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I don't think what you're asking for is possible. If you take the Fourier transform with the existing mask / window function you're going to get the spectral content of the underlying image convolved with the spectral response of the mask due to multiplication in the time domain being equivalent to convolution in the frequency domain. An alternative is to ...


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If you have a piecewise function, you need to separate the integral over that function into the same number of (sub-)intervals: $$\begin{align}\int_0^4x(t)e^{-jk\omega_0t}dt&=\int_0^1x(t)e^{-jk\omega_0t}dt+\int_1^2x(t)e^{-jk\omega_0t}dt+\int_2^4x(t)e^{-jk\omega_0t}dt\\&=\int_0^1(-t)e^{-jk\omega_0t}dt+\int_1^2e^{-jk\omega_0t}dt\end{align}$$ with $\...


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