ELEC6218 Signal Processing Statistical Signal Processing Coursework Submission Details This assignment contributes 10% of your final mark for the ELEC6218 Signal Processing mod- ule. You are required to produce a write-up, which needs to include the derivations, calculations, explanations and Matlab code, that are requested in the questions below. When you are finished, you need to submit the assignment at C-BASS https://handin.ecs.soton.ac.uk/handin/2021/ELEC6218/1/ in order to get a C-BASS re- ceipt before 4pm on Friday 08/01/2021. You only need to make an electronic submission. If you notice any mistakes in this document or have any queries about it, please email me at [email protected] Mohammed El-Hajjar Learning Outcomes 1. Apply maximum likelihood estimation technique; 2. Design an adaptive filter for system identification; 3. Implement your designs in Matlab for verification and testing. Table 1: Marking Scheme Accuracy of results: Are the obtained results correct? Is the formulation correct? 50% Interpretation of results: How well are the questions posed in the assignment answered? Do you answer all parts of the questions? Do you include the required derivations? Do you explain your derivations when requested? 50% 1 Question 1 [3 marks] A signal u is sent out, takes time τ to reach an object and is reflected back and received. In total, the signal is delayed by a time 2τ , attenuated by an amount a and is subject to noise n. Thus, the received signal is modelled by y(t) = au(t− 2τ) + n(t). (1) You know u and y, but because of the noise you are uncertain about τ , which is what you want to know, to determine the distance of the object from the sender/receiver. You also don’t know the noise, but you can assume that u and n are uncorrelated. Explain how to find τ . Use Matlab to apply your method on the data available from “https://secure.ecs.soton.ac.uk/notes/elec6218/ssp/coursework/delay.mat” which is the received signal y(t) in (1). Include your Matlab code in your solution. Hint : you can use the Matlab function load(’delay.mat’) to load the data, where you will get two vectors y and u. Question 2 [7 marks] In order to measure the range of an object, in radar a signal pulse is transmitted and then the time it takes to propagate to the object and return to the receiver is measured. Then, the round trip delay τ0 from the transmitter to the target and back is related to the range R as τ0 = 2R c , where c is the speed of light propagation. Estimation of the range is therefore equivalent to the estimation of the time delay, assuming that c is known. If we consider a sampled transmitted signal s[n], then the delay estimation is transformed from estimating τ0 to estimating n0, where n0 = τ0/∆, with ∆ being the time interval between samples. If s[n] is the transmitted signal, then the model for the received signal is x[n] = s[n− n0] + w[n], n = 0, 1, · · · , N − 1, where n0 is the delay to be estimated, w[n] is the White Gaussian noise with variance σ 2. It is assumed that s[n] is M samples in length and is non-zero for n = 0, 1, · · · , M − 1. Also, for all possible values of n0, the delayed signal s[n−n0] is contained in the observation interval 0 ≤ n ≤ N − 1. Since the signal is non-zero over the interval 0 ≤ n ≤ N − 1, x[n] can be written as x[n] = w[n] 0 ≤ n ≤ n0 − 1 s[n− n0] + w[n] n0 ≤ n ≤ n0 +M − 1 w[n] n0 +M ≤ n ≤ N − 1 where M is the length of the sampled signal and n0 is the delay in samples. 2 1. Find the Maximum Likelihood (ML) estimate of the delay n0, given that w[n] is Gaussian distributed with mean 0 and variance σ2. 2. Consider a signal that is a square pulse with height 1 and with M = 20. Also consider the case when N = 200, n0 = 50 and σ 2 = 0.25. Using your ML estimator in the first part, write a Matlab code to estimate the delay nˆ0, when the received signal is corrupted with noise w[n]. Include your Matlab code in your solution. 3 欢迎咨询51作业君