# Fourier transform of a sum

I have a function : $$C(t)=\left(1.42*\exp^{-1.192t}- 12.44*\exp^{-1.192t} +11.02 \right) u(t)$$ where u(t) is a unit-step function What is its fourier transform? a step by step breakdown would greatly be appreciated: notably how would you manage the fact that it is a summation, would the linearity of a fourier property apply? How would you compute the power and the phase of the sum?

would the linearity of a fourier property apply?

Of course, why not?

How would you compute the power and the phase of the sum?

Computing each term in complex numbers, summing and converting to power/phase representation

a step by step breakdown would greatly be appreciated

$$\begin{split} C(t)&=\left(1.42 \exp^{-1.192t}- 12.44 \exp^{-1.192t} +11.02 \right) u(t) \\ & = 1.42 \exp^{-1.192t} u(t) - 12.44 \exp^{-1.192t} u(t) +11.02 u(t) \end{split}$$

$$\mathcal{F}(C(t)) = 1.42 \mathcal{F}\left( \exp^{-1.192t} u(t)\right) - 12.44 \mathcal{F}\left(\exp^{-1.192t} u(t)\right) + 11.02\mathcal{F}\left( u(t)\right)$$

Can you go from here? And why aren't you summing your exponential terms?

• Yes I can take it from here on.....what do you mean but why not sum my exponentials they are not products here or do you mean something else? Commented Jul 20, 2015 at 18:34
• I just saw what you meant by summing , my mistake the second exponential exponent is -0,136*t...unfortunately I don't have the privileges to edit my own question per stackexchange Policy can somebody do it? Commented Jul 20, 2015 at 18:42
• So here is what I found as F(C(t)), could you please confirm if it checks out : F(C(t))= 1.42/(1.192+jw) - 12.44/(0.136+jw) +11.02 F(u(t)) Commented Jul 20, 2015 at 19:03

Fourier transform is a linear filter wolfram. Linearity means:

F[ax+by] = aF[x] + bF[y]

for any constants a and b, and for any signal/variable/function x and y. So of course, the linearity applies to your signal C(t).

For a signal of form

exp(-at)u(t)

its Fourier transform is readily available See Page 3.