Based on my tests so far is not just an approximation It s right on the money provided you exclude w N from the window function I m talking about its magnitude not the unimportant linear phase factor The symmetry issue for N even is also unimportant It only matters for the academic exercise of trying to create a function centered at n 0 whose DTFT is real valued But most actual windowing is done like Window function Window examples with an even number of points and a window centered at N 1 2 i e the visible peak is flat topped at both w N 2 1 and w N 2 1 So the DTFT generally has linear phase and therefore is not real valued