I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2\$f_1*f_2 = F_1 * F_2\$, where I'm assuming this means multiplication of two functions is equal to the convolution of their fourier transforms. I'm using f_1 = 0.5^n\$f_1 = 0.5^n\$ and f_2 = u(n)\$f_2 = u(n)\$.
So I can calculate the fourier transorm of u(n)\$u(n)\$ fine. It is (pi)(delta)(w) + 1/(jw))\$\pi\delta(\omega) + 1/(j\omega))\$. However, I cannot for the life of me figure out 0.5^n\$0.5^n\$. I tried to put it into the fourier transform integral integral of (0.5^t)/(e^(jwt))dtintegral of\$(0.5^t)/(e^{j \omega t})dt\$ from negative infinity to infinitynegative infinity to infinity, but I end up with 0.5t/(e^(jw))\$ 0.5t/e^{jw}\$, and when evaluated from negative infinity to infinitynegative infinity to infinity, I end up with infinity\$ \infty \$ as my answer, unless of course the integration is wrong.
Therefore, either the answer is infinity * (pi)(delta)(w) + 1/(jw)\$ \infty * \pi\delta(w) + 1/(j\omega)\$, which when convoluted would equal just the second function..? OR am I going about this problem completely wrong?
