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ECE 614 – Principles of Digital Communications Homework 2 Assigned on: 02/09/2016 Due by: 02/18/2016 2.1 Two subspaces S1 and S 2 of a Hilbert space are said to be orthogonal if every vector in S1 is orthogonal to every vector in S 2 . The sum of the two subspaces S1 S 2 is the subspace consisting of vectors that can be expressed as the sum of a vector in S1 and a vector in S 2 . Given two orthogonal subspaces S1 and S 2 of a Hilbert space H and an arbitrary vector x H , show that the projection x̂ of x onto S1 S 2 can be expressed uniquely as xˆ xˆ 1 xˆ 2 where x̂1 is the projection of x onto S1 and x̂ 2 is the projection of x onto S 2 . 2.2 Let 1 t , 2 t , 3 t be an orthonormal basis for a subspace S. Let us define three new waveforms f1 t , f 2 t , f 3 t according to f1 t A111 t A122 t A133 t f 2 t A211 t A222 t A233 t f 3 t A311 t A322 t A333 t Under what conditions will f1 t , f 2 t , f 3 t also be an orthonormal basis for S? 2.3 Consider the signals r t , s1 t and s2 t sketched below: 1 r(t) 2 t 1 t 2 1 3 t 1 s1(t) 2 1 3 t -1 1 s2(t) 2 1 3 t Find r̂ t , the projection of r t onto the space spanned by s1 t and s2 t . Carefully sketch r̂ t . 2.4 Let S st | st T st , for all t be the subspace of complex-valued periodic signals with fundamental period T. Show that the complex sinuisiod signal set k exp j 2 t , k Z (where Z is the integer set) forms an orthonormal basis of S under T the following inner product definition. xt , y t 1 T T 2 xt y t dt * T 2 Given st S , express st as linear combination of the basis signals. 2.5 Suppose you have a set of real-valued waveforms s1 t , s 2 t ,, s N t , and you want to find a basis for the span of their complex envelopes. The obvious approach would be to first downconvert each of the waveforms, and then apply the Gram-Schmidt procedure to the set of complex envelopes. Will we get the same answer if we first apply Gram-Schmidt, and then downconvert? Justify your answer