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Homework 1 Math / ECE 430 due Friday, Feb. 1, 2013 1. (a) Use angle-addition formulas to find A and B in terms of α and φ: α cos(ωt + φ) = A cos ωt + B sin ωt (b) Write α cos(ωt + φ) as a sum of complex exponentials. Give formulas for the frequencies and coefficients in terms of α, ω, and φ. 2. (a) Give an example of a system A for which zero input does not result in zero output. Is your system linear? (b) Show that linearity and causality together imply that whenever x(t) = 0 for t < t0 , Ax(t) = 0 for t < t0 . Why is causality by itself not sufficient? 3. For the system defined by mẍ + γ ẋ + kx = f (t), x(t) ≡ 0 for f (t) ≡ 0 (Here the dots denote differentiation with respect to t.) (a) Find the transfer function. (b) Find the energy spectrum. 1