I'm working on this problem where I need to find the Fourier Transform of $$ f(t)\approx f_P=\sum_{k\epsilon\ \mathbb{Z}}[\sum_{n=-N}^{N}\widehat {(f_{k})}[n]e^{2\pi int/T}]X_{[kT,(k+1)T]}(t) $$ Taking the Fourier Transform of the signal: $$ \widehat{f_P(n)}=\int_{-\infty}^{\infty}\sum_{k\epsilon\ \mathbb{Z}}[\sum_{n=-N}^{N}\widehat {(f_{k})}[n]e^{2\pi int/T}]X_{[kT,(k+1)T]}(t)e^{-2\pi it\omega} dt $$ $$ =\int_{kT}^{(k+1)T}\sum_{k\epsilon\ \mathbb{Z}}[\sum_{n=-N}^{N}\widehat {(f_{k})}[n]e^{2\pi it(n/T-\omega}] dt $$ I am not sure where to go from here, however.

Note that for $N=\lceil{T*\Omega}\rceil$ the desired result should be: $$ \widehat{f_P(n)}=\sum_{k\epsilon\ \mathbb{Z}}[\sum_{n=-N}^{N}\widehat {(f_{k})}[n]e^{2\pi i(k-(\frac{1}{2})T}(\omega-\frac{n}{T})(\frac{\sin (\pi(\frac{\omega T}{2}+\frac{n}{2}))}{\pi(\omega+\frac{n}{T})})] $$

Any help offered would be very appreciated!


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