Let $$x = \left(f(0), f(1/32), \ldots, f(31/32) \right)$$ $$f(x) = \sin(11(2\pi x)) + 5\sin(2(2\pi x + \pi/2)) + 3\cos(7(2\pi x))$$ What is $y = DFT(x)$?
So I was guided to use the equations: $$ \sin x = \frac{e^{ix} - e^{-ix}}{2i} , \ \cos x=\frac{e^{ix}+e^{-ix}}{2} $$
First, by definition $$y_{k}=A\left(\omega_{n}^{k}\right)=A\left(e^{2\pi ik/32}\right)=x_{0}+x_{1}e^{2\pi ik/32}+\cdots+x_{n-1}e^{2\pi ik\cdot31/32}$$
Next, we can use the suggested identities:
$$f\left(\frac{j}{32}\right)=\frac{1}{2i}\left(e^{i\cdot22\pi x}-e^{-i\cdot22\pi x}\right)+\frac{5}{2i}\left(e^{i\cdot4\pi x+\pi}-e^{-i\cdot4\pi x+\pi}\right)+\frac{3}{2}\left(e^{i\cdot14\pi x}+e^{-i\cdot14\pi x}\right)$$
Now, I tried to apply it on $y_k$ but things got messy a little bit.
Is there a better way evaluating $y$ vector?
I'd be glad if you could help re-arrange things.
Thanks!