Fourier transform of a signal sequence?

Question

Desparately, I am trying to calculate the Fourier transform of the following signal sequence. What can it be?

$$f(x)=\left(\frac{\sin(\pi x)}{x\ln(T)\sin[\pi\ln(x)/\ln(T)]}\right)^2$$

while $x > 0$.

Answer 1

Ignoring domain restrictions (e.g., $\log(x), x<0$), as I am not even sure this function is defined for all $x>0$, I think this is as good as you are going to get:

$$ f(x) = (g(x))^2 \Rightarrow F(\omega) = G(\omega)\ast G(\omega) $$

$$ g(x) = \alpha \cdot \operatorname{sinc}(x) \cdot h(x) $$

$$ h(x) = \dfrac{1}{\sin \left( \pi \log(x) / \log(T) \right) } $$

$$ G(\omega) = \alpha \cdot \operatorname{rect}(\omega) \ast H(\omega) $$

Good luck finding $H(\omega)$. In all likelihood you will have to do an asymptotic approximation. It doesn't look highly oscillatory, so nothing jumps out.