I want to calculate the DTFT of $$x[n] = \cos (\frac \pi 3 n) (n^4u[n+4] + n^2 u[n+2] - n^4 u[n-2] - n^2 u[n-4])$$
My first thought was to convert it to convolution since cosine decomposes to delta functions in the frequency domain $\Omega$ and use the property which states that $$n^m u[n + a] \leftrightarrow j^m \frac {d^m ( e^{j\Omega a} \mathcal {DTFT \{ u[n], n \to \Omega \})}} {d \Omega^m} $$ which turns out to be very frustrating. So I want a faster way to do it.
Thanks in advance
$n^4[u(n+4)-u(n-2)]$ is zero for $n<-4$ and $n>1$.
Similarly, $n^2[u(n+2)-u(n-4)]$ is zero for $n<-2$ and $n>3$.
Hence, the signal is composed of seven nonzero values. Therefore, the Fourier transform can be written as sum of seven appropriately shifted and weighted complex exponentials.