This question is about a simple linear filter. Suppose we have a chemical with concentration C(t). The chemical is being degraded at the rate aC for some constant a > 0, and is being made at the oscillating rate f (t) = a + e iωt

. It can then be shown that C(t) = 1 +



a + iω

e iωt

.

a. Why did we specify that a > 0?

b. Can you choose  so that f (t) is always positive?

c. What is the real part of C(t)?

d. Set a = 1 and  = 0.5. For a selection of values of ω plot the real parts of f and C on the same graph to compare the production rate to the output. What happens as ω gets large?