ELEC6218 Signal Processing

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ELEC6218 Signal Processing

Statistical Signal Processing Coursework

Submission Details

This assignment contributes 10% of your nal 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 nished, you need to submit the assignment at C-BASS

https://handin.ecs.soton.ac.uk/handin/2122/ELEC6218/1/ before 4pm on Tuesday 11/01/2022. 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 meh@ecs.soton.ac.uk.

Mohammed El-Hajjar

Learning Outcomes

1. Apply maximum likelihood estimation technique;

2. Implement your estimator in Matlab for veri cation 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%

Amplitude and frequency estimation of a sinusoidal signal

Consider a sinusoidal signal x[n] embedded in White Gaussian noise (WGN), which can be represented as

x[n] = A cos(2πf0n) + w[n], n = 0, 1, 2, . . . , N – 1,

where A > 0 is the amplitude, 0 < f0 < 0.5 is the frequency and w[n] is the WGN with mean 0 and variance σ 2 . 1. Find the maximum likelihood estimate (MLE) of the amplitude A and the frequency f0 of the sinusoidal signal. 2. Consider a sinusoid x[n] = 1.25 cos(2 × π × 0.15 × n) + w[n], with w[n] having a variance σ 2 = 0.1 and N = 200. (a) Write a Matlab code that implements your MLE estimate in part 1 above to nd A and f 0 , which represent the estimates of the A and f0 . (b) Include in your write up the output values of your Matlab code for A and f 0 . Hint: An idempotent matrix A has the following characteristic: A2 = A. In your analysis, you will get a matrix of the form╱I - H(HT H) 1 HT、. This will be idempotent matrix.