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.
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