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Since I gave you a wrong hint first, let me try to make it up. You have the correct transfer function and the rest is (I think) just to slog through the algebra. Here is an outline how this can be approached. We can use the geometric sum property to simply things a bit 4\cdot H(\omega) = 1 + e^{-j\omega} + e^{-j2\omega} e^{-j2\omega} = \frac{1-e^{-4j\...
I think that phase shift is what causes the DCT-II to have different boundary properties compared to the DCT-I, as shown on the DCT Wikipedia page. It means that it is even around $n=-1/2$ and $N-1/2$. I suspect that this helps it to be sparser, and might also be important for other applications.