Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa da Costa University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code 70.919-970, Brasília - DF de Brasília Homepage:Universidade http://www.pgea.unb.br/~lasp Laboratório de Processamento de Sinais em Arranjos 1 Exercises (1) 1) Assume that you have a Uniform Linear Antenna Array (ULA) that captured the following samples of narrowband signals: The rows of X indicates the antenna and the column indicates the snapshot. The wave fronts are planar. 1.1) Compute the sample covariance matrix and its eigenvalues and eigenvectors. 1.2) Estimate the model order of X. 1.3) Estimate the direction(s) of arrival(s) of the impinging wavefronts. 1.4) Estimate the transmitted signals S. 1.5) Estimate the noise amplitude. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 2 Exercises (2) 2) For the previous question, use DS and CAPON to make a beamforming and estimate the transmitted signals. Compare the results of DS and CAPON and also compare with the estimation done in 1.4. 3) What is the importance of using the joint diagonalization for the R-D ESPRIT? Show an example where a joint diagonalization is used and where is not used. 4) Explain mathematically the Forward Backward Averaging. 5) Given the first mode unfolding of the tensor below: 5.1) Compute the rank of the tensor. 5.2) Compute the EVD of the covariance matrices of the first unfolding. 5.3) Represent the tensor using outer products. 5.4) Represent the tensor using the identity tensor. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 3 Exercises (3) 6) Show the four subspaces of a matrix using the SVD. 6.1) How can you use the noise left singular values to estimate the Direction of Arrival? 7) Explain the properties of the TRINICON to estimate the signals. 8) Explain the difference between linear mixtures and convolutive mixtures. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 4 Solution (1) 1.1) Compute the sample covariance matrix and its eigenvalues and eigenvectors. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 5 Solution (2) 1.2) Estimate the model order of X. By applying Akaike Information Criterion, we obtain that: Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 6 Solution (3) 1.3) Estimate the direction(s) of arrival(s) of the impinging wavefronts. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 7 Solution (4) 1.4) Estimate the transmitted signals S. 1.5) Estimate the noise amplitude. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 8 Solution (5) 2) Estimate the transmitted signals S via CAPON and DS. Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 9