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HW1 - Boston University College of Engineering Electrical...

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Boston University, College of Engineering Electrical and Computer Engineering Department EC515 Digital Communications Fall 2009 Problem Set 1 Out: 9/2/2009 Due: 9/11/2009 Linear Algebra Problem 1. This problem tests your understanding of basis functions. 1. We are given three vectors (a, b, c), (0, b, 0) and (a, 0, c). Construct a vector space, V, of smallest dimension containing all the vectors. Construct the basis vectors of this space and express the three vectors as a linear combination of the basis vectors. 2. We are given a 2 dimensional Euclidean space with basis vectors (1, 0), (0, 1) and a vector (1/3, 2/3). We now choose vectors (1/ 2, 1/ 2), (0, 1). Does this form a basis for the 2 dimensional Euclidean space? If so, what are the coordinates of the vector in the old and new basis? Can you come up with a general rule for coordinate transformation based on this example? 3. Check whether the space of periodic discrete time signals, s[n], with period, N, forms a vector space. What is its dimension? Find the Fourier series coefficients of the DT signal.
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