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2 complex numbers 40 points consider the complex number c...

Question: 2 complex numbers 40 points consider the complex number c...

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2. Complex Numbers (40 points). Consider the complex number c= Rr(c) +jlm(c). where Re(c) is known as the real part of e and Imlc) is known as the imaginary part of c (a) Explicitly show that c can be written as where Im(ch Relc) θ = arctan 2 (b) Using (a), explicitly show that e can be written as and that its complex conjugate c. = Re()-jlm(c) can be written as (c) Now, consider the complex pole p a j8, its complex conjugate p-j8 and the transfer function G(s) = s--+ s-r. Then, show that the impulse response associated with G(s), g(t), can be written as ged) 10 (d) For p +j8 and p-j8, show that (e) For p = a+)3, p. = o_jß and c = a +jhc = a-jb, show that (f) Using (e), show that g(t) can be re-written as (g) Using the previous developments, show that g(t) can be re-written as (h) For u(t) 1t 10), find gu (t)

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