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/1920/ELEC6218/1/ in order to get a C-BASS re- ceipt before 4pm on Tuesday 07/01/2020. You also need to print out your write-up, staple or bind it together with your C-BASS receipt and submit it at the ECS reception before the same deadline of 4pm on Tuesday 07/01/2020. 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 [4 marks] In digital communication systems, a pilot signal s[n] is transmitted prior to data transmission to determine the attenuation of the channel. The amplitude of the received signal must be estimated to determine the necessary gain to be used to compensate for the channel attenuation. The received signal can be modelled as: x[n] = A · s[n] + w[n], n = 0, 1, · · · , N − 1, where A is the amplitude to be estimated (−∞ < A < +∞), s[n] is the known signal, and w[n] is the White Gaussian Noise with variance σ2. 1. Find the maximum likelihood estimator (MLE) of the amplitude A. 2. Given that s[n] = 0.9n with A = 1 and N=20 and the noise variance is σ2 = 1/16, write a Matlab code that implements your MLE of part 1 above. Include the Matlab code and the output value of the estimate of A in your write up. Note: In your Matlab code, use ’randn(‘state’, 0)’ to intialise the random number gener- ator to generate the same noise realisation. Question 2 [6 marks] System identification is primarily responsible for determining a discrete estimation of the trans- fer function for an unknown system. The same input x[n] is applied to both the adaptive filter and the unknown system and then their outputs are compared as shown in Figure 1. The output of the adaptive filter y[n] is subtracted from the output of the unknown system, which is the desired signal d[n]. The resulting difference is an error signal e[n] used to update the filter coefficients of the adaptive system until the error is close to zero. Adaptive Filter Unknown System + − x[n] y[n] d[n] e[n] Figure 1: System Identification model. In this exercise you are requested to write a Matlab code that implements the Least Mean Square (LMS) algorithm in order to estimate the filter coefficients of an unknown system. 2 The unknown system is a fourth order autoregressive model represented by d[n] = 4∑ i=0 w[i]x[n− i], where w represents the unknown system coefficients. Use the following Matlab function ‘un- knownSystem.m’ available here to model the unknown system of Figure 1. 0 1 2 3 4 5 6 7 frequency 0 5 10 15 20 m a gn itu de 0 1 2 3 4 5 6 7 frequency -4 -2 0 2 4 ph as e Figure 2: Unknown system magnitude and phase response. 1. The magnitude and frequency response of the unknown system are shown in Figure 2. Update the Matlab code ‘unknownSystem.m’ in order to plot the magnitude and phase responses of the unknown system as in Figure 2. Include your Matlab code and the generated plots. 2. Write a Matlab function that implements the LMS algorithms for finding the unknown system coefficients. Include your Matlab code in your write up and also include the adaptive filter estimated coefficients and the last value of error e from your Matlab code and plot the mean square error versus iteration index. You can use the Matlab function ‘systemIdentification.m’ available here as a starting point for your function. 3
