Discrete consine transform(DCT-II)

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DTFS/FFT (We will not detail into the difference between these two at this moment.) use $e^{jwt}$ as the base. And if $x(t)$ is real and even, $X(\omega)$ is real and even.
We are using FFT, we use assume the signal is periodic, we may introduce high frequency at the boundary. So we can build a real periodic signal to handle the boundary issue.
Assume the original signal is of length N. Goal is to build a even periodic signal and perform FFT on the new signal.
- Filp signal along the boundary. Then the start of the signal is the same as the end of the siganl -> solve the boundary issue. -> N becomes 2N.
- What about the value at $t = 0$? Add 0 between the original signal, so that we construct an even signal. -> 2N becomes 4N.
We perform FFT on this 4N signal to get 4N complex values. Then we get N real values for free.
It is used in JPEG compression.

dct