     Discrete Fourier Transform multiplier information for a DCT type 2.
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The factor of 2 in an unnormalized Discrete Cosine Transform of Type II
(DCT-II) arises because it mathematically maps an N-point finite sequence
into an even, symmetrical sequence of length 2N. The 2N-Point Symmetry.
The DCT is closely related to the Discrete Fourier Transform (DFT).

To avoid complex numbers, the DCT operates on a signal that has been
mirrored to create an even-symmetric sequence.
In a DFT: An N-point signal wraps around periodically.
In a DCT-II: The signal is extended to a length of 2N by mirroring it.

Where the "2" comes from when computing the Fourier coefficients of a
2N-point symmetric signal, the positive and negative frequency components
merge.
Because the extended signal is perfectly symmetrical, these two symmetric
halves provide redundant information.
By taking the summation over only the original N points instead of the full
2N-point symmetric sequence, you essentially drop half the data.
Multiplying the result by 2 acts as a scaling factor to account for this
missing half, ensuring that the magnitude of the signal's energy is properly
conserved.

The Trade-off. Because of this "2" multiplier in the forward DCT, the
Inverse DCT (the DCT-III) is scaled by 2/N.
This is why many libraries, (such as SciPy's dct), let you choose the
norm="ortho" parameter.
This orthogonal normalization divides by sqrt(2N) instead of using the
raw "2", which balances the forward and inverse equations perfectly so no
extra constant is required.

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