TELE 9753 Advanced Wireless Communications
Tutorial 2 – Wireless Channel
Question 1
Consider an indoor wireless LAN with fc = 900MHz, cells of radius 10m, and omnidirectional antennas with Gl = 1. Under the free-space path loss model, what transmit power is required at the access point such that all terminals within the cell receive a minimum power of 10µW. How does this change if the system frequency is 5GHz?
Question 2
Consider the set of empirical measurements of Pr/Pt given in the table below for an indoor system at 900MHz.
a) Find the path loss exponent n that minimizes the MSE between the simplified path loss model given by Pr dBm = Pt dBm + K dB − 10n log10 (d/d0) and the empirical dB power measurements. Assume that d0 = 1m and K is determined from the free space path gain formula at this d0 given by K
b) Find the received power at 150m for the simplified path loss model with this path loss exponent and a transmit power of 1mW (0dBm).
Question 3
Consider a channel with Rayleigh fading and average received power Pr = 20 dBm. Find the outage probability that the received power is below 10 dBm.
Question 4
A mobile receiver is moving at a speed v and is receiving signals arriving along two reflected paths which make angles θ1 and θ2 with the direction of motion. The transmitted signal is a sinusoid at frequency f.
a) Is the above information enough for estimating (i) the coherence time Tc; (ii) the coherence bandwidth Wc? If so, express them in terms of the given parameters. If not, specify what additional information would be needed.
b) Consider an environment in which there are reflectors and scatterers in all directions from the receiver and an environment in which they are clustered within a small angular range. Using part a), explain how the channel would differ in these two environments.
Home Work
Requirements
1. Write down your full name, student number, and signature.
2. Detail the steps of your analysis/calculations/solutions. A single answer without derivations or expla nations is not acceptable.
3. Hand in a hard copy in the following class.
Question 1
The following table lists a set of empirical path loss measurements.
a) Find the parameters of a simplified path loss model plus log normal shadowing that best fit this data.
b) Find the path loss at 2 Km based on this model.
c) Find the outage probability at a distance d assuming the received power at d due to path loss alone is 10 dB above the required power for non-outage.
Question 2
Let us consider two Nakagami-m fading channels, both with average received power Pr = 30 dBm. The Nakagami-m fading parameter of the first channel is 2, and the Nakagami-m fading parameter of the second channel is 4. Calculate the outage probabilities that the received power is below 15 dBm in these two channels. Compare the obtained outage probabilities and justify which channel represents a better propagation environment for wireless communication.
Tips: The PDF of the instantaneous signal to noise radio subject to Nakagami-m fading is given by
(1)
where Γ(m) = (m − 1)! is the gamma function.
Question 3
Consider the propagation model as depicted in Fig. 1 where there is a reflected path from the ground plane between the transmit antenna and the receive antenna. The ground distance between the transmit antenna and the receive antenna is r.
Fig. 1. Illustration of a direct path and a reflected path off a ground plane.
a) Let r1 be the direct path length and r2 be the reflected path length (the path length from the transmit antenna to the ground plane plus the path length from the ground plane to the receive antenna). Prove that r2 − r1 is asymptotically equal to b/r and find the value of the constant b.
Tips: Recall that for x small, in the sense that
b) Assume that the received waveform. at the receive antenna is given by
Approximate the denominator r2 by r1 in the above equation and prove that much smaller than c/f. Find the value of β.
c) Explain why this asymptotic expression remains valid without first approximating the denominator r2 by r1 in the above equation.