```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
http://www.pgea.unb.br/~lasp
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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.
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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.
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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.
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Solution (1)
 1.1) Compute the sample covariance matrix and its eigenvalues and
eigenvectors.
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Solution (2)
 1.2) Estimate the model order of X.
 By applying Akaike Information Criterion, we obtain that:
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Solution (3)
 1.3) Estimate the direction(s) of arrival(s) of the impinging wavefronts.
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Solution (4)
 1.4) Estimate the transmitted signals S.
 1.5) Estimate the noise amplitude.
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Solution (5)
 2) Estimate the transmitted signals S via CAPON and DS.
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