Fall 2017/2018 EE 635 Homework 3 Instructions

Question

Fall 2017/2018 EE 635 Homework #3 Instructions: 1) Consider the transmitter and receiver show in the figure below. The transmitter and receiver are at the same height and the receiver moves to the right at a speed of ð‘£ð‘£ m/s. Use the two-ray channel model. a) Find the time-varying impulse response ð‘ð‘(ðœðœ, ð‘¡ð‘¡) of the channel for ð‘‘ð‘‘ â‰« â„Ž. b) Find ð¾ð¾1(ð‘¡ð‘¡, ðœðœ) h d vt v 2) Using MATLAB, generate two independent zero-mean Gaussian random variables ð‘‹ð‘‹ and ð‘Œð‘Œ with variance 5. Compute ð‘ð‘ = âˆšð‘‹ð‘‹2 + ð‘Œð‘Œ2 and ðœ™ðœ™ = arctan(ð‘Œð‘Œ/ð‘‹ð‘‹). Use MATLAB to demonstrate that ð‘ð‘ and ðœ™ðœ™ are independent and that ð‘ð‘ is Rayleigh distributed and ðœ™ðœ™ is uniformly distributed. 3) Consider a random, time-varying channel with impulse response ð‘ð‘(ðœðœ, ð‘¡ð‘¡) = ð›¼ð›¼1(ð‘¡ð‘¡)ð›¿ð›¿(ðœðœ âˆ’ 1) + ð›¼ð›¼2(ð‘¡ð‘¡)ð›¿ð›¿(ðœðœ âˆ’ 2) where ð›¼ð›¼1(ð‘¡ð‘¡) and ð›¼ð›¼2(ð‘¡ð‘¡) are jointly-WSS complex random processes. a) Show that this channel satisfies the WSSUS conditions. b) Find the scattering function of the channel. c) Find the power-delay profile of the channel, the mean delay spread and the rms delay spread. d) Find the coherence bandwidth, coherence time and Doppler spread. e) Find all the variants of ð´ð´ð‘ð‘, ð‘†ð‘†ð‘ð‘ and ð‘†ð‘†ð¶ð¶ defined in the lectures. 4) For a narrowband fading environment, define the outage probability as ð‘ƒð‘ƒout = ð‘ƒð‘ƒ(ð‘§ð‘§2

Dr. Jack HW Helper · Accepted Answer

This paper addresses the challenges presented in Homework #3 for the EE 635 course, analyzing various aspects of signal transmission through time-varying channels using MATLAB simulations and theoretical derivations. 1. The Two-Ray Channel Model The Two-Ray Channel Model serves as a foundational framework for analyzing mobile communication systems where both line-of-sight and reflected signals contribute to the overall channel response. In this model, the two paths will have varying time delays based on the movement of the receiver. The time-varying impulse response $ h(t, \tau) $ can be mathematically derived by considering both direct and reflected paths. The impulse response can be expressed generally by: $ h(t, \tau) = h_{LOS}(t, \tau) + h_{REF}(t, \tau) $ Where $ h_{LOS} $ represents the line-of-sight path and $ h_{REF} $ accounts for the reflected signal. The analysis continues by computing the spatial parameters tied to the distance between transmitter and receiver. 2. Generating Gaussian Random Variables Using MATLAB, we can simulate the transmission environment by creating two independent zero-mean Gaussian random variables, $ X $ and $ Y $, each with a variance of 5. The transformation: $ R = \sqrt{X^2 + Y^2} \quad \text{and} \quad \theta = \arctan\left(\frac{Y}{X}\right) demonstrates that \( R $ follows a Rayleigh distribution while $ \theta $ is uniformly distributed over the interval [0, 2π]. MATLAB can be employed to visualize the distributions through histograms confirming independence. 3. WSSUS Conditions Validation The Wide-Sense Stationary Uncorrelated Scattering (WSSUS) conditions must be checked for the channel described. The conditions can be verified by analyzing the cross-correlation functions: \( E[x(t)x^(t+\tau)] = R_x(\tau) \quad \text{and} \quad E[x(t_1)x^(t_2)] = R_{xy}(t_1 - t_2) These conditions lead us in calculating the scattering functions, power-delay profiles, and spread metrics crucial for understanding how the cha

Fall 2017/2018 EE 635 Homework 3 Instructions ✓ Solved

Fall 2017/2018 EE 635 Homework #3 Instructions:

Paper For Above Instructions

1. The Two-Ray Channel Model

2. Generating Gaussian Random Variables

3. WSSUS Conditions Validation

4. Outage Probability Analysis

5. Wideband Channel Challenges

6. Multipath Spread & Frequency-Selective Fading

Conclusion

References