# Fourier transform of a signal

I have the following signal and need to derive its Fourier transform: I just started learning Fourier transform and don't know how to solve this kind of question. I know though how to derive the Fourier transform for: $$cos(2\pi f_0t)$$

$$\mathcal{F}\{cos(2\pi ft)\}=\int_{-\infty}^{\infty}cos2\pi f_0*e^{-j2\pi ft}dt$$ $$=\int_{-\infty}^{\infty}\frac{1}{2}(e^{-j2\pi(f-f_0)t}dt +\int_{-\infty}^{\infty}\frac{1}{2}(e^{-j2\pi(f+f_0)t}dt$$ $$=\frac{1}{2}(\delta(f-f_0)+\delta(f+f_0))$$

UPDATE: I have used the Euler's relation to rewrite the signal like this: $$g_1(t)=\frac{1}{2}Ae^je^{j2\pi ft}+\frac{1}{2}Ae^{-j}e^{-j2\pi ft}$$ How do continue from here?

• All that changes is the limits of your integral, instead of doing the transform between -ve infinity and +ve infinity just go between where the signal is non-zero. This is because integrals are linear and you can break them up into sections and add them. – jramsay42 Feb 14 '18 at 0:09
• Also this is really a mathematics, how to integrate question, not an electrical engineering question. – jramsay42 Feb 14 '18 at 0:10
• You can write the Fourier transform as a sum of sines and cosines, then given the limits and an even or odd functions one part of the integral may goes away. – George Herold Feb 14 '18 at 0:21
• If no answer this may get migrated to math.stackexchange.com – user105652 Feb 14 '18 at 1:16

The FT of a product of functions is equal to the convolution of the FT of each function $$F(a \times b) = F(a) * F(b)$$