but the phase portion is very sensitive to Ri . Why is that?
(10 pts)
g. Therefore, assuming the same ~ constant, run a Monte-Carlo simulation with
1000 iterations in MATLAB on amplitude of IERx I with θ being uniformly distributed between [-π, π ]. Plot the histogram and simulate the mean and variance of IERx I. The distribution is Rayleigh. Calculate the average path loss from the histogram. How does the mean path loss compare to that in part f.? (10pts)
h. Now this scattering is a random phenomenon and you don’t really know the actual locations of them. However, you also want the link to work more than 95% of the time> Therefore, you need to send enough Tx power, such that for 95% of the time, SNRmin is established. From the Monte-carlo simulations in part 7, calculate the 95th percentile path loss (path loss is worse than this only 5% of the time). (10pts)
i. Re-calculate the minimum Tx power from part i) and compare this to part f). (8pts)
Problem 2 (Antenna Arrays):
In this problem, we will plot the current distribution across a simple dipole antenna with various lengths and study the 2D and 3D radiation pattern to understand how the length of the antenna affects the radiation pattern directivity, and current distribution across the antenna. The abstract formula for directivity and current distribution are derived in Antenna Theory textbook, Chapter 4. You are encouraged to review the details and the steps leading to these formulas.
A. Consider a thin (ideally zero diameter) dipole antenna centered at (x,y,z) = 0. For the dipole antenna of length “ L” oriented along the z-axis, the current flows in the z- direction with an amplitude which closely follows the following function:
Plot the normalized current distribution across the length of the antennas for L/λ= 0.1, 0.5, 1, 2, 5. Comment on the number of nulls in the distribution plot as the antenna size increases. (16pts)
B. Dipole structure is an “omnidirectional” antenna which means that it radiated power is only a function of elevation angle (θ) and does not change with azimuth angle (φ) and its directivity is given by:
2 , 0 < θ < π, 0 < φ < 2π
Where η = 120π = intrinsic impedance of free space, Io = Exciation current amplitude
Plot the 2D and 3D plots of directivity in dB scale for 0 < θ < π, and 0 < θ < π, 0 < φ < 2π, respectively for various lengths. Assume, L/λ = 0.1, 0.5, 1, 2, and 5 and compare the number of nulls and maximas with the current distribution plotted in part A and comment. (22pts)