I'm working on a problem:
Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w).
I am table to take the Fourier transform of $x(5t+3)$ and $\sin(2t)$ just fine using a table and basic time shifting and scaling properties:
$$ \mathscr{F}[x(5t+3)] = \frac{X(\frac{w}{5})}{5} e^{j\frac{3}{5}w}$$
$$ \mathscr{F}[\sin(2t)] = \frac{\pi}{j}[\delta{(w-2)}-\delta{(w+2)}]$$
But the question is then how to "combine" these two results? I was thinking of using the modulation property:
$$ \mathscr{F}[x(t)m(t)] = \frac{1}{2\pi}X(w)*M(w) $$ where $*$ denotes convolution.
However, I'd much rather prefer to note convolve these two function together as it would be quite a task (by hand at least). Surely there must be a better way?
EDIT: MSE gives me a magical ability to solve a problem that I otherwise would not figure out shortly after posting a question.
I have overlooked two things that are extremely useful here:
- Euler's inverse identity for sine $$sin(2t) = \frac{1}{2j}[e^{j2t}-e^{-j2t}] $$
- Frequency shift property of Fourier Transform $$ \mathscr{F}[x(t)e^{jw_0t}] = X(w-w_0) $$